Docstoc

Development of an Intermediate DOF Vehicle

Document Sample
Development of an Intermediate DOF Vehicle Powered By Docstoc
					Development of an Intermediate DOF Vehicle
Dynamics Model for Optimal Design Studies



                    by

             LAKEHAL FATIMA
Table of Contents

    List of Tables                                                                                                  viii



    List of Figures                                                                                                    x


1   Introduction                                                                                                     1
    1.1 Motivation for the Study ................................................................................1
    1.2 Historical Background....................................................................................3
         Vehicle Modeling ...........................................................................................6
         Driver Modeling........................................................................................... 29
         Model Parameter Measurement .................................................................... 39
              Direct Measurement ............................................................................ 40
              Parameter Identification....................................................................... 41
    1.3 Derivation Methodology and Overview of the Thesis.................................... 43

2   Equations of Motion - Sprung Mass                                                                                  47
    2.1 Introduction ................................................................................................. 47
    2.2 Sprung Mass Kinetic and Potential Energy Terms......................................... 48
    2.3 Euler Parameter Constraints ......................................................................... 54

3   Equations of Motion - Front Suspension and Wheels                                                                  56
    3.1 Introduction ................................................................................................. 56
    3.2 Front Spindle Kinetic and Potential Energy Terms ........................................ 59
    3.3 Front Wheel and Tire Rotational Energy Terms ............................................ 60
    3.4 Generalized Forces for Springs and Dampers................................................ 62
    3.5 Constraint Forces for the Control Arms ........................................................ 69
    3.6 Summary of Results...................................................................................... 73




                                                         v
4   Equations of Motion - Three Link Rear Suspension                                                                   77
    4.1 Introduction ................................................................................................. 77
    4.2 Unsprung Mass Kinetic and Potential Energy Terms..................................... 77
    4.3 Rear Wheel Rotational Energy Terms ........................................................... 79
    4.4 Rear Springs and Dampers ........................................................................... 82
    4.5 Panhard Rod and Trailing Link Constraints................................................... 84
    4.6 Summary of Results...................................................................................... 86

5   Equations of Motion - Steering System                                                                              89
    5.1 Introduction ................................................................................................. 89
    5.2 Rack and Pinion ........................................................................................... 89
    5.3 Four Bar Linkage ......................................................................................... 90
    5.4 Tie Rod Constraints...................................................................................... 96

6   Road Model                                                                                                         99
    6.1 Introduction ................................................................................................. 99
    6.2 Road Surface Coordinate System ............................................................... 100
    6.3 Location of the Tire to Road Contact Point ................................................ 102
    6.4 Velocity of the Tire to Road Contact Point................................................. 108
    6.5 Vehicle Position and Heading Angle ........................................................... 110
    6.6 Road Segment Models................................................................................ 112
        Linear Polynomial Road Segment ............................................................... 113
        Quadratic Polynomial Road Segment.......................................................... 114
        Cubic Polynomial Road Segment ................................................................ 115

7   Equations of Motion - Tire Model                                                                                  117
    7.1 Introduction ............................................................................................... 117
    7.2 Coordinate Systems.................................................................................... 117
    7.3 The Magic Formula Tire Model.................................................................. 119
        Support Forces........................................................................................... 120
        Tractive Forces .......................................................................................... 121
              Lateral Slip and Longitudinal Slip ...................................................... 122
              Magic Formula .................................................................................. 125
    7.4 Generalized Force and Moments................................................................. 127
        Front Tires ................................................................................................. 127
        Rear Tires .................................................................................................. 129




                                                         vi
8      Equations of Motion - Driver Model                                                                               131
       8.1 Introduction ............................................................................................... 131
       8.2 Steering Control......................................................................................... 132
           Driver Path Definition ................................................................................ 133
           Steering Profile Optimization and Cost Function Computation.................... 134
       8.3 Speed Control ............................................................................................ 138
           Driver Dynamics Block .............................................................................. 140
           Vehicle Dynamics Block............................................................................. 143
           Preview Compensation Block ..................................................................... 144

9      Results, Conclusions and Recommendations                                                                          150
       9.1 Introduction ............................................................................................... 150
       9.2 Measurement Process and Model Data ....................................................... 150
           Vehicle Data............................................................................................... 151
           Tire Data.................................................................................................... 156
           Track Data ................................................................................................. 160
           Driver Model Data ..................................................................................... 161
       9.3 Model Chassis Setup .................................................................................. 162
       9.4 Simulation Results...................................................................................... 165
           Optimizer and Cost Function Computation Setup ....................................... 165
           Optimal Steering Profile Configuration ....................................................... 167
           Optimal Velocity Profile Setup ................................................................... 168
           Speed Control Algorithm Performance ....................................................... 170
           Steering Control Performance..................................................................... 174
       9.5 Vehicle Optimization Results...................................................................... 177
       9.6 Recommendations for Future Research....................................................... 179

Bibliography                                                                                                          183

Appendix A - Useful Derivatives                                                                                 190
    Angular Velocity Derivatives .............................................................................. 190
    Transformation Matrix Derivatives...................................................................... 192

Appendix B - Wheel Inertia Estimate                                                                                  196
    Thin Cylindrical Disk .......................................................................................... 196
    Thin Walled Cylindrical Shell .............................................................................. 197
    Rotating Assembly Model ................................................................................... 198

Appendix C - Tire Data                                                                                                  199
    BFGoodrich Letter.............................................................................................. 199
    Tire Data Plots.................................................................................................... 200




                                                           vii
List of Tables
1.1   Vehicle Model Degrees of Freedom ........................................................................ 2
1.2   Computational Degrees of Freedom ........................................................................ 3
1.3   Identification of the Sub-Terms in the Equations of Motion................................... 45
1.4   Organization of the Vehicle Model Derivations ..................................................... 46


3.1   Front Suspension Kinetic and Potential Energy Terms........................................... 73
3.2   Wheel and Tire Rotational Energy Terms.............................................................. 73
3.3   Generalized Forces due to Spring or Damper ........................................................ 74
3.4   Generalized Forces due to the Control Arm Length Constraint.............................. 75
3.5   Generalized Forces due to the Control Arm Orthogonality Constraint ................... 76


4.1   Kinetic and Potential Energy Terms for the Motion of the Rear Suspension .......... 86
4.2   Kinetic Energy Terms for the Rotation of the Rear Wheels and Tires .................... 87
4.3   Generalized Forces Associated with a Rear Spring or Damper .............................. 87
4.4   Constraint Forces Associated with the Panhard Rod.............................................. 88


7.1   Desired Longitudinal Force Sign and Sign of the Longitudinal Slip...................... 124


9.1   NCSU Legends Car - Front Suspension Geometric Data..................................... 151
9.2   NCSU Legends Car - Rear Suspension Geometric Data,
      Spring Data and Damper Data ............................................................................ 152
9.3   NCSU Legends Car - Sprung Mass Geometric Data ........................................... 153
9.4   NCSU Legends Car - Model Mass and Inertia Properties .................................... 154
9.5   NCSU Legends Car - Suspension Spring and Damper Properties ........................ 155




                                                      viii
9.6    Miscellaneous Tire Model Parameters: Geometric Data, Slip Equation
       Parameters and Normal Force Characteristics ..................................................... 157
9.7           97
       Delft ’ Tire Model Parameters: Pure Longitudinal Slip Equation ..................... 158
9.8           97
       Delft ’ Tire Model Parameters: Pure Lateral Slip Equation .............................. 159
9.9           97
       Delft ’ Tire Model Parameters: Combined Slip Equations................................ 160
9.10 Driver Model Parameters .................................................................................... 163
9.11 Vehicle Setup Parameters ................................................................................... 164
9.12 Optimization and Cost Function Computation Parameters................................... 165
9.13 Vehicle Suspension Parameter Optimization Ranges ........................................... 177
9.14 Vehicle Suspension Parameter Optimization Results ........................................... 178




                                                         ix
List of Figures
2.1   Earth Fixed and Vehicle Sprung Mass Coordinate Systems ................................... 48


3.1   Schematic Showing Front of Sprung Mass and Control Arms ............................... 57
3.2   Schematic of Spindle and Control Arms ................................................................ 58
3.3   Schematic of a Generic Control Arm..................................................................... 70


5.1   Schematic of the Rack and Pinion Steering System ............................................... 90
5.2   Relationship between the P and S Coordinate Systems .......................................... 91
5.3   Relationship between the D and P Coordinate Systems ......................................... 92
5.4   Schematic of the Four Bar Linkage Steering System ............................................. 94


7.1   The Tire Model Coordinate System .................................................................... 118
7.2   Relationship between Tire Velocity Components................................................. 123


8.1   Driver Path for the Kenley, NC Race Track ........................................................ 133
8.2   Steering Profile for the Kenley, NC Race Track .................................................. 135
8.3   Driver Speed Controller Block Diagram.............................................................. 139
8.4   Effect of the Traction Control Gain Parameter on the Acceleration ..................... 143
8.5   The Simplified Preview-Follower Control System ............................................... 146




                                                      x
9.1    Schematic of the Kenley, NC Race Track............................................................ 161
9.2    Optimized Steering Profile for the Kenley, NC Simulation................................... 168
9.3    Optimized Velocity Profile for the Kenley, NC Simulation .................................. 169
9.4    Comparison of the Prescribed Velocity and the Actual Vehicle Velocity.............. 170
9.5    Vertical Acceleration of the Sprung Mass (Sprung Mass Coordinate System)...... 171
9.6    Longitudinal Wheel Slip Percentages .................................................................. 172
9.7    Vehicle Position and 9.0 Seconds (Exiting Turn 2).............................................. 173
9.8    Tire Normal Loads.............................................................................................. 174
9.9    Vehicle Lateral Position Error............................................................................. 175
9.10 Yaw Velocity...................................................................................................... 176
9.11 Steering Wheel Angle and Lateral Acceleration................................................... 177




                                                            xi
1 Introduction

1.1 Motivation for the Study
        The demands imposed by the optimal design process form a unique set of criteria

for the development of a computational model for vehicle simulation. Due to the large

number of simulations which must be performed to obtain an optimized design the model

must be computationally efficient. For a fixed execution time a faster simulation will, in

general, lead to a better design. A competing criteria is that the computational model

must realistically model the vehicle.

        Current trends in vehicle simulation codes have tackled the problem of realism by

constructing elaborate full vehicle models containing dozens if not hundreds of distinct

bodies. Each body in a model of this type is associated with six degrees of freedom.

Numerous constraint equations are applied to the bodies to represent the physical

connections. 1 While the formulation of the equations is not particularly difficult, and in

fact has been automated in several software packages, the resulting model requires a

considerable amount of computational time to run. This makes the model unsuitable for

the application of computational optimal design techniques.




1
                                                       s
 Details of this approach can be found in P. Nikravesh’ “Computer-Aided Analysis of Mechanical
Systems”.


                                                  1
                         Table 1.1 - Vehicle Model Degrees of Freedom

Body                     Degrees of     Constraint             Constraint
                         Freedom        Equations                 Type
sprung mass                  7              1      Euler Parameter normalization
rear suspension              7              5      EP norm, panhard rod, trailing links
front right suspension       7              5      EP norm, upper (2) and lower (2)
                                                   control arms
front left suspension          7            5      EP norm, upper (2) and lower (2)
                                                   control arms
wheels                         4          none     none
         Past research in the field of vehicle dynamics has produced numerous

computational models which are small enough and fast enough to satisfy the speed

demands of the optimal design process. These models typically use less than a dozen

degrees of freedom to model the vehicle. They do a good job of predicting the general

motion of the vehicle and they are useful as design tools but they lack the required

accuracy for optimal design.

         A model which bridges the gap between these two existing classes of models is

required for optimal design. This type of model combines element of both approaches to

obtain an accurate solution and yet still emphasize computational efficiency. This is the

type of model which is developed in this thesis. The model consists of eight bodies which

represent the sprung mass, the rear suspension, the left front spindle, the right front

spindle, and the four wheels. There are a total of twenty-eight dynamical degrees of

freedom which are distributed as shown in Table 1.1.

         The total number of computational degrees of freedom is summarized in the Table

1.2. The equations of motion are second order which means that for each dynamical

degree of freedom there are two computational degrees of freedom (obtained in


                                              2
                            Table 1.2 - Computational Degrees of Freedom

Body                               Dynamical                Constraint              Computational
                                   Degrees of               Equations                 Degrees of
                                    Freedom                                           Freedom
sprung mass                            7                       1                         15
rear suspension                        7                       5                         19
front right suspension                 7                       5                         19
front left suspension                  7                       5                         19
wheels                                 4                      none                        8
                                                             TOTAL                       80
converting the second order differential equations to pairs of first order differential

equations). The constraint equations introduce additional degrees of freedom in the form

of Lagrange multipliers which are necessary for determining the constraint forces. There

are a total of 80 computational degrees of freedom.


1.2 Historical Background 2
        The study of automobile stability and control is a relatively new field. Although

significant quantities of automobiles were being produced in the early 1900s few efforts

were made to quantify the handling issues. Much of the early development was done on a

“cut and try” basis and this methodology is reflected in the literature. The majority of the

effort prior to 1925 was expended in designing suspensions which would keep the tires in

contact with the ground as much as possible in order to enable more effective steering

control. This preoccupation with controllability is typical of the early work. Progress in

the area of automotive stability was not seen until the 1930s.




2
 Much of the historical information prior to the mid 1950s is from the following references: [ Segel,
1956a], [Milliken, 1956], [ Segel, 1956b] and [Whitcomb, 1956].


                                                    3
        In 1903 the Wright brothers successfully built their first airplane. In the same year

G. H. Bryan started his pioneering work on a mathematical theory of airplane stability

                                                                         s
which was a few years later [Bryan, 1911]. While the refinement of Bryan’ stability

                                                                    t
theory progressed steadily similar theories for the automobile didn’ appear until much

later. This delay was most likely due to the less pressing need to consider stability in

ground vehicles as compared to aircraft. The development of usable aircraft hinges on an

understanding of aerodynamics and how it affects the stability of an aircraft. This

understanding had been evolving with the use of scale models and wind tunnels. The slow

development of an automotive stability theory was also the result of a lack of

understanding of the role of the tire mechanics in the stability of an automobile.

        The emphasis on vehicle control between 1900 and 1930 led to kinematic studies

of suspension and steering geometries. These studies led to improved designs including

Akermann steering geometry. Much of the remaining development work was concerned

with the drivetrain, structure and performance of the automobile with one notable

exception: A general theory of ride dynamics (the motion of the automobile in its plane of

symmetry) was well established by 1925. However, very little, if any, progress had been

made in the areas of static and dynamic directional stability. This statement may seem a

little strange at first given that the equations for ride dynamics are similar to those

involved in a full stability analysis. The key difference lies in the need to understand the

mechanism of lateral force generation by the tire. Without this knowledge it is impossible

to obtain meaningful stability results.




                                               4
        In 1925 Broulheit published a paper in which the basic concepts of side-slip and

slip angle were recognized for the first time [Broulheit, 1925]. The recognition of these

concepts came about during attempts to explain the phenomenon of steering shimmy

which plagued vehicles of the time period. In 1931 Becker, Fromm, and Maruhn published

a text on the role of the tire in steering system vibrations and further developed the field of

tire mechanics [Becker, 1931]. This realization enabled further study of the problem of

automotive stability.

        During the 1930s the Cadillac Suspension Group of General Motors, under the

direction of Maurice Olley, developed the first independent front suspension used on an

American car. It was found that certain steering geometries led to a condition which the

group termed oversteer [Olley, 1937]. It was recognized that these geometries led to

                                                 s
vehicles which were unsafe at high speeds. Olley’ oversteer is recognized today as being

roll oversteer. Further investigation revealed that behavior similar to oversteer could be

induced by over loading or under inflating the rear tires. In 1934 Olley wrote an

unpublished report containing his findings and in which the proposition of oversteer /

understeer was stated and the idea of critical speed was first mentioned [Olley, 1934]. As

a result of this research Goodyear Tire and Rubber Company began rolling drum tests to

determine tire characteristics and in 1935 R. D. Evans published the results in a paper on

lateral tire characteristics [Evans, 1935]. This paper gave data on cornering force and self-

aligning torque.

        This work precipitated a period of extensive research at General Motors. The

concepts of steady-state directional stability and roll steer were explored. Further


                                               5
exploration of steady-state tire characteristics occurred and skidpad tests were used for

the first time. A fundamental understanding of the steady state tire characteristics was

developed and a qualitative understanding of the transient behavior was obtained. During

the period from 1939 to 1945 very little progress was made due to World War II.

        In 1950 Lind Walker summarized the current state of knowledge on the issue of

directional stability and introduced the concept of the ‘neutral steer line’ and the ‘stability

margin’ [Walker, 1950]. These concepts had already been established in aeronautical

circles and were suggested as criteria for steady state directional stability in automobiles.

The concept of using aerodynamics and tire characteristics to aid in achieving stability was

also proposed.


Vehicle Modeling
        By the middle of the 1950s the groundwork for a mathematical model of the

vehicle had been laid. A basic understanding of the tire enabled the creation of reasonably

accurate mathematical tire models.

        In 1956 William F. Milliken, David W. Whitcomb, and Leonard Segel of the

Cornell Aeronautical Laboratory, published the first major quantitative and theoretical

analysis of vehicle handling in a series of papers [Segel, 1956a][Milliken, 1956][Segel,

1956b][Whitcomb, 1956]. These papers formed the basis for research in the area of

automotive stability and control for the next three decades and are still frequently

referenced in the current literature.




                                               6
         s
Milliken’ paper [Milliken, 1956] provides a historical overview of the field from which

much of the above material was taken. In summarizing the progress made to date, Milliken

made the following statement,

       Thus, [the] major effort in handling research to date has been in the recognition of
       individual effects, their isolation, and examination as separate entities. This work
       naturally started out as qualitative and in some instances has become quantitative.
       It has been conceptual in character; it has been pioneering and not infrequently
       intuitive and inspired, but it can hardly be viewed as an end in itself. Rather, it is a
       substantial beginning. All the individual effects now known need quantitative
       analytical expression. More significant, however, is the need for comprehensive,
       integrated analysis methods, for such overall theories will enable the prediction of
       the actual motion by rationally and simultaneously taking into account all of the
       separate effects.

       Milliken also noted that, although a great deal of progress had been made in

understanding the tire, there was much to be done still. Although much has been learned

                             s
about tire modeling Milliken’ observation is still true today; dynamic data on tires is only

now becoming available. There were no universally accepted set of reference axes and

measured tire data of the period were typically confined to two or three of the possible six

force/moments. This made translation of the data from one set of axes to another difficult

if not impossible. It was also recognized that the effects of tire design on handling were

largely unknown and that there existed a need to perform testing on a wide variety of

common passenger car tires to determine the effects of the various design parameters. In

discussing of the future objectives of the Cornell Aeronautics Lab research program

Milliken emphasized the need to concentrate on the ‘objective analysis of car stability and



                                                   7
        .
control’ In the process he made the following distinctions between stability and control,

performance and ride.

       In general, an automobile has ‘six-degrees-of-motion’freedom, and stability and
       control may be thought of as those lateral motions out of the plane of symmetry
       involving rolling, yawing and sideslipping. (‘            ,
                                                     Performance’ by way of distinction,
       is concerned with fore-and-aft motions in the plane of symmetry, such as
       acceleration, speed, and braking, while ‘ride’is composed of the vertical and
       pitching motions in the plane of symmetry.)

       The second paper of the series, written by Leonard Segel, derives a set of

nondimensionalized linearized three degree of freedom equations for lateral and directional

motion [Segel, 1956b]. In accordance with the research goals outlined by Milliken the

emphasis of the model was put on modeling for analysis of stability and control. The

bounce and pitch degrees of freedom of the chassis were ignored and a fixed longitudinal

roll axis parallel to the ground was used. Segel also made several other simplifying

assumptions including constant forward velocity, fixed driving thrust divided equally

between the rear wheels, and that the lateral mechanical properties of the tires are

decoupled from the longitudinal mechanical properties at the speeds studied. The

unsprung mass was modeled as a single non-rolling lumped mass.

An experimental validation of the model was performed using a 1953 Buick Super four-

door sedan. The vehicle was put through both pulse steering input and step steering input

tests and the transient response for the three degrees of freedom included in the model

(lateral displacement, yaw and roll) were measured at a variety of constant forward

velocities. The theoretical predictions of the model are compared to the experimental data



                                                8
taken at 32 mph in a series of frequency response curves with the results showing good

correlation.

The final paper in the series, written by D. W. Whitcomb, draws a series of conclusions on

automobile stability and control using a two degree of freedom model (yaw and side-slip)

with experimentally determined parameters [Whitcomb, 1956]. Due to the lack of a roll

degree of freedom, Whitcomb was able to assume that the car has no width and that the

tires lay on the centerline of the vehicle (a “bicycle model”). A set of linearized differential

equations is derived using stability derivatives and the steady state and transient responses

are studied. In studying the yaw response of the vehicle at a constant vehicle side-slip

angle (same angle for both tires) he introduces the concept of the “static margin”.

        The static margin is an indication of the sense and amplitude of the yawing
        moment associated with the total tyre side force. It immediately determines the
        yawing moment that the tyres would provide in reacting an externally applied side
        force.

        In his summary of response characteristics Whitcomb recognizes the strong

influence of the static margin on vehicle stability. For vehicle with a negative static margin

it was recognized that a critical speed existed, which if exceeded, would lead to instability.

As noted by Milliken, there existed a need to quantify and refine the current knowledge of

the individual vehicle subsystems. Additionally, he recognized the need to combine these

refined models into improved full vehicle models. Progress towards achieving these goals

began to be made with research done in the early 1960s.




                                                 9
       In 1960 H. S. Radt and W. G. Milliken Jr. explored the motions of a skidding

automobile [Radt, 1960a][Radt, 1960b]. They used a relatively simple vehicle model with

yaw and lateral velocity as the only degrees of freedom. A tire model was incorporated

which included the effect of saturation of the side force in the presence of braking and

thrust forces via the concept of a friction circle. Results were presented for a series of

steady state and transient maneuvers on a low friction surface ( µ=0.3). A simple driver

control was also implemented to study skid recovery. The driver model was based on

feedback from heading angle with a first order lag. Results are presented for several gains

and lag time constants.

       In August of 1961 Martin Goland and Frederick Jindra published a paper which

they used a two degree of freedom (yaw and sideslip) vehicle model to study the

directional stability and control of a four wheeled vehicle [Goland, 1961]. The model is a

                            s
simplified version of Segel’ model with the main difference being that the roll degree of

freedom enters as a quasi-coordinate which is only used to calculate the vertical load on

the tires. The paper takes into account the effects of load transfer and the variation of the

cornering performance of the tires with vertical loading. Results are presented which show

how the stability of a vehicle changes as the center of mass is moved, the tire inflation

pressure is changed, and the tire tread width is changed. The effect of tread width and

inflation pressure on the tire properties is given by a simplified form of the semi-empirical

equations published by R.F. Smiley and W. B. Horne in the late 1950s [Smiley, 1958].

       Walter Bergman published a paper in 1965 in which he explored the nature of

vehicle understeer and oversteer. While the definitions of the terms were relatively well


                                              10
established for steady state maneuvers, they were not well established for the transient

case. Bergman discussed the many origins of understeer and oversteer behavior including

steering inputs, aerodynamic forces and inertia forces in the transient case. He noted that

understeer and oversteer could be recognized by considering the change in the yaw

velocity induced by a change in lateral acceleration. This definition is in accordance with

the standardized definitions of oversteer and understeer put forth by the Society of

Automotive Engineers [S.A.E., 1965]. Bergman also develops a six degree of freedom

vehicle model to explore understeer and oversteer behavior as well as vehicle stability. The

model consists of a sprung mass and a single unsprung mass. The position of the unsprung

mass is given with respect to an inertial coordinate system by a two dimensional vector

and a yaw angle. The location of the sprung mass is given relative to the unsprung mass in

terms of four vertical wheel displacements. Both masses are assumed rigid which implies

that one of the vertical displacements is redundant.

       In 1966 Segel published a paper in which the stability of a free control automobile

(i.e. a vehicle with torque input at the steering wheel as opposed to a steering angle input)

was studied [Segel, 1966]. He proposed a two degree of freedom quasi-linear (due to

Coulomb friction) model for the steering system. This steering model was added to his

three degree of freedom model which was discussed above. The model was validated by

comparing simulation output, performed on an analog computer, to experimental data. A

reasonably good correlation was demonstrated as long as the lateral acceleration of the

vehicle did not exceed 0.3 g. He was able to effectively model the stable and unstable




                                              11
vibrational modes of the combined vehicle and steering model and to relate them to

vehicle design parameters.

       In 1967 R. Thomas Bundorf of General Motors published a paper relating vehicle

design parameters to the characteristic speed and to understeer [Bundorf, 1967]. This

paper utilized the definitions of understeer and characteristic speed proposed by the SAE

publication Vehicle Dynamics Terminology [S.A.E., 1965]. Methods are proposed to

predict understeer quality in vehicle designs and for measuring understeer in existing

vehicles. It is noted that the characteristic speed is an attribute associated with a linear

vehicle model. Bundorf argued that under most normal driving conditions, which he

characterized as having lateral accelerations below 1/3 g, a vehicle can be accurately

modeled by a linear model. This condition led to the construction of a large diameter skid

pad at GM for measuring the characteristic speed; it was not possible to reach high

enough vehicle speeds for accurate measurement of the characteristic speed on the

existing small diameter pad without exceeding the 1/3 g limit on lateral acceleration.

Bundorf derived an expression for predicting the characteristic speed of a vehicle given the

design parameters. The vehicle model used in his derivation was a bicycle model with

Ackermann (no slip) steering. The paper also contains a discussion, written by A. G.

                  s
Fonda, of Bundorf’ results with several significant contributions and suggestions.

       D. H. Weir, C. P. Shortwell, and W. A. Johnson published a paper in 1968 which

they explored the role of vehicle dynamics on controllability [Weir, 1968]. Their results

were obtained using experimental data and simulation data obtained from a model which

combined elements of a nonlinear model developed by H. S. Radt in 1964 [Radt, 1964]


                                             12
          s
and Segel’ earlier models. The model consisted of two unsprung masses representing the

front and rear suspension assemblies respectively and a single sprung mass representing

the body of the vehicle. The dynamics of the vehicle were described by a linearized set of

equations in four degrees of freedom (roll of the sprung mass about a fixed axis, lateral

velocity, yaw rate, and axial velocity). The three masses were assumed to posses the same

yaw rate, axial velocity and side slip velocity. Provisions were made for a stationary tilted

roll axis. In accordance with the inclusion of the axial velocity as a degree of freedom,

aerodynamic loads on the vehicle, longitudinal tire forces generated by braking and

acceleration and rolling resistance were considered. Dynamic data for a number of

automobiles made by U.S. manufacturers is also presented and, as an example, the transfer

functions for a typical 1960s sedan were calculated. It was noted that the yaw, lateral

velocity and roll modes have undamped natural frequencies of approximately 6 rad/sec at

60 mph. The yaw and lateral velocity modes are highly damped and the roll mode is lightly

damped. The roll mode damping ratio was found to be approximately 0.2 to 0.3 and it was

found to be largely decoupled from the yaw and lateral velocity modes. Increasing vehicle

speed tends to lower the vibrational frequencies and decreases damping which leads to a

destabilization of the vehicle.

        By the early 1970s simulations of vehicle dynamics were becoming more complex

and realistic. This was primarily due to advances in computing technology. Prior to the

1970s most simulations were performed on analog computers. These machines were

capable of solving the vehicle dynamics problems in real time (since the differential

equations were modeled by equivalent electrical component networks) in a cost effective


                                             13
manner. Unfortunately it was very difficult to model nonlinear functions of more than one

variable on these machines. Since most tire models are nonlinear functions of more than

one variable the accuracy of the simulations was compromised by limitations in the

computing equipment. The advent of digital computers allowed researchers to create

models containing nonlinear functions. This allowed increased realism in the simulations,

however, the slow speed of the digital machines (typically 10 to 100 times slower than real

time) meant increased computing costs. In the early 1970s researchers designed simulation

codes which ran on hybrid computers which combined digital and analog computing

hardware [Murphy, 1970][Tiffany, 1970][Hickner, 1971]. The new computers made it

possible to run simulations at real time speeds and at the same time include nonlinearities

in the model. A number of papers on computing techniques and on models can be found in

the literature. A few of the more significant papers are discussed below.

       In the early 1970s a vehicle dynamics simulation for a hybrid computer was created

by the research staff at the Bendix Corporation Research Laboratories [Tiffany, 1970].

The model was based on the ten degree of freedom model created by R. R. McHenry and

N. J. Deleys at the Cornell Aeronautics Laboratory for the Bureau of Public Roads

[McHenry, 1968]. The BPR-CAL model was improved by adding four spin coordinates

for the wheels and three coordinates for the steering system model. The original BPR-

CAL model had six coordinates for the sprung mass, one vertical coordinate for each front

wheel, and one vertical and one rotational coordinate for the rear axle. The steering

                               s
system model is based on Segel’ model [Segel, 1966]. At the time of publication the

model had been partially validated by comparison of simulation results with the CAL


                                             14
model. The model was upgraded in 1971 to include a dynamically accurate model of a

four wheel anti-lock braking system [Hickner, 1971].

       In 1973 T. Okada et al described in a paper a seven degree of freedom model for

vehicle simulation [Okada, 1973]. The model was used to simulate vehicle handling at the

first stage of vehicle design. Five of the degrees of freedom were used to model the

vehicle (roll, yaw, pitch, lift and lateral position). The remaining two degrees of freedom

were used to model the steering system in a manner similar to that proposed by Segel. The

vehicle was assumed to move with constant velocity. A tractive force was applied to

maintain constant vehicle speed and compensate for the six components of aerodynamic

forces which could be applied to the model. A roll axis which moves vertically in

accordance with the wheel travel was included. The effects of roll steer, axle steer, caster,

camber, toe-in, and so on were approximated by linear functions based on wheel travel,

steer angle etc. The simulation could be run in three different modes: straight-running

(with lateral “wind gust” disturbances), stationary circular motion (skid pad), and a slalom

mode (to predict critical speed). Gyroscopic effects of the wheels were only included in

the straight-running simulation mode where vehicle speeds are high. Steady-state motion

was assumed in the skid pad simulation which lead to simplification of the equations of

motion and an essentially algebraic system of equations for determining the maximum

lateral acceleration. For the slalom course simulation transients of the motion of the

vehicle were neglected and constant forward speed was assumed. The path followed was

assumed to be periodic with length 2*L where L was the distance between the cones.

         s
Galerkin’ method was used to solve for the path. The critical speed was taken to be that


                                             15
speed at which a solution could no longer be obtained. Driver response time limitations

were considered as well as vehicle limitations in determining the critical speed.

        In 1973 Frank H. Speckhart published a paper in which he presented a vehicle

model containing fourteen degrees of freedom [Speckhart, 1973]. Six degrees of freedom

were assigned to the sprung mass, four degrees of freedom were associated with the

suspension movement at the four corners of the vehicle, and four rotational degrees of

freedom were assigned to the wheels. He used a Lagrangian approach in deriving his

equations. Models were presented for several different suspension configurations. The

sprung mass was restricted to pivot about a specified roll axis. It is likely that this is done

because the suspension models were relatively simple (two dimensional in the case of the

front independent suspension) and did not provide a sufficiently accurate representation of

the kinematics involved.

        As digital computers gradually displaced analog and hybrid machines, primarily as

a result of economic concerns, it became necessary to create vehicle dynamics models

which were completely digital. The combination of the cost of computer time and the

slower solution speed of the digital machines made it desirable to create computationally

efficient models.

        In 1973 Bernard published a paper detailing several time saving methods used in

the digital vehicle simulation code created for the Highway Safety Research Institute

[Bernard, 1973]. He noted that the important sprung mass motions tended to be in the low

frequency range (below 2 Hz) and that the significant wheel hop motions tended to be

below 10 Hz. This implied that one should be able to integrate the equations of motion


                                               16
with a relatively large time step (0.005 sec) and obtain accurate results. Unfortunately the

cycling of brake torque (as in an anti-skid system) could cause rapidly changing spin

derivatives for the wheel degrees of freedom. The relatively high frequency motion

required a much smaller time step on the order of 0.0001 second. Bernard proposed an

approximate method for dealing with the spin degrees of freedom which allowed the use

of the larger time step. This improvement in combination with the use of a specially

modified predictor-corrector integration scheme which only updated the wheel-hop

derivatives during the corrector phase led to a speed improvement of a factor of five.

       In early 1976 Frederick Jindra published an interim report for the NHTSA

                                                            s
describing a vehicle simulation model being created at John’ Hopkins University Applied

Physics Laboratory [Jindra, 1976]. The model was called the Hybrid Computer Vehicle

Handling Program (or HVHP) because it was run on a hybrid computer. The HVHP

model was derived from a refined version of the Bendix Research Laboratories (BRL)

model which was discussed above. The HVHP model was used extensively by Calspan in

their study on the influence of tire properties on passenger vehicle handling. The HVHP

model contained seventeen degrees of freedom distributed as follows: six for the sprung

mass, one for the vertical motion of each front wheel, two for the vertical motion and

rotational motion of the rear axle assembly, three for the steering system, and four

rotational degrees of freedom for the wheels. The program had an option to use an

independent rear suspension model. In this case the model contained two degrees of

freedom for the vertical displacements of the rear wheels. The steering system model was

a lumped mass model consisting of two degrees of freedom to represent rotation of the


                                             17
front wheels about their steering pivots and one degree of freedom for the translational

motion of the connecting steering rod and associated mass elements. Friction and

compliance in the steering mechanism were included. The rear unsprung mass was

assumed to pivot about a point which was constrained to move along the sprung mass

vertical axis. This constraint was an improvement over the traditional fixed pivot point and

fixed roll axis. No pivot point was assumed for the independent front suspension; the front

wheels were assumed to move vertically with respect to the sprung mass. Due to the

difficulty in representing nonlinear functions on the hybrid machine, piecewise linear

functions were used to describe the spring force, coulomb friction, damping coefficients,

roll stiffness, etc. The camber angle, caster angle, and toe angle were specified as

functions of the suspension deflection. Compliance coefficients were used to model the

change in camber angle and steer angle due to applied forces and moments at the tire.

Radial loading of the tire was computed using a point contact model.

       In 1977 Kenneth N. Mormon of Ford Motor Company presented a paper

[Morman, 1977] containing a detailed three degree of freedom model of the front

suspension. The model included the effects of lower control arm bushing compliance along

the axis of rotation (but not perpendicular to the axis of rotation) and compliance of the

ball joints connecting the tie rod ends to the steering knuckle. The model was derived

using a standard Lagrangian approach with constraint equations. A variety of displacement

type inputs were applied to the model; the results of the simulation matched experimental

results fairly well. In the original model all of the spring, dampers and bushings were

assumed to be linear. Improvements could likely be made by replacing the linear elements


                                             18
with appropriate nonlinear relations. It was also assumed that the sprung mass of the

vehicle forms inertial coordinate system.

       In 1981 W. Riley Garrot described an all digital vehicle simulation developed at the

University of Michigan [Garrot, 1981]. The model contained a total of seventeen degrees

of freedom distributed in a manner identical to the HVHP model discussed above. To

reduce computational costs the steering system was described statically and the wheel-spin

degrees of freedom were handled algebraically. The model contained numerous features

which could be turned on or off as desired. These features included an anti-lock braking

system, multiple tire models, optional activation of nonlinear kinematic terms, solid rear

axle or independent rear suspension and interactive capability. The program was

constructed in a modular fashion to enable future enhancements and upgrades. The

simulation consisted of two main parts: a vehicle model called IDSFC and a general-

purpose driver module called DRIVER. The driver module could be readily altered

without affecting the vehicle model. The driver model controlled steering, braking and

drive torque inputs to the vehicle model. It contained five preprogrammed open-loop

maneuvers and could accept user defined maneuvers using tabulated data or a user defined

subroutine. Various closed-loop control strategies were implemented including a

crossover model for path following and two types of preview-predictor models. Mixed

open-loop and closed-loop control could be used. Validation of the model was performed

by comparison with the validated HVHP model.

       In 1986 R. Wade Allen and several associates from Systems Technology Inc.

performed experimental tests and correlated the results with a computer model in order to


                                            19
validate a simplified lateral vehicle dynamics model and the associated tire modeling

procedure [Allen, 1986]. The tests consisted of a number of steady state skid pad runs and

several low amplitude sinusoidal steer frequency sweeps while negotiating a steady turn.

The tests were performed for a rear wheel driver 1980 Datsun 210 and a front wheel drive

1984 Honda Accord. Several types of tires were used on the Datsun including both radial

and bias ply tires. The physical parameters describing the vehicles and the tires were input

                                                                               s
into a simplified lateral handling model which was derived directly from Segel’ original

                                                                               s
model [Segel, 1956b] and which was discussed above in the review of D. H. Weir’ paper

[Weir, 1968]. A good correlation was obtained with the experimentally obtained data. The

model was also used to extrapolate vehicle behavior under combined cornering and

braking. In 1987 Allen published a revised model containing five degrees of freedom

[Allen, 1987a]. The new model added pitch and forward velocity degrees of freedom and

was called VDANL (Vehicle Dynamics Analysis : Non-Linear). It was also a nonlinear

model and, unlike earlier linear models, the solutions were obtained in the time domain

using numerical integration. Neither of the models approach the complexity of some of the

other more detailed models discussed above [Okada, 1973][Speckhart, 1973][Jindra,

1976]; the intent was to provide a simulation code which could be run on relatively

inexpensive desktop PCs and which could utilize the graphical output capabilities of those

PCs.

       In 1987 Andrez Nalecz presented the results of an investigation into the effects of

suspension design on the stability of vehicles and, in particular, how the design of the

suspension related to movement of the roll axis [Nalecz, 1987]. Twenty-five different


                                             20
suspension types were considered. A typical three degree of freedom lateral dynamics

model was used with the addition of a quasi-static pitch degree of freedom. The sprung

mass was assumed to rotate about a roll axis whose position varied as a function of body

roll. The location of the front and rear roll centers was found via a kinematic analysis of

the suspension in which the wheel contact patches were treated as revolute joints and were

allowed to move laterally along the ground (thus allowing for track width changes). It was

found that for certain types of suspensions, most notably the double wishbone and

MacPherson strut type systems, that the assumption of a fixed roll axis could not be

justified. In 1992 Nalecz published a second paper in which he described an eight degree

of freedom model called LVDS (Light Vehicle Dynamics Simulation) [Nalecz, 1992]. The

model consisted of a three degree of freedom lateral dynamics model coupled to a five

degree of freedom planar rollover model. The models are coupled through the inertia

terms and tire force terms. The lateral dynamics model was derived in the same manner as

      s
Segel’ original model. The rollover model consisted of sprung and unsprung masses

connected through the various elements of the suspension system. The model also

included aerodynamic effects; all six possible forces and moments are modeled. The effects

of lateral and longitudinal weight transfer were accounted for in determining the lateral

forces generated by the tires. The roll axis was modeled in the quasi-static fashion

discussed above.

       In the early 1990s R. Wade Allen and his associates at Systems Technology Inc.

published a number of papers in which they validated their VDANL simulation code

[Allen, 1992] and in which details of experimental studies and simulation runs involving


                                             21
of vehicle stability and vehicle rollover are presented [Allen, 1990][Allen, 1991][Allen,

1993]. VDANL and IDSFC (which is derived from the HVOSM simulation code) were

also put through a rigorous validation process by Gary J. Heydinger et al at Ohio State

University [Heydinger, 1990]. Both validations were carried out by comparing

experimental data to simulation data in the time domain and in the frequency domain. The

control inputs from the experimental tests were recorded along with the vehicle responses

for later use as simulation inputs. Sinusoidal frequency sweeps and step inputs were used

in the testing. Heydinger explored the use of pulse inputs which require shorter test runs

and could excite the same frequency range in a later paper [Heydinger, 1993]. In studying

vehicle stability and rollover stability the authors gathered model parameter data for a total

of 41 different vehicles of various types. The connection between load transfer distribution

(which is largely governed by the relative roll stiffness at the front and rear axles) and

vehicle stability was discussed in detail. Simulation results for a set of maneuvers were

plotted. The effects of braking, acceleration and throttle lift on stability in limit handling

situations was also discussed. A similar paper, also using the VDANL software, was

written by Clover and Bernard at Iowa State [Clover, 1993]. Details of the updated

vehicle dynamics model VDANL were presented in [Allen, 1991]. The biggest change in

the model was the removal of the fixed roll axis assumption and the addition of a front

suspension model which reflects camber change with body roll. The model also included

the effect of lateral deflection of the tire, wheel and suspension which decreases the track

width and affects rollover stability. In [Allen, 1993] the authors demonstrated that the

standard single lane change maneuver was sometimes inadequate for vehicle stability


                                              22
studies in that it failed to cause unstable behavior and that it did not adequately model the

large lateral displacements which could occur in real world accident avoidance maneuvers.

Simulation results for larger lateral displacements (with the same peak lateral acceleration)

demonstrated both spinout and rollover.

       By the early 1980s a shift in the vehicle modeling process was taking place. The

demand for accurate vehicle dynamics models combined with the difficulty in deriving the

equations of motion for large multibody systems led to the use of general multibody

simulation codes. A wide range of capabilities are present in modern MBS codes including

the ability to handle non-inertial reference frames, to incorporate flexible elements in the

model, to utilize generalized coordinates, and to symbolically generate the equations of

motion. Several reviews of multibody codes have been published in recent years, several of

which are discussed in more detail below. Additionally, brief descriptions of a few papers

utilizing MBS codes for vehicle dynamics simulations are presented below.

       In 1985 W. Kortüm and W. Schiehlen presented a paper [Kortüm, 1985] which

they presented the desirable qualities of an MBS program, discussed two contemporary

examples in some detail and utilized the two codes to generate some simple vehicle

models. The first code discussed was NEWEUL which generates the equations of motion

in symbolic form with the output being FORTRAN code. It had the capability of using

both Cartesian and generalized coordinates, non-holonomic constraints and moving

reference frames. The second program was MEDYNA which generates the equations of

motion in numerical form. It also had the capability of using generalized coordinates and




                                              23
moving reference frames. Both codes supported the use of closed loops (i.e. four bar

linkages).

        In 1993 W. Kortüm and R. S. Sharp published a supplement to the periodical

Vehicle System Dynamics in which the capabilities of 27 currently available multibody

simulation codes and general purpose vehicle simulation codes were reviewed [Kortüm,

1993]. The programs discussed include ADAMS, MEDYNA, NEWEUL, DADS,

AUTOSIM, and SIMPACK among others. Tables were presented which offer

comparisons of the capabilities of the various codes. Kortüm discussed the desirable traits

of a multibody code and gave a brief discussion of the contemporary numerical methods

which are most applicable to vehicle dynamics simulation. Sharp discussed the four models

which were used in benchmarking and evaluating the codes in his introduction.

        In 1994 R. S. Sharp wrote a paper in which he compared the capabilities of the

major multibody computer codes with emphasis on those which generate the equations of

motion symbolically [Sharp, 1994]. The codes reviewed were selected based on their

applicability to automotive simulation. He discussed the methods used by each code in

deriving the equations of motion with attention to the limitations of each method. In

particular he noted the limitations of each code with respect to the types of constraint

equations that could be handled. References to significant papers in the area of multibody

dynamics were given.

        R. J. Antoun discussed a vehicle dynamic handling computer simulation created

using the multibody code ADAMS ( Automatic Dynamic Analysis of Mechanical Systems)

in a paper which was published in 1986 [Antoun, 1986]. A model of a 1985 Ford Ranger


                                            24
pickup truck was created utilizing a combination of the standard ADAMS model definition

language and user written subroutines for non-standard system components such as the

tires. A detailed kinematic model of the front I-beam suspension and the rear leaf spring

suspension (using a three link approximation) was constructed. The effects of bushing

compliance were included in the model. Nonlinear shock absorbers were used. Excellent

agreement of simulation results with experimental data was obtained. Other studies were

made using models for a 1986 Bronco II and a 1986 Aerostar van. Using the respective

models the researchers were able to optimize the stabilizer bar dimensions and tire

characteristics at an early stage of the design process. The Bronco model contained 55

degrees of freedom. It was noted that the extensive graphical display capabilities of the

ADAMS program were invaluable in debugging the model geometry and in interpreting

the results.

        A paper describing a model built utilizing a program which automates the

generation of the equations of motion was presented in 1991 by C. W. Mousseau

[Mousseau, 1991]. The program, AUTOSIM, was used to create a 14 DOF vehicle

                                       s
model. The program used a form of Kane’ equations to derive the equations of motion

and applies extensive algebraic and programming optimizations to achieve high efficiency.

The user was responsible for choosing the generalized coordinates which describe the

configuration of the system. It was not necessary to use Cartesian coordinates and

numerous constraint equations to formulate the equations of motion. In generating the

vehicle model the location and orientation of the spindle was expressed in terms of four

generalized coordinates; the generalized coordinates were specified as prescribed cubic


                                            25
polynomial functions of the suspension deflection (a quasi-static approximation). The

cubic polynomials were obtained from a kinematic suspension model. The effects of

suspension geometry and suspension bushing compliance were included in the suspension

model which was also created using AUTOSIM. Integration of the resulting FORTRAN

model produced good correlation with measured data. The computational efficiency of the

resulting model allowed it to be used in real time in a driving simulator. In 1993 Michael

W. Sayers published a paper in which AUTOSIM was used to generate a number of

vehicle models [Sayers, 1993]. The simplest model possessed 4 degrees of freedom system

while the more complicated models contained 10 degrees of freedom. The emphasis in the

paper was on demonstrating the ease with which computationally efficient models can be

generated and tested.

       Yoshinori Mori et al at Toyota described a model created for simulation of active

suspension control systems in a paper presented in 1991 [Mori, 1991]. The vehicle model

was described using a simulation language. The control algorithms were coded in

FORTRAN and interfaced to the vehicle model. The vehicle model contained 20 degrees

of freedom. The unsprung masses were assigned three degrees of freedom each and the

sprung mass was given six degrees of freedom. Each of the front wheels was assigned a

single steer degree of freedom. The model also included a 19 degree of freedom drive-

train model. Provisions for front wheel drive, rear wheel drive and four wheel drive were

made. The road surface was modeled using a combination of a flat or undulating surface

and random input noise.




                                             26
        In 1989 a research group at the University of Missouri-Columbia began a DOT

sponsored project to study the effects of vehicle design on rollover propensity [Nalecz,

1988]. A nonlinear 14 degree of freedom vehicle model called the Advanced Dynamics

Vehicle Simulation (ADVS) was developed to carry out this research. The model was

derived using a Lagrangian approach and utilizes quasi-velocities to describe the angular

velocities. The degrees of freedom were utilized as follows: three translational and three

rotational for the sprung mass, two for the front suspension and two for the rear

suspension and one rotational for each wheel. To study vehicle-terrain interaction it was

necessary to model the body of the vehicle as well as the terrain [Lu, 1993]. The vehicle

body was represented by a set of massless, three-dimensional nodes which obey nonlinear

force-deflection curves. Each node was checked for interference with the terrain at each

time step of integration and its position was adjusted as necessary. The force resulting

from body-terrain interaction was applied to the vehicle dynamics model. The terrain was

modeled by a single curve which was extruded along the direction of travel. This

prevented the use of curved roadways and other such fully three-dimensional structures

but it simplified the body node-terrain interference calculation substantially.

        In 1993 the results of a program at Lotus Engineering to develop a vehicle

simulation code for studying the application of predominantly linear control algorithms to

the suspension of a nonlinear vehicle were published by J. G. Dickinson and A. J. Yardley

[Dickison, 1993]. Although commercial multibody simulation codes were available it was

desired to utilize a simpler model which did not require the large quantities of descriptive

data associated with the more complicated codes. The model which was presented in the


                                               27
paper utilized six degrees of freedom for the sprung mass. The front and rear suspensions

were modeled in a quasi-static fashion. Each wheel was assigned a ‘bump’ degree of

freedom which was measured relative to the sprung mass. The location of the

instantaneous pivot axis was determined from a look-up table based on the value of the

bump variable. Since the motion was handled in a quasi-static fashion the pivot axis

location, camber angles, wheel hub location, toe angle, effective spring rate, effective

damper velocities and so on could be calculated off-line. The front suspension was

modeled in the same fashion but adds a steering swivel axis and two degree of freedom

steering system. The tires were modeled using the Pacejka curve fits to measured tire data.

The longitudinal force at the tires was set by the driver acceleration input. The lateral

force was reduced accordingly by utilizing a standard friction ellipse. Wheel angular

velocities were apparently not included as degrees of freedom in the model. The authors

claimed a speed advantage of a factor of three over more complicated models generated

using standard multibody codes and hoped to increase the advantage to a factor of six in

later versions of the software.

       In 1996 Michael R. Petersen and John M. Starkey described a relatively detailed

straight line acceleration vehicle model for predicting vehicle performance [Petersen,

1996]. The model included longitudinal weight transfer effects, tire slip, aerodynamic

drag, aerodynamic lift, transmission and driveline losses and rotational inertias of the

wheels, engine and driveline components. A manual transmission was assumed with 100%

clutch engagement. Shifts were simulated by disengaging the clutch completely, assuming

that the engine torque is zero during the shift, changing the gear ratio, and then reapplying


                                              28
the full torque of the motor. Shifts occurred when the applied torque at the rear wheels in

the next gear exceeded the torque at the rear wheels in the current gear, or alternatively,

when redline was reached. After validating the model the authors conducted sensitivity

analyses to determine which design parameters most strongly affected vehicle

performance.


Driver Modeling
       Beginning in the early 1960s an increasing emphasis on vehicle safety created a

push toward modeling vehicles under the more demanding conditions associated with

crash avoidance maneuvers. In order to accurately represent the reactions of the vehicle

under these circumstances it was necessary to include the driver as an integral part of the

model. While this fact had been recognized in the early 1960s it was not until the late

1960s that increasing computational power and an improved understanding of vehicle

dynamics and driver behavior made it practical to model the driver and vehicle together.

       In 1968 David H. Weir and Duane T. McRuer of Systems Technology Inc.

published the first in a long series of papers on modeling driver steering control (lateral

                                                                      s
control) [Weir, 1968b]. The vehicle dynamics were modeled using Segel’ equations

[Segel, 1956b]. The equations were Laplace-transformed and the analysis was performed

in the frequency domain. The transfer functions relating the motion variables to the inputs

                     s                                            s
were taken from Weir’ earlier paper [Weir, 1968a]. Although Segel’ steering system

model was available [Segel, 1966], the lack of dynamic data on vehicle steering systems

made its use impractical. Consequently, a pure gain was used to describe the steering




                                             29
system dynamics. The driver model was divided into four subsystems: quasi-linear

compensatory control, pursuit control, precognitive control and a remnant.

       The quasi-linear compensatory control consisted of a describing function with

parameters which were adjusted to fit the situation and the system, an additive remnant

and a set of adjustment rules. The form of the driver model, of the describing function and

of the parameter adjustment rules was derived from extensive experiments involving

human operators. It was noted that the parameter adjustment rules could be eliminated by

considering the combined response of the vehicle/driver system. In this case an

approximate crossover model was found to represent driver/vehicle behavior adequately.

This simplification was a result of experimental studies involving human drivers which

found that drivers adjust their behavior to obtain an approximately invariant form for the

combined vehicle-driver response function.

                                                        s
       The pursuit control subsystem modeled the driver’ ability to see the roadway

ahead. This is in contrast to the compensatory subsystem in which the driver reacted to

errors in the current position of the vehicle. The details of pursuit control are not

mentioned except to note that experimental evidence indicates that the magnitude of the

feedforward describing function was approximately equal to the inverse of the magnitude

of the vehicle response function. Thus the command path and the actual vehicle path are

nearly identical. It was also noted that compensatory control was often used in

combination with pursuit control to regulate errors in path following.

       The precognitive control model attempted to mimic learned driver responses. A

common example of this type of maneuver is pulling out and pulling in while passing


                                             30
another automobile. Weir and McRuer note that these types of maneuvers do not involve

a feedback based on position information or a feedforward based on the desired path. The

maneuver is initiated by the driver in response to stimuli other than those involved in

pursuit and compensatory control. No other results are presented by Weir and McRuer

beyond defining the nature of precognitive control.

       The driver remnant component of the model accounted for the portion of the

       s
driver’ output which was not linearly correlated with the input. It was modeled as a

random input which was described by an experimentally obtained power spectral density.

It was noted that the major source of this remnant is due to variation of the parameters of

the driver describing function. The remnant could be neglected for vehicles which

demonstrated good response characteristics.

       Following discussion of the various model components them authors presented the

results of a guidance and control analysis of the potential loop closures for compensatory

control. A number of multiloop structures were considered. The best multiloop feedback

structures were considered to be those which demonstrated good frequency response and

                                                       s
required minimal driver attention. Based on the author’ analysis it was concluded that a

feedback structure based on heading angle and lateral acceleration gave the best results. A

review of perceptual experiments performed by other authors [Gordon, 1966a][Gordon,

                                                    s
1966b][Crossman, 1966] corroborates Weir and McRuer’ conclusions.

       In a later paper Weir and McRuer reviewed data from experiments on the

directional response of vehicles subjected to cross wind gust disturbances. Driver/vehicle

describing functions were measured for several test drivers. The results support Weir and


                                              31
       s                                  s
McRuer’ earlier assertion that the driver’ steering outputs could be explained as

functions of lateral position and heading angle or alternatively as functions of path angle

and path rate.

       In 1975 Errol R. Hoffman presented a paper [Hoffman, 1975] in which he

reviewed the state of the knowledge of human control of road vehicles. He covered lateral

control and longitudinal control of automobiles and motorcycles. The areas of research

reviewed in the paper were divided into the following major categories: lateral control of

automobiles, lateral control of motorcycles, longitudinal control of automobiles and

                                                                            s
combined lateral and longitudinal control. The relevant portions of Hoffman’ review of

the literature in the areas of lateral and longitudinal control of automobiles is summarized

below. The work done in the area of lateral control was divided it into four sub-groupings:

lateral control vehicle dynamics, perceptual studies, mathematical models of driver

steering control and vehicle characteristics and driver/vehicle performance.

       Hoffman classified the work done on lateral control vehicle dynamics category into

the following three categories: fixed control, free control and vehicle-driver interface

variables. Fixed control occurs when steering wheel input angle is specified directly. Free

control occurs when the steering wheel input is in the form of a specified torque. The

                                                                  s
majority of the research up to the time of publication of Hoffman’ review had been

performed on the fixed control mode; very little work had been performed using free

control. Hoffman noted that, in reality, a human driver uses a combination of the these

two types of control. He also noted that the proportion of each type of control varies with

                                              s
the type of maneuver being performed. Hoffman’ third grouping under lateral control


                                              32
vehicle dynamics category encompassed research done on driver/vehicle interface

variables. Driver/vehicle interface variables are defined as the quantities which relate

steering wheel input (either angle or torque) to vehicle response. Typically they are

approximations to the actual output and are used in determining gains in the control

algorithms. Again, the majority of the existing work concentrated on identification of the

gains associated with fixed control (i.e. neutral steer path curvature vs. steering wheel

angle, etc.). Very little work had been performed relating steering force to vehicle

response for the free control mode.

                              s
       At the time of Hoffman’ review a number of papers on driver perception of the

roadway had been published. Several papers suggested that the driver uses the perceived

velocity field to guide the vehicle. Later studies of driver eye movements indicated that

peripheral vision is used to monitor steering control for tracking and directional guidance

while central vision is used for obstacle avoidance. Studies of driver steering control

movements indicated that vehicle yaw rate and inertial lateral deviation are the most

probable control cues used by the driver.

       Hoffman reviewed a variety of mathematical models available in the literature at

the time. He included a brief review of the quasi-linear model proposed by McRuer et al

which was discussed in above. He also briefly discussed the predictive models of Kondo

and Ajimine and of Yoshimoto [Kondo, 1968][Yoshimoto, 1969]. These models were

single loop models which used estimated position and heading data as feedbacks. Hoffman

also mentioned an optimal control model outlined by Roland and Sheridan [Roland,

1966][Roland, 1967] which was useful in course planning situations. He noted that the


                                             33
                                                                              s
primary difficulty in using this type of model lies in determining the driver’ cost

weighting. Several other types of models were briefly reviewed.

The research done in the area of vehicle characteristics and driver/vehicle performance

was primarily concerned with relating driver/vehicle response to vehicle design parameters

and vice versa. A considerable amount of work had been performed at the time of

        s
Hoffman’ review. The majority of the papers were concerned with determining the

optimal vehicle characteristics which maximized driver/vehicle dynamic performance.

Steering force, stability factor, and steering gain and sensitivity were among the design

parameters considered.

       Hoffman noted that the state of the knowledge for longitudinal control was

considerably less developed and that very little work had been performed for combined

lateral and longitudinal driver control models. A good deal of work had been presented

regarding the drivers perceptual processes for longitudinal control and a number of driver

models had been created. Some work related to the car-following control task had been

presented. Combined models containing both lateral and longitudinal driver controls were

essentially nonexistent. The lack of full vehicle models which included the interface

variables for braking and acceleration largely prevented the application of driver models to

vehicle dynamics simulations.

       In 1977 Duane T. McRuer et al presented a paper which reviewed the progress

made to date in driver modeling with emphasis on quasi-linear models for lateral control

[McRuer, 1977]. He noted that the driver can be modeled in terms of three types of

controllers which, based on the position and velocity of the vehicle and information from


                                             34
the road ahead of the vehicle, regulate steering wheel angle. The three controllers were the

same ones introduced in earlier papers by the same authors: compensatory, pursuit and

precognitive. Under normal driving circumstances any one of these control modes could

be used and sometimes they were combined. In the later part of the paper McRuer

presented some results which verified some of the assumptions made in the paper and

assisted in quantifying the model.

       In the same issue George A. Bekey presented a paper [Bekey, 1977] in which he

discussed a variety of driver models used for car following tasks. The models assumed that

the positions and velocities of the lead car and of the following car were known by the

driver. The first part of the paper was devoted to discussion of models using classical

control structures. It was found that, in general, linear models performed quite well for

small disturbances in the neighborhood of the steady state condition. The nonlinear models

functioned better than the linear models and they also performed well under transient

conditions. The second part of the paper discussed models derived from optimal control

approaches to the problem. A quadratic criterion function was used which was based on

spacing error, velocity error and control effort. The control algorithm was determined by

minimizing this function. The resulting control model did not include driver reaction time,

neuromuscular dynamics or vehicle nonlinearities. Incorporation of time delays and vehicle

nonlinearities into the model and experimentally determining the gains provided improved

the results. Optimal stochastic models which included the effects of input and output noise

provided additional improvement. Two other classes of models were reviewed: look ahead

models and finite state models. The look ahead model was similar to the above models but


                                             35
with the addition that the driver “averages” the behavior of several lead vehicles to

determine a course of action. The finite state model arranges the following task into four

specific conditions. Driver action was based on the current condition and a set of rules

was formulated for switching modes. Both of these models provided acceptable results.

       In 1978 David H. Weir and Richard J. DiMarco presented a paper [Weir, 1978] in

which the results of vehicle handling tests from three sources were correlated to vehicle

design parameters and then evaluated quantitatively and subjectively. The original tests

were conducted by Systems Technology, Inc. (STI), the Highway Safety Research

Institute (HSRI), and the Texas Transportation Institute (TTI). The STI study included

data obtained with an expert test driver as well as data from typical drivers. The tests

included unexpected obstacle avoidance maneuvers, lane change maneuvers and double

lane change maneuvers. The results were correlated with a simplified two degree of

freedom analytical model in order to obtain estimates of the vehicle parameters. The

drivers also gave subjective evaluations of the handling of the various vehicles and these

results were used to determine the limits of acceptability on the vehicle design parameters.

The results for the expert test driver are analyzed and presented separately.

       Edmund Donges wrote a paper in 1978 in which he described a two level model of

driver steering behavior [Donges, 1978]. The model consisted of a compensatory

submodel in parallel with an anticipatory submodel (which was equivalent to McRuer’s

pursuit submodel). The mathematical forms of the submodels were derived using

parameter identification techniques. The experimental data was obtained using a driving

simulator and a number of typical test drivers. The vehicle dynamics model consisted of


                                             36
two components. The first component modeled the lateral dynamics of the vehicle and the

second modeled the longitudinal dynamics of the vehicle. Both components were

extremely simplified. The parameter estimates of both of the driver submodels showed

significant dependence on vehicle speed and on the curvature of the roadway. Comparison

of the performance of the simulated driver and of the real drivers showed good agreement.

          In 1979 R. Wade Allen and Duane T. McRuer expanded on their compensatory-

precognitive-pursuit quasi-linear model [Allen, 1979]. Previous papers had concentrated

on the compensatory and precognitive aspects of the human controller. This paper

analyzed more recent driving simulation results from a pursuit control point of view and

proposed a pursuit control model with roadway preview. The curvature of the roadway

was provided as an input in addition to the standard heading angle and lateral position

inputs.

          During the 1980s R. Wade Allen, Henry T. Szostak, Theodore J. Rosenthal, et al

published a series of papers [Allen, 1982, 1986, 1987a, 1987b] based on work done for

the NHTSA [Allen, 1988]. The first paper, published in 1982, reviewed and expanded on

previously published driver steering control models. Experimental data were used to

validate the model structure. The second paper, published in 1986, presented test methods

and modeling procedures for identifying the directional handling characteristics of

vehicles. Computer modeling of the vehicle dynamics was used to extrapolate vehicle

response beyond the typical steady-state tests done in previous papers. The third paper,

published in 1987, presented linear dynamic models, nonlinear dynamic models and

numerical procedures to implement them on a typical microcomputer. Both front and rear


                                            37
wheel drive vehicles were analyzed in the paper. The fourth paper, also published in 1987,

presented an updated driver control model. The primary feedback was based on perceived

curvature error in the vehicle path with a secondary feedback to control lateral position

error. Simulation results obtained by combining the updated driver model with a modified

version of the nonlinear vehicle model from the previous paper are presented. The

simulation results from several accident avoidance type maneuvers were presented

including results which demonstrated the transition to oversteer under combined braking

and cornering.

        In September of 1993 A. Modjtahedzadeh and R. A. Hess published a paper which

presented a simple model of driver steering control [Modjtahedzadeh, 1993]. The model

was capable of producing driver/vehicle steering responses which compared favorably

with experimental data. The results of a computer simulation of a lane keeping task on a

curved roadway were provided for both two-wheel steer vehicles and four-wheel steer

vehicles.

        In 1993 K. Guo and H. Guan published a brief review and quantitative comparison

of the various approaches used in implementing driver/vehicle/road closed-loop direction

control systems [Guo, 1993]. The review includes a discussion of compensation tracking

models and preview tracking models. Utilizing optimal control theory, the authors develop

a theoretical framework for designing an optimal preview controller for use with vehicle

simulation codes. The resulting controllers are compared to the other types of controllers

reviewed in the paper.




                                            38
        In 1996 R. Wade Allen et al of Systems Technology Inc. published a paper which

discussed an updated driver control model [Allen, 1996]. The lateral position control

model was updated to include a feedforward based on curvature error. This addition to the

model enhanced stability at higher lateral accelerations. The paper also discussed a driver

speed control model. When the roadway is straight the driver model follows a

preprogrammed speed profile (i.e. speed limits). In the event that a curve lies ahead the

speed control model is capable of utilizing the road curvature information to maintain safe

or comfortable speeds through curves, reducing speed if necessary, based on a user

selected maximum lateral acceleration value. Model responses were shown for several test

cases which demonstrated improved controller stability over earlier models. The lateral

position control model exhibited some oscillatory instability as the vehicle approached its

cornering limits.


Model Parameter Measurement
        As vehicle models have increased in complexity it has become more difficult to

obtain accurate values for the parameters which describe the model including inertial

parameters and suspension geometry parameters. Additionally, for the more complex

vehicle models, it is necessary to obtain sufficient information to describe environmental

inputs to the model such as road surface profile and wind velocity information. There are

two approaches to determining these parameters: direct measurement and parameter

identification. Modern simulation codes should use a combination of these methods.

Several research papers on these approaches are described below.




                                             39
          In addition to the difficulties associated with measuring parameters there is the

problem of validating the input. The large number of parameters which describe even a

moderately complex vehicle model can make it difficult to discover errors. Recently two

papers were published which discuss this problem [Bernard, 1994][Gruening, 1996]. The

authors note that there are several potential sources of error including erroneous

measurements, misinterpretation of the input parameter format, and typographical

mistakes in data entry. To detect these errors it is suggested that simulations be run for a

series of standard maneuvers and that the results be checked against closed form solutions

associated with simpler models (this only applies to low lateral acceleration maneuvers

which exhibit essentially linear behavior). Additionally, the authors suggest that a number

of basic vehicle performance metrics be computed. While these methods may not be

completely foolproof they can reliably detect errors in the input data and in the model

itself.

Direct Measurement
       A number of papers have been published dealing with measuring various vehicle

model parameters. Several road profile measurement systems have been developed which

allow data to be acquired while traveling at freeway speeds. Tetsushi Mimuro et al

describe a system of this type which use four laser displacement transducers attached to

the bottom of a vehicle [Mimuro, 1993]. Several devices for measuring suspension

parameters have been developed to facilitate vehicle modeling. Some require removal of

the suspension components from the vehicle (e.g. [Bell, 1987]) while others can obtain

measurements without disassembly as done in [Chrstos, 1991]. A facility for the


                                              40
measurement of vehicle inertial parameters is described in [Heydinger, 1995]. The system

is capable of measuring center of gravity location, roll, pitch and yaw inertias and the roll-

yaw product of inertia.

Parameter Identification
     The application of parameter identification techniques to vehicle models is a recent

development. A search of the literature revealed a total of four papers on the subject. A

literature search performed by the authors of one of the papers found even fewer papers

on the subject. Brief reviews of the papers are presented below.

        Richter, Oberdieck and Zimmerman appear to have published the first paper in

which a form of parameter identification was used to improve the fit of a vehicle model to

experimental data [Oberdieck, 1979]. In their paper they utilized a simple bicycle model

with a nonlinear tire side force coefficient model in order to improve model performance.

All of the model parameters are easily obtained except for the side force coefficients which

include axle kinematics and elasticities in addition to the tire characteristics. The nonlinear

side force model contained three unknown constants for each axle and a seventh

parameter which specified the maximum obtainable lateral acceleration. A least squares fit

method was used in combination with a genetic search algorithm to obtain the best fit to

the experimental data. For the purpose of demonstrating the technique the authors utilized

a complex 27 degree of freedom model to generate the ‘experimental’ data instead of

instrumenting a real vehicle.

        Y. Lin and W. Kortüm published a paper in 1991 on parameter identification in the

time domain [Lin, 1991]. They presented a method which is applicable to linearized


                                               41
models containing possibly nonlinear forcing terms. It was assumed that the model is linear

in the unknown parameters. A least-squares type cost function was utilized and a closed

form solution was obtained for the optimal set of parameters. The authors presented an

example of the technique using a four degree of freedom bicycle model. A simulation was

run using a set of parameters selected by the authors and a colored noise input. The

parameter identification algorithm was then applied to the model using the specified input

and the simulation output. The parameter identification yielded results which correlated

very well with the known values.

        In 1993 Feng Huang et al. published a paper in which parameter estimation was

used to determine model parameters [Huang, 1993]. The authors of this paper were only

able to locate one other author who had previously published a paper on using parameter

estimation in vehicle handling dynamics [Oberdieck, 1979]. Experimentally obtained data

from sinusoidal sweep steering tests was reduced to obtain the frequency response

functions between the various input and output variables. The parameter estimation

process involved fitting the model responses in the frequency domain to the experimentally

obtained frequency response functions. Both models used in this paper were linear which

restricted their use to situations in which the lateral acceleration was small. The first model

was a simple two degree of freedom bicycle model. The second model added an artificial

roll axis and tire camber effects and contained three degrees of freedom.

        Yi Kyongsu and Karl Hedrick presented a paper in mid-1993 in which they used a

sliding observer to estimate unmeasured states during the parameter identification process

[Kyongsu, 1995]. The authors consider nonlinear systems of equations which are linear in


                                               42
the unknown parameters. A sufficient condition for convergence of the algorithm was

given. In the example given the observer was derived from a quarter car model and was

used to identify the parameters of a half car (bicycle) model. Experimental data was

obtained from a laboratory based half-car test rig and was used to validate the parameter

identification process. Comparison of the measured damper force-velocity relationship and

the identified force-velocity function shows good agreement.


1.3 Derivation Methodology and Overview of the Thesis
       The equations of motion of the vehicle are derived using a slight variation on

         s                      s
Lagrange’ formulation. Lagrange’ equation is typically written


                                 d ∂  ∂
                                       L     L
                                         −    = Qk
                                 dt  ∂&k  ∂ k
                                       q     q                                          (1.3.1)
                                L ≡T − V
where L is Lagrangian and Qk is the generalized force associated with the generalized

coordinate qk. The Lagrangian L is defined as the difference between the kinetic energy of

the system (T) and the potential energy of the system ( V). The generalized forces are those

forces not included in the equation via the potential energy term V due to their

nonconservative nature. These forces consist of the forces generated by the tires, by the

aerodynamics of the vehicle, and by the suspension dampers. The potential energy of the

system is the result of gravitational potential energy and the energy stored in the

suspension springs and tires (modeled as springs in the vertical direction). Substituting the

definition of the Lagrangian into the equation gives the result




                                              43
                       d  ∂  d ∂  ∂
                             T          V     T ∂ V
                               −        −   +    = Qk                             (1.3.2)
                       dt  ∂&k  dt  ∂&k  ∂ k ∂ k
                             q          q     q   q
The potential energy associated with the vehicle depends only on the position of the

bodies (i.e. the qk) which implies that the second term of the equation above must be zero.


                              d ∂  ∂
                                    T     T ∂ V
                                      −   +    = Qk                                 (1.3.3)
                              dt  ∂&k  ∂ k ∂ k
                                    q     q   q
The equation can be broken down further by dividing the kinetic and potential energy

terms into sub-terms associated with the various subsystems of the vehicle model.

             T = TSM + TRS + TRT + TLF + TRF + TFT
            V = VSM + VRS + VLF + VRF                                                 (1.3.4)
            Qk = QRS,k + QLF,k + QRF,k + QT,k + QC ,k + QRP,k + QTR ,k + QCA ,k

The sub-terms are identified in Table 1.3. Substituting these expressions into Equation

1.3.2 and rearranging gives


     d ∂ SM  d  ∂ RS  d  ∂ RT  d  ∂ FL  d  ∂ FR  d ∂ FT 
          T          T          T          T          T           T
             +        +        +        +        +        
     dt  ∂&k  dt  ∂&k  dt  ∂&k  dt  ∂&k  dt  ∂&k  dt  ∂&k 
           q          q          q          q          q          q
        ∂ SM ∂ RS ∂ RT ∂ FL ∂ FR ∂ FT ∂ SM ∂ RS ∂ FL ∂ FR
         T    T    T    T    T    T    V    V    V    V
    −       −    −    −    −    −    +    +    +    +                                 (1.3.5)
        ∂k
         q    ∂k
              q    ∂k
                   q    ∂k
                         q   ∂k
                              q   ∂k
                                  q    ∂k
                                        q   ∂k
                                             q   ∂k
                                                  q   ∂k
                                                       q
    − QRS,k − QLF,k − QRF,k − QT,k − QC,k − QRP,k − QTR ,k − QCA ,k = 0




                                                44
               Table 1.3 - Identification of Sub-Terms in the Equations of Motion

Term          Description
TSM           Kinetic energy of the sprung mass
TRS           Kinetic energy of the rear suspension including the terms associated with
              the mass of the rear wheels and tires.
TRT           Kinetic energy associated with the rotational motion of the rear tire &
              wheel assemblies
TLF           Kinetic energy of the left front suspension
TRF           Kinetic energy of the right front suspension
TFT           Kinetic energy associated with the front wheels and tires.
VSM           Gravitational potential energy of the sprung mass
VRS           Gravitational potential energy of the rear suspension including the rear tires
              and wheels
VLF           Gravitational potential energy of the left front suspension, wheel and tire
VRF           Gravitational potential energy for the right front suspension, wheel and tire
QRS,k         Generalized forces associated with the rear suspension dampers, springs
QLF,k         Generalized forces associated with the left front suspension damper and
              spring
QRF,k         Generalized forces associated with the right front suspension damper and
              spring
QT,k          Generalized forces associated with all tire forces
QC,k          Generalized forces associated with the normalization constraint on the
              Euler parameters.
QRS,k         Generalized constraint forces associated with the rear panhard rod and the
              trailing links.
QTR,k         Generalized constraint forces associated with steering linkage attachments
              to the front suspension
QCA,k         Generalized constraint forces associated with the upper and lower control
              arms for the front suspension


                                                                           &      &
                                                                                  &
The first group of terms gives rise to scalar expressions in terms of qk , qk and qk . The

                                                                       &
remaining terms give rise to scalar expressions containing only qk and qk . The equations

                             &
                             &
of motion are coupled in the qk terms but it is possible to solve for the generalized

accelerations algebraically. Note that to simulate the motion of the vehicle using numerical




                                              45
                     Table 1.4 - Organization of the Vehicle Model Derivations

  Chapter        Vehicle Component                              EOM Terms
      2          Sprung Mass                                    TSM, VSM, QC,k
      3          Front Suspension and Wheels                    TLF, TRF, TFT, VLF, VRF, QLF,k,
                                                                QRF,k, QCA,k
      4          Three Link Rear Suspension and Wheels          TRS, TRT,VRS
      5          Steering System                                QTR,k
      6          Road Model
      7          Tire Model                                     QT,k
      8          Driver Model


integration, the system of equations must be formulated and solved each time the

integrator requests a function evaluation.

          The detailed derivations of the terms in the preceding equation are provided in the

following chapters. As mentioned before, the terms are grouped by function. Table 1.4

shows the organization of the derivations.




                                                46
2 Equations of Motion - Sprung Mass

2.1 Introduction
                                 s
       The motion of the vehicle’ sprung mass is expressed in terms of seven generalized

coordinates: three Cartesian displacements and four Euler parameters. A normalization

constraint on the Euler parameters produces the desired six degree of freedom system.

The terms in the equations of motion associated with the sprung mass are found by

expressing the position of the sprung mass in terms of these generalized coordinates,

differentiating to obtain the velocity of the sprung mass, finding the angular velocity of the

mass in terms of the Euler parameters, formulating the kinetic energy and the potential

energy of the body, and finally, differentiating the energy expressions to obtain the

relevant terms in the equations of motion.

       A partial schematic of the vehicle is shown in Figure 2.1. The coordinate system E

represents an Earth fixed coordinate system which is assumed to be inertial. The SM

coordinate system is rigidly attached to the vehicle sprung mass and its origin is at the

center of mass of the sprung mass.

                                                                          $
       The unit vectors of the SM coordinate system are oriented with the s1 axis pointing

             $
forward, the s2 axis pointing out of the driver’ side of the vehicle and the
                                                s                                       $
                                                                                        s3 axis

pointing upwards. The s1 − s3 plane is chosen so that it is parallel to the symmetry plane of
                      $ $




                                              47
                                                                   $
                                                                   s3
                                                                        $
                                                                        s2



                                                               SM
             $
             e3                                                                        $
                                                                                       s1
                                            rSM/E
                   $
                   e2

         E                   $
                             e1


                  Figure 2.1: Earth Fixed and Vehicle Sprung Mass Coordinate Systems




the sprung mass; this is done to allow simplification of the steering model which is

discussed in a later chapter. The position of the origin of the SM coordinate system with

respect to the origin of the inertial coordinate system E is given by the vector rSM/ E .


2.2 Sprung Mass Kinetic and Potential Energy Terms
        Since the position of the vehicle is typically expressed with respect to the roadway

surface and the data describing the roadway surface is expressed in terms of the Earth

fixed coordinate system, it is desirable to represent the position of the vehicle in terms unit

vectors associated with the E coordinate system:

                                  E
                                      rSM/E = x e1 + y e2 + z e3
                                                $      $      $                             (2.2.1)

The superscript at the upper left of the r indicates that the vector rSM/ E is written in terms

of the unit vectors of coordinate system E. The subscript is intended to be read as “SM

with respect to E” so E rSM/E is the position of point SM with respect to point E written in




                                                     48
terms of the unit vectors of the E coordinate system. The three degrees of freedom x, y

and z represent the position of the sprung mass. Since the E system is inertial the velocity

of the sprung mass can be obtained by direct differentiation giving

                               E
                                   vSM /E = E r SM/ E = x e1 + y e2 + z e3
                                              &         &$ &$         &$                 (2.2.2)

           The angular orientation of the sprung mass can be described in several ways. In the

past models of this type have ignored some of the rotational degrees of freedom which

leads to an essentially one dimensional or two dimensional description for the angular

position and angular velocity. This approach typically leads to a simple form for the

equations of motion. While this approach can yield closed form solutions it compromises

the accuracy of the model. In order to capture the subtleties of the motion it is necessary

to utilize a representation which encompasses all three rotational degrees of freedom. This

can be achieved by representing the orientation of the SM coordinate system with respect

to the E system in terms of a rotation matrix. Several forms of rotation matrices have been

used in the past with the most common ones being those based on Euler angle sequences.

While this approach works well there are problems with the rotation matrix becoming

singular at certain orientations of the body which can make solution of the equations of

motion difficult. A better solution is to express the rotation matrix in terms of Euler

Parameters. 1 The rotation matrix is shown below.




1
                               s
    See Chapter 6 of Nikravesh’ text for a detailed explanation of Euler Parameters.


                                                         49
[E CSM ] = [G ][L]T
              βSM ,0 + βSM,1 − 1
                   2        2
                                    2     βSM,1βSM, 2 − βSM, 0 βSM,3 βSM,1βSM, 3 + βSM, 0 βSM, 2 
                                              2        2                                         (2.2.3)
         = 2 βSM,1βSM, 2 + βSM, 0 βSM, 3   βSM, 0 + βSM, 2 − 1 2    βSM, 2 βSM,3 − βSM, 0 βSM,1 
             β β                                                      βSM, 0 + βSM,3 − 1 
                                                                          2         2
              SM,1 SM,3 − βSM, 0 βSM, 2 βSM, 2 βSM,3 + βSM, 0βSM,1                         2    

                               − βSM,1     βSM, 0     − βSM,3       βSM, 2 
                                                                           
                        [G ] = − βSM, 2    βSM,3          βSM, 0   − βSM,1 
                               − βSM,3
                                          − βSM, 2        βSM,1     βSM, 0 
                                                                            
                                                                                                   (2.2.4)
                              − βSM,1     βSM, 0      βSM,3        − βSM, 2 
                                                                            
                        [L] = − βSM, 2    − βSM,3     βSM, 0        βSM,1 
                              − βSM,3
                                          βSM, 2      − βSM,1       βSM, 0 
Note that there are four Euler parameters (βSM,i). In order for the rotation matrix to be

normal a constraint on the Euler parameters is required.

                                    2        2       2        2
                                   βSM, 0 + βSM,1 + βSM, 2 + βSM, 3 = 1

The normalization constraint reduces the number of rotational degrees of freedom from

four to the expected three degrees of freedom.

       The angular velocity of the sprung mass is written in terms of the Euler parameters

as follows.

                                                                                 &
                                                                               β 
                                         − β1         β0        β3     − β2   &0 
                                   &                                         β 
               SM
                    ω SM/E   = 2[L]β = 2 − β2        − β3       β0      β1   &1                (2.2.1)
                                                                                β
                                         − β3
                                                      β2       − β1     β0   &2 
                                                                              
                                                                               β3 
It is important that the angular velocity be written in terms of the sprung mass centroidal

coordinate system (SM) so that the inertia tensor is constant with respect to the frame of

reference.




                                                      50
        With these considerations in mind the kinetic energy can be written as

                    1                              1
            TSM =     msm ( SM v T )( SM v SM/E ) + SM ω SM/E [SM J SM ] SM ω SM/ E
                                 SM/E
                                                         T
                                                                                                     (2.2.2)
                    2                              2
The velocity of the sprung mass written in terms of the body fixed coordinate system is

related to the velocity in the inertial coordinate system by a simple coordinate

transformation matrix:

               SM
                    vSM /E = [ E CSM ]T E v SM/E
                             β2 + β1 − 1
                                0
                                     2
                                        2          β1β2 − β0β3 β1β3 + β0β2 
                                                                                                   (2.2.3)
                [E CSM ] = 2 β1β2 + β0β3          β2 + β2 − 1 β2β3 − β0β1 
                                                     0    2  2
                             β1β3 − β0β2
                                                  β2β3 + β0β1 β2 + β2 − 1 
                                                                 0    3  2 

                                                     E
Substituting this result and the definition of           vSM/E into the expression for the kinetic

energy and simplifying gives

         1                                                1
    TSM =  mSM ( E v T [ E CSM ])([E CSM ]T E v SM/E ) + SM ω SM/E [SM J SM ]SM ω SM/E
                     SM/E
                                                                 T
         2                                                2
                                                                                                     (2.2.4)
         1                              1
        = mSM ( E v SM/E )( E v SM/E ) + SM ω SM/E [SM J SM ]SM ω SM/E
                     T                        T
         2                              2
Given the form of the kinetic energy equation it is possible to find the terms in the

equations of motion which are derived from the kinetic energy of the sprung mass. The

relevant terms in the equations of motion are given by the expression


                                              d  ∂ SM  ∂ SM
                                                   T      T
                                 ETSM ,qk =           −                                            (2.2.5)
                                              dt  ∂&k  ∂ k
                                                    q     q
Rather than expanding the expression for TSM and differentiating directly it is preferable to

differentiate TSM and then substitute the derivatives of the linear velocity and the angular

velocity into the result.




                                                     51
                                          T
 ∂ SM 1     ∂E vSM /E                                                        T  ∂ v SM /E 
                                                                                    E
  T
 ∂k
  q
     = mSM 
      2     ∂k 
                q
                                             (   E
                                                      v SM/E ) + mSM ( v SM/E ) 
                                                                1
                                                                2
                                                                      E

                                                                                  ∂k 
                                                                                      q
                                                                                             
                                      T
                                                                                                                             (2.2.6)
         1 ∂SM ω SM/ E  SM                                        ∂ ω SM /E 
                                                                      SM

                         [ J SM ]( ω SM/ E ) + ( ω SM/E ) [ J SM ]
                                               1 SM       T SM
        +                         SM
                                                                               
         2 ∂k  q                             2                    ∂k q
The resulting terms are scalars which allows them to be transposed without affecting the

equation. Note that the inertia tensor is symmetric so it is also unaffected by transposition.

Thus, the first and second terms are equal and that the third and fourth terms are equal.


       ∂ SM                   T  ∂ v SM/E                                                     ∂SM ω SM /E 
                                   E
            = mSM ( E vSM/ E )                                    (        ω SM/ E ) [SM J SM ]
        T                                                                            T
                                           +                          SM
                                                                                                                            (2.2.7)
       ∂k
        q                        ∂k 
                                     q                                                           ∂k q       
                                &
Differentiation with respect to qk leads to a result identical in form. To complete the

                                                                                      &
Lagrangian the preceding result needs to be differentiated with respect to time (with qk

substituted for qk).


  d ∂ SM  d                                 T ∂ v                                                  ∂SM ω SM/ E 
                                (                     )                     (                    )
                                                   E
       T                                                                                      T
           = mSM                     v SM/ E            +                       ω SM / E [SM J SM ]            
                                    E                 SM/ E                     SM

  dt  ∂&k  dt 
        q                                         ∂&k 
                                                     q                                                   ∂&kq       
                                                      T
                     ∂E vSM / E   d E                                                                  d ∂E v SM/ E 
                                                                                     (                 )
                                                                                                       T
              = mSM                   v SM/ E  + mSM                                 E
                                                                                             v SM/ E                    
                     ∂&k   dt
                          q                                                                               dt  ∂&k 
                                                                                                                  q
                                                                                                                             (2.2.8)
                                              T
                  ∂ ω SM/ E  SM
                        SM
                                        d SM          
               +             [ J SM ]      ω SM / E 
                  ∂&k
                     q                 dt            
                                                   d  ∂SM ω SM / E 
                   (                  )
                                          T SM
               +       SM
                            ω SM/ E        [ J SM ]                
                                                   dt  ∂&kq        
                                                                                     &
Substituting the generalized coordinates x, y and z and their derivatives for qk and qk into

the preceding equations gives the following results:




                                                                       52
                       d ∂ SM  ∂ SM
                            T     T
         ETSM ,x =            −
                       dt  ∂& 
                             x    ∂x
                                     ∂ vSM/ E                 T d  ∂ v SM /E 
                                    T
                        d E
                                        E                               E
                  = mSM     v                 + mSM ( vSM/E )
                                                         E
                                                                                
                         dt SM/ E   ∂& 
                                          x                        dt  ∂& 
                                                                          x                       (2.2.9)
                                    T  ∂ v SM/E 
                                               E
                  − mSM ( E vSM/ E )            
                                       ∂ x
                  = mSM &
                        &
                        x

                       d  ∂ SM  ∂ SM
                            T      T
         ETSM , y =            −     = mSM &
                                             &
                                             y                                                   (2.2.10)
                       dt  ∂  ∂
                             &
                             y      y

                       d ∂ SM  ∂ SM
                            T      T
         ETSM ,z =             −    = mSM &
                                            z&                                                   (2.2.11)
                       dt  ∂ 
                             z&    ∂z
Considering the rotational coordinates and applying Equations A.10 and A.11 gives

                       E
                           d  ∂ SM  ∂ SM
                                T      T
         ETSM ,βi =            & −
                                        β
                           dt  ∂ i  ∂ i
                                 β
                                          T
                    ∂SM ω SM / E  SM       E d SM          
                  =      &        [ J SM ]        ω SM / E                                   (2.2.12)
                    ∂i  β                  dt              
                                                   E d ∂SM ω SM/ E   ∂SM ω SM / E 
                       (              )
                                       T
                   +       SM
                                ω SM/ E [SM J SM ]
                                                   dt  ∂&          −              
                                                           βi        ∂i  β        
                                                                                       

The derivatives of          SM
                                 ω SM / E are calculated in Appendix A. The first term in the preceding

expression is the only one containing second derivatives of the Euler parameters (due to

the   d SM                                          &
                                                    &
             ω SM / E term) and it is linear in the βi . The remaining terms are nonlinear functions
      dt


of the Euler parameters and their first derivatives.

         The potential energy of the sprung mass consists only of a gravitational potential

energy term. The terms associated with the springs and dampers which support the sprung

mass are included in the appropriate generalized force terms. This is done so that nonlinear




                                                          53
springs and dampers can be implemented without significantly revising the model. The

mass terms associated with the suspension and wheels are lumped into the suspension

potential energy terms and will be derived in a later section. It is assumed that gravity acts

                $
parallel to the e3 axis. Given these considerations the potential energy for the sprung mass

is simply

                                    VSM = mSM gz                                        (2.2.13)

        The terms in the equations of motion associated with the sprung mass potential

energy are

                                                 ∂ SM
                                                  V
                                    EVSM ,qk =                                          (2.2.14)
                                                 ∂kq
Clearly the only nonzero derivative is the one associated with the z degree of freedom.

                                    EVSM ,z = mSM g                                     (2.2.15)




2.3 Euler Parameter Constraints
        The normalization constraint on the Euler parameters used to represent the angular

orientation must be incorporated into the equations of motion in some manner. While it is

possible to eliminate one of the βi from the equations of motion using the constraint

equation this leads to complicated nonlinear equations and the symmetry of the

transformation matrix is destroyed. Although it is not as efficient, it is more elegant, to

determine a set of constraint forces which can be applied to the model which will ensure

that the constraint equation is satisfied at all times. The method of Lagrange multipliers




                                                 54
will be used to determine the constraint forces 2. The constraint forces are given by the

equation

                                            QC ,βi = λC a βi                                       (2.3.1)

where

                                         ∂ 2
                                a βi =
                                         β
                                         ∂i
                                            (β0 + β12 + β22 + β23 − 1)                             (2.3.2)


Evaluating Equation 3.39 gives

                   a β0 = 2β0       a β1 = 2β1         a β2 = 2β2        a β3 = 2β3                (2.3.3)

           Note that there is now one additional unknown λC (the constraint force) which

must be determined. It will be necessary to determine the value of λC for each integration

function evaluation.

           The λC s can be determined by appending them to the acceleration vector and

augmenting the equations of motion with the second derivative of the constraint

equations. The first and second derivatives of the normalization constraint equation are as

follows.

                                d 2
                                dt
                                   {β0 + β12 + β22 + β32 − 1 = 0}                                  (2.3.4)
                                     &        &       &
                                ⇒ β0β0 + β1β1 + β2β2 + β3β3 = 0&

                    d
                       { &
                    dt 0 0
                                &      &      &
                       β β + β1β1 + β2β2 + β3β3 = 0            }                                   (2.3.5)
                        &&     &
                               &      &
                                      &      &
                                             & & &2 & &
                    ⇒ β0β0 + β1β1 + β2β2 + β3β3 + β2 + β1 + β2 + β2 = 0
                                                   0         2    3




2
                   s
    See Meirovitch’ “Methods of Analytical Dynamics”, Chapter 2 for derivation of the equations.


                                                       55
3 Equations of Motion - Front Suspension and Wheels

3.1 Introduction
         The relatively complicated geometry of the front suspension creates a dilemma. To

obtain an accurate model it is necessary to represent the linkages between the spindle and

the sprung mass as exactly as possible. The best way of doing this is to allow the spindle

to have six degrees of freedom and then limit its motion via constraint equations. This type

of model is not optimal if one is concerned about computational efficiency. The opposing

approach is to assign two degrees of freedom (steering angle and spring length for

instance) to the spindle. This approach leads to a potentially more efficient model but the

difficulty in deriving the appropriate equations of motion is prohibitive. For this reason the

first modeling technique discussed is chosen in spite of computational costs.

         A schematic of the sprung mass and the upper and lower control arms on the left

side of the vehicle is shown in Figure 3.1. The spindle is omitted for clarity. The upper and

lower control arms are located with respect to the sprung mass by the vectors

SM                SM
     rUC/SM and        rLC/SM respectively. Each control arm has a coordinate system associated

with it. The UC coordinate system is associated with the upper control arm and has the

unit vectors u1 = SM u UA and u 2 . The vector
             $       $        $                    SM
                                                        $
                                                        u UA is a constant unit vector (with respect

to the sprung mass coordinate system) which points along the direction of the rotation axis



                                                  56
                                                                                                 $
                                                                                                 s3

                                      $
                                      s1

                  $
                  u1                UC       SM
                                                  rUC/SM
         SM
              $
              u UA                                                                          SM


                       $
                       l1      $
                               u2
                                      LC              SM
                                                            rLC/SM
           SM   $
                l LA
                                                                               $
                                                                               s2
                                 $
                                 l2
                            Figure 3.1: Schematic Showing the Front of the Sprung
                                          Mass and the Control Arms



                                                                                 $
of the control arm. The origin of the coordinate system is located such that the u 2 unit

vector points at the ball joint where the control arm connects to the spindle. The

coordinate system for the lower control arm (LC) is set up analogously with unit vectors

$ = SM $ and $ . The control arms are assumed to be massless. This avoids the need for
l1     l UA  l2

the control arms to be modeled as separate bodies which would increase the number of

degrees of freedom of the model. The mass of the control arms can be accounted for by

distributing it to the spindle and the sprung mass.

        A schematic showing the spindle and the control arms is shown in Figure 3.2. The

                                                                                   $
spindle has a coordinate system attached at it’ mass center (SP) with unit vectors p i . The
                                               s

upper and lower ball joints (UJ and LJ respectively) which connect the spindle to the

                                                       SP                 SP
control arms are located by the vectors                     rUJ /SP and        rLJ /SP . The coordinate system




                                                       57
associated with the right spindle is                                                                UC
                                                                      $
                                                                      u1
                                                                 $
                                                                 p3
oriented in the same fashion as the

coordinate system for the left spindle
                                                                           SP                       $
                                                                                                    u2
                                                                 SP             rUJ /SP
(i.e. both y-axes point to the driver’s         $
                                                p1

left and both x-axes point straight
                                                                                SP
                                                                                     rLJ /SP
            $
ahead). The p 2 axis is parallel to the                     $
                                                            p2                                      LC
                                                                      $
                                                                      l1
axis of rotation of the wheel and tire.

This is done so that the inertia tensor                                                        $
                                                                                               l2
associated with the wheel, tire and
                                                      Figure 3.2: Schematic of the Spindle
brake disc (or drum) retains its                             and the Control Arms

symmetry and its constant value with

respect to the SP coordinate system.

         The spring and damper are assumed to be connected to the lower control arm. The

location of the point of attachment is specified relative to the LC coordinate system by the

         LC
vector        rLM / LC . The subscript LM indicates the lower mount. The upper mount for the

spring and damper is assumed to be located on the sprung mass at a location given by the

                     SM
constant vector           rUM /SM . The subscript UM indicates the upper mount. The forces

produced by the front springs and dampers are applied to the model via generalized forces.

         The front suspension is modeled using a 7 degree of freedom spindle whose

motion is constrained. There are a total of five constraint equations: four physical

constraints which represent the control arms and one normalization constraint on the Euler



                                                58
parameters. This leaves two unconstrained degrees of freedom which can be roughly

equated with steer angle and vertical position of the wheel with respect to the sprung

mass. The steer angle degree of freedom is eliminated by another constraint equation

associated with the steering system model which is discussed in a later chapter.


3.2 Front Spindle Kinetic and Potential Energy Terms
         The terms in the differential equations associated with the motion of the spindle are

calculated below. The kinetic energy terms associated with the linear motion of the spindle

include the mass of the wheel and tire assembly. The rotational kinetic energy of the

spindle, but not of the wheel and tire assembly, is also included.

         The position of the spindle relative to the inertial coordinate system is given by the

vector

                            E
                                r SP/ E = xSP e1 + ySP e 2 + zSP e 3
                                              $        $         $                        (3.2.1)

The angular orientation is given by the four Euler parameters βSP,i . The form of the kinetic

and potential energy expression for the spindle are identical to the formulations for the

sprung mass. The resulting terms in the equations of motion are identical in form as well.

                                        ETSP ,x = mSP &SP
                                                      &
                                                      x
                                                                                          (3.2.2)

                                        ETSP , y = mSP &SP
                                                       &
                                                       y                                  (3.2.3)

                                        ETSP ,z = mSP z&
                                                      &SP                                 (3.2.4)




                                                    59
                                                      T
                             ∂SP ω               d           
             ETSP ,βSP,i   =  & SP/ E  [SP J SP ] SP ω SP/ E 
                              ∂       
                              βSP,i               dt         
                                                                                                     (3.2.5)
                                                             E d  ∂SP ω         ∂SP ω SP/ E 
                               (                  ) [ J SP ]                                   
                                                   T SP
                           +       SP
                                        ω SP/ E                          SP/ E
                                                                                − 
                                                             dt  ∂&
                                                                      βSP,i   ∂ SP,i 
                                                                                      β
                                                                                                
The terms in the equations of motion associated with the spindle potential energy are

                                                          EVSP ,z = mSP g                            (3.2.6)


3.3 Front Wheel and Tire Rotational Energy Terms
          The front wheels are assumed to be rigidly affixed to the spindle/steering knuckle

                                                     $
assembly and to rotate about an axis parallel to the p 2 axis of the spindle coordinate

system. The terms in the equations of motion associated with the kinetic energy of the

front wheels due to rotational motion are derived here. Due to the rotational symmetry of

                                            $
the wheel assembly and the alignment of the p 2 axis with the axis of rotation it is not

necessary to utilize a distinct coordinate system for the wheels; the inertia tensor is

constant in the SP coordinate system. The kinetic energy for one of the wheels can be

written

                                                  1 SP T
                                   TWH =              ω wheel / E [SP J wheel ]SP ω wheel / E        (3.3.1)
                                                  2
The angular velocity vector of the wheel with respect to the SP coordinate system has

constant direction but variable magnitude. The total angular velocity of the wheel includes

the angular velocity of the SP coordinate system.

                                          SP                              &
                                               ω wheel / E = SP ω SP/ E + φ        SP
                                                                                        $
                                                                                        p2           (3.3.2)
                                                                           wheel




                                                                      60
where φwheel is the angular velocity degree of freedom (scalar) associated with the rotation

of the wheel. The value is time dependent and is dictated by the interaction of the vehicle

model and the tire model. The terms in the equations of motion related to the rotational

motion of the wheel are given by

                                                             E
                                                                 d  ∂ WH  ∂ WH
                                                                      T      T
                                          ETWH ,qk =                     −                                               (3.3.3)
                                                                 dt  ∂&k  ∂ k
                                                                       q      q
Substituting for the kinetic energy and differentiating gives

                                                     T
                       ∂ ω wheel / E  SP
                         SP
                                                    E d SP             
          ETWH ,qk   =                [ J wheel ]        ω wheel / E 
                       ∂&k q                      dt                 
                                                                                                                           (3.3.4)
                                                                 E d  ∂ ω wheel / E  ∂ ω wheel / E 
                         (                      )
                                                                         SP               SP
                                                    T SP
                     +       SP
                                  ω wheel / E        [ J wheel ]                     −             
                                                                 dt  ∂&k  q               ∂k
                                                                                             q        
The only degrees of freedom which generate non-zero results for the expression above are

the βSP,i and φ . The derivatives of the wheel angular velocities are calculated as
               wheel



follows


                 ∂SP ω wheel / E ∂SP ω SP/ E                             ∂SP ω wheel / E ∂SP ω SP/ E
                                =                                                       =                                  (3.3.5)
                   ∂ SP,i
                     β             β
                                  ∂ SP,i                                   ∂&SP,i
                                                                             β            ∂&SP,i
                                                                                           β

                                  ∂SP ω wheel / E                       ∂SP ω wheel / E SP
                                                  =0                        &          = p2$                               (3.3.6)
                                      φ
                                    ∂ wheel                               ∂ φ     wheel


           E                      E
               d SP
               dt
                    ω wheel / E =
                                    d
                                    dt
                                                (   SP
                                                                 ) &
                                                                   &
                                                         ω SP/ E + φ wheel
                                                                             SP
                                                                                  p2 +
                                                                                  $       SP            &
                                                                                               ω SP/ E ×φwheel
                                                                                                                 SP
                                                                                                                      $
                                                                                                                      p2   (3.3.7)


                                    E
                                        d  ∂SP ω wheel / E  E d ∂SP ω SP/ E 
                                                           =                
                                        dt  ∂&SP,i  dt  ∂&SP,i 
                                                                                                                           (3.3.8)
                                               β                 β          




                                                                       61
                             d ∂SP ω wheel / E
                        E                        E                       SP

                                   &
                                   φ
                             dt ∂ wheel
                                               =
                                                 d SP
                                                 dt
                                                      p2 =
                                                      $
                                                           d
                                                           dt
                                                                                   (   SP
                                                                                            p2 +
                                                                                            $    )    SP
                                                                                                           ω SP/ E ×SP p 2
                                                                                                                       $
                                                                                                                                             (3.3.9)
                                               = SP ω SP/ E ×SP p 2
                                                                $

For qk = βSP,i and substituting for the derivatives of                                                     SP
                                                                                                                ω wheel / E , using the expressions

above, the result becomes


                   E d  ∂ ω T  ∂ ω T  SP
                                                                                            (                                )
                           SP          SP
    ETWH ,βSP,i =           SP/ E
                                    −     SP/ E 
                                                  [ J wheel ]                                   SP             &
                                                                                                     ω SP/ E + φ SP p 2
                                                                                                                      $
                   dt  ∂&
                         βSP,i 
                                      ∂ SP,i 
                                         β
                                                                                                                wheel
                                                
                                                                                                                                           (3.3.10)
                 ∂ ω SP/ E  SP        E d                                                                                   
                                                               (                   )
                   SP T
              +            [ J wheel ]                          SP
                                                                        ω SP/ E          &
                                                                                         &                            &
                                                                                       + φwheel SP p 2 + SP ω SP/ E × φ SP p 2 
                                                                                                   $                         $
                 ∂&        
                 βSP,i                 dt                                                                                   
                                                                                                                       wheel



For qk = φwheel the expression simplifies as follows


                   (                      )                    (                                                )
                                           T
       ETWH ,φ =       SP
                            ω SP/ E ×SP p 2 [SP J wheel ]
                                        $                          SP             &
                                                                        ω SP/ E + φ SP p 2
                                                                                         $
                                                                                   wheel

                                              SP d                                                                                       (3.3.11)
               +   (   SP
                                )
                            p T [SP J wheel ]
                            $2
                                              dt
                                                      (   SP             &
                                                               ω SP/ E + & wheel SP p 2 +
                                                                         φ )        $                       SP             &
                                                                                                                 ω SP/ E × φ SP p 2 
                                                                                                                            wheel
                                                                                                                                  $
                                                                                                                                    


3.4 Generalized Forces for Springs and Dampers
           To calculate the generalized forces associated wi th the front suspension it is

necessary to calculate the virtual work done by the spring and the damper. The virtual

work done by the spring or damper is equal to the force exerted by the spring or damper

multiplied by the change in length of the spring or damper due to a virtual displacement of

the system:


                                                            δ
                                                 δ = E FSD ⋅ E LSD
                                                  W                                                                                          (3.4.1)
E
    FSD is the force vector due to the spring or damper. The vector δ LSD is the change in
                                                                     E




length due to a virtual displacement of the upper and lower spring mounts. To determine




                                                                              62
the length vector is it necessary to find the position of the upper and lower mounts in

terms of the generalized coordinates.

          Since the spring and damper are typically attached to the lower control arm for

type of vehicle being considered the derivation below is carried out using the lower

control arm vectors defined in Figure 3.1. The unit vector                                                                        E   $ is given in terms of the
                                                                                                                                      l1

sprung mass coordinate system as part of the vehicle model specification and is determined

trivially. The       E   $ unit vector is found by determining the vector from the origin of the
                         l2

control arm coordinate system to the ball joint on the spindle ( E l BJ / CA ), subtracting those

components of the resulting vector which are parallel                                                            E   $ and normalizing the result.
                                                                                                                     l1

                                                             E   $ = [ C ]SM $
                                                                 l1          l LA                                                                          (3.4.2)
                                                                      E SM


               E
                   l BJ / CA =   (   E
                                         r SP/ E +   [C ]E           SP
                                                                          SP
                                                                               r LJ /SP − E r SM / E −           [C ]E       SM
                                                                                                                                  SM
                                                                                                                                       r LC/SM   )         (3.4.3)


                                      E   $ =
                                          l2
                                                     E
                                                         l BJ / CA −       (    E T
                                                                                   l
                                                                                  BJ /CA
                                                                                         E
                                                                                           1
                                                                                             E
                                                                                               1
                                                                                                    $
                                                                                                    l) $l
                                                     E
                                                         l BJ / CA        −(    E T
                                                                                   l
                                                                                  BJ /CA
                                                                                         E
                                                                                           1
                                                                                             E
                                                                                               1
                                                                                                    $) $
                                                                                                    l l

                                                                                   (                            ) $l
                                                                                                                                                           (3.4.4)
                                                                 E
                                                                     l BJ / CA −       E T
                                                                                        l       E   E       $
                                                                                                            l
                                                                                         BJ /CA   1   1
                                              =
                                                                                                (                            )
                                                                                                                             2
                                                         E T
                                                             l     E
                                                                               l            −       E T
                                                                                                        l     E          $
                                                                                                                         l
                                                           BJ / CA   BJ /CA                           BJ / CA   1


Using these definitions the position of the lower spring mount can be written as

                                 E
                                     r LM / E = E r SM / E +              [C ]
                                                                            E      SM
                                                                                            SM
                                                                                                 r LC/SM + E rLM / LC                                      (3.4.5)

where E rLM / LC = LC x LM / LC E $1 +
                                  l                  LC
                                                          y LM / LC E $2 +
                                                                      l                LC
                                                                                            z LM / LC E $3 is a constant vector (in the control
                                                                                                        l

arm coordinate system) which locates the spring or damper mount with respect to the

origin of the LC coordinate system. To simplify the development of the equations it is




                                                                                        63
assumed that the lower mounts are in the plane of the control arm (i.e.                                                          LC
                                                                                                                                      z LM / LC = 0 ).

              SM
Note that          r LC/SM is a constant vector with respect to the SM coordinate system. The

upper mount is typically attached directly to the sprung mass so its position can be written

as

                                      E
                                          r UM / E = E r SM / E +    [C ]
                                                                       E     SM
                                                                                     SM
                                                                                          r UM/SM                                             (3.4.6)

The length of the spring or the damper is then

        E
            LSD = E r UM / E − E r LM / E
                 = E r SM / E + [C ]  E    SM
                                                  SM
                                                       r UM /SM − E r SM / E −       [C ]
                                                                                        E    SM
                                                                                                     SM
                                                                                                          r LC/SM − E rLM / LC                (3.4.7)

                 = [ CSM
                    E         ]( rSM
                                            UM /SM   −     SM
                                                                r LC/SM )−   LC
                                                                                  x LM / LC E $1 −
                                                                                              l      LC
                                                                                                          y LM / LC E $2
                                                                                                                      l
Differentiating to obtain the virtual displacement gives


                                    [ ( r −
                                  ∂ E CSM ] SM
                      δ LSD = ∑ 
                       E
                                  ∂
                                                                                        SM
                                                                                             r LC/SM )δ SM, i
                                                                                                       β
                               i    βSM,i 
                                                 UM /SM
                                                                                                                                              (3.4.8)

                                  −   LC              E
                                                       (
                                           x LM / LC δ $1 −
                                                        l        )   LC
                                                                                    (
                                                                          y LM / LC δ $2
                                                                                     E
                                                                                       l     )
The derivatives of the unit vectors of the lower control arm coordinate system are


                                                             [  $ δ
                                                           ∂ E CSM ] SM
                                          δ $1 =
                                           E
                                             l     ∑       
                                                            ∂
                                                                      l LA βSM, i
                                                                                                                                             (3.4.9)
                                                       i    βSM, i 




                                                                           64
                                                                                                     ) (                                                                             ) $l )
                                                                                                                     −

                                                                 (                                                                                          (
                                                                                                                             1

                
        δ $2 = δ E l T / CA E l BJ / CA −
         E
          l                                                              E T
                                                                             l    E              $
                                                                                                 l
                                                                                                         2
                                                                                                                            2
                                                                                                                                     E
                                                                                                                                          l BJ / CA −           E T
                                                                                                                                                                    l     E   E  $
                                                                                                                                                                                 l
                 BJ
                                                                          BJ /CA   1
                                                                                                              
                                                                                                                                                                 BJ / CA   1   1



                                                                     (                               )                                (                         (                        ) $l )
                                                                                                                     −       3

        δ $2 = − E l T / CA E l BJ / CA −                                                                     2 ⋅ El
                                                                                                         2
                                                                                                 $                                                                                   $
                                                                                                                       BJ / CA −
         E                                                               E T       E                                                                                E T       E   E
           l                                                                 l                   l                                                                      l            l
                 
                  BJ                                                      BJ / CA   1
                                                                                                              
                                                                                                                                                                     BJ / CA   1   1


               ⋅(( l   E T
                         BJ /CA     −    (   E T
                                                 l
                                               BJ / CA
                                                       E
                                                         1
                                                                         $
                                                                         l   ) $l )δl
                                                                                 E
                                                                                         1
                                                                                                 E
                                                                                                         BJ / CA                  −   (      E T
                                                                                                                                               l
                                                                                                                                               BJ / CA
                                                                                                                                                       E
                                                                                                                                                         1
                                                                                                                                                           E T  $
                                                                                                                                                                l
                                                                                                                                                             BJ /CA     )l           δ $1
                                                                                                                                                                                      E
                                                                                                                                                                                        l     )
                                                                                                                 −

                                                                 (                                   )
                                                                                                                         1

               + E l T / CA E l BJ / CA −                                                               
                                                                                                     2                   2
                                                                     E T
                                                                         l     E             $
                                                                                             l
                 
                  BJ                                                  BJ / CA   1
                                                                                                         
                                                                                                         
                (
               ⋅ δ l BJ / CA − δ l T / CA E $1 + E l T /CA δ $1 E $1 −
                  E             E
                                   BJ    (  l        BJ
                                                            E
                                                              l l                                                                )                 (   E T
                                                                                                                                                        lBJ / CA
                                                                                                                                                                 E
                                                                                                                                                                   1
                                                                                                                                                                            $ δ$) )
                                                                                                                                                                            l E l1
                                                                                                             −

                                                             (   ) {δl
                                                                                                                     1


        δ $2 = E l T / CA E l BJ / CA −                                                                 
                                                                                                     2               2
         E
           l                                                     E T
                                                                         lE                  $
                                                                                             l                                       E
               
                BJ                                                BJ /CA   1
                                                                                                         
                                                                                                                                            BJ / CA


             − δl T
                 E
                   (        E$
                               l + ElT
                           BJ /CA            1            δ$ ) $ − ( l
                                                           l l
                                                         BJ / CA
                                                                       $ )δ $
                                                                       l    lE
                                                                                     1
                                                                                         E
                                                                                             1
                                                                                                              E T
                                                                                                                BJ / CA
                                                                                                                        E
                                                                                                                          1
                                                                                                                                                         E
                                                                                                                                                                1

                  $T E                                                                                                                                                 −

                       (                             )−  l l − ( l $l ) 
                                                                                                                                                                            1
                                                                                                                                                                    2       2

               −  E l 2 δ l BJ /CA                                  E T       E                                                     E T       E

                                                       
                                                                      BJ / CA
                                                                              
                                                                                BJ / CA                                               BJ / CA   1



                (
               ⋅ E l T / CA E $1
                     BJ       l          )(      E T
                                                     l
                                                   BJ / CA           δ $1
                                                                      E
                                                                        l        )) $l } E
                                                                                             2                                                                                                    (3.4.10)

Simplifying a bit further gives the final result.


      E      1  E
     δ $2 = 
        l
             mag 
                               {
                   δ l BJ /CA − δ l BJ / CA $1 + l BJ / CA δ $1 $1 −
                                  E T       E
                                              l E T          (
                                                             E
                                                               l El                                                                                )                (   E T
                                                                                                                                                                            l
                                                                                                                                                                          BJ /CA
                                                                                                                                                                                 E
                                                                                                                                                                                   1      )
                                                                                                                                                                                         $ δ$
                                                                                                                                                                                         l E l1

                T E                1  E T                                                                                                                 $ 
                 (
             −  E $2 δ l BJ /CA − 
               
                   l                             )
                                          l BJ / CA $1
                                    mag 
                                                    E
                                                      l                          (                                           )(   E T
                                                                                                                                         l
                                                                                                                                    BJ /CA             δ $1 E l 2 
                                                                                                                                                        E
                                                                                                                                                          l
                                                                                                                                                             
                                                                                                                                                                   
                                                                                                                                                                    )                             (3.4.11)
                                                                                                                                                                   
                                                                                                     −

                                                         (                                   )
                                                                                                          1


     mag ≡ E l T / CA E l BJ / CA −                                                                 
                                                                                                 2        2
                                                             E T
                                                                 l     E                 $
                                                                                         l
           
            BJ                                                BJ / CA   1
                                                                                                     
                                                                                                     
where


          δ l BJ /CA = (1 0 0)δSP + ( 0 1 0)δSP + ( 0 0 1)δSP
           E
                               x             y             z
                           − (1 0 0)δSM − ( 0 1 0)δSM − ( 0 0 1)δSM
                                     x             y             z                                                                                                                                (3.4.12)

                                         [
                                        ∂ E CSP ]SP                                                                                    [
                                                                                                                                      ∂ E CSM ]SM
                           +   ∑
                               i         ∂ SP,i
                                          β
                                                              β
                                                    r LJ /SP δ SP,i −                                                ∑       i          β
                                                                                                                                       ∂ SM,i
                                                                                                                                                          β
                                                                                                                                                  r LC/SM δ SM,i

Substitution of the expression for δ l BJ / CA into the expression for δ $2 produces an
                                    E                                   E
                                                                          l

unwieldy result. Rather than write the complete expression it is preferable to break it


                                                                                                         65
down into components based on the degree of freedom. The complete expression for δ $2
                                                                                  E
                                                                                    l

is obtained by summing the individual terms.


    δSM ⇒ − 
     x
             1 
             mag 
                       {
                   (1 0 0) −         ( $l (1
                                        E T
                                          1                )
                                                    0 0) E $1 −
                                                           l             ( $l (1
                                                                           E T
                                                                             2                    ) }
                                                                                          0 0) E $2 δSM
                                                                                                 l x       (3.4.13)



    δSM ⇒ − 
     y
             1 
             mag 
                       {
                   ( 0 1 0) −        ( $l ( 0
                                        E T
                                          1                )
                                                    1 0) E $1 −
                                                           l             ( $l ( 0
                                                                           E T
                                                                             2                    ) }
                                                                                          1 0) E $2 δSM
                                                                                                 l y       (3.4.14)



    δSM ⇒ − 
     z
             1 
             mag 
                       {
                   ( 0 0 1) −        ( $l ( 0
                                        E T
                                          1                )
                                                    0 1) E $1 −
                                                           l             ( $l ( 0
                                                                           E T
                                                                             2                    ) }
                                                                                           0 1) E $2 δSM
                                                                                                  l z      (3.4.15)



     δSP ⇒ 
      x
            1 
            mag 
                       {
                  (1 0 0) −          ( $l (1
                                       E T
                                         1                 )
                                                   0 0) E $1 −
                                                          l              ( $l (1
                                                                          E T
                                                                            2                     ) }
                                                                                          0 0) E $2 δSP
                                                                                                 l x       (3.4.16)



     δSP ⇒ 
      y
            1 
            mag 
                       {
                  ( 0 1 0) −         ( $l ( 0
                                       E T
                                         1                 )
                                                   1 0) E $1 −
                                                          l              ( $l ( 0
                                                                          E T
                                                                            2                     ) }
                                                                                          1 0) E $2 δSP
                                                                                                 l y       (3.4.17)



     δSP ⇒ 
      z
            1 
            mag 
                       {
                  ( 0 0 1) −         ( $l ( 0
                                       E T
                                         1                 )
                                                   0 1) E $1 −
                                                          l              ( $l ( 0
                                                                          E T
                                                                            2                     ) }
                                                                                          0 1) E $2 δSP
                                                                                                 l z       (3.4.18)



                     1 
           δ SM,i ⇒ 
            β              α − α T E $1 + E l T / CA γE $1 −
                     mag 
                               { (    l        BJ        l  )             (   E T
                                                                               lBJ / CA
                                                                                        E
                                                                                          1
                                                                                           $ γ
                                                                                           l  )
                      T         1 E T                                               
                       (        )
                   −  E l2 α − 
                     
                         $             l    ( E$
                                                l
                                 mag  BJ / CA 1
                                                               )(   E T
                                                                     lBJ / CA    ) $  β
                                                                                γE l 2 δ SM,i
                                                                                        
                                                                                                           (3.4.19)
                                                                                        
                         [
                        ∂ E CSM ]SM                        [ SM
                                                          ∂ E CSM ]
                α ≡−                  r LC/SM , γ≡                                 $
                                                                                   l LA
                            β
                           ∂ SM,i                           β
                                                           ∂ SM,i

                           1 
                  β
                 δ SP,i ⇒       α−
                           mag 
                                    { (      E   $ Tα E $ −
                                                 l1  )  l1     (     $ α E$ δ
                                                                    E T
                                                                     l2       ) }
                                                                          l 2 βSP,i

                           [
                          ∂ E CSP ]SP
                                                                                                           (3.4.20)
                       α≡             r LJ /SP
                            β
                           ∂ SP,i




                                                     66
        The final step in determining the virtual work is calculating the magnitude and

direction of the force vector. The force exerted by the spring is typically a function

(possibly nonlinear) of the length. The force exerted by the damper is a potentially

nonlinear function of the time derivative of the length. The length vector for the spring or

damper was already determined to be

                 E
                     LSD = [ CSM ] SM r UM /SM −
                            E     (                                         SM
                                                                                 r LC/SM )−       LC
                                                                                                       x LM / LC E $1 −
                                                                                                                   l        LC
                                                                                                                                 y LM / LC E $2
                                                                                                                                             l          (3.4.21)

Taking the time derivative of this expression gives


                     LSD =[ CSM ] SM r UM/SM −
                             & (                                                 r LC/SM )−
                     &                                                                                             &                         &
                 E
                           E
                                                                            SM                    LC
                                                                                                       x LM / LC E $1 −
                                                                                                                   l        LC
                                                                                                                                 y LM / LC E $2
                                                                                                                                             l          (3.4.22)

where

                                                                   E   $ = [ C ]SM $
                                                                       &
                                                                       l1    &     l LA                                                                 (3.4.23)
                                                                            E SM



             E   &  1  E&
                 $ =
                 l2        l
                     mag  BJ / CA
                                    −      {                       (&
                                                                    l  E T
                                                                         BJ /CA
                                                                                E
                                                                                  1
                                                                                     $ + ElT
                                                                                     l            E& E$
                                                                                                   $ l −
                                                                                           BJ / CA l1  1      )         (   E T
                                                                                                                              l
                                                                                                                              BJ / CA
                                                                                                                                      E
                                                                                                                                        1
                                                                                                                                          $
                                                                                                                                          E
                                                                                                                                          l 1 )&
                                                                                                                                               $
                                                                                                                                               l


                                                                                                   )(                     )
                                                                                                                     & E $ 
                             T                  1  E T
                             (
                          −  E $2 E &BJ / CA −      )
                                                       l BJ/ CA E $1        (                                       $  l 
                                                                                                                           2
                                                                                                        E T       E
                                l l                                l                                      l          l                                  (3.4.24)
                                                mag                                                    BJ / CA   1
                                                                                                                          

                           [l                                      (                 2 −
                                                                                         )]
                                                                                              1
                                                                                              2
            mag ≡            E T       E
                                                l          −           E T
                                                                        l        E   $
                                                                                     l
                               BJ / CA   BJ / CA                         BJ / CA   1



                     E   &
                         l BJ / CA =   (   E
                                               r SP/ E +
                                               &           [C ]
                                                            &
                                                               E       SP
                                                                            SP
                                                                                 r LJ /SP − E r SM / E −
                                                                                              &               [C ]
                                                                                                              E
                                                                                                               &
                                                                                                                   SM
                                                                                                                            SM
                                                                                                                                  r LC/SM   )           (3.4.25)



Once    E
            LSD and              E   &
                                     LSD are known the force generated by the spring or damper may be

                                                                                                                                                   E
calculated. In most cases the length of the spring is not given directly by                                                                            LSD . An

example of this is a coil-over shock assembly. For this type of assembly the preload on the

spring is set by adjusting a ring which moves up and down the body of the damper on a




                                                                                         67
thread. For modeling purposes the preload on the spring can be specified by providing the

distance along the coil over shock assembly from the appropriate mount to the end of the

spring as an input. If the force-displacement curve of the spring is modeled as a

polynomial containing linear and cubic terms the expression for the spring force can be

written as


                               ∆ L = fS −             (   E
                                                              LS − pS   )                             (3.4.26)
                               E
                                   FS   (   E
                                                   )(                   $
                                                LS = k 0 ∆ L + k1∆ L3 E LS           )
where fs is the free length of the spring, ps is the preload length as described above, ELS is

the magnitude of the vector ELS,                  E   $
                                                      LS is the unit vector along the direction of ELS, ∆L is

the amount of compression applied to the spring, and k0 and k1 are the linear and cubic

stiffness coefficients.

        The force generated by the damper is typically dependent on the rate of change of

the length of the damper. The amount of damping is usually dependent on whether the

damper is extending (rebound) or compressing (jounce). The force generated by the

damper can be modeled as follows:

                          E
                              FD   ( L )= − (c
                                    E&
                                        D                 R0
                                                                E   &          &3 $
                                                                    LD + cR1 E LD E L D  )            (3.4.27)

where

                                        d
                                                  [L                  ]=
                                                                       1
                               E   &
                                   LD =               E   T E
                                                                LD
                                                                           2   E   $ &
                                                                                   LT E L D           (3.4.28)
                                                          D                         D
                                        dt
      $
and E L D is the unit vector along the direction of ELD.




                                                                      68
                                     δ
         The virtual work δ = E FSD ⋅ E LSD is obtained by combining the above results.
                           W

The generalized forces are the coefficients of the corresponding virtual displacements. The

results are summarized in Table 3.3 at the end of the chapter.


3.5 Constraint Forces for the Control Arms
         The motion of the front spindle is constrained by the action of the control arms and

the action of the steering linkage. The control arms and steering linkage are modeled using

constraint equations. The discussion of the steering system constraints is covered in the

chapter on the steering model. The constraint equations for the control arms are derived

below.

         There are a total of four control arms (two per side) used in the typical SLA front

suspension used on most race cars. The derivation procedure is identical for each of the

control arms. To eliminate redundancy the following derivation is done for a generic

control arm. The vectors describing the generic control arm are depicted in Figure 3.3.

         Each control arm can be represented by two constraint equations. The first

constraint forces the ball joint attachment point (BJ) on the spindle to remain a fixed

distance (i.e. the length of the control arm LCA) from the origin of the control arm

coordinate system (CA) which is affixed to the sprung mass. This can be written

mathematically as




                                              69
              E
                  r BJ / CA = E r SP/ E +   [C ]E      SP
                                                            SP
                                                                 r BJ /SP − E r SM / E −       [C ]
                                                                                               E   SM
                                                                                                            SM
                                                                                                                 r CA /SM        (3.5.1)


                                          [r                    r BJ /CA ] − LCA = 0
                                                                           1
                                            E       T       E              2
                                                    BJ / CA                                                                      (3.5.2)

The second condition is that the same vector be perpendicular to the axis of rotation of the

control arm. This can be expressed mathematically by noting that the inner product of the

vectors must be zero:

                                            E
                                                rBJ /CA [ CSM ]SM c CAA = 0
                                                 T
                                                         E
                                                                  $                                                              (3.5.3)

The generalized constraint forces are of the form

                                                        QCA,qi = λCA a qi                                                        (3.5.4)

where


                         ∂E T E                                                            ∂          
                                 [                                   ]−   LCA = E rBJ /CA  E r BJ /CA 
                                                                     1

              a qi =         rBJ /CA r BJ / CA                      2
                                                                               $T                                               (3.5.5)
                         ∂i 
                          q                                                               ∂q          
                                                                                                        i

or



                                                                                                                            $
                                                                                                                            s3

                                                                $
                                                                s1



                           $
                           p3                                                                                               SM
                                      c1 = c CAA
                                      $ $
                                                                               SM
                                                                                    rCA / SM
                                                      CA
                           SP                                BJ                                                  $
                                                                                                                 s2
         $
         p1                          SP
                                          rBJ /SP           $
                                                            c2

                                $
                                p2


                                 Figure 3.3: Schematic of a Generic Control Arm



                                                                           70
                     ∂
            a qi =
                     ∂i
                     q
                          (   E
                                  rBJ /CA [ CSM ]SM c CAA
                                   T
                                           E
                                                    $                    )
                                                                                                                                  (3.5.6)
                    ∂ T                                     ∂       
                 =  E rBJ /CA [ CSM ]SM c CAA + E rBJ / CA 
                                          $          T
                                                                 [ CSM ] SM c CAA
                                                                        $
                   ∂ i                                     ∂i       
                                 E                                E
                     q                                         q

           {
For qi ∈ xsp , ysp , zsp , xsm , ysm , z sm           }the        ∂ E
                                                                  ∂i
                                                                  q   BJ / CAr         term evaluates to


 ∂ E                                             ∂ E                                                ∂ E
     r
∂xsp BJ /CA
            =        (1       0 0)                   r
                                                ∂ysp BJ /CA
                                                            =                    (0    1 0)              r
                                                                                                   ∂z sp BJ / CA
                                                                                                                 =   (0   0 1)
                                                                                                                                  (3.5.7)
 ∂ E                        ∂ E                          ∂ E
     r BJ / CA = − (1 0 0)       r BJ / CA = − ( 0 1 0)       r       = − ( 0 0 1)
∂xsm                       ∂y sm                        ∂z sm BJ / CA

           { }
For qi ∈ βSP,i the            ∂ E
                              ∂i
                              q        r
                                  BJ / CA           term evaluates to


                                    ∂ E                ∂            
                                   ∂ SP,i
                                    β
                                          r BJ / CA = 
                                                       ∂ SP,i
                                                         β
                                                               [ CSP ] SP r BJ /SP
                                                                E
                                                                     
                                                                     
                                                                                                                                  (3.5.8)


           {
For qi ∈ βSM, i the  }            ∂ E
                                  ∂i
                                  q     r
                                      BJ / CA        term evaluates to


                            ∂ E                 ∂            
                          ∂ SM, i
                           β
                                  r BJ /CA = − 
                                                ∂ SM,i
                                                  β
                                                        [ CSM ] SM r CA /SM
                                                         E
                                                              
                                                              
                                                                                                                                  (3.5.9)


In order to determine the Lagrange multipliers it is necessary to append the second

derivative of the constraint equations to the equations of motion. The first and second

derivatives are calculated below. For the length constraint on the control arm:

                                   d
                                        {[ r                 r BJ /CA ] − LCA = 0            }
                                                                         1
                                            E    T       E               2
                                                 BJ / CA
                                   dt
                                        [r                  r BJ/ CA ]
                                                                     −   1

                                   ⇒        E   T       E                    E
                                                                                 rBJ / CA E r BJ / CA = 0
                                                                                  T
                                                                                            &
                                                                         2
                                                BJ / CA                                                                          (3.5.10)

                                   ⇒ E rBJ / CA E r BJ /CA = 0
                                        T
                                                  &


                                       { rBJ / CA r BJ / CA = 0}
                                    d E T E
                                    dt
                                                     &
                                                                                                                                 (3.5.11)
                                    ⇒ E rBJ / CA E & BJ / CA + E rBJ / CA E r BJ /CA = 0
                                         T
                                                   &
                                                   r             &T         &



                                                                                 71
For the orthogonality constraint:



                     dt
                         {
                     d E T
                          rBJ / CA [ CSM ]SM c CAA = 0
                                     E
                                              $                      }                                                         (3.5.12)
                     ⇒ E rBJ / CA [ CSM ]SM c CAA + E rBJ /CA [ CSM ]SM c CAA = 0
                         &T        E
                                            $          T
                                                               E
                                                                 &      $


  dt
     {
  d E T
       rBJ / CA [ CSM ]SM c CAA + E rBJ /CA E CSM SM c CAA = 0
        &         E
                           $         T        &      $[ ]                                }                                     (3.5.13)
  ⇒ E &BJ / CA [ CSM ]SM c CAA + 2 E rBJ / CA E CSM SM c CAA + E rBJ / CA
      rT
      &         E
                         $           &T         &      [ ]
                                                       $          T
                                                                                              [  &
                                                                                                 &
                                                                                               E C SM    ]    SM
                                                                                                                   c CAA = 0
                                                                                                                   $

The derivatives of the vector E r BJ / LC are

               E
                   r BJ / CA = E r SP/ E +
                   &             &           [C ]
                                             E
                                              &
                                                 SP
                                                      SP
                                                           r BJ /SP − E r SM / E −
                                                                        &            [C ]
                                                                                     E
                                                                                      &
                                                                                         SM
                                                                                              SM
                                                                                                   r CA /SM                    (3.5.14)

               E
                   r BJ / CA = E &SP/ E +
                   &
                   &             &
                                 r           [C ]
                                             E
                                              &
                                              &
                                                 SP
                                                      SP
                                                           r BJ /SP − E & SM / E −
                                                                        &
                                                                        r            [C ]
                                                                                     E
                                                                                      &
                                                                                      &
                                                                                         SM
                                                                                              SM
                                                                                                   r CA /SM                    (3.5.15)




                                                                    72
3.6 Summary of Results

   Table 3.1 - Front Suspension Kinetic and Potential Energy Terms                                          ( ( )−
                                                                                                            d
                                                                                                            dt
                                                                                                                   ∂ SP
                                                                                                                    T
                                                                                                                    &
                                                                                                                    qk
                                                                                                                                   ∂ SP
                                                                                                                                    T
                                                                                                                                    qk    +
                                                                                                                                              ∂ SP
                                                                                                                                               V
                                                                                                                                               qk    )
Generalized                                                                     Term
Coordinate
    xSP                                                                           &
                                                                                  &
                                                                             mSP xSP
    ySP                                                                           &
                                                                             mSP &SP
                                                                                  y
    z SP                                                                 mSP &SP + mSP g
                                                                             &
                                                                             z
    βSP,i                                                   T
                                     ∂SP ω SP/ E  SP
                                                   [ J SP ] SP ω SP/ E 
                                                              d
                                    
                                     ∂&                               
                                     βSP,i                 dt         

                                                                    E d  ∂SP ω         ∂SP ω SP/ E 
                                        (               )
                                                 ω SP/ E [SP J SP ]                                   
                                                         T
                                    +       SP
                                                                                SP/ E
                                                                                       − 
                                                                    dt  ∂&           
                                                                          βSP,i   ∂ SP,i β
                                                                                                       



            Table 3.2 - Wheel and Tire Rotational Energy Terms                                       ( ( )− )
                                                                                                       d
                                                                                                       dt
                                                                                                            ∂ SP
                                                                                                             T
                                                                                                             &
                                                                                                             qk
                                                                                                                            ∂ SP
                                                                                                                             T
                                                                                                                             qk


Generalized                                                                     Term
Coordinate
   φ
                   (                               )                (                                        )
                                                    T
     SP                SP
                            ω SP/E ×SP p 2 [SP J wheel ]
                                       $                                 SP            &
                                                                              ω SP/E + φ SP p 2
                                                                                              $
                                                                                        wheel

                                                  SP d                                                                                  
                   +    (   SP
                                    )
                                 pT [SP J wheel ]
                                 $2
                                                  dt
                                                                (   SP
                                                                                  )&
                                                                         ω SP/ E + & wheel SP p 2 +
                                                                                   φ          $                  SP             &
                                                                                                                      ω SP/ E × φ SP p 2 
                                                                                                                                 wheel
                                                                                                                                       $
                                                                                                                                         
    βSP,i
                  E d  ∂ ω T  ∂ ω T  SP
                                                                                          (                                           )
                          SP          SP
                            SP/E
                                   −     SP/ E 
                                                 [ J wheel ]                                  SP            &
                                                                                                   ω SP/E + φwheel SP p 2
                                                                                                                      $
                  dt  ∂&
                           βSP,i   ∂ SP,i 
                                        β
                                               
                    ∂ ω SP/E  SP        E d                                                                                               
                                                                     (                )
                      SP  T
                   
                 +           [ J wheel ]
                              
                                                                         SP            &
                                                                              ω SP/E + & wheel SP p 2 +
                                                                                       φ          $                   SP            &
                                                                                                                           ω SP/E × φ SP p 2 
                                                                                                                                           $
                    ∂&SP,i 
                        β
                                                                                                                                     wheel
                                           dt                                                                                               




                                                                73
                  Table 3.3 - Generalized Forces due to Spring or Damper QSD, k                            (       )
Generalized                                               Generalized Force
Coordinate
                                       LC y LM / LC E T
                                                                     { (                                          ) }
xSP , ySP , zSP
                                     −
                                       mag  SD
                                                      F α − E $1Tα E $1 − E $2 α E $2
                                                               l      l      lT     l   )          (
                             α = (1 0 0) for xSP , α = (0 1 0) for ySP and α = (0 0 1) for zSP

                                   LC y LM / LC E T
                                                                 { (                                             ) }
xSM , ySM , zSM
                                  
                                   mag  SD
                                                  F α − E $1Tα E $1 − E $2 α E $2
                                                           l      l      lT     l       )          (
                         α = (1 0 0) for xSM , α = (0 1 0) for ySM and α = (0 0 1) for zSM
     βSP,i
                                        LC y LM / LC E T
                                      −
                                        mag  SD
                                                       F α−     { (        E   $ Tα E $ −
                                                                                l1      )
                                                                                       l1              (   E T
                                                                                                             2
                                                                                                               $ α E$
                                                                                                               l  ) }
                                                                                                                    l2

                                                                  [
                                                                 ∂ E CSP ]SP
                                                         α≡                  r LJ /SP
                                                                   β
                                                                  ∂ SP,i
     βSM,i
                               [
                             ∂ C ]                                                                           [
                                                                                                             ∂ C ]   
                    E
                        FSD  E SM 
                         T

                             ∂ SM,i 
                                β
                                          (   SM
                                                   r UM /SM −   SM
                                                                     r LC/SM −  )       LC
                                                                                             x LM / LC E FSD  E SM SM $LA
                                                                                                          T

                                                                                                                β
                                                                                                              ∂ SM,i 
                                                                                                                        l

                      LC y LM / LC E T
                    −
                      mag 
                                              { (
                                     FSD α − α $1 + l BJ / CA γ $1 −
                                               TE
                                                  l E T         E
                                                                  l                 )          (   E T
                                                                                                       l
                                                                                                     BJ / CA
                                                                                                             E
                                                                                                               1
                                                                                                                   $ γ
                                                                                                                   l   )
                       T         1  E T                                            
                         (
                    −  E $2 α − 
                      
                          l       )
                                  mag 
                                               (
                                        l BJ / CA $1
                                                  E
                                                    l             )(   E T
                                                                        lBJ /CA     γE $2 
                                                                                      
                                                                                        )
                                                                                        l
                                                                                           
                                                                                           
                                                [
                                               ∂ E CSM ]SM               [ SM $
                                                                        ∂ E CSM ]
                                       α ≡−                r LC/SM , γ≡           l LA
                                                 β
                                                ∂ SM,i                    β
                                                                         ∂ SM, i




                                                          74
                                                                               (
            Table 3.4 - Generalized Forces due to Control Arm Length Constraint QCA, k   )
Generalized                                      Generalized Force
Coordinate
    xSP                                         λ E rBJ / CA (1 0 0)
                                                 CA
                                                    $T
    ySP                                         λ E rBJ / CA ( 0 1 0)
                                                 CA
                                                    $T
    z SP                                        λ E rBJ / CA ( 0 0 1)
                                                 CA
                                                    $T
    xSM                                        − λ E rBJ / CA (1 0 0)
                                                  CA
                                                     $T
    ySM                                        − λ E rBJ / CA (0 1 0)
                                                  CA
                                                     $T
    zSM                                        − λ E rBJ / CA (0 0 1)
                                                  CA
                                                     $T
    βSP,i
                                                          [ 
                                                        ∂ C ]
                                           λ E rBJ / CA  E SP  SP r BJ /SP
                                               $T
                                                           β
                                                         ∂ SP,i 
                                            CA


    βSM,i
                                                          [
                                                        ∂ C ]   
                                         − λ E rBJ / CA  E SM  SM r CA /SM
                                               $T
                                                           β
                                                         ∂ SM,i 
                                            CA




                                                75
                                                                              (
    Table 3.5 - Generalized Forces due to Control Arm Orthogonality Constraint QCA, k   )
Generalized                                 Generalized Force
Coordinate
    xSP                                λCA,2 (1 0 0)[ CSM ]SM c CAA
                                                     E
                                                              $
    ySP                                λCA,2 ( 0 1 0)[ CSM ]SM c CAA
                                                      E
                                                               $
    z SP                               λCA,2 ( 0 0 1)[ CSM ]SM c CAA
                                                      E
                                                               $
    xSM                                    [  c − (1 0 0)[ C ]SM c 
                                          ∂ E CSM ] SM
                       λCA,2 E rBJ / CA 
                                 T
                                          ∂x
                                                     $ CAA
                                                                        E SM
                                                                                $ CAA 
                             
                                             SM                                    
                                                                                      
    ySM                                    [  c − ( 0 1 0)[ C ]SM c 
                                          ∂ E CSM ] SM
                       λCA,2 E rBJ / CA 
                                 T
                                          ∂y
                                                     $ CAA
                                                                        E SM
                                                                                $ CAA 
                             
                                              SM                                   
                                                                                      
    zSM                                    [  c − ( 0 0 1)[ C ]SM c 
                                          ∂ E CSM ] SM
                       λCA,2 E rBJ / CA 
                                 T
                                          ∂z
                                                     $ CAA
                                                                        E SM
                                                                                $ CAA 
                             
                                             SM                                    
                                                                                      
    βSP,i
                                          [ ]
                                            ∂ E CSP  SP
                                    λCA,2  ∂              [      ]
                                                       r BJ /SP E CSM SM c CAA
                                                                         $
                                            βSP,i 
    βSM,i
                           [  r [ C ]SM c + E r T  ∂[ CSM ] SM c
                         ∂ E CSM ] SM
                  λCA,2 
                         ∂
                                                  $ CAA          E       $ CAA
                           βSM,i 
                                       CA /SM E SM       BJ / CA 
                                                                    β     
                                                                  ∂ SM,i 




                                            76
4 Equations of Motion - Three Link Rear Suspension

4.1 Introduction
        The equations derived in this chapter are for a rear suspension geometry which

consists of a solid rear axle supported by a three or four link mechanism (in the absence of

                                                          t
bushing compliance the fourth link is redundant and needn’ be explicitly modeled). This

geometry is similar to the layout used by the Legends series race cars and some dragsters.

The motion of the rear suspension is expressed in terms of three position coordinates and

four Euler parameters. As in the previous section the terms in the equations of motion

associated with the rear unsprung mass are found by formulating the kinetic energy and

the potential energy of the body and differentiating the energy expressions with respect to

the generalized coordinates.


4.2 Unsprung Mass Kinetic and Potential Energy Terms
        In order to calculate the kinetic energy associated with the rear suspension of the

vehicle model it is first necessary to calculate the velocity of the origin of the centroidal

coordinate system associated with the rear suspension. The velocity is found by

differentiating the position vector which is written as

                            E
                                r RS/ E = x RS e1 + y RS e 2 + z RS e 3
                                               $         $          $                   (4.2.1)




                                                      77
       The velocity vector can be expressed in terms of several different coordinate

systems. The Earth fixed inertial coordinate system (E) is chosen so that the expressions

for velocity, position and acceleration have simple dependence on the variables x, y, z and

their derivatives and to simplify computations of tire to road contact.

Differentiating Equation 4.2.1 gives

                                 E
                                     v RS/ E = ( x RS
                                                 &      &
                                                        y RS        z RS )
                                                                    &                          (4.2.2)

The kinetic energy of the rear suspension is

                      1                                1
              TRS =     mRS ( E v T E )( E v RS/ E ) +
                                  RS/
                                                             RS
                                                                  ω T E [RS J RS ]RS ω RS/ E
                                                                    RS/                        (4.2.3)
                      2                                2
       The mass of the rear suspension includes the mass of the rotating components

(wheels, tires and brakes) as well as the mass of the rear axle, differential and suspension.

The inertia tensor [RS J RS ] does not included the inertia of the rotating components. The

terms related to the angular velocity of the wheels are considered separately below.

       The form of the kinetic energy for the rear suspension is identical to the

expressions obtained for the sprung mass and the front unsprung masses. The resulting

terms in the equations of motion are identical as well.

                                         ETRS ,x RS = mRS &RS
                                                          &
                                                          x                                    (4.2.4)

                                         ETRS , yRS = mRS & RS
                                                          &
                                                          y                                    (4.2.5)

                                         ETRS , z RS = mRS &RS
                                                           z&                                  (4.2.6)




                                                        78
                                                     T
                           ∂ ω
                             RS                 d           
           ETRS ,βRS,i   =  & RS/ E  [RS J RS ] RS ω RS/ E 
                            ∂       
                            βRS,i               dt         
                                                                                                 (4.2.7)
                                                           d ∂RS ω         ∂RS ω RS/ E 
                              (                  )[ J RS ]                               
                                                 T RS
                          +       RS
                                       ω RS/ E                       RS/ E
                                                                           − 
                                                                  βRS,i   ∂ RS,i 
                                                           dt  ∂&              β
                                                                                         

The derivatives of            RS
                                   ω RS/ E which appear in the above expression are calculated in

Appendix A.

        The potential energy of the rear suspension consists only of a gravitational

                                                                       $
potential energy term. It is assumed that gravity acts parallel to the e3 axis. Given that this

is the case the potential energy is simply

                                                         VRS = mRS gz RS                         (4.2.8)

The terms in the equation of motion associated with the rear suspension potential energy

are

                                                                      ∂ RS
                                                                       V
                                                         EVRS ,qk =                              (4.2.9)
                                                                      ∂k
                                                                       q
Clearly the only nonzero derivative is the one associated with the zRS degree of freedom.

                                                         EVRS ,z = mRS g                        (4.2.10)


4.3 Rear Wheel Rotational Energy Terms
        The rear wheels are assumed to be rigidly affixed to the rear suspension and to

rotate about the     RS
                          $
                          r2 axis of the RS coordinate system. The kinetic energy and potential

energy terms associated with the translational motion of the rear wheels were derived by

lumping mass of the wheels, tires and brakes with the rest of the rear suspension. The

kinetic energy associated with the rotational motion of the wheels must be treated



                                                                      79
separately due to the relative rotation between the wheels and the RS coordinate system.

Due to the rotational symmetry of the wheels it is not necessary to utilize a distinct

coordinate system for each wheel; the inertia tensor is constant in the RS coordinate

system. The kinetic energy for a rear wheel can be written

                                      1
                            TRT =         RS
                                               ω T / E [RS J wheel ]RS ω wheel / E
                                                 wheel                                    (4.3.1)
                                      2
The angular velocity vector associated with the wheel has constant direction but variable

magnitude with respect to the RS coordinate system. The total angular velocity of the

wheel includes the angular velocity of the RS coordinate system.

                               RS                              &
                                    ω wheel / E = RS ω RS/ E + φ         RS
                                                                              $
                                                                              r2          (4.3.2)
                                                                wheel


where φwheel is the angular degree of freedom (scalar) associated with the rear wheel. The

value is time dependent and is dictated by the interaction of the vehicle and tire models.

The terms in the equations of motion are given by

                                                  E
                                                      d  ∂ RT  ∂ RT
                                                           T      T
                                    ETRT ,qk =                −                         (4.3.3)
                                                      dt  ∂&k  ∂ k
                                                            q     q
Considering just one of the wheels, substituting for the kinetic energy and differentiating

gives

                                                                        T
                       E d  ∂ ω wheel / E  ∂ ω wheel / E  RS
                               RS              RS
          ETRT ,qk   =                    −               [ J wheel ]RS ω wheel / E
                       dt  ∂&k  q              ∂k
                                                  q         
                                                                                          (4.3.4)
                                           T
                        ∂ ω wheel / E  RS
                          RS
                                                     E d RS             
                     +                 [ J wheel ]        ω wheel / E 
                        ∂&k q                      dt                 




                                                             80
The only degrees of freedom which generate non-zero results for the expression above are

the βRS,i and φ wheel. The derivatives of the wheel angular velocities are calculated as

follows


                   ∂ RS ω wheel / E ∂ RS ω RS/ E                           ∂RS ω wheel / E ∂RS ω RS/ E
                                   =                                                      =                                                    (4.3.5)
                      ∂ RS,i
                        β              β
                                      ∂ RS,i                                 ∂&RS,i
                                                                               β            ∂&RS,i
                                                                                             β

                                  ∂ RS ω wheel / E                        ∂ RS ω wheel / E RS
                                                   =0                          &          = r2$                                                (4.3.6)
                                      φ
                                     ∂ wheel                                  φ
                                                                             ∂ wheel


                                    (                                     )                         (                                     )
    E                        RS
        d RS                  d                        &                                                               &
             ω wheel / E =              RS
                                             ω RS/ E + φwheel
                                                                   RS
                                                                        r2 +
                                                                        $      RS
                                                                                    ω RS/ E ×           RS
                                                                                                             ω RS/ E + φwheel
                                                                                                                                RS
                                                                                                                                     $
                                                                                                                                     r2
        dt                    dt
                                                                                                                                               (4.3.7)

                                    (               )
                           RS
                              d                        &
                                                       &                                     &
                         =              RS
                                             ω RS/ E + φ wheel
                                                                   RS
                                                                        r2 +
                                                                        $      RS
                                                                                    ω RS/ E ×φwheel
                                                                                                                RS
                                                                                                                     $
                                                                                                                     r2
                              dt

                                                                                                                                               (4.3.8)

                     E
                         d ∂ RS ω wheel / E E d RS      RS


                         dt ∂ wheel
                               φ&          =
                                              dt
                                                   $2 =
                                                   r
                                                           d RS
                                                           dt
                                                              ( r2 )+
                                                                $                              RS
                                                                                                    ω RS/ E ×RS r2
                                                                                                                $
                                                                                                                                               (4.3.9)
                                                  = ω RS/ E × r2
                                                    RS
                                                              $     RS



For qk = βRS,i and substituting for the derivatives of                                              RS
                                                                                                         ω wheel / E , using the expressions

above, the result becomes


              E d ∂RS ω T  ∂ ω T  RS
                                                                                    (                                     )
                                   RS
 ETRT ,qk   =           RS/ E
                                −     RS/ E 
                                              [ J wheel ]                               RS             &
                                                                                             ω RS/ E + φ RS r2$
              dt  ∂&
                      βRS,i     ∂ RS,i 
                                     β
                                                                                                        wheel
                                            
                                                                                                                                              (4.3.10)
              ∂RS ω T E  RS         RS d                                                                                
              
            +       RS/

               ∂&RS,i 
                  β
                         [ J wheel ]
                                     dt
                                                          (   RS
                                                                   ω RS/ E     )   + & wheel RS r2 + RS ω RS/ E × φ RS r2 
                                                                                     &
                                                                                     φ          $                 &
                                                                                                                   wheel
                                                                                                                         $
                                                                                                                           

             {
For qk ∈ φ , φright
          left               }the expression simplifies as follows



                                                                         81
                 (                      )                      (                             )
                                           T
     ETRT ,φ =       RS
                          ω RS/ E ×RS r2 [ RS J wheel ]
                                      $                            RS             &
                                                                        ω RS/ E + φ RS r2$
                                                                                   right

                                         RS d                                                                     (4.3.11)
             +   RS
                      r2T [ RS J wheel ]
                      $
                                         dt
                                                    (   RS             &
                                                                       &)
                                                             ω RS/ E + φ RS r2 +
                                                                         wheel
                                                                               $        RS            &
                                                                                             ω RS/ E ×φ RS r2 
                                                                                                       wheel
                                                                                                             $
                                                                                                               


4.4 Rear Springs and Dampers
       The forces exerted by the springs and dampers which connect the rear suspension

to the chassis of the vehicle depend on the length and on the rate of change of the length

of the springs. The positions of the upper attachment points to the chassis are given by

four vectors which are constant with respect to the sprung mass coordinate system SM.

Likewise, the lower attachment points are given by four vectors which are constant with

respect to the rear unsprung mass coordinate system RS.

       To determine the terms in the equations of motion which are generated in

connection with the spring and damping elements it is necessary to determine the

magnitude and direction of the force and then to find the generalized forces. The length of

the spring or damper under consideration is

             E
                 L i = E r SM / E +   [ CSM ]SM rupper /SM − E r RS/ E − [ C RS ] RS rlower / RS
                                       E                                  E                                          (4.4.1)

The rate of change of the length is required to determine the force exerted on the system

by the dampers.

             E   &
                 L i = E r SM / E +
                         &            [C ]
                                       &
                                       E       SM
                                                    SM
                                                         rupper /SM − E r RS/ E −
                                                                        &           [C ]
                                                                                    E
                                                                                     &
                                                                                        RS
                                                                                                 RS
                                                                                                      rlower / RS    (4.4.2)

The forces generated by the springs and by the dampers are determined in the same

manner as was done for the front suspension in the previous chapter. For the springs,




                                                                            82
                                     ∆ L = fS −              (   E
                                                                     LS − pS       )                                                                         (4.4.3)
                                     E
                                         FS    (   E
                                                            )(                 $
                                                       LS = k 0 ∆ L + k1∆ L3 E LS                         )
where


                                                                  [L                      ]
                                                                                              1
                                                       E
                                                           LS =      E       TE                   2
                                                                             S    LS                                                                         (4.4.4)

and for the dampers,

                              E
                                  FD    ( L )= − (c
                                         E&
                                                 D                R0
                                                                         E   &          &3 $
                                                                             LD + cR1 E LD E L D                )                                            (4.4.5)

where

                                             d
                                                           [L                  ]=
                                                                                  1
                                    E   &
                                        LD =                E     T E
                                                                         LD
                                                                                      2       E   $ &
                                                                                                  LT E L D                                                   (4.4.6)
                                                                  D                                D
                                             dt
The generalized forces are found by determining the virtual work done by the springs and

dampers. The total virtual work is


                      δ =
                       W            ∑        E
                                                             δ
                                                 Fspring, i ⋅ E L i +                     ∑           E
                                                                                                          Fdampers, j ⋅ E L j
                                                                                                                       δ                                     (4.4.7)
                                  i ∈springs                                      j∈ dampers


The virtual displacement δ L i is the displacement at the spring or damper caused by an
                          E




infinitesimal contemporaneous perturbation of the generalized coordinates. The

displacement is equivalent to the total derivative of the expression for the length of the

spring or damper.

 E
            (
δ L i = δE r SM / E +   [C ]
                          E   SM
                                        SM
                                             rupper /SM − E r RS/ E −              [C ]   E           RS
                                                                                                               RS
                                                                                                                    rlower / RS   )
δ L i = (1 0 0)( δSM − δRS ) + ( 0 1 0)( δSM − δRS ) + ( 0 0 1)( δSM − δRS )
 E
                  x     x                 y    y                  z     z                                                                                    (4.4.8)

                 ∂                     SM                                                    ∂                                 RS
        +   ∑ ∂ [ C ]
               β
              
                     
                     
                     
                          E   SM                   rupper /SM δ SM, j −
                                                              β                       ∑ ∂ [ C ]
                                                                                         β
                                                                                        
                                                                                               
                                                                                               
                                                                                               
                                                                                                                      E    RS                      β
                                                                                                                                        rlower / RSδ RS, j
            j     SM, j                                                                   j                   RS, j




                                                                               83
                                                             β          β
The generalized forces are equal to the coefficients of the δ SM,j and δ RS, j terms in the

expression for the virtual work δ . The generalized forces for a single spring or damper
                                 W

are given below.


                                       QRS,xSM = E F ⋅(1 0 0)
                                       QRS,ySM = E F ⋅( 0 1 0)                                    (4.4.9)

                                       QRS,zSM = E F ⋅( 0 0 1)


                                                  [  r
                                                ∂ E CSM ] SM
                                               
                               QRS,βSM,j = E F ⋅           upper /SM
                                                ∂                                              (4.4.10)
                                                βSM, j 

                                                        (
                                      QRS,x RS = − E F ⋅ 1 0 0)
                                      QRS,y RS = − E F ⋅(0 1 0)                                  (4.4.11)

                                      QRS,z RS = − E F ⋅(0 0 1)

                                                   [ 
                                                 ∂ C ]
                                                 
                               QRS,βRS,j = − E F ⋅ E RS RS rlower / RS
                                                  ∂                                            (4.4.12)
                                                  βRS, j 


4.5 Panhard Rod and Trailing Link Constraints
        The rear suspension is located laterally by a panhard rod and longitudinally by

multiple trailing links. These mechanical constraints are represented by an equation which

forces the attachment points of the links to remain at a fixed distance from each other. The

constraint equation for the panhard rod is derived below. The constraints for the trailing

links are identical in form.

The vector which lies along the panhard rod is given by the expression

             E
                 p = E r SM/ E + [ CSM ]SM rpanhard /SM − E r RS/ E − [ C RS ]RS rpanhard / RS
                                  E                                    E                          (4.5.1)




                                                          84
The locations of the points of attachment of the panhard rod to the chassis and to the rear

                                                            SM                      RS
suspension are given by the constant vectors                     rpanhard /SM and        rpanhard /RS . For a panhard

rod of fixed length L, the following constraint equation can be written.


                                      [ p ⋅ p] −
                                                   1
                                       E       E
                                                        L=0
                                                   2
                                                                                                                 (4.5.2)

The generalized forces associated with the panhard rod constraint are of the form

                                           QC ,βi = λC a βi                                                      (4.5.3)

where


                          ∂  E E 12                   − 1      ∂ p
                                                                  E
                 aβ i =      [ p ⋅ p] − L = [ p ⋅E p] 2  E p ⋅
                                           E                       
                          ∂i
                          β                                     β
                                                                  ∂i
                                                                                                                 (4.5.4)
                         ∂p
                          E
                     = p⋅
                       $  E
                          β
                          ∂i
                                                                                                             E
        $
where E p is the unit vector along the panhard rod. Evaluating the derivatives of                             p and

substituting gives


                                 QC , xSM = λP E pT ⋅(1 0 0)
                                                 $
                                 QC , ySM = λP E p T ⋅( 0 1 0)
                                                 $                                                               (4.5.5)

                                 QC , zSM = λP E p T ⋅( 0 0 1)
                                                 $


                                                 [ 
                                               ∂ C ]
                                           $  E SM  SM rpanhard /SM
                          QC ,βSM,i = λP E pT ⋅ ∂                                                              (4.5.6)
                                                βSM, i 

                                                  $ (
                                QC , x RS = − λ E pT ⋅ 1 0 0)
                                               P

                                QC , y RS = − λ E p T ⋅(0 1 0)
                                               P
                                                  $                                                              (4.5.7)

                                QC , z RS = − λ E p T ⋅(0 0 1)
                                               P
                                                  $




                                                       85
                                                 [ 
                                               ∂ C ]
                         QC ,βRS,i = − λ E pT ⋅ E RS 
                                           $                                   RS
                                                                                    rpanhard / RS                    (4.5.8)
                                                  β
                                                ∂ RS,i 
                                        P



The Lagrange multiplier λP is determined by appending the second time derivative of the

constraint equation to the equations of motion. The first and second time derivatives of

the constraint equation are calculated below.

                                     d
                                          {[   ]                           }
                                                              1
                                               E
                                                   p ⋅E p 2 − L = 0
                                     dt
                                     ⇒ [ p ⋅ p] (                          )
                                                          −   1
                                           E         E            E
                                                                      p ⋅E p = 0
                                                                           &
                                                              2
                                                                                                                     (4.5.9)

                                     ⇒ E p ⋅E p = 0
                                              &


                                             { p ⋅ p = 0}
                                          d E E
                                          dt
                                                   &
                                                                                                                    (4.5.10)
                                          ⇒ E p⋅ p + E p⋅ p = 0
                                                E
                                                  & &E&
                                                  &
The derivatives of the panhard rod vector are computed as follows.

           E
               p = E r SM/ E + [ CSM ]SM rpanhard /SM − E r RS/ E − [ C RS ]RS rpanhard / RS
                                E                                    E
           E
               p = E r SM/ E +
               &     &           [C ]
                                 E
                                  &
                                     SM
                                           SM
                                                   rpanhard /SM − E r RS/ E −
                                                                    &            [C ]
                                                                                  &
                                                                                    E    RS
                                                                                              RS
                                                                                                    rpanhard / RS   (4.5.11)
           E
               p = E & SM/ E
               & r
               &     &         + [C ]
                                 E
                                  &
                                  &
                                     SM
                                           SM
                                                   rpanhard /SM − E r RS/ E
                                                                    &
                                                                    &          − [C ]
                                                                                  &
                                                                                  &
                                                                                    E    RS
                                                                                              RS
                                                                                                    rpanhard / RS


4.6 Summary of Results

       Table 4.1 Kinetic and Potential Energy Terms for the Motion of the Rear Suspension

Generalized           Term
Coordinate
     xSM                  &
                          &
                      mRS x RS
     ySM                   &
                      mRS & RS
                          y
     zSM              mRS ( &RS + g )
                            z&




                                                                      86
     βRS,i          ∂ ω RS/ E  RS
                                           T

                                [ J RS ] RS ω RS/ E 
                      RS
                                           d
                   
                    ∂&                             
                    βRS,i               dt         

                                                    d ∂RS ω         ∂RS ω RS/ E 
                       (               )
                                 ω RS/ E [RS J RS ]                               
                                        T
                   +        RS                                RS/ E
                                                                    − 
                                                    dt  ∂&                        
                                                     βRS,i   ∂ RS,i β




          Table 4.2 Kinetic Energy Terms for the Rotation of the Rear Wheels and Tires

Generalized        Term
Coordinate
    βRS,i           E d ∂ ω T  ∂ ω T 
                                                                                        (                              )
                           RS          RS
                            RS/ E
                                    −     RS/ E  RS                                       RS             &
                                                                                                 ω RS/ E + φ RS r2$
                    dt  ∂&
                          βRS,i 
                                      ∂ RS,i 
                                         β
                                                  [ J wheel ]                                               wheel
                                                
                      ∂ ω T E  RS
                        RS
                                             E d RS                                                                               
                   +  &
                      ∂
                            RS/
                                [ J wheel ]
                          βRS,i             dt
                                                 ( ω RS/ E ) + &&wheel RS r2 +
                                                               φ          $                                 RS            &
                                                                                                                 ω RS/ E ×φ RS r2 
                                                                                                                           wheel
                                                                                                                                 $
                                                                                                                                   
                               
     φwheel
                   (                           )                   (                                )
                                               T
                       RS
                            ω RS/ E ×RS r2 [RS J wheel ]
                                        $                              RS             &
                                                                            ω RS/ E + φ RS r2$
                                                                                       right

                                               RS d                                                                      
                   +   RS
                            r2T [ RS J wheel ]
                            $
                                               dt
                                                         (   RS             &
                                                                            &)
                                                                  ω RS/ E + φwheel RS r2 +
                                                                                      $            RS            &
                                                                                                        ω RS/ E ×φ RS r2 
                                                                                                                  wheel
                                                                                                                        $
                                                                                                                          




              Table 4.3 Generalized Forces Associated with a Rear Spring or Damper

Generalized        Generalized Force
Coordinate
 xSM, ySM, zSM     E
                       F ⋅(1 0 0),                 E
                                                       F ⋅( 0 1 0),              E
                                                                                     F ⋅( 0 0 1)
     βSM,i                 [  r
                         ∂ E CSM ] SM
                   E    
                       F⋅          
                         ∂ βSM, j  upper /SM
                                  
  xRS, yRS, zRS    − E F ⋅(1 0 0), − E F ⋅(0 1 0), − E F ⋅(0 0 1)
     βRS,i                [  r
                        ∂ E CRS ] RS
                       
                   − EF⋅           lower / RS
                        ∂        
                        βRS, j 




                                                                   87
                  Table 4.4 Constraint Forces Associated with the Panhard Rod

Generalized       Generalized Constraint Force
Coordinate
 xSM, ySM, zSM     λP E p T ⋅(1 0 0), λP E pT ⋅( 0 1 0), λP E pT ⋅( 0 0 1)
                        $                  $                  $
     βSM,i                    [ 
                            ∂ C ]
                        $  E SM  SM rpanhard /SM
                   λP E pT ⋅ ∂      
                             βSM, i 
  xRS, yRS, zRS    − λ E pT ⋅(1 0 0), − λ E p T ⋅(0 1 0), − λ E p T ⋅(0 0 1)
                      P
                         $               P
                                            $                P
                                                                $
     βRS,i                        − 1     ∂                                                         
                   − λ P [ p ⋅E p]  E p ⋅
                          E         2

                                      
                                          ∂ RS,i
                                             β
                                                   [ C RS ]
                                                    E
                                                          
                                                          
                                                              (   RS
                                                                       rRS/ piv +   RS
                                                                                                     )
                                                                                         rpanhard / RS 
                                                                                                       
                                                                                                       




                                              88
5 Equations of Motion - Steering System

5.1 Introduction
        The steering system model is tightly integrated with the chassis model and with the

front suspension model. Since the input to the steering model is an angular displacement

of the steering wheel it is not necessary to consider the inertia of the steering wheel. For

this reason, and also to reduce the total number of degrees of freedom, the steering system

is modeled as a quasi-static system. If the driver model were to interact with the steering

wheel via a prescribed torque it would be better to include the inertia of the wheel in the

model. Two types of steering systems are commonly used. The simpler of the two is the

rack and pinion type steering system. The second type is based on a four bar linkage.


5.2 Rack and Pinion
The rack and pinion steering system is modeled almost trivially. A schematic of the system

is shown Figure 5.1. The location of the center of the steering rack relative to the sprung

                                                   SM
mass coordinate system is given by the vector           R R /SM . The steering rack coordinate

system in this case is simply a displaced version of the sprung mass coordinate system; the

$      $
r1 and r2 unit vectors are parallel to the corresponding unit vectors of the sprung mass

coordinate system. The locations of the tie rod connection points on the rack are given by

the relations



                                             89
                                                         w
                                                          $
                                                          r1
                                      $
                                      r2

                                                              R
                                                               SM
                                                                    R R / SM

                                θsw                                  $
                                                                     s1


                                                                     SM
                                                   $
                                                   s2


                  Figure 5.1 Schematic of the Rack and Pinion Steering System




                                                           w
                       SM
                            R LTR/SM = SM R R /SM + G θsw + s 2
                                                                $
                                                            2
                                                                                       (5.2.1)
                                                                  w
                       SM
                            R RTR /SM =   SM
                                               R R /SM   + G θsw −  s 2
                                                                       $
                                                                   2
where G is the gain of the rack and pinion with units of length/angle. The remainder of the

steering linkage is represented by a constraint equation which forces the steering knuckle

joint on the left and right spindles to remain a fixed distance from the two points given

above.


5.3 Four Bar Linkage
The four bar linkage type of steering mechanism is considerably more difficult to model

than the rack and pinion type mechanism. To facilitate modeling the components of the

four bar steering mechanism are assumed to be composed of rigid bodies. The gearbox is

modeled as a simple gear reduction without losses. The gearbox output drives the pitman


                                                         90
arm which is rigidly attached to the drag link. The pitman arm, idler arm and drag link are

modeled as a planar four bar linkage. The position of the four bar linkage is completely

specified by the steering wheel angle input. The drag link is connected to the steering

knuckles via tie rods which are represented by constraint equations.

       The plane of the four bar linkage is typically not aligned with the                    s1 − s 2 plane of
                                                                                              $ $

the sprung mass coordinate system. Given the axis of rotation of the pitman arm (and

assuming a parallel axis of rotation for the idler arm) a coordinate system P (for pitman

arm) can be defined which contains the four bar linkage in its                 p1 − p 2 base plane. A
                                                                               $ $

schematic showing the relationship between the sprung mass and pitman arm coordinate

systems is shown in Figure 5.2. Note that the view shown in the figure is not necessarily in

the s1 − s 2 plane. The orientation of the coordinate system is chosen such that p1 lies in
    $ $                                                                          $




                  $
                  p1


                                      P
                                          $
                                          rPA / IA
  $
  p2          P                                                                 P
                                                                                    $
                                                                                    rIJ /IA
                       P
                           $
                           rPJ / PA                                                           Idler Arm
        Pitman Arm                                   Drag Link

                                                                            Right Tie Rod
               Left Tie Rod                                                                            $
                                                                                                       s1
                                                             SM
                                                                  $
                                                                  rPA /SM
                                                                                                       $
                                                                                                       s3
                                                                                    $
                                                                                    s2                S

              Figure 5.2: Relationship Between the P and the S Coordinate Systems



                                                     91
                                                                                $
                                                                                p1

                   $
                   d1

                                        Pitman Arm
                                                                    $
                                                                    p2         P
 $
 d2
                                  D
                                      $
                                      rLTR / D                                           Idler Arm
               D                                              D
                                                                  $
                                                                  rRTR/ D




               Figure 5.3: Relationship Between the D and the P Coordinate Systems



the s1 − s3 plane and p 2 is equal to s 2 . Note that this choice only allows the plane of the
    $ $               $               $

                                         $
four bar linkage to be rotated about the s 2 axis relative to the sprung mass coordinate

system (i.e. only fore-aft tilt of the pitman/idler arm rotational axis is allowed). The

          $
vectors P rPJ / PA and   P
                             $
                             rIJ / IA represent the pitman arm and the idler arm respectively; the

points PJ and IJ refer to the pitman arm joint and the idler arm joint while points PA and

IA refer to the rotational axes of the pitman arm and the idler arm.

        A second coordinate system D, attached to the drag link, is defined in order to

                                                          $
locate the tie rod joints which lie on the drag link. The d 2 axis is chosen to lie along the

       $
vector rPJ /IJ (from the idler arm joint on the drag link to the pitman arm joint on the drag

                       $                                   $
link). The unit vector d1 is chosen to be perpendicular to d 2 while at the same time

remaining in the p1 − p 2 plane; it’ direction is chosen to point in roughly the same
                 $ $                s

             $
direction as p1 (in the direction of forward motion). A schematic showing the relationship




                                                     92
between the P and D coordinate systems is shown in Figure 5.3. The locations of the tie

rod joints on the drag link are given by the vectors                   D
                                                                           $              $
                                                                           rLTR / D and D rRTR / D .

        The first step in modeling the system is to determine the unit vectors of the P

coordinate system which defines the plane of the four bar linkage. The origin of the P

coordinate system is on the axis of rotation of the pitman arm and located in the plane of

rotation of the pitman arm to drag link joint. Given that the axes of rotation for the pitman

and idler arms are assumed to be parallel to the s1 − s3 plane the following constraints
                                                 $ $

                         $
must be satisfied by the p1 unit vector.

                               SM
                                    p1 ⋅SM p3 = SM p1 ⋅SM a axis = 0
                                    $      $       $      $                                                (5.3.1)

                                         SM
                                                  p1 ⋅SM s 2 = 0
                                                  $      $                                                 (5.3.2)


                                                       p1 = 1
                                                       $                                                   (5.3.3)

The unit vector    SM
                        $
                        a axis represents the axes of rotation of the pitman arm and idler arm.

Applying the constraints gives the following results.

                                                                                                       1
                                                                                                   2
                  SM
                       p1 = η 0 −
                       $                (   SM
                                                  $
                                                  a axis, x
                                                              )       
                                                              η; η ≡ 
                                                               
                                                                                   1                

                                                                              (                    )
                                            SM
                                                  $
                                                  a axis, z                      SM               2      (5.3.4)
                                                                                       $
                                                                      1 +                          
                                                                                       a axis, x

                                                                                                   
                                                                                  SM
                                                                                       $
                                                                                       a axis, z


                                                  SM
                                                         p2 = s2
                                                         $ $                                               (5.3.5)

                                             SM
                                                    p 3 = SM a axis
                                                    $        $                                             (5.3.6)

        To locate the tie rod joints on the drag link it is first necessary to determine the

orientation of the drag link and of the idler arm.




                                                                  93
                                $
                                p1                                                                              $
                                                                                                                p1


           θP + θP0                                                                          θI + θI0
                                                                             SM
                                                                                  $
                                                                                  rPA / IA

                       SM
                            $
                            rPJ / PA
                                                                                                         SM
                                                                                                               $
                                                                                                               rIJ / IA
                                                           L DL


                  Figure 5.4: Schematic of the Four Bar Linkage Steering System



                                                                                                        SM
         The geometry of the problem is shown in Figure 5.4. The vectors                                     rPJ / PA and

SM
     rIJ /IA represent the position of the pitman arm and of the idler arm respectively. The

angles θP0 is the between the p1 unit vector and the pitman arm for zero steer angle
                              $

(straight ahead steering). θI0 is defined in the same manner but for the idler arm.                                   θP is

related to the steering wheel angle through a gear reduction. The vectors which represent

the pitman arm and the idler arm can be written in terms of the previously defined angles.

                 SM
                      rPJ / PA = L P [ θP + θP0 )SM p1 + sin( θP + θP0 )SM p 2 ]
                                     cos(           $                      $                                       (5.3.7)

                  SM
                       rIJ / IA = L I [ θI + θI0 )SM p1 + sin( θI + θI0 )SM p 2 ]
                                      cos(           $                      $                                      (5.3.8)


         The unknown angle θI , and hence the location of the drag link, can be obtained

writing a constraint equation which enforces connectivity between the element of the four

bar linkage.

                                   SM
                                        rPJ / PA +   SM
                                                          rPA /IA −   SM
                                                                           rIJ /IA = L DL                          (5.3.9)



                                                                      94
Expanding the equation gives the following result:


                    [
                   L I L P cos(θP + θP0 )+                             SM
                                                                                           ]
                                                                              rPA / IA,1 cos(θI + θI0 )
                  + L [ sin( θ
                      L     I            P               P    + θP0 )+       SM
                                                                                  rPA / IA,2]sin( θ + θ I         I0   )
                  = (L + L +                                                                 − L )
                                                                                                                                 (5.3.10)
                    1
                    2
                                    2
                                    P
                                                 2
                                                 I
                                                         SM
                                                              rPA / IA,1 +
                                                               2             SM     2
                                                                                   rPA / IA,2          2
                                                                                                       DL

                  + L [ r   P
                                        SM
                                                PA / IA,1     cos(θP + θP0 ) +             SM
                                                                                                rPA / IA,2 sin( θP + θP0 ) ]
Making the following definitions


                       [
                 α = L I L P cos(θP + θP0 )+                                  SM
                                                                                   rPA /IA,1    ]
                 β = L [ sin( θ
                       L        I           P             P   + θP0 )+       SM
                                                                                  rPA /IA,2     ]
                 γ (L + L +                                                                                  )
                                                                                                                                 (5.3.11)
                   =    1
                        2
                                        2
                                        P
                                                     2
                                                     I
                                                          SM
                                                               rPA / IA,1 +
                                                                2             SM
                                                                                    rPA /IA,2 − L2DL
                                                                                     2



                   + L [ r          P
                                         SM
                                                 PA /IA,1      cos(θP + θP0 ) +               SM
                                                                                                   rPA /IA,2 sin( θP + θP0 ) ]
and substituting gives

                                                α cos(θI + θI0 ) + β sin( θI + θI0 ) = γ                                         (5.3.12)

which can be solved for θI by application of Newton’ method. Using an initial guess of
                                                    s

θI = 0 appears to work well.

          Once the locations of the pitman arm and of the idler arm are known the vector

SM
     rPJ / IJ can be determined

                                                SM
                                                     rPJ / IJ = SM rPJ / PA +        SM
                                                                                          rPA / IA −   SM
                                                                                                            rIJ / IA             (5.3.13)

The unit vectors of the D coordinate system can be determined at this point.

                                                                                    SM
                                                                       $                 rPJ /IJ
                                                                  SM
                                                                       d2 =         SM
                                                                                                                                 (5.3.14)
                                                                                         rPJ /IJ




                                                                                         95
                                SM   $     (    ) $            )      ) $
                                     d1 = cos( φ SM p1,1 sin( φ cos( φ SM p1,3                                )
                                               −              SM
                                                                    p1,1 SM d 2,1 + SM p1,3 SM d 2,3 
                                                                    $       $          $       $                                              (5.3.15)
                                     φ= arctan 
                                                                           SM $
                                                                                                     
                                                                                                     
                                                                                d  2, 2             

                                                      SM    $
                                                            d 3 = SM p 3 = SM a axis
                                                                     $                                                                        (5.3.16)

Finally, the locations of the tie rod joints can be expressed as

  SM
       rLTR/SM =    (    D                  $
                             rLTR / PJ,1 SM d1        D                 $
                                                          rLTR/ PJ,2 SM d 2         D                  $ )
                                                                                        rLTR / PJ,3 SM d 3 +   SM
                                                                                                                    rPJ / PA +   SM
                                                                                                                                      rP/SM   (5.3.17)

  SM
       rRTR / P =   (   D                 $
                            rRTR/ PJ,1 SM d1         D                  $
                                                         rRTR / PJ,2 SM d 2        D                  $  )
                                                                                       rRTR / PJ,3 SM d 3 +    SM
                                                                                                                    rPJ / PA +   SM
                                                                                                                                      rP/SM   (5.3.18)


5.4 Tie Rod Constraints
          Given the locations of the tie rod joints (for either the rack and pinion steering

system or the four bar link steering system) found in the previous sections a pair of

constraint equations can be written to connect the steering mechanism with the steering

knuckles on the spindles. The constraint equation simply expresses the fact that the

distance between the ball joint on the steering knuckle and the tie rod joint on the steering

mechanism is a constant. That is,

                    E
                        r KJ /TJ = E r SP/ E +   [C ]
                                                  E       SP
                                                                SP
                                                                     r KJ /SP − E r SM / E −   [C ]
                                                                                               E    SM
                                                                                                         SM
                                                                                                               r TJ /SM                        (5.4.1)


                                                 [r                 r KJ / TJ ] − LTR = 0
                                                                             1
                                                 E    T       E              2
                                                      KJ / TJ                                                                                  (5.4.2)

where TJ indicates the tie rod joint, KJ indicates the steering knuckle joint and LTR is the

length of the tie rod.

          The generalized constraint forces are of the form

                                                            QSP ,qi = λSP a qi                                                                 (5.4.3)




                                                                              96
where


             ∂E T E                                                    − 1             ∂           
                [ rKJ / TJ r KJ / TJ ] − LTR  = [ rKJ /TJ E r KJ /TJ ] 2  E rKJ / TJ  E r KJ / TJ 
                                       1

    a qi =
                                       2
                                               E T                             T
             ∂i
             q                                                                       ∂ i
                                                                                          q           
                                                                                                                            (5.4.4)


             {
For qi ∈ xsp , ysp , zsp , xsm , ysm , z sm     }the        ∂ E
                                                            ∂i
                                                            q   KJ / TJ r        term evaluates to


  ∂ E                                      ∂ E                                                 ∂ E
      r
 ∂xsp KJ / TJ
              =       (1   0 0)                 r
                                          ∂y sp KJ / TJ
                                                        =                   (0    1 0)              r
                                                                                              ∂z sp KJ / TJ
                                                                                                            =   (0   0 1)
                                                                                                                            (5.4.5)
  ∂ E                        ∂ E                         ∂ E
      r KJ / TJ = − (1 0 0)      r KJ / TJ = − ( 0 1 0)       r       = − ( 0 0 1)
 ∂xsm                       ∂ysm                        ∂z sm KJ / TJ

             { }
For qi ∈ βSP,i the          ∂ E
                            ∂i
                            q   r
                                KJ / TJ     term evaluates to


                              ∂ E                ∂            
                             ∂ SP,i
                              β
                                    r KJ / TJ = 
                                                 ∂ SP,i
                                                   β
                                                         [ CSP ] SP r KJ /SP
                                                          E
                                                               
                                                               
                                                                                                                            (5.4.6)


             {
For qi ∈ βSM, i the }       ∂ E
                            ∂i
                            q     r
                                KJ / TJ      term evaluates to


                             ∂ E                 ∂             
                           ∂ SM, i
                            β
                                   r KJ /TJ = − 
                                                 ∂ SM, i
                                                   β
                                                          [ CSM ] SM r TJ /SM
                                                           E
                                                                
                                                                
                                                                                                                            (5.4.7)


In addition to the generalized forces the second time derivative of the constraint equation

is required in order to solve for the Lagrange multiplier. The first derivative is calculated

as follows

                               d
                                    {[ r               r KJ / TJ ] − LTR = 0            }
                                                                    1
                                      E    T       E                2
                                           KJ / TJ
                               dt
                                    [r                r KJ / TJ ]
                                                               −    1

                              ⇒       E   T       E                     E
                                                                            rKJ / TJ E r KJ / TJ = 0
                                                                             T
                                                                                       &
                                                                    2
                                          KJ / TJ                                                                           (5.4.8)

                              ⇒ E rKJ / TJ E r KJ / TJ = 0
                                   T
                                             &
and the second derivative is




                                                                            97
                                     { rKJ / TJ r KJ / TJ = 0}
                                  d E T E
                                  dt
                                                    &
                                                                                                            (5.4.9)
                                 ⇒ E rKJ / TJ E r KJ / TJ + E rKJ / TJ E & KJ / TJ = 0
                                     &T         &              T
                                                                         &
                                                                         r
The first and second time derivatives of the position vector are simply

             E
                 r KJ /TJ = E r SP/ E +
                 &            &           [C ]
                                          E
                                           &
                                              SP
                                                   SP
                                                        r KJ /SP − E r SM / E −
                                                                     &            [C ]
                                                                                  E
                                                                                   &
                                                                                      SM
                                                                                           SM
                                                                                                r TJ /SM   (5.4.10)

             E
                 r KJ /TJ = E r SP/ E +
                 &
                 &            &
                              &           [C ]
                                          E
                                           &
                                           &
                                              SP
                                                   SP
                                                        r KJ /SP − E & SM / E −
                                                                     &
                                                                     r            [C ]
                                                                                  E
                                                                                   &
                                                                                   &
                                                                                      SM
                                                                                           SM
                                                                                                r TJ /SM   (5.4.11)




                                                                 98
6 Road Model

6.1 Introduction
          The road model is a critical component of the vehicle simulation. If the complete

simulation is to accurately represent reality the road model must accurately represent the

terrain. In addition to modeling the large scale features of a particular road course the road

model must also be capable of representing the small scale features such as bumps and

other road surface irregularities. On the other hand, it is undesirable for the road model to

be so detailed that massive amounts of data are required to generate a complete road

course.

          It is necessary to balance the competing demands of accuracy and a minimal data

set to obtain a useful road model. The model described below is an effective compromise

in attaining these opposing goals. The model utilizes parametric polynomial equations in

three dimensions to describe the path followed by the centerline or the road. Polynomials

of various orders are used depending on the nature of the boundary conditions on a

particular segment of the road. In addition to specifying the endpoints (and the tangent

vectors at the endpoints for the higher order polynomials) it is also necessary to provide

the angle of the road surface normal with respect to the vertical. This surface normal angle

specifies the lateral tilt of the road. It is assumed that the road is flat in the lateral direction

(no crown or concavity of the road surface).


                                                 99
        The modeling methodology laid out in the preceding paragraph is sufficient for

describing the large scale features of the roadway and perhaps even some of the larger

bumps on the road surface. The description of small scale features requires an addition to

the model. The texture associated with the road surface can be simulated by a filtered

white noise function. A relatively small number of parameters are required to describe the

average amplitude of the road surface irregularities as a function of their size. Prior to

actually running the simulation, the irregularities can be generated for each road segment

and tabulated for quick lookup. The number of entries in the lookup table is determined by

the size of the smallest irregularity being modeled.


6.2 Road Surface Coordinate System
        As discussed above the road is modeled by specifying a parametric function which

gives the location of the road centerline and by specifying an angle which gives the tilt of

the road surface from the vertical. It is useful to determine a set unit vectors (in terms of

the inertial coordinate system) which defines a road surface coordinate system. This is

most easily done by utilizing an intermediate coordinate system T. The intermediate

coordinate system is defined such that the x-axis unit vector             E   $
                                                                              t x lies along the tangent to

the path. The y-axis unit vector      E   $
                                          t y is perpendicular to the x-axis vector and is parallel to

the x-y plane of the inertial system.

                                E   $        ′               ′
                                    t x = E rP/ E ( s ) / E rP/ E ( s )                              (6.2.1)




                                                        100
           E
where          rP/E ( s) is the parametric equation specifying the location of the road center line

and   E
           ′
          rP/E ( s) is the tangent to the path (obtained by differentiating                E
                                                                                               rP/E ( s) with respect

to s). The y-axis unit vector is of the form

                                              E
                                                  t      (
                                                  $ y = t y1      t y2     0 )                                (6.2.2)

The orthogonality condition with tx dictates that

                                        $
                                       E TE
                                        tx
                                                  $ y = t x1 t y 1 + t x 2 t y 2 = 0
                                                  t                                                           (6.2.3)

and the normalization condition requires that


                                                      t y1 + t y 2 = 1
                                                        2      2
                                                                                                              (6.2.4)

Solving for the components of ty gives

                                                    2
                                                  tx2                         t 
                              t y1 = ±                              t y 2 = −  x1 t y1                      (6.2.5)
                                              t x1 + t x 2
                                                2      2
                                                                              t x 2 
In the event that tx2 is much less than tx1 (or even if it is zero) then the alternative solution

below may be used.


                                        t                                       2
                                                                                t x1
                             t y1   = −  x 2 t y 2             t y2    =± 2                                 (6.2.6)
                                         t x1                             t x1 + t x22
The sign of the solution is chosen such that the z-axis of the intermediate coordinate

system points up (i.e. in roughly the same direction as the z-axis of the inertial coordinate

system).

                                        E   $
                                            t z = E $ x ×E $ y
                                                    t      t             tz3 > 0                              (6.2.7)

           The road surface coordinate system is obtained from the intermediate coordinate

system via a simple rotation which accounts for the lateral tilt of the road surface:



                                                                  101
                              E         $
                                  nx = Etx
                                  $
                              E                                $
                                  n y = cos(θ)E $ y − sin( θ)E t z
                                  $             t                                                      (6.2.8)
                              E                  $
                                  n z = sin( θ)E t y + cos(θ)E $ z
                                  $                            t

Note that θ may be a function of the free parameter.


6.3 Location of Tire to Road Contact Point
         In order to model the interaction of the road surface and the vehicle model it is

necessary to determine the location of the point of contact between the tire and the road.

The contact point is known to lie in the plane of the road and is assumed to lie in the plane

of the wheel. The intersection of these planes is a line. The point on this line which

minimizes the distance to the center of the wheel is taken to be the contact point. These

conditions form a constrained minimization problem which can be solved using the method

of Lagrange multipliers.


                                   f ≡ min E rCP/E − E rWC/E                                           (6.3.1)


                             c1 ≡ ( E rCP/ E − E rWC/ E ) E n W = 0
                                                             T
                                                            $                                          (6.3.2)


                             c2 ≡E rCP/ E = α E n y ( s )+ E rP/ E ( s )
                                                $                                                      (6.3.3)

                                             E
The variables are defined as follows:            rCP/E is the unknown location of the contact point,

E
    rWC/E is the location of the wheel center with respect to the inertial coordinate system,

E
    nW is the normal to the plane of the wheel, α is an unknown constant which is equal to
    $

the distance of the contact point from the road center line, s is the unknown parameter

                                                 E
giving the location along the path, and              $
                                                     n y ( s ) is the normalized y-axis vector of the road


surface coordinate system. Rather than minimizing the expression                  E
                                                                                      rCP/E − E rWC/ E , which


                                                       102
contains a square root, it is simpler to minimize                                        (   E
                                                                                                 rCP/E − E rWC/E )
                                                                                                                      T
                                                                                                                          (   E
                                                                                                                                  rCP /E − E rWC /E ). The

system of equations can be further simplified by substituting the last equation into the first

                                                                                                                                       E
two equations; for the moment though, the first equation is left in terms                                                                  rCP/ E to allow

easier manipulation.


                           f ≡ min ( E rCP/ E − E rWC / E )                 (    rCP/ E − E rWC/ E )
                                                                           T E




                                   (                                                         )
                                                                                                 TE
                           c1 ≡ α E n y ( s )+ E rP/ E ( s )− E rWC / E
                                    $                                                                 nW = 0
                                                                                                      $                                            (6.3.4)

Applying the method of Lagrange multipliers gives a system of three of equations

                                                              ∇
                                                       ∇ f + λ c1 = 0                                                                              (6.3.5)

                                                                 c1 = 0                                                                            (6.3.6)

This system of equations has two unknowns: α and s. Thus, the del operator is defined as


                                                       ∇ = ( ∂α
                                                              ∂           ∂ T
                                                                          ∂s )                                                                     (6.3.7)

Evaluating the derivatives in the preceding equations gives


                                       f = 2( E rCP/ E − E rWC/ E )                  (                       )
                                  ∂                                              T       ∂E
                                  ∂α                                                     ∂α CP/ Er                                                 (6.3.8)

                                  ∂
                                  ∂s   f = 2( E rCP/ E − E rWC/ E )
                                                                                 T
                                                                                     (   ∂E
                                                                                         ∂s CP/ Er           )                                     (6.3.9)


                                 (α                                                  )
                                                                                         TE
               ∂
               ∂α   c1 =   ∂
                           ∂α
                                       E
                                           n y ( s )+ E rP/E ( s )− E rWC/E
                                           $                                                     n W = E nT E n y ( s)
                                                                                                 $       $W $                                     (6.3.10)


                                             (                                                           )
                                                                                                         TE
                           ∂
                           ∂s   c1 ≡ ∂s α E n y ( s )+ E rP/ E ( s )− E rWC/ E
                                     ∂
                                            $                                                                    $
                                                                                                                 nW

                                       [(                                                        )]
                                                                                                                                                  (6.3.11)
                                                                 ) (
                                                                                                     T
                                   = α           ∂E
                                                 ∂s   n y ( s) +
                                                      $                 ∂E
                                                                        ∂s   r
                                                                           P/ E          (s)             E
                                                                                                             $
                                                                                                             nW

where

                            ∂E
                            ∂α    r
                               CP/ E          =   ∂
                                                  ∂α   (α   E
                                                                n y ( s )+ E rP/E ( s ) = E n y ( s )
                                                                $                           $    )                                                (6.3.12)




                                                                        103
                              ∂E
                              ∂s   r
                                 CP/ E     =α   (   ∂E
                                                    ∂s   n y ( s) +
                                                         $      ) (       ∂E
                                                                          ∂sr ( s)
                                                                             P/E       )                              (6.3.13)

              ∂E                   ∂E
The terms     ∂s
                   $
                   n y ( s ) and   ∂s  r
                                      P/ E   ( s ) depend on the form of the parametric function being

used to model the road segment. Expressions for these terms are derived in sections

below.

Assembling terms gives the final system of equations which can be solved to determine the

Lagrange multiplier, α, and s. The location of the contact point is determined trivially from

α and s.


                        [(                                            )            ]
                                                                                   T
                 f 0 ≡ 2 α E n y ( s )+ E rP/ E ( s )− E rWC/ E + λE n W
                             $                                       $                 E
                                                                                           n y ( s) = 0
                                                                                           $                          (6.3.14)



         [(                                         )            ] (α(                     ) (                 ))
                                                                  T
    f1 ≡ 2 α E n y ( s )+ E rP/ E ( s )− E rWC / E + λE n W
               $                                        $                  ∂E
                                                                           ∂s
                                                                                n y ( s) +
                                                                                $                ∂E
                                                                                                 ∂s
                                                                                                    r
                                                                                                    P/ E   (s ) = 0   (6.3.15)



                                   [                                        ]
                                                                             TE
                           f 2 ≡ α E n y ( s )+ E rP/ E ( s )− E rWC/ E
                                     $                                            nW = 0
                                                                                  $                                   (6.3.16)

The three equations above are typically nonlinear for anything but the simplest of road

segment geometries. To obtain the solution it is necessary to apply a multidimensional

               s
form of Newton’ method or a similar algorithm. Most of the efficient algorithms require

evaluation of the Jacobian. The terms of the Jacobian are computed below:

                                             ∂f 0
                                                  = 2E n T E n y
                                                       $y $                                                           (6.3.17)
                                             ∂α

                     ∂En y ∂ErP/ E                                                                          ∂En y
     ∂ f1
                                                    [(                                                    ]
                        $                                                                                       $
                                                                                            )
                                                                                                          T
          = 2 E n T α
                $y         +         + 2 α E n y + E rP/ E − E rWC/ E + λE n W
                                              $                             $                                         (6.3.18)
     ∂α              ∂s     ∂s                                                                               ∂s

                                              ∂f 2 E T E
                                                  = nW n y
                                                    $    $                                                            (6.3.19)
                                              ∂α




                                                              104
                                                   ∂f 0 E T E
                                                       = nW n y
                                                         $    $                                                       (6.3.20)
                                                   ∂λ

                                    ∂f1 E T  ∂ n y ∂ErP/ E 
                                                  $
                                                E

                                       = n W α
                                         $          +                                                                (6.3.21)
                                    ∂λ        ∂s     ∂s 

                                                       ∂f 2
                                                            =0                                                        (6.3.22)
                                                       ∂λ

                                           T
             ∂En y ∂ErP/ E                                                                            ∂En y
    ∂f 0
                                                          [(                                        ]
                $                                                                                         $
                                                                                          )
                                                                                                    T
         = 2α     +                          E
                                                   n y + 2 α E n y + E rP/ E − E rWC/ E + λE n W
                                                   $           $                             $                        (6.3.23)
    ∂s       ∂s     ∂s                                                                                 ∂s

                                                      T
           ∂ f1     ∂En y ∂ErP/ E   ∂En y ∂ErP/ E 
                       $                 $
                = 2α     +         α     +        
           ∂s       ∂s     ∂s   ∂s         ∂s 
                                                                                                                      (6.3.24)
                                                                              ∂2 E n y ∂2 E rP/ E 
                   [(                                                  ]
                                                                                    $
                                                          )
                                                                         T
               + 2 α E n y + E rP/ E − E rWC/ E + λE n W
                       $                             $                       α        +           
                                                                              ∂s        ∂s 2 
                                                                                    2




                                    ∂f 2 E T  ∂ n y ∂ErP/ E 
                                                   $
                                                 E

                                        = n W α
                                          $          +                                                               (6.3.25)
                                    ∂s         ∂s     ∂s 

The direction vector     E
                             $
                             n y ( s ) has not yet been determined. The direction vector for the x-

axis of the road coordinate system is given by (recalling the results from the preceding

section)

                              E                $             ′               ′
                                  n x ( s )= E t x ( s )= E rP/ E ( s ) / E rP/ E ( s )
                                  $                                                                                   (6.3.26)

where the prime indicates differentiation with respect to s. Let                                E   $
                                                                                                    t y ( s ) be a vector

perpendicular to   E   $ x ( s ) and oriented such that it lies in the x-y plane of the inertial
                       t

coordinate system (again, following the development in the first section). If                             E   $ x ( s ) has the
                                                                                                              t

form




                                                               105
                                           t x ( s ) = (t x1 ( s ) t x 2 ( s ) t x 3 ( s ))
                                           $
                                       E                                                   T
                                                                                                                                            (6.3.27)

then


                                            (                                    )
                                                                                 T
                           E   $
                               t y ( s ) = t y1 ( s ) t y 2 ( s ) 0
                                                         1                                                                                  (6.3.28)
                                       =                                    (−   t x 2 ( s ) t x 1 ( s ) 0)
                                                                                                                T

                                                t x21 ( s ) + t x 2 ( s )
                                                                2



and


                     t z ( s ) = (t z1 ( s ) t z 2 ( s ) t z 3 ( s ))
                     $
                 E                                                   T

                                                                                                                                            (6.3.29)
                                                                (                                                          )
                                                                                                                           T
                                               $
                               = E $ x ( s )×E t y ( s ) = − t x 3t y 2
                                   t                                                 t x 3 t y1   t x1t y 2 − t x 2 t y1

where the tx,i depend on the form of the parametric equation used for a particular road

segment. The tx,i are derived for several types of road segments in the following section.

The particular orientation of                    E   $
                                                     t y ( s ) is irrelevant (to the left of                        E   $ x ( s ) or to the right of
                                                                                                                        t

E   $
    t x ( s ) ); the only result will be a change of sign of α in the solution. Applying the rotation

to account for the tilt of the road gives


                                                        t y1 cos(θ ) + t z1 sin( θ ) 
                                                                                     
                                       E
                                           n y ( s ) = t y 2 cos(θ ) + t z 2 sin( θ )
                                           $                                                                                                (6.3.30)
                                                                                     
                                                               t z 3 sin( θ )        
where θ is typically a function of s.

The first and second derivatives of                        E
                                                               $
                                                               n y ( s ) are also required:


                            ∂t ys1 cos(θ ) − t y1 sin( θ ) ∂θ + ∂t zs1 sin( θ ) + t z1 cos(θ ) ∂θ 
            ∂E             ∂t∂                              ∂s       ∂                         ∂s
                                                                                                    
                                                             ∂θ      ∂t z 2
               n y ( s ) =  ∂s cos(θ ) − t y 2 sin( θ ) ∂s + ∂s sin( θ ) + t z 2 cos(θ ) ∂θ 
               $              y2
                                                                                                 ∂s
                                                                                                                                            (6.3.31)
            ∂s                                                                                     
                                              ∂t z 3                          ∂θ
                                                      sin( θ ) + t z 3 cos(θ ) ∂s                   
                                               ∂s                                                  




                                                                            106
                     ∂2 t y1 cos(θ ) − 2 ∂t y 1 sin(θ ) ∂θ − t cos(θ ) ∂θ 2 − t sin( θ ) ∂2 θ 
                     ∂s 2                  ∂s           ∂s     y1         ∂s     y1          ∂s 2             ( )
                     ∂2 t                                                                            
                     ∂2 t
                                              ∂t z          ∂θ              ∂ 2
                                                                              θ                 ∂2 θ
                     + ∂s 2z 1 sin( θ ) + 2 ∂s1 cos(θ ) ∂s − t z1 sin( θ ) ∂s + t z1 cos(θ ) ∂s 2 
                                                                                                      
                                                                                                                 ( )
                                           ∂t            ∂θ                ∂θ 2                ∂θ
                                                                                                                ( )
                                                                                                2

    ∂ E
      2              ∂s y2 2 cos(θ ) − 2 ∂ys2 sin( θ ) ∂s − t y 2 cos(θ ) ∂s − t y 2 sin( θ ) ∂s 2 
         n y ( s) =  2
         $                                                                                                                                         (6.3.32)
    ∂s 2            + ∂ t 2z 2 sin(θ ) + 2 ∂t z 2 cos(θ ) ∂θ − t sin( θ ) ∂θ 2 + t cos(θ ) ∂2θ 
                     ∂s                      ∂s            ∂s     z2       ∂s      z2           ∂s 2           ( )
                     ∂ t z 3 sin( θ ) + 2 ∂t z 3 cos(θ ) ∂θ − t sin( θ ) ∂θ + t cos(θ ) ∂ θ 
                                                                                                                ( )
                          2                                                     2                2


                     ∂s 2                  ∂s            ∂s     z3        ∂s     z3           ∂s 2 
                    
                                                                                                     
                                                                                                      
                                                                                                     
where


                    [                ] [                      ]
              θ( s ) = θ0 + θ1′ 2(θ1 − θ0 ) s 3 + 3(θ1 − θ0 ) − 2θ0 + θ1′2 + θ0 s θ0
                        ′ −                                       ′ s         ′+
              ∂θ
                                                                ]s
                 = [θ ′ 3θ ′ 6(θ − θ )]s + [ (θ − θ ) − 4θ ′ 2θ ′ + θ ′
                   3 +     −                6               +           2                                                                           (6.3.33)
              ∂s
                         0          1             1             0                   1        0             0          1          0


              ∂2 θ
              ∂s 2
                        [                                           ]
                   = 6 θ0 + θ1′ 2(θ1 − θ0 ) s + 6(θ1 − θ0 )− 4θ0 + 2θ1′
                        ′ −                                    ′

The derivatives of      E   $       $
                            t x , E t y and                 E   $
                                                                t z are also required. The derivatives of                                 E   $
                                                                                                                                              t x depend on

the functional form of the particular type of road segment begin considered. The

                 $         $
derivatives of E t y and E t z have a specific functional dependence on the derivatives of                                                             E   $
                                                                                                                                                           tx

and can be determined here.


                             (                                  )
                                 ∂t x 2        ∂t x1                T
              ∂E $    −                                     0               t x1 ∂t xs1 + t x 2   ∂t x 2

                                                                                                           (−                        0)
                                  ∂s            ∂s                                ∂                ∂s
                 ty =                                                   −
                                                                                                                                      T
                                                                                                                tx2       t x1                      (6.3.34)
              ∂                                                                [                  ]
                                                                                                  3
               s                        t + t
                                          2
                                          x1
                                                       2
                                                       x2                      t  2
                                                                                  x1   + t   2
                                                                                             x2
                                                                                                      2




                                                                                 107
                                 (                                    )
                                       2             2                    T
                                      ∂ tx 2        ∂ t x1
                2
              ∂ E$     −                                          0                      t x1 ∂t xs1 + t x 2   ∂t x 2
                                       ∂s 2          ∂s 2
                                                                                                                         (−                                      )
                                                                                               ∂                ∂s               ∂t x 2         ∂t x 1           T
                  ty =                                                         − 2                                                                           0
                                                                                              [                ]
                                                                                                                                  ∂s             ∂s
              ∂2
                                                                                                               3
               s                              t x21 + t x22                                   t2
                                                                                               x1   + t x22
                                                                                                                   2




                                       ( )+t      + ( )
                                          2                                        2
                                                      ∂t x 1 2                                      ∂t x 2 2
                                  t x1 ∂ st2x 1 +                                 ∂ tx 2

                                                        (− t                                                                                    0)
                                       ∂               ∂s                     x 2 ∂s 2               ∂s
                              −
                                                                                                                                                    T
                                                                                                                                  t x1                                          (6.3.35)
                                         t + t ]
                                         [
                                                                                 3                                      x2
                                                             2            2          2
                                                             x1           x2



                              + 3
                                  ( t + t ) ( − t t 0)
                                           ∂t x 1
                                       x 1 ∂s
                                                              ∂t x 2 2
                                                          x 2 ∂s                                               T


                                     [ +t ]
                                                              5                          x2         x1
                                               2        2         2
                                     t         x1       x2



                                                                              ∂t
                                                        − ∂t xs3 t y 2 − t x 3 ∂ys2                                                     
                                 ∂E $                      ∂
                                                                              ∂t y
                                                                                                                                        
                                                         ∂t x 3
                                    t z ( s) =            ∂s y1
                                                                t + t x 3 ∂s1                                                                                                  (6.3.36)
                                 ∂s             ∂t x 1                                                                          ∂t y 1 
                                                t y 2 + t x1 ∂t y 2 − ∂t x 2 t y1 − t x 2                                              
                                                ∂s             ∂s       ∂s                                                       ∂s 



                                                                                                                                                             
                                                  2                                                                      2
                                                                            ∂t         ∂t
                                            − ∂ st 2x 3 t y 2 − 2 ∂t xs3 ∂ys2 − t x 3 ∂s y2 2
                                                  ∂                    ∂                                                                                      
              2
            ∂ E$                               2
                                               ∂ tx3                ∂t x 3 ∂t y 1    ∂2 t y 1                                                                 
                 tz =                               2 t y1 + 2 ∂       s ∂s + t x 3 ∂s 2
                                                                                                                                                                                (6.3.37)
            ∂s 2                               ∂s                                                                                                            
                       ∂2 t x 1 t + 2 ∂t x 1 ∂t y 2 + t ∂2 t y 2 − ∂2 t x 2 t − 2 ∂t x 2                                            ∂t y 1           ∂2 t y1 
                                                                                                                                              − t x 2 ∂s 2 
                       ∂s 2 y 2        ∂s ∂s              x 1 ∂s 2         ∂s 2  y1         ∂s                                       ∂s




6.4 Velocity of the Tire to Road Contact Point
            The velocity of the tire to road contact point is required by the tire model. The

velocity can be found by differentiating the system of equations developed in the preceding

section.

d
   f0 ⇒
dt                                                                                                                                                                               (6.4.1)

[(                           )                                    ]                      [(                                      )                       ](             )
                                                                      T                                                                                  T
2    E
         &       E
                  &         & $       &
         rCP/ E − rWC / E + λ n W + λ n W
                                      $
                                      E               E                   E
                                                                              n y + 2 rCP/ E − rWC / E + λ n W
                                                                              $                E
                                                                                                           $   E                            E                 ∂E
                                                                                                                                                                     ny s = 0
                                                                                                                                                                     $ &
                                                                                                                                                              ∂s



          d
             f1 ⇒
          dt
          [( r&                           )                                      ] (α(                    ) (                         ))
                                                                                     T
           2 E
                             &         & $         &
                         − E rWC / E + λE n W + λE n W
                                                   $                                          ∂E
                                                                                                     ny +
                                                                                                     $                 ∂E
                                                                                                                             r                                                   (6.4.2)
                 CP/ E                                                                        ∂s                       ∂s P/ E


          + [( r                              )               ] (α (                          ) (                        ) (                                 )s&) = 0
                                                                  T                                         2                               2
             2   E
                     CP/ E   − E rWC/ E + λE n W
                                             $                   &              ∂E
                                                                                ∂s       ny + α
                                                                                         $                ∂ E
                                                                                                          ∂s 2
                                                                                                                       ny s +
                                                                                                                       $ &                ∂ E
                                                                                                                                          ∂s 2
                                                                                                                                                  r
                                                                                                                                               P/ E




                                                                                           108
             d
                        [r                        ]                 [r                           ]    &
                                                  TE                                             TE
                f2 ⇒     &
                         E
                             CP/ E   − E rWC/ E
                                         &                nW +
                                                          $            E
                                                                           CP/ E   − E rWC / E        nW = 0
                                                                                                      $                     (6.4.3)
             dt
In the preceding section the position of the contact point was specified in terms of a point

on the road centerline and the y-axis unit vector road coordinate system. Differentiating

that expression gives

                         E
                             rCP/ E = α E n y + α
                             &        & $                 (   ∂E
                                                              ∂s   ny s +
                                                                   $ & ) (         ∂E
                                                                                   ∂s  r
                                                                                      P/ E   )s&                            (6.4.4)

The derivatives of the contact point velocity with respect to α and s are
                                                              &     &

                                ∂E
                                ∂&
                                 s
                                      &
                                      r
                                   CP/ E     =α       (   ∂E
                                                          ∂s   ny +
                                                               $    ) (     ∂E
                                                                            ∂s P/ Er    )                                   (6.4.5)

                                              ∂E
                                              ∂&
                                               α CP/ E
                                                      &
                                                      r        = En y
                                                                  $                                                         (6.4.6)

Once the position of the contact point has been determined using the procedure from the

preceding section the only remaining unknowns in the differentiated system of equations

              &
are α , s and λ. Unlike the problem of the preceding section the unknowns appear
    & &

linearly in the equations and standard linear systems techniques can be applied to

determine the solution. A little manipulation of the three equations gives

                                                                       T
                                                             
                                                             
                                                             
                          $y $
                        2E n T E n y                                      α 
                                                                             &
                                                                          &
                                                                                       [                       ]
                                                                                                               T
                                                                                              E&
                          $      $                                         λ = 2 rWC / E − λ n W
                                                                                    &          $                       $
                         E T E                                                    E                                E
                          nW n y                                                                                       ny   (6.4.7)
                                                             
       ( (          ) (
       2α ∂ E n + 2 ∂ E r             ))                                 s 
                                     T
               $y                      E
                                         $
                                         ny                                 &
          ∂s             ∂s   P/ E
                                                              
      
        [(                      )             ](              
                                                                   )
                                           T
      + 2 E rCP/ E − E rWC/ E + λE n W
                                     $                ∂  E
                                                           $
                                                           ny 
                                                     ∂s      




                                                                   109
                                ((                               ))
                                                                                                  T
           
                         $y     ∂
                                     $       ∂
                                              ) (
                     2 E n T α ∂s E n y + ∂s E rP/ E
                                                                                               
                                                                                               
                                                                                              
                           [(                          )              ](        )
                                                         T
                    + 2 E rCP/ E − E rWC/ E + λE n W ∂s E n y
                                                    $        ∂
                                                               $                                     α 
                                                                                                        &
                                                                                              
           
           
                              $
                             E T       ∂
                                         ((
                                          $         ) (
                              n W α ∂s E n y + ∂s E rP/ E
                                                   ∂
                                                         ))                                    
                                                                                               
                                                                                                       &
                                                                                                      λ
                                                                                                      s 
                ((          ) (               )) ( ( ) (    ))                                         &
                                        T
           2 α ∂s E n y + ∂s E rP/ E
                ∂
                     $        ∂
                                          α ∂s E n y + ∂s E rP/ E
                                              ∂
                                                 $          ∂
                                                                                                                  (6.4.8)
                                                                                              
                [(
           + 2 E r − E r
                                          )            ]( (                )(                ))
                                               T
                                                      ∂2           2
                                     + λE n W α ∂s 2 E n y + ∂s 2 E rP/ E
                                          $                $      ∂
           
                   CP/ E     WC / E                                                           
                                                                                               


                [                        ] (α(             ) (             ))
                                         T
                             &
           = 2 E rWC/ E − λE n W
                 &           $                   ∂E
                                                      ny +
                                                      $          ∂E
                                                                      r
                                                 ∂s              ∂s P/ E

           ⋅
                                                 T
                    E
                      $y $
                      nT E nW                       α 
                                                       &
                                                    & E T E
                                                                                [r                       ]    &
                                                                                                         TE
                                                     λ= rWC/ E n W −
                                                           &     $                          − E rWC/ E        $
                                                                                E
                       0                                                                                    nW   (6.4.9)

      [(                            )]
                                                                                    CP/ E
     α              ) (                 nW         s 
                                     T
           ∂E
                $
                ny   + ∂s E rP/ E
                         ∂             E
                                         $            &
          ∂s                               

6.5 Vehicle Position and Heading Angle
       In order for the vehicle to be driven along the road it is necessary to implement

some sort of steering control. Many of the steering control algorithms in use today require

feedback of the vehicle location error and the vehicle heading error (or sometimes the

vehicle orientation error) with respect to the road. Expressions for the vehicle position and

the vehicle heading are developed below.

       The position of the vehicle relative to the road is a somewhat difficult to quantify

in that there are numerous points on the vehicle which may be used as a reference. To

simplify matters a single point on the sprung mass can be chosen as a reference point. For

most situations it seems reasonable to chose a point in the longitudinal plane of symmetry

of the vehicle, somewhere in the vicinity of the center of gravity. Varying the fore/aft




                                                             110
location of the point will likely have an effect on the stability and or sensitivity of the

control algorithm and so the point should be chosen carefully. The chosen point will most

likely be a significant distance above the road surface. Since the distance of the reference

point above the road surface is of little concern in a steering control algorithm it is

desirable to determine only the lateral components of the position. This can be done by

projecting the reference point onto the road surface. Finding the location of the projected

point is a good deal easier than finding the contact point between the road and the tire. A

set of equations can be written expressing the desired constraints on the projected point:

                                     E
                                          rPRJ / E = α E n y + E rP/ E
                                                         $                            (6.5.10)


                             (                                 )
                                                                  TE
                                 E
                                     rREF/ E − E rPRJ / E              nx = 0
                                                                       $              (6.5.11)


                             (                                 )
                                                                  TE
                                 E
                                     rREF/ E − E rPRJ / E              ny = 0
                                                                       $              (6.5.12)

The first equation simply specifies that the projected point lies on the road surface while

the second and third equations specify that the vector between the reference point and the

projected point is normal to the surface. Substituting the first equation into the second and

eliminating the term which contains the product of orthogonal unit vectors gives


                                 (                            )
                                                               TE
                                     E
                                         rREF/ E − E rP/ E          nx = 0
                                                                    $                 (6.5.13)

This equation no longer contains the unknown α although for most types of road segment

models it is strongly nonlinear in s. Once s has been determined the third equation can be

used to determine α.


                                           (                           )
                                                                       TE
                                 α=            E
                                                   rREF/ E − E rP/ E        $
                                                                            ny        (6.5.14)




                                                              111
The velocity of the projected point is determined by differentiating the preceding equations

with respect to time:

                                  E
                                      rPRJ / E = α E n y + α
                                      &          & $                        (   ∂E
                                                                                ∂s       ny s +
                                                                                         $ &    ) (          ∂E
                                                                                                             ∂s P/ E     r       )s&                   (6.5.15)


             (                        (              )s&)                       (                                        )(                 )
                                                          T                                                                  T
                 E
                     v REF/ E −           ∂E
                                          ∂s r
                                             P/ E
                                                              E
                                                                  nx +
                                                                  $                 E
                                                                                        rREF/ E − E rP/ E                        ∂E
                                                                                                                                 ∂s    nx s = 0
                                                                                                                                       $ &             (6.5.16)



                      (                     (                 )s&)                          (                                     )(              )
                                                                   T                                                               T
            α=
            &             E
                              v REF/ E −        ∂E
                                                ∂s   r
                                                   P/ E
                                                                        E
                                                                            ny +
                                                                            $                   E
                                                                                                    rREF/ E − E rP/ E                  ∂E
                                                                                                                                       ∂s
                                                                                                                                                $ &
                                                                                                                                                ny s   (6.5.17)

                                           &
Rearranging the second equation to isolate s gives

                                                                   E
                                                                               $
                                                                        vT E E nx
                              s=
                                   [                                                                                                   )]
                              &                                          REF/
                                                                                                                                                       (6.5.18)
                                   (                  )              (                                           )(
                                                                                                                     T
                                          ∂E T
                                          ∂s r
                                             P/ E
                                                  E
                                                          nx −
                                                          $              E
                                                                             rREF/ E − E rP/ E                           ∂E
                                                                                                                         ∂s
                                                                                                                                 $
                                                                                                                                 nx

The second and third equations can be solved trivially once α and s have been determined.

The velocity of the projected point is then found using the first equation.

        It is also useful to determine the error in the heading angle. The heading angle

error is determined by projecting the longitudinal axis of the sprung mass coordinate

system onto the road surface finding the angle between it and the tangent vector to the

road.


                                                     s x = [ CSM ]1 0 0)
                                                                  (                                          T
                                                 E
                                                     $      E


                                                 E   $
                                                     hx =
                                                               E
                                                                   sx −
                                                                   $      (         E
                                                                                        $z $
                                                                                        n T E sx      )s
                                                                                                       $ E
                                                                                                                 x

                                                                         −(                         s )s
                                                                                                                                                       (6.5.19)
                                                               E
                                                                   $
                                                                   sx               E
                                                                                        $
                                                                                        n   TE
                                                                                            z  x
                                                                                                 E
                                                                                                   x
                                                                                                    $ $

                                                                $T $
                                                     ψ = acos E n x E h x(                           )
6.6 Road Segment Models
        A number of road segment models are formulated in the following sections. The

terms which are required in the equations from the preceding sections are calculated for


                                                                                        112
each type of segment. All of the segments derived below are based on polynomial

functions of various orders to match different types of boundary conditions. It is possible

to base road segments on other types of functions.

Note that for all road segment models the free parameter (denoted by s) is assumed to

range from zero to one. Also note that the equation expressing the tilt of the road surface

is linear for all road segment models and takes the form

                                     θ( s ) = ( θ1 − θ0 )s + θ0                            (6.6.1)

where θ0 is the tilt of the road at the beginning of the segment ( s=0) and θ1 is the tilt of the

road at the end of the road segment (s=1).


Linear Polynomial Road Segment
The linear polynomial road segment is based on a parametric equation of the form

                                        E
                                            rP/ E ( s ) = a s + b                          (6.6.2)

There are two unknown vectors which can be determined via a pair of boundary

conditions. The boundary conditions are as follows:

                              E
                                  rP/ E ( 0) = r0         E
                                                              rP/ E (1) = r1               (6.6.3)

Solving for the unknowns gives

                                      a = r1 − r0             b = r0                       (6.6.4)

The tangent vector is obtained by differentiation with respect to s

                                         E   $ x ( s ) = r1 − r0
                                             t                                             (6.6.5)
                                                         r1 − r0

The overall path length along the segment is determined trivially in this case


                                             L = r1 − r0                                   (6.6.6)



                                                         113
Quadratic Polynomial Road Segment
The quadratic polynomial road segment is based on a parametric equation of the form

                                        E
                                            rP/ E ( s ) = a s 2 + b s + c                                          (6.6.7)

The first two boundary conditions are identical to those for the linear road segment. The

remaining boundary condition allows matching the tangent vector at one of the endpoints.

The choice of which endpoint to use leads to two possible cases. For the first case the

boundary conditions are

                       E
                           rP/ E ( 0) = r0              E
                                                             ′
                                                            rP/ E (0) = t 0       E
                                                                                      rP/ E (1) = r1               (6.6.8)

and for the second case the boundary conditions are

                        E
                            rP/ E ( 0) = r0        E
                                                       rP/ E (1) = r1             E
                                                                                       ′
                                                                                      rP/ E (1) = t1               (6.6.9)

Solving for the unknowns gives

                     a = r1 − r0 − t 0                  b = t0           c = r0          ( case 1)                (6.6.10)

               a = t1 − (r1 − r0 )            b = 2(r1 − r0 ) − t 0                     c = r0       ( case 2 )   (6.6.11)

where the tangent vector is obtained by differentiation with respect to s

                                              E   $ x ( s ) = 2a s + b
                                                  t                                                               (6.6.12)
                                                              2a s + b

The derivatives of E $ x ( s ) can be shown to be
                     t


                                     ∂E $
                                        t x (s) =
                                                              [(
                                                           $T t
                                                  2 a − E tx a E $x           ) ]                                 (6.6.13)
                                     ∂s                2a s + b

                          2
                        ∂ E$
                             t x (s) = −
                                         2          (   ∂E T
                                                        ∂s x
                                                              $   ) $         (
                                                                             $T
                                                              t a E tx + 4 E tx a          )∂E
                                                                                            ∂s x
                                                                                                 $
                                                                                                 t
                                                                                                                  (6.6.14)
                        ∂s 2
                                                                      2a s + b

The overall path length along the segment is determined by evaluating the following

integral


                                                                   114
                                                 s

                                    L( s ) = ∫        E
                                                           ′             ′
                                                          rP/TE ( s ) E rP/ E ( s ) ds                               (6.6.15)
                                                 0

While a closed form solution to this integral may be possible, numerical integration

appears to be preferable.


Cubic Polynomial Road Segment
The cubic polynomial road segment is based on a parametric equation of the form

                                     E
                                         rP/ E ( s ) = a s 3 + b s 2 + c s + d                                       (6.6.16)

A total of four boundary conditions are required to determine the coefficients. The first

two boundary conditions force the function to pass through the starting and ending points

and are identical to those for the linear road segment and the quadratic road segment. The

remaining two boundary conditions force the tangent vector at the end points to match a

specified value. The boundary conditions are

             E
                 rP/ E ( 0) = r0     E
                                          ′
                                         rP/ E ( 0) = t 0             E
                                                                          rP/ E (1) = r1       E
                                                                                                    ′
                                                                                                   rP/ E (1) = t1    (6.6.17)

Solving for the unknowns gives

       a = t1 + t 0 − 2(r1 − r0 )             b = − t1 − 2t 0 + 3(r1 − r0 )                c = t0           d = r0   (6.6.18)

where the tangent vector is obtained by differentiation with respect to s


                                              $ x ( s ) = 3a s + 2b s + c
                                                               2
                                          E
                                              t                                                                      (6.6.19)
                                                          3a s 2 + 2b s + c

The derivatives of E $ x ( s ) can be shown to be
                     t


                             ∂E $
                                t x (s) =
                                                                  (
                                          (6as + 2b)− E $ T (6as + 2b) E t x
                                                          tx             $                 )                         (6.6.20)
                             ∂s                   3a s 2 + 2b s + c




                                                                  115
          2
        ∂ E$
             t x (s) =
                       6a −   (     t (6as + 2b)+ 6 E $ T a E t x − 2 E $ T (6as + 2b)
                                    $
                                  ∂E T
                                  ∂s x                tx      $  )      tx  (        )(   ∂E
                                                                                          ∂s
                                                                                            $
                                                                                            tx   )   (6.6.21)
        ∂s 2
                                                       3a s + 2b s + c
                                                            2



The overall path length along the segment is determined by evaluating the following

integral in the same manner as was done for the quadratic road segment.

                                           s

                                   L( s ) = ∫   E
                                                     ′             ′
                                                    rP/TE ( s ) E rP/ E ( s ) ds                     (6.6.22)
                                           0

There is no readily available closed form solution for the resulting integral. Numerical

integration appears to be the only viable means to obtain a solution.




                                                            116
7 Equations of Motion - Tire Model

7.1 Introduction
       The tire model plays an important role in the performance of the vehicle model.

With the exception of aerodynamic forces which are only relevant at higher speeds, all of

the interactions of the vehicle with the external environment occur through the tire.

Accurate modeling of the tire is crucial to an accurate vehicle simulation.

       In keeping with current modeling practices the model for horizontal force

generation is decoupled from model for vertical force generation with the exception of the

dependence of horizontal force on normal load. The tire model is thus divided into two

components. The first component models the vertical support characteristics of the tire

utilizing a simple nonlinear spring. Damping in the tire is neglected due to its small

magnitude relative to the damping in the shock absorber. The second model component

handles the generation of lateral and longitudinal forces using the normal force determined

by the vertical model.


7.2 Coordinate Systems
       There are two coordinate systems which are utilized in tire modeling. They are

depicted in Figure 7.1. The wheel coordinate system W is chosen such that the plane of

symmetry of the wheel lies in the x-z plane. The x-axis lies in the ground plane. The tire



                                             117
model coordinate system T is defined                                          $
                                                                              wz       $z
                                                                                       t

by choosing $ z normal to the road
            t
                                                                Wheel Plane
surface at the point of contact and

choosing $ x to point in the direction
         t

of the wheel heading ( not necessarily                                                      T, W
                                                           $
                                                           wx
the direction of the wheel velocity).                        $
                                                             tx                                    $y
                                                                                                   t
The forces and moments which are
                                                               Figure 7.1: The Tire Model Coordinate Systems

determined by the tire model are

referred to the T coordinate system.

       Before proceeding with the actual tire modeling it is necessary to determine the

unit vectors of the W and T coordinate systems. The T coordinate system is closely

related to the road surface coordinate system described in the chapter on road modeling.

The difference is a single rotation about the road surface normal ( $ z axis) to bring the $ x
                                                                    t                      t

                                                                               $
axis into alignment with the wheel heading vector. Given a set of unit vectors ri which

define the road surface coordinate system and the unit vectors for the spindle coordinate

       $
system p i the T coordinate system unit vectors are determined as follows:

                                             E   $ z = E rz
                                                 t       $                                              (7.2.1)


                             E   $
                                 tx =
                                        E
                                            px −
                                            $      (   E
                                                           pT E $ z
                                                           $x t    )$tE
                                                                          z

                                                  −(             $ ) $
                                                                                                        (7.2.2)
                                        E
                                            $
                                            px         E
                                                           $
                                                           p   TE
                                                               x t t
                                                                  z
                                                                    E
                                                                      z


                                        E   $ y = E $ z ×E $ x
                                            t       t      t                                            (7.2.3)




                                                           118
                                          $
It is assumed in the above equations that p x points in a direction which is roughly in the

direction of travel. The equations above have a singularity when p x coincides with $ z .
                                                                 $                  t

        The W coordinate system is trivially derived from the unit vectors of the spindle

coordinate system and the unit vectors of the T coordinate system:

                                          E
                                              wx = E $x
                                              $      t                                   (7.2.4)

                                          E
                                              wy = E py
                                              $      $                                   (7.2.5)

                                    E
                                        w z = E w x ×E w y
                                        $       $      $                                 (7.2.6)

        It is also useful to calculate the camber angle of the wheel for use in the horizontal

tire force calculations. The camber angle is simply the angle between the      $
                                                                               w z unit vector

and the $ x − $ z plane. The magnitude can be found by taking the inverse cosine of the
        t t

inner product since the w z unit vector lies in the $ y − $ z plane. The sign is determined by
                        $                           t t

looking at the component of w z along $ y . Camber is taken to be positive for positive
                            $         t

rotations about the $ x axis (using the right hand rule).
                    t



                          =
                                                              (
                             E t T E w z ≥ 0, - cos-1 E $ z E w z
                             $y $
                         γ  E TE
                                                          tT $        )
                                                          (       )
                                                                                         (7.2.7)
                                $ $
                             t y w z < 0, cos
                                                  -1 E $ T E
                                                       t z wz$
                            

7.3 The Magic Formula Tire Model
        The tire itself is modeled using a variation on the popular Magic Formula Tyre

Model [Bakker, Pacejka]. The formula is based on a function whose behavior

approximates the shape of the curves obtained from experimental measurements on tires.

   s
It’ parameters are determined so as to fit the curve to a particular set of experimental



                                                    119
data. The function has the added benefit that the coefficients describing the shape of the

curve have simple interpretations. The tire model can be divided into two sub-models: one

for the vertical (support) force and one for the horizontal (tractive) forces. The only

coupling between the two sub-models is the dependence of the tractive forces on the

normal load. The support force has no dependence on the tractive force.


Support Forces
       The support force generated by the tire can be modeled by a nonlinear spring

which is placed between the hub of the wheel and the point of contact between the tire and

the road surface. Damping effects are similarly modeled using a nonlinear viscous type

damping element. The vector from the tire-to-ground contact point to the center of the

wheel must be found in order to determine the length of the spring. The time derivative of

the same vector must also be determined in order to find the magnitude of the damping

force. The calculations for determining the position and velocity of the point of contact

between the tire and the road are derived in the chapter on road modeling. The direction

of the force is assumed to be along the surface normal and camber effects are ignored

(although they appear in the tractive force model).

                                                                 E
       Once the location of the contact point                        rCP/ E is known the generalized forces

associated with the tire can be determined. The vector between the contact point and the

wheel center is given by

                           E
                               rWC/ CP = E rWC / E − E rCP/ E = Re E n WC /CP
                                                                     $                               (7.3.1)




                                                        120
where   E
            $
            n WC/ CP is the unit vector along                E
                                                                 rWC/ CP and Re is the effective radius of the tire

                                     E
(equal to the magnitude of               rWC/ CP ). The time derivative of this vector is simply

                                          E
                                              rWC/ CP = E rWC/ E − E rCP/ E
                                              &           &          &                                                (7.3.1)

The rate of change in the length of the vector is determined as follows:

                                d
                                     [r                          ]=
                                                                  1
                           &
                           Re =       E       T      E
                                                        r
                                                                      2       E
                                                                                  $ WC      &
                                                                                  n T /CP E rWC/ CP                   (7.3.1)
                                              WC /CP   WC /CP
                                dt
Thus, the force exerted by the spring on the wheel center is

               E
                           [
                   Ftire,z = k 0 ( R0 − Re )+ k1 ( R0 − Re )
                                                                          3
                                                                              ] t$ − [ R + c R ] t$
                                                                                  E
                                                                                     c &
                                                                                      z   0
                                                                                             &
                                                                                              e       1
                                                                                                          3
                                                                                                          e
                                                                                                              E
                                                                                                                  z   (7.3.2)

where R0 is the free length of the spring (unloaded wheel radius).


Tractive Forces
        The Magic Formula Tyre Model requires as inputs the longitudinal slip, the lateral

slip and the normal force on the tire. The normal force is easily obtained from the vertical

force model as discussed previously. The longitudinal and lateral slip angles are somewhat

more difficult to obtain and are discussed below. The output of the Magic Tyre Formula

consists of a system of forces and moments at the center of the contact patch. In order to

keep the tire model to vehicle model interface as general as possible it is desirable to

eliminate any references to point(s) of contact. It is preferable to return a system of forces

and moments applied at the wheel center. This also eliminates the need to determine the

virtual displacements associated with any points of contact. It is trivial to relocate a system

of forces and moments; the forces and moments are modified as follows:

                                                    E
                                                        Fwc = E Fcp                                                   (7.3.3)




                                                                 121
                              E
                                  M wc = E M cp + E rCP/ WC ×E Fcp                       (7.3.4)

where a WC subscript indicates the force or moment as applied at the wheel center and a

CP subscript indicates the force or moment as applied at the contact point.

Lateral Slip and Longitudinal Slip
      The SAE definition of longitudinal slip velocity is ω-ω0 where ω is the actual

angular velocity of the tire and ω0 is the angular velocity of a free-rolling tire moving with

the same linear velocity as the driven or braked tire. The longitudinal slip      percentage is

defined as the ratio of the slip velocity to the free rolling angular velocity:

                                                ω
                                           κ=      −1                                    (7.3.5)
                                                ω0
The SAE definition works well for forward velocities but breaks down when negative

velocities are considered. The desired result is for the longitudinal force to have the same

sign as the tractive force under all conditions of braking and acceleration for both forward

motion and backward motion of the vehicle. The SAE definition can be modified to

product the proper results by taking the magnitude of the angular velocity in the

denominator:

                                                ω
                                          κ=       −1                                    (7.3.6)
                                                ω0
or written in terms of linear velocities and extending to negative velocities,

                                           Vx − Vr    V
                                    κ= −           = − sx                                (7.3.7)
                                             Vx        Vx
where Vsx = Re (ω 0 − ω ) is also referred to as the longitudinal slip velocity, Vr = Re ω is

the forward speed of rolling, V x = Re ω 0 is the actual forward velocity of the tire and Re is


                                                   122
the effective radius of the tire. The                                                Vr
                                              Vsy
relationships between the Vi are
                                                                                           V
                                                               Vs
shown in Figure 7.2. Table 7.1

shows that the preceding expression                                            α

produces the desired sign for κ
                                                             Vsx                                   Vx
under all conditions of forward and

reverse motion.                                        Figure 7.2: The Relationship between the
                                                        Components of the Tire Velocity Vector

        A problem arises due to the

singularity in the slip percentage that occurs at Vx=ω0=0. This singularity is encountered

when the linear velocity of the tire goes to zero. The solution to this problem is to derive a

first order differential equation for longitudinal slip following the example of Bernard and

Clover1 [Bernard, 1995],

                                          Vx      V
                                    κ+
                                    &        κ = − sx                                             (7.3.8)
                                          B        B
where Vx is the longitudinal component of the wheel velocity, and B is an experimentally

determined parameter with units of distance which characterizes the first order lag. Note

that the steady state solution to the preceding equation is identical to the SAE definition of

slip. Note that the differential equation has good behavior for small values of Vx including

Vx=0.




1
                    s
 Bernard and Clover’ definition of longitudinal slip is the negative of the SAE definition used by
         s                                                   s
Pacejka’ Magic Formula Tyre Model. Bernard and Clover’ results have been modified to use the SAE
definition of slip.


                                                 123
            Table 7.1 - Desired Longitudinal Force Sign and Sign of Longitudinal Slip

                                              Vr < Vx                        Vr > Vx
                                             (Braking)                    (Acceleration)
            Vx > 0                        Desired: Fx < 0                Desired: Fx > 0
       (Forward Motion)                     Sign of κ -
                                                     :                     Sign of κ +
                                                                                    :
            Vx < 0                        Desired: Fx > 0                Desired: Fx < 0
       (Reverse Motion)                     Sign of κ +
                                                     :                     Sign of κ -
                                                                                     :


        Bernard and Clover found that oscillations occur in the lateral and longitudinal slip

at low velocities. To eliminate these oscillations it is necessary to include a damping term

which is only activated when the velocity is below a certain threshold (0.15 m/s is

suggested) and changes sign from one integration time step to the next. The form of the

damping used by Bernard and Clover is


                                                         Fz Cs
                                 Fdamping ,x = 2ζ Vx                                        (7.3.9)
                                                          gB
The term Fz/g can be viewed as the portion of the vehicle mass supported on the wheel in

question, Cs is the longitudinal stiffness, ζ is the damping coefficient and Vx is the

longitudinal velocity of the tire.

        The lateral slip angle can be treated with a similar approach. The SAE definition

for slip angle is the angle between the vector defined by the intersection of the wheel plane

and the road plane and the velocity vector of the center of the contact patch. It is

necessary to modify the SAE definition to handle negative velocities as was done for the

longitudinal slip. With this modification the slip angle is given by

                                                   Vsy
                                       tan α = −                                           (7.3.10)
                                                    Vx



                                                   124
where u and v are the lateral and longitudinal components of the wheel velocity vector.

                             s
Modifying Bernard and Clover’ derivation for slip angle yields


                               d            V          V
                                  (tan α )+ x tan α = − sy                                       (7.3.11)
                               dt           b           b
Note that the steady state solution matches the modified SAE definition of slip angle. The

parameter b is the relaxation length for the tire which controls slip angle lag. A damping

force is also applied to the wheel in the lateral direction to eliminate oscillations in the

lateral velocity which occur at small velocities.


                                                          Fz Cα
                                  Fdamping , y = 2ζ Vsy                                          (7.3.12)
                                                           gb

Magic Formula
        The Magic Formula Tyre Model 2 is a complete tire model in that it allows

determination of all six of the forces and moments generated by the tire: lateral force,

longitudinal force, normal force, rolling resistance, overturning moment and self-aligning

torque. The normal force model specified by the Magic Formula Tyre model consists of a

simple linear spring and linear damper. This component of the model has been modified to

include nonlinearities as discussed in the preceding section. The driving/braking moment is

modeled as part of the power train and braking submodel. The equations for each of the

forces or moments are based on one of the two Magic Formulas:




2
                                                                                                97
 The version of the Magic Formula Tyre model used here is the static version of the Delft Tyre ’ model
which is designed to handle the combined slip case. See Pacejka (1997) for a complete description of the
model. The previous version of the Magic Formula Tyre model is described in Pacejka (1992) and in
      s                                                                                          s
Genta’ text. Additional information on even earlier forms of the model can be found in Bakker’ papers.


                                                    125
                  Y = y + SV

                            [        {                          ]
                                                                }
                   y = D sin C arctan Bx − E( Bx − arctan ( Bx ))                     (7.3.13)

                   x = X + SH

                  Y = y + SV

                            [         {                         ]
                                                                }
                   y = D cos C arctan Bx − E( Bx − arctan ( Bx ))                     (7.3.14)

                   x = X + SH

       In both of the formulas the parameter D is the maximum value the force or

moment apart from the small effect due to the Sv term. For the sin() form of the formula

the product BCD gives the slope of the curve at σ + S h = 0 which, in the case of the

lateral force model, corresponds to the initial cornering stiffness. Sv and Sh are introduced

to allow for non-zero forces and moments at zero slip. This can occur due to asymmetries

            s
in the tire’ construction (ply steer, conicity, etc.). The parameter C is called the shape

factor and limits the range of the arguments in the sin() function. This leaves the factor B

to control the slope of the curve at the origin and hence it is referred to as the stiffness

factor. For the cos() form of the formula the product BC controls the breadth of the peak.

The parameter C controls the value of the asymptote of the function as x goes to positive

or negative infinity and thus acts to shape the flanks of the curve. This leaves the

parameter B to control the peak width. As the Magic Tyre Formula has evolved it has

become necessary to express the parameters B through E as functions of other variables

such as normal load ( Fz) and camber angle ( γ in order to obtain the required levels of
                                             )

accuracy (see Pacejka, 1997 for details). The combined slip case is handled by an

additional set of formulas, also based on the form of the ‘magic’ formula, which take as



                                            126
arguments the forces and moments for the pure slip cases. Equations are also given for

rolling resistance force, overturning moment and normal load determination.


7.4 Generalized Forces and Moments
        Once the forces and moments applied at the wheel center by the tire have been

determined the generalized forces and the generalized moments can be determined. The

calculation of the generalized forces is slightly different for the front and for the rear tires

due to the differences in the suspension geometry.


Front Tires
        The position of the center of the wheel for the front tires is specified relative to the

spindle/steering knuckle assemble. The position of the wheel center is

                                E
                                    rWC/ E = E rSP/ E + [ CSP ] rWC/SP
                                                         E
                                                              SP
                                                                                          (7.4.15)

where the SP subscript indicates the spindle and WC indicates the wheel center.

Computation of the generalized forces gives the following results:


              δ rWC/ E = (1 0 0) xSP + (0 1 0) ySP + (0 0 1) zSP
               E
                                δ             δ             δ
                                   [  r δ
                                 ∂ E CSP ]SP                                            (7.4.16)
                        +   ∑
                            i      β
                                   ∂ SP,i 
                                               WC/SP βSP, i




                                        δ = E Fspring δ rWC / E
                                         W      T      E
                                                                                          (7.4.17)

and


                                       QxSP = E Fspring (1 0 0)
                                                  T
                                                                                          (7.4.18)


                                       Q ySP = E Fspring (0 1 0)
                                                   T
                                                                                          (7.4.19)


                                       QzSP = E Fspring (0 1 0)
                                                  T
                                                                                          (7.4.20)




                                                      127
                                                  [
                                                ∂ C ]
                            QβSP,i = E Fspring  E SP SP rWC /SP
                                         T
                                                                                                (7.4.21)
                                                   β
                                                ∂ SP,i 
The virtual work performed by an applied moment can be shown 3 to be of the form


                                   δ =2 E M T [ G SP ] β
                                    W       WC E     δ                                          (7.4.22)

The component of the moment generated by the tire which lies along the axis of rotation

                                          s
of the wheel is responsible for the wheel’ rotation. The remaining components are

transferred to the spindle via the wheel bearings. Dividing the moment into these

components gives

                             E
                                                [ E
                                 M ROT = (0 1 0) E CSP ] M WC                                   (7.4.23)

and

                                    E
                                        M SP = E M WC − E M ROT                                 (7.4.24)

The generalized forces are thus


                           QβSP,0 =2 E M SP [ G SP ]1 0 0 0)
                                                   (
                                         T                          T
                                             E                                                  (7.4.25)


                           QβSP,1 =2 E M SP [ G SP ]0 1 0 0)
                                                   (
                                         T                          T
                                             E                                                  (7.4.26)


                           QβSP,2 =2 E M SP [ G SP ]0 0 1 0)
                                                   (
                                         T                          T
                                             E                                                  (7.4.27)


                           QβSP,3 =2 E M SP [ G SP ]0 0 0 1)
                                                   (
                                         T                          T
                                             E                                                  (7.4.28)


                                            QΩ = E M T
                                                     ROT                                        (7.4.29)

where




3
                                                                 s
 See Nikravesh, p.290 for a similar problem. Extending Nikravesh’ derivation to a force couple gives the
desired result.


                                                     128
                                       − βSP,1                βSP,0            − βSP,3            βSP,2 
                            [ G SP ]= − βSP,2
                             E                               βSP,3              βSP,0
                                                                                                          
                                                                                                  − βSP,1                                        (7.4.30)
                                       − βSP,3
                                                            − βSP,2             βSP,1             βSP,0 

Rear Tires
       The position of the center of the wheel for the rear tires is specified relative to the

rear suspension coordinate system RS which is in turn located relative to the sprung mass.

The position of the wheel center is

             E
                 rWC / E = E rSM / E +   [C ]
                                         E   SM
                                                       SM
                                                            rpiv /SM +     [ C ](
                                                                            E        RS
                                                                                             RS
                                                                                                  rRS/ piv +     RS
                                                                                                                      rWC/ RS     )               (7.4.31)

where WC indicates the wheel center. Computation of the generalized forces gives

δ rWC/ E = (1 0 0) xSM + (0 1 0) ySM + (0 0 1) zSM
 E
                  δ             δ             δ
                        [
                      ∂ E CSM ]SM                                           [
                                                                            ∂ E C RS ]
                                                                                                     (                                 )
                                                                                                                                                  (7.4.32)
         +   ∑
                i       β
                                 rpiv /SM δ SM,i +
                        ∂ SM, i 
                                            β                      ∑
                                                                      i       β
                                                                              ∂ RS,i 
                                                                                                        RS
                                                                                                              rpiv / RS +   RS
                                                                                                                                 rWC/ RS δ RS,i
                                                                                                                                          β


                                     δ =
                                      W       (   E
                                                                          δ      )
                                                      Fspring + E Fdamper ⋅ E rWC / E                                                             (7.4.33)

The generalized forces are simply the coefficients of the virtual displacements in the

expression for the virtual work:


                                             QxSM = E Fspring (1 0 0)
                                                        T
                                                                                                                                                  (7.4.34)


                                             Q ySM = E Fspring (0 1 0)
                                                         T
                                                                                                                                                  (7.4.35)


                                             QzSM = E Fspring (0 0 1)
                                                        T
                                                                                                                                                  (7.4.36)


                                                         [
                                                       ∂ C ]   
                                    QβSM,i = E Fspring  E SM  SM rpiv /SM
                                                 T
                                                                                                                                                  (7.4.37)
                                                          β
                                                        ∂ SM,i 

                                                  [
                                                ∂ C ]
                            QβRS,i = E Fspring  E RS 
                                         T

                                                ∂ RS,i 
                                                   β
                                                                       (   RS
                                                                                rRS/ piv +   RS
                                                                                                  rWC/ RS       )                                 (7.4.38)




                                                                    129
Dividing the moment into components gives

                         E
                                            [ E
                             M ROT = (0 1 0) E C RS ] M WC                   (7.4.39)

                                E
                                    M SP = E M WC − E M ROT                  (7.4.40)

Computing the generalized forces gives


                       QβRS,0 =2 E M T [ G RS ]1 0 0 0)
                                               (                   T
                                     WC E                                    (7.4.41)


                       QβRS,1 =2 E M T [ G RS ]0 1 0 0)
                                              (                    T
                                     WC E                                    (7.4.42)


                       QβRS,2 =2 E M T [ G RS ]0 0 1 0)
                                              (                    T
                                     WC E                                    (7.4.43)


                       QβRS,3 =2 E M T [ G RS ]0 0 0 1)
                                              (                    T
                                     WC E                                    (7.4.44)


                                        QΩ = E M T
                                                 ROT                         (7.4.45)

where


                             − βRS,1       βRS,0      − βRS,3    βRS,2 
                  [ G RS ]= − βRS,2
                   E                       βRS,3       βRS,0
                                                                         
                                                                 − βRS,1    (7.4.46)
                             − βRS,3
                                          − βRS,2      βRS,1     βRS,0 




                                                 130
8 Equations of Motion - Driver Model

8.1 Introduction
        The driver model is a critical component of the simulation. It is responsible for

controlling the steering, acceleration and braking inputs to the vehicle. In a racing

situation, lap time is strongly dependent on the ability of the driver to operate the vehicle

at or near the limits of handling and performance. This skill must be reflected in the driver

model. Therefore, the degree of skill with which the computer performs the driving task

plays a significant role in the accuracy and usefulness of the simulation.

        There are several classes of driver models which appear frequently in the literature,

either alone, or in combination with one another. These include quasi-linear compensatory

controllers, pursuit mode controllers and precognitive controllers. In general, quasi-linear

compensatory controllers, which utilize feedback loops based on the current vehicle

position and orientation, are unable to perform adequate lane keeping control due to the

                                                                                  s
lag time in the vehicle response. Pursuit controllers attempt to model the driver’ ability to

see the roadway ahead and can thus initiate control input prior to reaching a turn or

                                                                             s
arriving at a braking point. Precognitive controllers model the human driver’ ability to

execute learned maneuvers on command. This is typically implemented by specifying a

steering wheel angle directly without regard to feedback quantities. A number of these




                                              131
driver control approaches were tried before the final design was chosen for both the

steering control case and for the throttle and brake control case.


8.2 Steering Control
        The first several steering control attempts utilized pursuit type controllers. The

most successful of these was based on the optimal preview-follower theory of Guo and

Guan (Guo, 1993). This controller worked reasonably well under steady state velocity

conditions, low lateral accelerations and on flat road surfaces. The weakness of this

                           s
controller, however, is it’ dependence on knowing the transfer function which relates the

lateral acceleration of the vehicle to the steering angle. While this transfer functions can be

readily determined for a simple linear vehicle model with ideal tires, it must be generated

computationally for more complex vehicle models containing significant nonlinearities.

The lateral response transfer function for both the linear model and the nonlinear model is

strongly dependent on vehicle speed, lateral acceleration and road bank angle. It was

deemed impractical to characterize the lateral response transfer function for the nonlinear

vehicle model over the broad range of operating conditions necessary for the simulation.

Characterization of the vehicle is even more difficult during the optimal design process

because changes made to the vehicle suspension setup during the optimization process can

produce significant changes in the response functions. This necessitates a re-tuning of the

driver controller during the optimization process.

        The inherent problems with compensatory and pursuit controllers discussed above

made it necessary to try a different approach to the driver control. The resulting controller



                                              132
                  Figure 8.1 - The Driver Path for the Kenley, NC Race Track



fits into the precognitive controller category in that the steering angle is specified as a

                        s
function of the vehicle’ position along the roadway and is specified without regard to

feedback quantities. On the other hand, the steering angle vs. road position curve is

obtained via minimization of a cost function based on the accumulated lateral position

error. In this sense the controller can be thought of as an optimal controller which can

adapt to changes in the vehicle setup and which can operate at the limits of adhesion. The

details of the driver path definition, of the cost function computation, and of the

optimization process are discussed below.


Driver Path Definition
       The driver path is specified relative to the centerline of the road. The lateral offset

of the path is specified at the beginning, middle and end of each road segment. A quadratic


                                             133
polynomial is fitted through the three points to form a smooth path through the segment.

The current version of the driver path formulation does not attempt to match tangents at

the road segment boundaries. The driver path segments are adjusted by eye to obtain

reasonable continuity of the driver path tangent vector between road segments. The driver

path, indicated by the dark line, for the Kenley, NC race track is shown in Figure 8.1. The

positioning of the driver path is based on the experience of the user of the software and

should approximate the line followed by real drivers when racing on the real track. At

present no optimization of the driver path is performed by the computer.

       The steering curve is described as a series of points which consist of the steering

angle and the position of the point along the track. The position coordinate is specified as

a distance from the start of the track as measured along the centerline of the road. The

lower bound on position is zero and the upper bound on the position is the length of the

track as measured along the centerline. The steering angle between points is determined

via linear interpolation. A sample steering profile utilizing ten data points is shown in

Figure 8.2.


Steering Profile Optimization and Cost Function Computation
       As mentioned before, an optimal steering profile is obtained via the minimization

of a cost function based on lateral position error. It is necessary to include the parameters

which describe the steering profile in the optimization process, along with those

parameters of the vehicle model which are being optimized, since the handling

characteristics of the vehicle may change as part of the optimization. This has the

unfortunate side effect of significantly increasing the dimension of the optimal design


                                             134
                     45.0

                     40.0

                     35.0

                     30.0

                     25.0
   Angle (degrees)




                     20.0

                     15.0

                     10.0

                      5.0

                      0.0

                      -5.0

                     -10.0
                             0   200       400       600     800           1000      1200   1400      1600       1800       2000
                                                                   Track Position (ft)
                                   Steering Wheel Angle    Unconstrained SW Profile Point    Constrained SW Profile Point


                                 Figure 8.2 - The Steering Profile for the Kenley, NC Race Track



search space and of slowing the optimization process. On the other hand, it allows the

                                                   s
simulation to consistently drive the vehicle to it’ limits.

                        Ordinarily, two parameters are used to describe each steering profile point:

position and steering wheel angle. For the ten point steering profile shown in Figure 8.2,

this leads to a 20 parameter optimization space. The size of the optimization space is

further increased by the addition of the speed profile parameters (discussed below) and by

the addition of vehicle design parameters. The number of parameters being optimized can

rapidly become unwieldy for any but the simplest of road courses. Fortunately, the number

of parameters can be significantly reduced by taking advantage of symmetry and/or




                                                                     135
periodicity in the steering profile and also by fixing the position of those steering points for

which optimization of both the position and the steering wheel angle is unnecessary.

        The cost function used to optimize the steering profile is computed by integrating

the magnitude of the instantaneous error in lateral position with respect to position along

the track and then dividing by the length of the track.

                                   1 L
                                       y veh ( x ) − y path ( x ) dx
                                   L∫
                           C.F.=                                                           (8.2.1)
                                     0


The resulting value is the average lateral position error of the vehicle measured over the

entire lap.

        The lateral position of the vehicle is determined by projecting the vehicle’s

reference point (usually taken to be a point near the geometric center of the vehicle) onto

the road surface and then finding the distance between the projected point and the road

                                                             s
centerline along a line perpendicular to the road centerline’ tangent vector. The lateral

position of the driver path is determined by evaluating the interpolating polynomial which

defines the driver path at the position corresponding to the point on the road centerline

which was just determined.

        The computation of the cost function can be facilitated by integrating it along with

the equations of motion. Integrating the cost function in the time domain requires that the

variable of integration in Equation 8.2.1 be changed:

                              dx
                                 = v veh      ⇒      dx = v veh dt                         (8.2.2)
                              dt

                                1 T
                                    y veh ( t ) − y path ( t ) v veh ( t )dt
                                L∫
                        C.F.=                                                              (8.2.3)
                                  0




                                                    136
where vveh is the velocity of the vehicle at the time (or position) in question. Note that

integrating the path error in time without the velocity weighting term would lead to a cost

function which de-emphasizes errors in the sections of the track which are traversed

quickly. An even weighting is far more desirable.

       In the event that the vehicle crashes before the completing the desired number of

laps or before reaching the end of the course, a penalty is computed and added to the cost

function value as computed up to the time of the crash. In designing the function for the

penalty term computation there are two desirable characteristics which should be satisfied

if possible. The first characteristic is that the penalty for a crash near the end of the road

course should carry less weight than the penalty for a crash near the beginning of the

course. This property aids the optimization of the steering profile by encouraging the

optimizer to move towards a path which results in the vehicle completing the entire road

course. The second desirable characteristic of the penalty term is that, in the worst case, it

should be at least as large as the maximum possible cost function value which could be

obtained by a vehicle which successfully complete the road course. A penalty function

which satisfies both of these conditions can be obtained by integrating the maximum

possible lateral position error for the remainder of the distance around the track. The

penalty term is computed by integrating the function


                                                ymax ( x ) dx
                                         xend
                                P.F.=   ∫
                                        xcurr
                                                                                         (8.2.4)




                                                  137
where xcurr is the current position (the position at the time of the crash), xend is position at

the end of the road course and ymax is the maximum distance between the driver path and

the edge of the road as defined below:


                                      1 w + y path ( x ),   y path ( x ) ≥ 0
                       y max ( x ) =  2                                                  (8.2.5)
                                      2 w − y path ( x ),   y path ( x ) < 0
                                       1




8.3 Speed Control
        The speed control system is similar to the steering control system in that a

prescribed velocity profile is defined which specifies the desired vehicle velocity as a

function of position along the track. Unlike the steering control system the profile is not

used as a direct input to the vehicle model. Instead, the goal of the speed controller is to

modulate the throttle and brake to follow the speed profile as precisely as possible. The

controller utilizes a single point preview strategy to anticipate changes in the vehicle

velocity. The use of preview allows the controller to begin responding to a rapid change in

the velocity profile before the vehicle actually arrives at the point where the change

occurs, thus reducing error in tracking the velocity profile. The controller also

incorporates traction control and anti-lock brake control features which operate by

limiting the longitudinal slip at the tires. This is done in order to prevent wheel lockup or

excessive wheel spin which, if left uncontrolled, can cause numerical problems with the

integration process.

        The speed profile is specified in the same manner as the steering profile: A series of

speed/position data points defines the basic curve and linear interpolation is used to obtain




                                                     138
                                           Aeff
                                      s


    Vdes             Veff                          Ain                                         Vveh
                                                                         V ( s)
                                                                 Flong            Aveh 1
             P( s)                  Gerr                 C( s)
                                                                                           s



                        Figure 8.3 - The Driver Speed Controller Block Diagram



speed values between the data points. The parameters describing the speed profile are

typically included in the optimization process, the goal being to minimize the lap time by

maximizing the vehicle speed. Note that, since the velocity profile can change during the

optimization process, it may be necessary to dynamically update the initial conditions used

for each simulation.

           The block diagram for the speed control algorithm is shown in Figure 8.3. The

form of the control was generated by considering the types of information available to the

speed controller and the types of input (external inputs and feedbacks) which could be

applied to the vehicle model. The available inputs and feedbacks were then combined to

produce the desired vehicle model control inputs. The detailed discussion of the driver

control algorithm, presented below, follows the logical progression used in its

development: identifying the available inputs and feedback quantities, identifying and

generating the control input to the vehicle model, and finally, the adding the preview

functionality.




                                                  139
        The sole external input consists of the prescribed velocity profile (designed by Vdes

in the block diagram). By differentiating the velocity profile with respect to time the

desired acceleration at any point along the track may also be determined. Since the

velocity profile is specified as a function of track position it is necessary to apply the chain

rule for differentiation to find the acceleration at a particular point along the track.

                                           d
                              Ades ( x ) =   Vdes ( x )
                                          dt
                                          dx d
                                        =           Vdes ( x )                             (8.3.6)
                                          dt dx
                                                    d          
                                        = Vdes ( x ) Vdes ( x )
                                                     dx        
        The desired acceleration is used as the primary input to the velocity controller.

Under ideal circumstances it would be the only necessary input since tracking the desired

acceleration exactly should yield the desired velocity curve. In practice it is necessary to

included an additional feedback loop to provide some error correction ability.

        There are a number of potentially useful feedback terms generated by the vehicle

model including vehicle speed, wheels speed, longitudinal slip, lateral slip and so on. The

most useful of these is, of course, the longitudinal velocity of the vehicle, designated by

                                       s
Vveh in the block diagram. The vehicle’ longitudinal velocity is subtracted from the

previewed desired longitudinal velocity to generate an error value which is multiplied by a

gain Gerr (which has units of s-1). The resulting value is then added to the previewed

acceleration value (the preview block is discussed in greater detail below) and provided as

a command input to the driver dynamics control block.




                                                  140
Driver Dynamics Block
        The dynamics of the driver are represented by the control block C(s). This block

converts the command acceleration to a force to be applied to the vehicle via the tires and

implements the traction control and anti-lock brake features. Unlike many of the driver

control models found in the literature, there is no need to model the lags and the frequency

response limitations of a human driver in this application. In fact, modeling these delays

would most likely degrade the tracking performance of the controller. In light of these

considerations, the conversion portion of the driver control block consists of a simple

gain, equal to the effective mass of the vehicle, which converts the acceleration command

input to a force to be applied to the vehicle via the rear tires, in the case of acceleration, or

via all four tires, in the case of braking.

                                 Facc _ brk ( t ) = M eff ⋅ Ain ( t )                      (8.3.7)

        The manner in which the force is applied to the vehicle depends on whether the

force represents an accelerating force or whether it represents a braking force. In the

acceleration case, the longitudinal force is divided equally between the rear wheels (since

the rear differential is locked in a Legends car) and converted to an applied moment by

dividing by the current wheel radius. In the braking case, the force is apportioned between

the front and rear of the car using a fixed brake bias constant and then divided equally

between the left and right wheels. Again, the force is converted to an applied moment by

dividing by the wheel radius.

        The traction control portion of the driver control block is somewhat more

complex. The goal of the traction control algorithm is to limit wheel spin under conditions


                                                      141
of excessive drive torque. This is accomplished by reducing the drive torque in proportion

to the degree of excessive wheel slip.

        To design a useful controller it is first necessary to understand the physical

significance of the longitudinal slip and to determine the possible range of longitudinal slip

values. The range of the longitudinal slip variable under conditions of acceleration can be

determined by considering the steady state solution to the differential equation for

longitudinal slip introduced in the chapter on tire modeling:

                                         Vx      V
                                   κ+
                                   &        κ = − sx                                      (8.3.8)
                                         B        B
which has the steady state solution

                                            Vx − Vr
                                      κ=−                                                 (8.3.9)
                                              Vx
where Vx is the actual longitudinal velocity of the wheel center and Vr is the velocity of the

wheel center if it were in a free rolling state (i.e. Vr = Rω where R and ω are the current

wheel radius and angular velocity). Under conditions of acceleration (either forward or

reverse) the magnitude of the rolling velocity Vr is greater than the magnitude of the actual

velocity Vx. This condition produces values of κ greater than zero. If the angular velocity

of the wheel is zero (locked wheel under braking) κ is negative one. Under conditions of

extreme acceleration where Vr is much greater than Vx, κ approaches positive infinity. A

value of κ equal to one is obtained when Vr = 2Vx or the wheel is spinning at twice the

free rolling angular velocity.




                                               142
                                           1.1
                                           1.0
                                           0.9

     Acceleration Gain Factor (unitless)
                                           0.8
                                           0.7
                                           0.6
                                           0.5
                                           0.4
                                           0.3
                                           0.2
                                           0.1
                                           0.0
                                                 0.0            0.5         1.0               1.5         2.0          2.5         3.0
                                                                               Longitudinal Slip (unitless)
                                                           Gtc = 0.0       Gtc = 0.05     Gtc = 0.1       Gtc = 0.25   Gtc = 0.5
                                                           Gtc = 1.0       Gtc = 1.5      Gtc = 2.0


                                           Figure 8.4 - The Effect of the Traction Control Gain Parameter on the Acceleration



                                           To control wheel spin under acceleration it is desirable to gradually reduce the

magnitude of the acceleration command input, and thus the applied torque, as the amount

of wheel spin increases. The reduction should begin at the desired slip threshold and

increase until the applied torque goes to zero at infinite slip. A gain factor which can be

applied to the command acceleration, thus reducing the drive torque, and which has the

desired characteristics is


                                                                            10
                                                                              .              ,κ + Coffset ≤ 10
                                                                                                             .
                                                                 
                                                       Gaccel   =            1
                                                                                             ,κ + Coffset > 10
                                                                                                             .                     (8.3.10)
                                                                  1 + GTC (κ + Coffset − 1)
                                                                 
The constants GTC and Coffset control the rate of decrease of the command acceleration and

the threshold at which the traction control begins to take effect respectively. Figure 8.4

shows a plot of the Gaccel as a function of longitudinal slip for several values of GTC.


                                                                                        143
Vehicle Dynamics Block
       Ideally, the tires would immediately generate the desired longitudinal force upon

                                                                    t
application of an appropriate driving torque. In reality this doesn’ happen for two

reasons. The first reason is that the longitudinal force generated by a tire lags the

application of a driving or braking torque. This lag is modeled by the first order

longitudinal slip equation in the tire model and the lag is characterized by the longitudinal

relaxation length for the tire. These lags are represented in the block diagram of the speed

controller by the vehicle dynamics block V(s):

                                                  1
                                   V (s) =                                              (8.3.11)
                                             τ lag s + 1

where τ lag is the time constant associated with slip equations. Note that the time lag may

vary with vehicle velocity. The second reason that the longitudinal force may not track the

applied torque exactly is that the applied torque may exceed the tractive capabilities of the

tire. The traction control algorithm does nothing to alleviate this problem since all it does

it to limit wheel spin under conditions of excessive acceleration and it does not actually

enhance the tractive capabilities of the tires. These two effects are not modeled in the

controller block diagram.


Preview Compensation Block
       The preview block is used to partially compensate for the lags inherent in the

vehicle response discussed above. As indicated by the name, the function of the preview

block is to generate an input to the remainder of the vehicle speed controller which is

based on profile information from the section of road ahead of the vehicle. This enables



                                                144
                                                    s
the controller to anticipate changes in the vehicle’ velocity and thus reduce lags in the

response.

                                                    s
       There are number of ways to model the driver’ ability to see the road (or in this

case the velocity profile) ahead of the vehicle. The majority of the models utilize a

weighted average of the road profile or of the velocity profile information directly ahead

of the vehicle. The weighted average usually takes a form similar to

                                            τ2


                            Veff ( t ) =
                                           ∫ w(τ )V ( t + τ )dτ
                                           τ1
                                                           des
                                                                                          (8.3.12)
                                                 τ2
                                              ∫ w(τ )dτ
                                                 τ1


where Vdes(t) is the prescribed velocity profile, Veff(t) is the effective velocity input

provided to the control algorithm (the previewed velocity) and w(τ) is a weighting

function. The integration limits define the boundaries of the preview interval. The simplest

type of preview function to implement is the so-called single point preview. Single point

preview is used in this application. The single point preview form is obtained by inserting

the weighting function


                                                  (
                                      w ( τ ) = δ t + Tp         )                        (8.3.13)

where δis the Dirac delta function, into the preceding equation. The result is


                                                       (
                                   Veff ( t ) = Vdes t + Tp           )                   (8.3.14)

where Tp is the preview time. The transfer function for the preview block can be obtained

by computing the Laplace transform of the preceding result.


                            { (
                Veff ( s) = L Vdes t + Tp    )}= e L{V ( t )}= e
                                                      Tp s
                                                                     des      Vdes ( s)
                                                                           Tp s
                                                                                          (8.3.15)




                                                      145
                V des                                      V eff                         V veh
                                  P(s)                                        F (s)

                   Figure 8.5 - The Simplified Preview-Follower Control System



                                                            s
The resulting transfer function can be expanded in a Taylor’ series to get a polynomial

form which is used in later computations.


                                  Veff ( s)                               Tp2
                        P( s) =                =e          = 1 + Tp s +         s2 + K
                                                    Tp s

                                  Vdes ( t )                              2                      (8.3.16)

                             = 1 + P s + P2 s 2 + K
                                    1

        At this point, the form of the preview function has been chosen. It is still necessary

to tune the preview block to provide the best velocity tracking performance by

determining the proper value for Tp. In order to obtain an optimal controller, it is

necessary to consider the dynamics of the combined controller-vehicle system. The

dynamics of the system can be depicted using a simplified block diagram like the one

shown in Figure 8.5. The dynamic response of the preview function is represented by the

P(s) (preview) block and the dynamic response of the combined vehicle and control

network is represented by the F(s) (follower) block. For an ideal control system the

product of the preview transfer function and of the follower transfer function should be

unity for all values of s. This would produce an output equal to the input (i.e. perfect

tracking of the velocity profile). In reality, it is sufficient for this equality to be satisfied

within a particular low frequency range (usually below 5-10 Hz) [Guo, 1993].

                                               P( s) ⋅ F ( s) ≈1                                 (8.3.17)



                                                             146
If the inverse of the follower transfer function, which represents the combined controller

response and vehicle response, is given by

                           F − 1( s) = 1 + Ps + P2 s 2 + P3s3 + K
                                            1                                        (8.3.18)

then the product of the preview function and the follower function (which represents the

complete system) is unity within the low frequency range and perfect control is achieved.

In practice it has been shown that matching the coefficients beyond third order can lead to

controller stability problems [Guo, 1993]. On the other hand, matching an insufficient

number of coefficients can lead to significant tracking errors.

       The transfer function for the controller shown in Figure 8.3 can be shown to be


                                               1
                                                ( Gerr + s)V ( s)
                         vveh ( s)             s
                                   = F ( s) =                                        (8.3.19)
                         veff ( s)                  1
                                               1 +   Gerr V ( s)
                                                    s
where V(s) is the block representing the vehicle dynamics and where the effects of traction

control and anti-lock brake control on the system are ignored. Setting the product of the

preview transfer function and the follower transfer function equal to one gives the

following result


                       1                              1
                        ( Gerr + s)V ( s)P( s) = 1 +   Gerr V ( s)               (8.3.20)
                       s                              s
                     s
Inserting the Taylor’ series expansion of the preview transfer function (keeping only

constants and terms linear in s) derived earlier and canceling terms gives


                                     [                  ]
                               V ( s) 1 + Gerr Tp + Tp s = 1                         (8.3.21)




                                                147
The transfer function used to model the vehicle response to acceleration commands was

assumed to be a simple first order lag:

                                                  1
                                   V ( s)=                                            (8.3.22)
                                              1 + τ lag s
Inserting this result into the preceding equation and collecting terms based on the order of

s gives two equations:

                                          Tp = τ lag                                  (8.3.23)

                                      Gerr Tp = 0                                     (8.3.24)

       The first equation provides a means of selecting the best preview time for a given

                         s
vehicle once the vehicle’ response has been characterized. The second equation indicates

that Gerr should be zero (since Tp is non-zero according to the first equation). This

effectively removes the velocity feedback loop from the controller. While this might work

for a vehicle whose response matches the vehicle dynamics model V(s) exactly, the real

vehicle requires velocity error feedback to correct deviations from the desired response.

The deviations are primarily due to limitations of the tires at high slip values and to the

action of the traction control and anti-lock braking control algorithms at high acceleration

                             t
values. Even though Gerr can’ be set to zero it is desirable to satisfy the condition as

nearly as possible by keeping the product of the preview time and the gain reasonably

small (so that the acceleration command generated by the feedback loop is small compared

to the acceleration command from the prescribed velocity profile).

       The preview time can be determined once an estimate for the lag constant τ lag has

been computed. The lag constant can be estimated given the relaxation length of the tire


                                                 148
and the velocity of the vehicle. The differential equation for the longitudinal slip of the tire

is of the form

                                          Vx      V
                                   κ+
                                   &         κ = − sx                                     (8.3.25)
                                          B        B
where the time constant for the exponential solution is

                                                  B
                                        τ lag =                                           (8.3.26)
                                                  Vx
For a typical longitudinal velocity on the order of 100 ft/sec and a typical relaxation length

of 0.3 feet the lag time is found to be

                                    τ lag = 0.003 sec                                     (8.3.27)

Inspection of simulation results seems to back up this result with lag time values on the

order of 0.1 seconds or less.

        The gain Gerr controls the responsiveness of the controller to errors in the vehicle

velocity. Selecting a large gain produces rapid responses but can interfere with the desired

acceleration feed forward. Selecting a gain which is too small causes long delays in

correcting velocity errors. Experimentation has shown that taking Gerr ≈ 0.5 works well

when simulating the NCSU Legends car. Note that the product of the of the velocity error

gain and the preview time is small (on the order of 0.01) which comes close to satisfying

condition 8.3.24.




                                                  149
9 Results, Conclusions and Recommendations

9.1 Introduction
        The ultimate goal of the vehicle dynamics research program at NCSU is to

demonstrate the applicability of optimal design techniques used in other automotive

applications (e.g. valvetrain and cam design [Etheridge, 1998 and Kim, 1990]) to vehicle

dynamics. While complete success has not yet been achieved, a great deal of progress has

been made. A vehicle model with an intermediate number of DOF has been developed and

a computer program has been written to solve the equations of motion. The computer

simulation has demonstrated the necessary computational efficiency to allow optimal

design to be performed. The vehicle model and the associated computer simulation have

been partially validated by modeling the NCSU Legends race car. The following sections

discuss the measurement of the NCSU Legends Car, present the vehicle data used in the

simulation, discuss the chassis setup procedure applied to the model, and finally, discuss

the simulation results.


9.2 Measurement Process and Model Data
        The first step of the simulation and/or optimization process is to determine the

geometric parameters and physical parameters which describe the system being modeled.

The system in this case consists of the vehicle itself, the tires, the road surface and the



                                            150
driver. The measurement processes used to obtain the data and the data itself are

presented in the following sections for each of the components of the system.


Vehicle Data
       The vehicle data primarily consists of the physical data describing the positions of

the various joints connecting the components which make up the vehicle and the mass and

inertia properties of those components. The vehicle measurement process is described

below and is followed by the vehicle data.

       The first step in the vehicle measurement process is to locate the origins of the

various body fixed coordinate systems. The use of centroidal body fixed coordinate

systems in the derivation of the equations of motion requires that the origin of the each of

the coordinate systems used to measure the car be located at the center of gravity of the

respective bodies. Thus, it is necessary to compute the location of the center of gravity of

each model component prior to measuring the vehicle.

       The orientation of the body fixed coordinate systems are subject to a few

restrictions which result from additional conditions imposed on the system during the




                         Table 9.1 - Front Suspension Geometric Data

Parameter Description           Left Front (ft)                Right Front (ft)
Wheel Center Location           {0, 0.0833, 0 }                {0, -0.0833, 0 }
Lower Ball Joint Location       {0, -0.375, -0.315 }           {0, 0.375, -0.406 }
Upper Ball Joint Location       {-0.015, -0.41, 0.3 }          {-0.015, 0.41, 0.3 }
Steering Knuckle Location       {0.325, -0.323, -0.305}        {0.325, 0.281, -0.333}
Control Arm Spring Mounts       {0.125, 1.15, 0 }              {-0.125, 1.13, 0 }
Control Arm Damper Mounts       {0.125, 1.15, 0 }              {-0.125, 1.13, 0 }
Upper Control Arm Length        0.677                          0.688
Lower Control Arm Length        1.33                           1.35


                                              151
derivation of the equations of motion. The x-axis of the sprung mass coordinate system

should be parallel to the longitudinal plane of symmetry of the vehicle. This requirement

provides the lateral symmetry required by the steering system model. The x-axis need not

be coincident with that plane. The y-axis of the unsprung mass coordinate systems should

be parallel to the axes of rotation of the wheels. This requirement is necessary to preserve

the rotational symmetry of the inertia tensor of the wheels in the unsprung mass body fixed

coordinate systems.

        Based on these considerations, the coordinate systems used for the four masses are

defined as follows. The sprung mass x-axis points forward from the center of gravity

                                                                            s
parallel to the plane of lateral symmetry. The y-axis points out the driver’ side door and

the z-axis points up. The remaining three unsprung mass coordinate systems are oriented




           Table 9.2 - Rear Suspension Geometric Data, Spring Data and Damper Data

Parameter Description                                          Value (ft)
Suspension Link Lengths:
    Left Trailing Link                                         1.23
    Center Trailing Link                                       0.775
    Right Trailing Link                                        1.2
    Panhard Rod                                                2.19
Suspension Link Mounting Locations:
    Left Trailing Link                                         { -0.0167, 1.29, -0.271 }
    Center Trailing Link                                       { -0.133, -0.0933, 0.354 }
    Right Trailing Link                                        { 0.00833, -0.883, -0.312 }
    Panhard Rod                                                { -0.192, 1.23, -0.283 }
Spring and Damper Mounting Locations:
    Left Side Spring                                           { 0.292, 1.67, -0.275 }
    Left Side Damper                                           { 0.292, 1.67, -0.275 }
    Right Side Spring                                          { 0.292, -1.26, -0.282 }
    Right Side Damper                                          { 0.292, -1.26, -0.282 }
Wheel Center Locations:
    Left Side                                                  { 0, 2.33, 0 }
    Right Side                                                 { 0, -1.83, 0 }


                                            152
in approximately the same manner. The y-axis for each of the unsprung mass coordinate

systems is parallel to the axis of rotation of the wheels.

        With the location of the origin and the orientation of the coordinate axes having

been determined for each of the four body fixed coordinate systems, the measurement

process can be begun. Since it is rarely convenient to measure the locations of points on




                            Table 9.3 - Sprung Mass Geometric Data

Parameter Description                                          Value (ft)
Spring Mounting Locations:
        Left Front (Upper)                                     { 3.31, 1.18, 0.771 }
        Right Front (Upper)                                    { 3.29, -1.29, 0.792 }
        Left Rear (Upper)                                      { -3.32, 1.08, 1 }
        Right Rear (Upper)                                     { -3.29, -1.19, 1 }
Damper Mounting Locations:
        Left Front (Upper)                                     { 3.31, 1.18, 0.771 }
        Right Front (Upper)                                    { 3.29, -1.29, 0.792 }
        Left Rear (Upper)                                      { -3.32, 1.08, 1 }
        Right Rear (Upper)                                     { -3.29, -1.19, 1 }
Control Arm Coordinate System Origin:
        Left Front (Upper)                                     { 3.13, 0.9, 0.344 }
        Left Front (Lower)                                     { 3.1, 0.26, -0.281 }
        Right Front (Upper)                                    { 3.13, -1.04, 0.354 }
        Right Front (Lower)                                    { 3.1, -0.385, -0.271 }
Control Arm Axes of Rotation:
        Left Front (Upper)                                     { 1, 0, 0 }
        Left Front (Lower)                                     { 1, 0, 0 }
        Right Front (Upper)                                    { -1, 0, 0 }
        Right Front (Lower)                                    { -1, 0, 0 }
Rear Suspension Mounting Points:
        Left Trailing Link                                     { -2.41, 1.25, -0.167 }
        Center Trailing Link                                   { -2.95, -0.354, 0.438 }
        Right Trailing Link                                    { -2.39, -1.29, -0.167 }
        Panhard Rod                                            { -4.05, -1.22, 0.0417 }
Reference Point Locations:
        Left Front Passenger Compartment Frame Corner          { 1.73, 0.750, -0.4583 }
        Right Front Passenger Compartment Frame Corner         { 1.73, -0.883, -0.4583 }
        Left Rear Passenger Compartment Frame Corner           { -2.73, 1.18, -0.4583 }
        Right Rear Passenger Compartment Frame Corner          { -2.73, -1.31, -0.4583 }
        Vehicle Center                                         { 0, 0, 0 }


                                              153
                Table 9.4 - NCSU Legends Car Model Mass and Inertia Properties

Parameter Description                   Value
Chassis (Sprung Mass) Mass              23.3 slugs
Chassis Inertia Tensor                  {{ 110, 0, 0 },{ 0, 775, 0 },{ 0, 0, 775 }} slug-ft2
Left Front Spindle (Unsprung) Mass      2.07 slugs
Left Front Spindle Inertia Tensor       {{ 2, 0, 0 },{ 0, 2, 0 },{ 0, 0, 2 }} slug-ft2
Left Front Wheel Inertia Tensor         {{ 4.5, 0, 0 },{ 0, 8.97, 0 },{ 0, 0, 4.5 }} slug-ft2
Right Front Spindle (Unsprung) Mass     2.07 slugs
Right Front Spindle Inertia Tensor      {{ 2, 0, 0 },{ 0, 2, 0 },{ 0, 0, 2 }} slug-ft2
Right Front Wheel Inertia Tensor        {{ 4.5, 0, 0 },{ 0, 8.97, 0 },{ 0, 0, 4.5 }} slug-ft2
Rear Axle Assembly (Unsprung) Mass      6.74 slugs
Rear Axle Inertia Tensor                {{ 30, 0, 0 },{ 0, 10, 0 },{ 0, 0, 30 }} slug-ft2
Left Rear Wheel Inertia Tensor          {{ 4.5, 0, 0 },{ 0, 8.97, 0 },{ 0, 0, 4.5 }} slug-ft2
Right Rear Wheel Inertia Tensor         {{ 4.5, 0, 0 },{ 0, 8.97, 0 },{ 0, 0, 4.5 }} slug-ft2


the vehicle with respect to the center of gravity of each mass, several reference points on

the chassis, rear suspension and spindles were chosen to ease the measurement process.

The measurements were made with respect to these reference points and then converted to

the body fixed coordinate systems. The chassis reference points on the NCSU Legends car

were taken to be the bottom edge of the inside corners of the frame rails which make up

the boundaries of the passenger compartment floor. These are roughly the same points

used to check chassis height during the vehicle setup process.

        The measurements were taken by placing the vehicle on jack stands above a large

sheet of paper. A plum bob was used to find the projection of key points onto the ground

plane. A combination square was used to determine the height of the joints above the

ground plane. The measurements for the front spindles were taken with the spindles still

attached to the vehicle but with the springs and the shocks removed from the vehicle. The

wheels were supported so that their position relative to the chassis was roughly the same

as when the vehicle was sitting on the ground. This was done primarily to reduce



                                              154
                      Table 9.5 - Suspension Spring and Damper Properties

Parameter Description                           Left Side               Right Side
Front Suspension
    Spring Free Length                          0.833 ft                0.833 ft
    Spring Preload Length                       0.24 ft                 0.35 ft
    Spring Rate (Linea r)                       2340 lb/ft              2460 lb/ft
    Spring Rate (Cubic)                         0 lb/ft 3               0 lb/ft 3
    Damping Coeff. (Jounce, Linear)             150 lb-s/ft             150 lb-s/ft
    Damping Coeff. (Jounce, Cubic)              0 lb(s/ft)3             0 lb(s/ft)3
    Damping Coeff. (Rebound, Linear)            150 lb-s/ft             150 lb-s/ft
    Damping Coeff. (Rebound, Cubic)             0 lb(s/ft)3             0 lb(s/ft)3
Rear Suspension
    Spring Free Length                          0.833 ft                0.833 ft
    Spring Preload Length                       0.462 ft                0.433 ft
    Spring Rate (Linear)                        2400 lb/ft              2100 lb/ft
    Spring Rate (Cubic)                         0 lb/ft 3               0 lb/ft 3
    Damping Coeff. (Jounce, Line ar)            150 lb-s/ft             150 lb-s/ft
    Damping Coeff. (Jounce, Cubic)              0 lb(s/ft)3             0 lb(s/ft)3
    Damping Coeff. (Rebound, Linear)            150 lb-s/ft             150 lb-s/ft
    Damping Coeff. (Rebound, Cubic)             0 lb(s/ft)3             0 lb(s/ft)3


measurement error by aligning the spindle so that it was roughly parallel to the ground,

thereby making the ground plane parallel to the y-axis of the spindle coordinate system.

The rear axle was supported in the same manner. The geometric vehicle data is presented

in Table 9.1, Table 9.2 and Table 9.3.

        The mass properties of the various bodies were determined as follows. The rear

axle was weighed by removing the springs and shocks and weighing each wheel with the

chassis supported on jack stands. The front wheel and spindle assemblies were weighed in

a similar manner. Inertias for the wheels were estimated using geometric data from the

wheels and tires and using the appropriate material densities (see Appendix B for a more

detailed explanation). The inertia of the sprung mass was estimated by extrapolating the

data found in (Garrot, 1988) to the appropriate total vehicle weight. The mass and inertia




                                             155
properties for the various bodies which comprise the vehicle model are presented in Table

9.4.

       Spring rates were assumed to be linear and were computed based on the physical

dimensions of the springs (Shigley, 1989). Damping rates were estimated using measured

data from Winston Cup car shock absorbers which were scaled by the ratio of the vehicle

masses. This scaling can be justified if one models the vehicle as a simple one degree of

freedom mass-spring-damper system. The damping ratio for such a system is written as

                                              C
                                      ζ =                                               (9.2.1)
                                            2 Mω n
Equating the damping ratios for the Winston Cup car and the Legends Car and assuming

that the natural frequencies are the same gives

                                      CWC      CLC
                              ζ =           =                                           (9.2.2)
                                    2 MWCω n 2 M LCω n
or


                                          M 
                                    CLC =  LC CWC                                     (9.2.3)
                                           MWC 

Tire Data
       The tire data consists of the geometric parameters which describe the tire as well

as data describing the tractive properties of the tire. The tire data used for simulating the

NCSU Legends car came from several sources. The geometric data was obtained by direct

measurement of the tire. The tire equivalent spring stiffness was estimated based on values

which appear in the literature. Damping in the tire was neglected. The relaxation length

values and low speed damping threshold were set based on the recommendations of


                                               156
               Table 9.6 - Miscellaneous Tire Model Parameters: Geometric Data,
                    Slip Equation Parameters and Normal Force Parameters.

Parameter Name      Parameter Description                                         Value
Fz0                 nominal wheel load                                            350 lbs
R0                  tire radius (no load)                                         0.938 ft
K0                  radial tire stiffness (linear coefficient)                    18000 lb/ft
K1                  radial tire stiffness (cubic coefficient)                     0 lb/ft 3
C0                  radial tire damping (linear coefficient)                      0 lb-s/ft
C1                  radial tire damping (cubic coefficient)                       0 lb(s/ft)3
RLX                 longitudinal relaxation length                                0.3 ft
RLY                 lateral relaxation length                                     3.0 ft
DCX                 longitudinal slip low speed damping coefficient               0.8
DCY                 lateral slip angle low speed damping coefficient              0.8
DAMPVEL             low speed damping threshold                                   0.5 ft/s


Bernard and Clover (Bernard, 1995). Table 9.6 summarizes the values used for these

parameters.

       The lateral and longitudinal force generation characteristics were modeled using

tire data taken from an information packet supplied by BFGoodrich for use by the

collegiate Legends car racing teams. The data packet is reproduced in Appendix C. The

tires are designated as BFGoodrich Comp TA HR4 “Legends Edition” and are derived

from a passenger car tire design. Tire data was provided for three tire pressures (15 psi,

25 psi and 35 psi) and four wheel loads (1125 lbs, 900 lbs, 675 lbs and 450 lbs). The tire

data packets consists of the following plots: Cornering Stiffness versus Load, Lateral

Force versus Slip Angle and Aligning Moment versus Slip Angle. No data was provided

for longitudinal force versus longitudinal slip and no data was provided for combined slip

operation. Under typical racing conditions the wheel loads for the NCSU Legends car are

in the 150 lb to 750 lb range. Thus, the most useful data sets are those that were taken at

675 lbs and 450 lbs normal load. The 450 lb normal load data curves were used since they



                                                157
                             97
          Table 9.7 - Delft ’ Tire Model Parameters: Pure Longitudinal Slip Equation

Parameter Name      Parameter Description                                               Value
P_CX1               longitudinal force curve shape factor                               1.5
P_DX1               longitudinal coefficient of friction                                1.2
P_DX2               longitudinal coefficient of friction - normal load dependence       0
P_EX1               longitudinal force curvature factor                                 0
P_EX2               longitudinal force curvature factor - normal load dependence        0
P_EX3               longitudinal force curvature factor - normal load dependence        0
P_EX4               longitudinal force curvature factor - longitudinal slip asymmetry   0
P_HX1               longitudinal force horizontal offset                                0
P_HX2               longitudinal force horizontal offset - normal load dependence       0
P_KX1               initial longitudinal force stiffness                                20
P_KX2               initial longitudinal force stiffness - normal load dependence       1
P_KX3               initial longitudinal force stiffness - normal load dependence       0
P_VX1               longitudinal force curve vertical offset                            0
P_VX2               longitudinal force curve vertical offset - normal load dependence   0


are most representative of the load conditions seen by the tire. No attempt was made to

model the variation of the tire model parameters with normal load.

        The Magic Formula Tire Model, discussed in Chapter 7, consists of several types

of curves which are fitted to the available tire data. One of the advantages of using the

Magic Formula Tire Model is that the constants which appear in the equations have

physical significance and can be easily obtained from tire performance plots, such as the

ones in Appendix C. Due to the lack of more detailed data on the Legends car tires a

number of the model parameters were set to zero. These parameters are primarily

responsible for modeling the more subtle dependencies of the tire forces on such things as

normal load variation and tire camber angles. The parameters which are responsible for

modeling the slight asymmetries between the positive and negative slip regions were also

set to zero.




                                                158
                                97
             Table 9.8 - Delft ’ Tire Model Parameters: Pure Lateral Slip Equation

Parameter Name       Parameter Description                                               Value
P_CY1                lateral force curve shape factor                                    1.207
P_DY1                lateral coefficient of friction                                     0.95
P_DY2                lateral coefficient of friction - normal load dependence            0
P_DY3                lateral coefficient of friction - camber angle dependence           0
P_EY1                lateral force curvature factor                                      -0.932
P_EY2                lateral force curvature factor - normal load dependence             0
P_EY3                lateral force curvature factor - camber angle dependence            0
P_EY4                lateral force curvature factor - camber angle dependence            0
P_HY1                lateral force horizontal offset                                     0
P_HY2                lateral force horizontal offset - normal load dependence            0
P_HY3                lateral force horizontal offset - camber angle dependence           0
P_KY1                initial cornering stiffness                                         20.05
P_KY2                initial cornering stiffness - normal load dependence                1
P_KY3                initial cornering stiffness - camber angle dependence               0
P_VY1                lateral force curve vertical offset                                 0
P_VY2                lateral force curve vertical offset - normal load dependence        0
P_VY3                lateral force curve vertical offset - camber angle dependence       0
P_VY4                lateral force curve vertical offset - camber and normal load dep.   0


        The parameters for the longitudinal pure slip curve and the lateral pure slip curve

are shown in Table 9.7 and Table 9.8 respectively. Since the steering system model isn’t

sensitive to aligning torque, no effort was made to model the aligning torque and all of the

associated coefficients are set to zero. The overturning moment and the rolling resistance

were also ignored and their associated coefficients were set to zero.

        The coefficients in the preceding tables describe the lateral and longitudinal force

generation characteristics of the tires under pure slip conditions (i.e. acceleration/braking

or cornering, but not both at the same time). It is also necessary to characterize the

relationship between lateral and longitudinal force generation under conditions of

combined slip (simultaneous acceleration/braking and cornering). Due to the lack of data

on combined slip tire properties, the general shapes for the combined slip curves were

estimated based on plots obtained from the literature for other tires [e.g. Gillespie, 1992].


                                                 159
The parameters for the combined slip equations are shown in Table 9.9.


Track Data
        The road surface data provides the geometric description of the road surface. The

data for the road model used for the simulation results presented in this chapter was

obtained from measurements made on the Kenley, NC race track. The measurements were

made by pacing off the track dimensions. The bank angles were measured using a

protractor and a bubble level. Although the dimensions of the runoff apron were

measured, and are included in the drawing below, the runoff apron was not included in the

                                             t
final road surface model since the car doesn’ normally drive it. A widened version of the

track model, utilizing a 100 ft track width instead of the measured 65 ft width, was created

and used during optimization to make it easier for the optimizer to find valid steering

profiles.




                                97
             Table 9.9 - Delft ’ Tire Model Parameters: Combined Slip Equations

Parameter Name      Parameter Description                                           Value
R_BX1               longitudinal force - longitudinal slip dependence               1.0
R_BX2               longitudinal force - longitudinal slip dependence               0.5
R_CX1               longitudinal force - minimum value coefficient                  9
R_HX1               longitudinal force - horizontal offset                          0
R_BY1               lateral force - lateral slip dependence                         16.5
R_BY2               lateral force - lateral slip dependence                         0
R_BY3               lateral force - lateral slip dependence                         0
R_CY1               lateral force - minimum value coefficient                       1.04
R_HY1               lateral force - horizontal offset                               0
R_VY1               lateral force - vertical offset                                 0
R_VY2               lateral force - vertical offset, normal load dependence         0
R_VY3               lateral force - vertical offset, camber dependence              0
R_VY4               lateral force - vertical offset, lateral slip dependence        0
R_VY5               lateral force - vertical offset, longitudinal slip dependence   0
R_VY6               lateral force - vertical offset, longitudinal slip dependence   0


                                                 160
                     Figure 9.1 - The Schematic of the Kenley, NC Race Track




Driver Model Data
       The driver data consists of the gains and other parameters which describe the

       s
driver’ response to command inputs. The driver model parameters for the simulation

were set as shown in Figure 9.1. The minimum and maximum steering angle parameters

specify the range of optimization for the steering profile points. The optimization range is

asymmetric about zero because all of the turns on the road course are left turns. The

minimum and maximum velocity parameters specify the optimization range for the velocity

profile points. The delay constant parameter (DM_VEL_T_DELAY) is the time constant

for a first order ODE which is used to filter the steering input function before it is applied

to the vehicle. The filter acts to remove the high frequency components of the steering

                                                                      s
input function which keeps the integration algorithm from reducing it’ step size



                                             161
unnecessarily. The roles of the remaining constants were discussed in the chapter on driver

modeling.


9.3 Model Chassis Setup
        Once the vehicle measurements have been made it is usually necessary to make fine

adjustments to the suspension in the same way that a real race car is set up prior to a race.

The setup process includes setting the toe angle, camber angle and caster angle for the

front wheels, setting the track width for the front suspension, squaring the rear axle with

respect to the chassis, setting the left side and right side wheel base, setting the cross

weight and finally, setting the frame heights.

        The vehicle setup process for a real car is performed in several steps. It is

frequently necessary to repeat the sequence of steps because an adjustment made to one

part of the car can upset an adjustment made elsewhere. The cross weight adjustments are

very sensitive to irregularities in the surface, thus, all adjustments should be made with the

car sitting on a flat and level surface.

        The first step is to set the ride height to the desired value. This is done by adjusting

the position of the spring seat on each of the four coil-over shocks used on the NCSU

Legends car. This effectively lengthens or shortens the shock body. Lengthening the shock

body raises the associated corner of the car. To set the ride height, all four shocks are

adjusted until the desired frame heights are achieved at each of the four corners of the car.

        The next step is to align the rear axle. This is done by adjusting the trailing links to

square the rear axle with respect to the chassis and by changing the length of the panhard



                                                 162
                             Table 9.10 - Driver Model Parameters

Parameter Name              Parameter Description                                        Value
DM_STEER_MAXANGLE           maximum allowed steering angle for steering profile opt.     1.0472 rad
DM_STEER_MINANGLE           minimum allowed steering angle for steering profile opt.     -0.5236 rad
DM_VEL_MAXSPEED             maximum allowed speed for velocity profile optimization      140 ft/s
DM_VEL_MINSPEED             minimum allowed speed for velocity profile optimization      40 ft/s
DM_VEL_T_DELAY              time constant for high frequency filter for steering angle   0.05 sec
DM_VEL_T_PREVIEW            preview (look ahead) time                                    0.003 sec
DM_VEL_LAMBDA               velocity error feedback loop gain factor                     2.0
DM_VEL_M_EFF                command acceleration gain (effective vehicle mass)           70 slugs
DM_VEL_BRAKEBIAS            front/rear brake bias constant                               0.37
DM_VEL_MAXACCEL             maximum allowed acceleration                                 100 ft/s2
DM_VEL_MAXDECEL             maximum allowed deceleration                                 -30 ft/s2
DM_VEL_STTC_OFFSET          traction control threshold                                   0.8
DM_VEL_STTC_GAIN            traction control gain factor                                 0.25


                    s
rod to set the axle’ lateral position with respect to the chassis. It is also possible, and

sometimes desirable, to apply a small amount of steering angle to the rear axle by

adjusting the relative lengths of the trailing links. The rear wheel track width can be

increased by using shims between the wheel mounting flange and the wheel rim but is

otherwise fixed.

       The next step is to align and to position the front wheels relative to the rear axle

and the chassis. The camber angle, the caster angle and the toe angle for each of the front

wheels is set by adjusting the lengths of the control arms and strut rods. The wheel base

can be adjusted by lengthening or shortening the three trailing links supporting the rear

axle or by adjusting the lengths of the strut rods supporting the front spindles. The track

width is set by increasing or decreasing the lengths of the control arms.




                                               163
        Once the basic vehicle setup geometry has been achieved, the cross weights can be

set and the frame heights can be rechecked. To set the cross weight, the four wheels of the

vehicle are placed on scales and the lengths of the shocks on opposite corners of the car

are adjusted together to achieve the desired weight distribution. It may be necessary to

iterate between adjusting the frame height and the cross weight. If the suspension

geometry changes significantly due to raising or lowering the car it may also be necessary

to reset the suspension geometry.

        The setup process for the model follows a nearly identical sequence of steps. The

simulation code used to generate the results discussed in this chapter also has the

capability of solving for the equilibrium position of the system. By placing the model on a

flat road surface and finding the equilibrium position, the wheel weights and the front and

rear suspension alignment parameters can be determined. By iteratively adjusting the

suspension link lengths, in the same manner as discussed in the preceding paragraphs, the

desired vehicle setup may be achieved. The vehicle setup parameters used for the NCSU

Legends car simulation are shown in Table 9.11. The suspension link lengths presented in



                               Table 9.11 - Vehicle Setup Parameters

Parameter Description                             Value
Cross Weight                                      50.5%
Left Front and Right Front Frame Heights          3.625”
Left Rear and Right Rear Frame Heights            3.875”
Left Front Camber                                 3.0 degrees (out at top)
Right Front Camber                                -5.0 degrees (in at top)
Front toe angle                                   0.20” out
Front Track Width                                 50.9”
Rear Track Width                                  49.9”
Left Side Wheel base / Right Side Wheel base      73” ± ¼”
Rear Axle Steer Angle                             0.0 degrees


                                               164
the suspension geometry tables presented earlier in this chapter are the result of

performing this setup process on the vehicle model.


9.4 Simulation Results
        Prior to starting a vehicle optimization it is necessary to establish a baseline for

comparison purposes. The baseline run is identical to the vehicle optimization run in that it

optimizes the steering profile and the velocity profile to obtain the best possible lap time

and the minimum path tracking error. The difference is that the baseline optimization run

does not include vehicle design parameters as degrees of freedom. The optimizer setup,

the steering profile configuration and results, the velocity profile configuration and results,

and the simulation results are discussed in the following sections.


Optimizer and Cost Function Computation Setup
        The optimizer parameters were set as shown in Table 9.12. The cost function value

is computed as

             CF = WPosition CFPosition + WVelocity CFVelocity + WLap _ Time CFLap _ Time   (9.4.4)

where the CFi represent the cost function components and the Wi are weighting



              Table 9.12 - Optimization and Cost Function Computation Parameters

Parameter Description                                   Value
Iffco_RMaxH                                             0.5
Iffco_StartH                                            0.5
Iffco_RMinH                                             2.0e-006
Iffco_Fscale                                            100
Iffco_Restarts                                          4
CFW_PathErr                                             1.0
CFW_VelErr                                              0.001
CFW_LapTime                                             1.0
CFW_LapPenalty                                          40.0


                                                     165
parameters. The cost function for the baseline optimization run utilized all three cost

function components (position, velocity and lap time) using the weights shown in the

table. The cost function scale parameter was set to a value of 100 which is approximately

equal to the weighted sum of the maximum values of the cost function components:

                  CFMAX = 10(66.8)+ 0.001(100.0) + 10(40.0) ≈
                           .                        .        100                          (9.4.5)

The maximum values for the cost function components were obtained as follows. The

maximum lateral position error is equal to the maximum possible lateral position penalty as

discussed in section 9.1. The maximum velocity error is a conservative estimate based on

experience with the simulation code. The maximum lap time cost is specified by the user

via the CFW_LapPenalty parameter (see Table 9.12) and represents a time to be assigned

to laps which are not completed.

        The velocity error cost function component, which is equal to the time integral of

the instantaneous error between the vehicle velocity and the desired velocity, is weighted

               s
to minimize it’ contribution to the overall cost function. Experience has shown that using

a larger weight for this cost function component can prevent the optimizer from improving

the lap time. This is because increases in the peak speeds can lead to increased velocity

                                  s                              s
errors when the car is nearing it’ acceleration limits and/or it’ tractive limits. Setting the

weight to zero would be fine except that the simulation code uses the cost function

component weight as a flag to determine which parameters (in this case the speed profile

parameters) to include in the optimization. Thus, it is necessary to use a small, but

nonzero, value for the cost function weight.




                                               166
        For a reasonable well optimized steering profile, for the Kenley, NC track, the lap

time cost function component is roughly one order of magnitude larger than the path error

cost function component. The weights for both of these components are set to unity. This

has the tendency to emphasize lap time in the cost function minimization process.

Experience has shown that preferentially weighting the lap time is necessary in order to

achieve the fastest possible laps. It is believed that this is a result of the increase in the path

error cost function component, which normally occurs when changes are made to the

velocity profile, which has the tendency to discourage the optimizer from changing the

velocity profile.


Optimal Steering Profile Configuration
        The optimized steering profile for the Kenley, NC track, which is shown in Figure

9.2, consists of 20 steering angle points. Fourteen of these points are fixed with respect to

the track so that their position is not included in the optimization process. The remaining

six points can be repositioned by the optimization algorithm as necessary. There are a total

of 26 optimization degrees of freedom (2 DOF for each of the 6 unconstrained points plus

1 DOF for the remaining 14 constrained points) for this steering profile.

        The large number of optimization degrees of freedom makes the solution process

very slow. To alleviate this problem, the first optimization run was made using a version

of the steering profile which took advantage of the periodicity of the steering profile. The

second half of the steering profile was made identical to the first half, with the exception

of being offset to the second half of the track. This reduced the number of optimization

parameters to 13 and enabled a solution to be found relatively rapidly. This solution was


                                                167
                      45.0

                      40.0

                      35.0

                      30.0

                      25.0
    Angle (degrees)




                      20.0

                      15.0

                      10.0

                       5.0

                       0.0

                       -5.0

                      -10.0
                              0   200      400        600     800         1000        1200   1400      1600       1800       2000
                                                                    Track Position (ft)

                                    Steering Wheel Angle    Unconstrained SW Profile Point    Constrained SW Profile Point


                                  Figure 9.2 - The Steering Profile for the Kenley, NC Simulation



used as an initial starting point for the full 26 parameter aperiodic steering profile

optimization run.


Optimal Velocity Profile Configuration
                       The optimized velocity profile for the Kenley, NC track, which is shown in Figure

9.3, consists of four points. The symmetry of the track allows for periodicity of the

velocity profile. Accordingly, the second pair of velocity profile points is based on a

shifted image of the first pair of points. This reduces the number of optimization

parameters from a total of eight parameters (speed and position for each of the four

points) to four parameters. Unlike the steering profile, the position for all of the velocity

profile points is free to be optimized. This allows the optimizer to pick the best point at



                                                                    168
which to begin braking for the corner and the best point at which to begin accelerating out

of the corner.

                      The velocity profile used as a starting point for this optimization was the result of

numerous prior simulation and optimization runs and, as such, is somewhat atypical of a

normal initial velocity profile. In the absence of prior knowledge of the vehicle’s

performance limits, one starts with a velocity profile with moderate speeds which are

guaranteed to allow the car to complete a lap. Including the lap time term in the overall

cost function allows the optimizer to gradually increase the speeds at which the car

navigates the road course until it is no longer possible for the car to remain on the desired

path.




                    75.0



                    70.0



                    65.0
   Velocity (mph)




                    60.0



                    55.0



                    50.0



                    45.0
                           0   200    400      600        800          1000       1200    1400       1600   1800   2000
                                                                Track Position (ft)
                                            Desired Long Vel         Aperiodic Points    Periodic Points


                               Figure 9.3 - The Velocity Profile for the Kenley, NC Simulation



                                                                   169
 Speed Control Algorithm Performance
                            The driver speed control performance is quite good, as shown in Figure 9.4, which

 compares the prescribed velocity profile and the actual vehicle velocity profile. The best

 lap time obtained for the baseline simulation was 21.4 seconds. This is slower than the lap

 times obtained for the real vehicle on the Kenley, NC track. The best times obtained to

 date by NCSU Legends car team members are in the 17.4 second to 18.0 second range.

                            The discrepancy in the lap times is not due to the inability of the model to match

 the speeds attained by the real vehicle. The maximum velocity on the back stretch and the

 minimum velocity in the corner were measured for the real Legends car using a radar gun.

 The minimum speed was approximately 40 mph to 44 mph, depending on the driver. The




                 75



                 70



                 65
Velocity (mph)




                 60



                 55



                 50



                 45
                      0.0                5.0                  10.0                      15.0       20.0
                                                                Time (sec)

                                                    Vehicle Velocity         Prescribed Velocity


                  Figure 9.4 - A Comparison of the Prescribed Velocity and the Actual Vehicle Velocity



                                                                       170
                           20.0




                           15.0
   Acceleration (ft/s^2)




                           10.0




                            5.0




                            0.0




                           -5.0
                                  0.0        5.0             10.0           15.0            20.0
                                                              Time (sec)

   Figure 9.5 - The Vertical Acceleration of the Sprung Mass (Sprung Mass Coordinate System)



maximum speed was between 71 mph and 78 mph. The optimized speed profile for the

simulation, shown in Figure 9.4, matches the measured speed ranges quite well. It is

possible that the sawtooth pattern used for the prescribed velocity profile does not

represent the velocity profile of the real vehicle very well. The real vehicle may be

spending a greater amount of time at the high speeds, thus reducing the lap time.

                             Another possible explanation for the discrepancy is that the combined slip

characteristics of the tire model may not be correct. The combined slip behavior was

modeled in an ad hoc fashion due to the lack of tire data for the combined slip loading.

The real Legends car may exhibit greater cornering power, allowing it to exit the corner at

greater speeds, again reducing lap times.




                                                             171
                    0.30


                    0.25


                    0.20


                    0.15
  Slip (unitless)




                    0.10


                    0.05


                    0.00


                    -0.05


                    -0.10


                                                              10.0




                                                                                     15.0




                                                                                                        20.0
                            0.0




                                           5.0




                                                              Time (sec)

                                      LF Long Slip    RF Long. Slip        LR Long. Slip    RR Long. Slip


                                    Figure 9.6 - The Longitudinal Wheel Slip Percentages



                       One final possible explanation is that the track dimensions could be in error. As

                                                           s
mentioned earlier, the track was measured by pacing off it’ dimensions. An over-estimate

of 10%, which would not be out of the question given the methods used, could easily

cause the lap times to be 2 seconds slower. The measured bank angles at the midpoints of

the corners have been confirmed to be within half a degree. The bank angles at the corner

entrance, corner exit, and at the midpoint of the straight away have not been confirmed

and could be error.

                       Aside from the slow lap times, the performance of the driver speed controller is

very good. There are two deviations from the prescribed velocity profile which occur at




                                                              172
                 Figure 9.7 - The Vehicle Position at 9.0 seconds (Exiting Turn 2)



9.0 seconds and at 19.5 seconds. These deviations are a result of the traction control

algorithm reducing the drive torque in order to limit wheel spin.

       The longitudinal wheel slip percentages are shown in Figure 9.6. The traction

control threshold was set to activate at a longitudinal slip of 0.2. The rear wheel slip

percentages exceed the traction control threshold value at 9.0 seconds and at 19.5 seconds

which matches the deviations from the velocity profile.

       The reason for the loss of traction can be seen in Figure 9.5 which contains a plot

of the vertical acceleration of the vehicle. The vertical acceleration takes on negative

values at the times in question. This is due to a slight crest in the track surface which

                                                                   s
occurs at the exit of each of the turns. A snapshot of the vehicle’ position at 9.0 seconds


                                               173
                      800

                      700

                      600
   Normal Load (lb)


                      500

                      400

                      300

                      200

                      100
                            0.0           5.0                 10.0                 15.0                20.0
                                                               Time (sec)
                                           Left Front      Right Front      Left Rear     Right Rear


                                                Figure 9.8 - The Tire Normal Loads



is shown in Figure 9.7. Note that a widened version of the Kenley, NC track is shown in

the figure. The use of a widened track makes it easier for the optimizer to find an optimal

steering profile. The location of the driver path is, in actuality, very close to the inside of

the curve when viewed on the unmodified track. The plot of the normal loads on each tire

(Figure 9.8) shows similar evidence of a lack of normal force leading to the loss of

traction.


Steering Control Performance
                        The performance of the steering controller can be analyzed by looking at the lateral

position error of the vehicle with respect to the prescribed driver path over the course of

the simulation. This plot is shown in                     Figure 9.9. The overall performance of the driver

controller is quite good with an average tracking error of approximately 2.5 ft. The worst

case tracking error is approximately 9 ft and occurs at approximately 18 sec. The reason



                                                                174
for this large position error becomes apparent upon inspection of the related vehicle

performance plots. The most interesting of these plots are the yaw velocity plot, the lateral

acceleration plot, and the steering wheel angle plot.

                                   The yaw velocity plot shows peaks at approximately 6.0 seconds and at

approximately 16.5 seconds. The car is in the middle section of the turns from 4 seconds

to 8 seconds (turns 1 and 2) and from 15 seconds to 19 seconds (turns 3 and 4). The yaw

velocity peaks are significantly higher than the average yaw velocity through the turn

which indicates that there is a loss of traction. When viewing an animation of the vehicle’s

motion, it is possible to see the rear end of the car slip toward the outside of the corner at

the times noted above.




                                  10.0


                                   9.0


                                   8.0


                                   7.0
    Lateral Position Error (ft)




                                   6.0


                                   5.0


                                   4.0


                                   3.0


                                   2.0


                                   1.0


                                   0.0
                                         0.0      5.0                10.0              15.0       20.0
                                                                      Time (sec)
  Average Position Error = 2.84 ft

                                                Figure 9.9 - The Vehicle Lateral Position Error




                                                                     175
                               40


                               35


                               30
    Angular Velocity (deg/s)



                               25


                               20


                               15


                               10


                               5


                               0


                               -5
                                    0.0           5.0               10.0                 15.0          20.0
                                                                     Time (sec)


                                                        Figure 9.10 - The Yaw Velocity



                                Inspection of the lateral acceleration plot ( Figure 9.11) provides addition

confirmation of the loss of traction at the rear wheels. The lateral acceleration plot shows

two strong peaks at 6.0 seconds and at 17.0 seconds. These peaks have magnitudes of

                     s
approximately 1.25 G’ which exceeds the tractive capabilities of the tires by a significant

margin.

                                The steering angle plot ( Figure 9.11) again confirms the loss of traction. The plot

shows two minima, located in the center of the turns, at 6.2 seconds and at 16.9 seconds.

The steering wheel angle drops to approximately 4 degrees at both minima. The steering

angle at the beginning and end of each of the turns is approximately 42 degrees. The

minima in the steering profile occur because the driver is counter-steering the car to




                                                                     176
                                      45

                                      40

                                      35
   or Lateral Acceleration (ft/s^2)
    Steering Wheel Angle (deg)
                                      30

                                      25

                                      20

                                      15

                                      10

                                       5

                                       0

                                       -5

                                      -10
                                            0.0                5.0                     10.0                     15.0                      20.0
                                                                                        Time (sec)
                                                               Lateral Acceleration                    Steering Wheel Angle
                                                               Unconstrained SW Profile Points         Locked SW Profile Points


                                                  Figure 9.11 - The Steering Wheel Angle and the Lateral Acceleration



compensate for the loss of traction at the rear wheels in order to remain on the desired

driver path.


9.5 Vehicle Optimization Results
                                       The baseline optimization results generated above were used as a starting point for

a relatively simple vehicle setup optimization. The left rear spring rate and the left rear

spring free length were selected as the optimization parameters. Modification of these

                               s                             s
parameters affects the vehicle’ cross weight and the vehicle’ weight transfer



                                                    Table 9.13 - Vehicle Suspension Parameter Optimization Ranges

Parameter Name                                           Parameter Description                   Lower Bound                      Upper Bound
DP_RS_LSP_FL                                             left rear spring free length            0 inches                         14.4 inches
DP_RS_LSP_K0                                             left rear spring rate                   166.7 lbs/in                     250.0 lbs/in


                                                                                        177
characteristics, and thus, affects the oversteer/understeer behavior of the car during

cornering.

       The optimization was run using the parameter set which was used for the baseline

optimization. The only change was the addition of the two vehicle suspension parameters

mentioned above. The optimization ranges for the suspension parameters are shown in

Table 9.13. The initial values and optimized values for the suspension parameters are

shown in Table 9.14.

       As can be seen from the optimization results, the changes were minimal. Although

the initial setup is known to be a good setup for the real vehicle when running on the real

track, it is unlikely that this setup is the best possible setup. Also, given the lack of

accurate tire data and other unavoidable discrepancies between the model and the real

vehicle, there are most likely some handling differences between the model and the real

vehicle which would lead to differences in the optimal configuration. Also of note is the

fact that the car demonstrated a tendency to oversteer in the apex of the corners during

the baseline optimization as noted in the preceding section. Based on these observations, it

seems clear that the vehicle optimization failed to improve the vehicle setup.

       This result is not entirely unexpected given the difficulty of simultaneously

optimizing the steering profile and the velocity profile which has been encountered in the




                Table 9.14 - Vehicle Suspension Parameter Optimization Results

Parameter Name       Parameter Description                Original Value   Optimized Value
DP_RS_LSP_FL         left rear spring free length         9.996 inches     9.997 inches
DP_RS_LSP_K0         left rear spring rate                200 lbs/in       200.06 lbs/in


                                                    178
past and which was noted in the preceding section. The explanation for this behavior is

that making significant changes to the vehicle setup (or to the prescribed velocity profile)

                   s
causes the vehicle’ path to deviate from the desired path by a significant amount. If the

steering profile has been reasonably well optimized already, this leads to an increase in the

path error cost function which discourages the optimizer from making changes to the

vehicle setup unless it is able to simultaneously correct the steering profile to eliminate the

driver path tracking error. This problem, and possible solutions, are discussed in the

following section.


9.6 Recommendations for Future Research
        The vehicle model itself seems to be capable of reproducing the fundamental

behavior of the NCSU Legends car on the Kenley, NC track. At this point no detailed

experimental validation of the model has been performed. It is recommended that the car

be fitted with a data acquisition system so that comparisons can be made to the model

predictions. The data acquisition system should include sensors for recording the same

type of data from the real car as was presented in the preceding sections for the model car.

This list of sensors includes the following devices: steering wheel angle position sensor,

three axis accelerometer, three axis angular velocity sensor, wheel velocity sensors (all

four wheels), and finally an engine RPM sensor.

        Another useful sensor would be a high sample rate high accuracy global

positioning system or another system with equivalent functionality. This would allow

determination of the driver path around the track as well as providing a better estimate of



                                              179
vehicle velocity which, in combination with the wheel velocities, could be used to compute

the longitudinal slip ratios for the wheels. By driving the vehicle around the inner

periphery and outer periphery of the track at low speeds it could also be used for

measuring the track itself.

        At present, no engine model is included in the simulation. The current arrangement

provides whatever power is necessary to achieve the desired acceleration, up to the limits

of adhesion of the tires. While it is not an unreasonable simplification when applied to the

Legends car, it would not be terribly difficult to limit acceleration based on the available

engine power at the current engine RPM (this can be calculated based on rear wheel

speed, gear ratio, etc.).

        Another area of possible impro vement deals with the braking system. The current

algorithm simply limits the maximum deceleration of the vehicle to avoid lockup. Applying

a more intelligent control algorithm, perhaps one similar to the currently implemented

traction control algorithm, could improve braking performance entering the corners which

would allow the optimizer to maintain straight away speed a bit longer or to increase the

maximum speed.

        The braking system on the real car includes a device which causes the brake

pressure at the rear wheels to lag the brake pressure applied to the front wheels. This can

assist in preventing rear wheel lockup when the brakes are applied rapidly. This aspect of

the braking system is not modeled in the simulation. It would also be useful to include

brake bias as a parameter in the optimization to extract the maximum possible

performance from the braking system.


                                             180
       The biggest improvements can be made in the driver control algorithms. The use of

the optimizer to handle the steering and throttle control tasks, in addition to optimizing the

vehicle design parameters, interferes with the optimization process and can prevent the

code from improving the performance of the vehicle. The large number of parameters

being optimized with this approach severely degrades the performance of the optimizer. It

is highly desirable to separate the driver control task from the vehicle parameter

optimization. One possible approach would be to break the road course into a fairly fine

mesh of segments which can be traversed sequentially using a shooting-method approach:

Given that the vehicle is on course at the beginning of the current segment, determine the

steering input required to arrive at the beginning of the next segment. An interpolating

polynomial of fairly low order could be used to generate the steering profile over the

length of the segment. The steering angle at the beginning of the segment is known (from

traversing the preceding segment). The steering input at the end of the segment, and

perhaps one or two other parameters, could be treated as the unknowns in the shooting

problem. The velocity at which the vehicle traverses the segment could be handled in a

similar manner.

       This approach has the advantage of guaranteeing that the car will either compete

the lap in an acceptable manner or that it will fail to complete the lap. Either way, the task

of driving the vehicle has been removed from the main optimization loop. The only

potential problem with this approach is that the vehicle response typically lags the steering

input by a significant amount (see Figure 9.11 - The Steering Wheel Angle and the Lateral

Acceleration for an example of this). When the length of the segments gets too small the


                                              181
steering inputs from the current segment may have a strong effect on the handling of the

vehicle when it is traversing the succeeding segment. This could lead to more and more

over-compensation as the vehicle crosses from one segment to the next and result in

instability of the driver controller. It may be possible to improve the stability to some

extent by enforcing a path tangency criterion in addition to a path offset criterion as the

vehicle traverses the segment.




                                            182
Bibliography
Allen, R. Wade and McRuer, Duane T. (1979). “The Man/Machine Control Interface -
Pursuit Control”, Automatica, 15:683-686.

Allen, R. Wade. (1982). “Stability and performance analysis of automobile driver steering
control”. SAE Technical Paper Series , Paper #820303.

Allen, R. Wade, Henry T. Szostak, Theodore J. Rosenthal and Donald E. Johnston.
(1986). “Test Methods and Computer Modeling for the Analysis of Ground Vehicle
Handling”, SAE Technical Paper Series , Paper #861115.

Allen, R. Wade, Theodore J. Rosenthal, Henry T. Szostak. (1987a). “Steady State and
Transient Analysis of Ground Vehicle Handling”, SAE Technical Paper Series , Paper
#870495.

Allen, R. Wade, Henry T. Szostak, and Theodore J. Rosenthal. (1987b). “Analysis and
computer simulation of driver/vehicle interaction”, SAE Technical Paper Series , Paper
#871086.

Allen, R. Wade, Theodore J. Rosenthal, Henry T. Szostak. (1988). “Analytical Modeling
of Driver Response in Crash Avoidance Maneuvering Volume I: Technical Background”,
U.S. Department of Transportation, NHTSA, DOT HS 807 270.

Allen, R. Wade, Henry T. Szostak, Theodore J. Rosenthal, and David H. Klyde. (1990).
“Field Testing and Computer Simulation Analysis of Ground Vehicle Dynamic Stability”,
SAE Technical Paper Series , Paper #900127.

Allen, R. Wade, Henry T. Szostak, Theodore J. Rosenthal, David H. Klyde, and K. J.
Owens. (1991). “Characteristics Influencing Ground Vehicle Lateral/Directional Stability”,
SAE Technical Paper Series , Paper #910234.

Allen, R. Wade, Theodore J. Rosenthal, David H. Klyde, K. J. Owens and Henry T.
Szostak. (1992). “Validation of Ground Vehicle Computer Simulations Developed for
Dynamics Stability Analysis”, SAE Technical Paper Series , Paper #920054.

Allen, R. Wade, Theodore J. Rosenthal. (1993). “A Computer Simulation Analysis of
Safety Critical Maneuvers for Assessing Ground Vehicle Dynamic Stability”, SAE
Technical Paper Series , Paper #930760.




                                           183
Allen, R. Wade, Theodore J. Rosenthal, Jeffrey R. Hogue. (1996). “Modeling and
Simulation of Driver/Vehicle Interaction”, SAE Technical Paper Series , Paper #960177.

Antoun, R. J., P. B. Hackert, M. C. O’Leary, and A. Sitchin. (1986). “Vehicle Dynamic
Handling Computer Simulation - Model Development, Correlation, and Application Using
ADAMS”, SAE Technical Paper Series , Paper #860574.

Bakker, E., L. Nyborg and H.B. Pacejka. (1987). “Tyre Modeling for Use in Vehicle
Dynamics Studies”, SAE Paper #870421.

Bakker, E., H.B. Pacejka and L. Lidner. (1989). “A New Tire Model with an Application
in Vehicle Dynamics Studies”, SAE Paper #890087.

Becker, G. , H. Fromm and H. Maruhn. (1931) “Schwingungen in Automobillenkungen”
(“Vibrations of the Steering Systems of Automobiles”), Krayn, Berlin.

Bekey, G. A., G. O. Burnham and J. Seo. (1977) “Control Theoretic Models of Human
Drivers in Car Following”, Human Factors, 19(4):399-413.

Bell, Steven C., W. Riley Garrott, John R. Ellis, Y. C. Liao. (1987). “Suspension Testing
Using the Suspension Parameter Measurement Device”, SAE Technical Paper Series ,
Paper #870577.

Bernard, James E. (1973). “Some time-saving methods for the digital simulation of
highway vehicles”. Simulation, 21(6):161-165, December 1973.

Bernade, James E. and C. L. Clover. (1994). “Validation of Computer Simulations of
Vehicle Dynamics”, SAE Technical Paper Series , Paper #940231.

Bernard, James E. and C.L. Clover. (1995). “Tire Modeling for Low-Speed and High-
Speed Calculations”, SAE Paper #950311.

Bryan, G. H. (1911). “Stability in Aviation”, Macmillan & Co., London.

Broulheit, G. (1925). “La Suspension de la Direction de la Voiture Automobile: Shimmy
et Danadinement’ (“The Suspension of the Automobile Steering Mechanism: Shimmy and
Tramp”), Société des Ingénieurs Civils de France, bulletin 78, 1925.

Bundorf, R. Thomas. (1967). “The influence of vehicle design parameters on characteristic
speed and understeer”. Transactions of the Society of Automotive Engineers, SAE
670078.

Clover, Chris L. and James E. Bernard. (1993). “The Influence of Lateral Load Transfer
Distribution on Directional Response”, SAE Technical Paper Series , Paper #930763.




                                           184
Crossman, E. R. F. W., H. Szostak and T. L Cesa. (1966). “Steering Performance of
Automobile Drivers in Real and Contact-Analog Simulated Tasks”, Human Factors
Society Tenth Annual Meeting , Oct. 1966.

Chrstos, Jeffrey P. (1991). “A Simplified Method for the Measurement of Composite
Suspension Parameters”, SAE Technical Paper Series , Paper #910232.

Dickson, J. G. and A. J. Yardley. (1993). “Development and Application of a Functional
Model to Vehicle Development”, SAE Technical Paper Series , Paper #930835.

Donges, Edmund. (1978). “A Two-Level Model of Driver Steering Behavior”, Human
Factors, Vol. 20(6), pp. 691-707.

Etheridge, Mark C. (1998). “Preliminary Performance of Carbon-Carbon Valves in High
                                            s
Speed Pushrod Type Valve Trains”, Master’ Thesis, North Carolina State University,
Raleigh, NC.

Evans, R. D. (1935). “Properties of tires affecting riding, steering, and handling”. Journal
of the Society of Automotive Engineers, 36(2):41.

Garrot, W. Riley, Douglas L. Wilson and Richard A. Scott. (1981). “Digital Simulation
For Automobile Maneuvers”, Simulation , pp. 83-91, September 1981.

Garrot, W. Riley, Michael W. Monk and Jeffrey P. Chrstos. (1988). “Vehicle Inertial
Parameters - Measured Values and Approximations”, SAE Technical Paper Series , Paper
#881767.

Genta, Giancarlo. (1997). “Motor Vehicle Dynamics: Modeling and Simulation”, ISBN
9810229119.

Gillespie, Thomas D. (1992). “Fundamentals of Vehicle Dynamics”, Society of
Automotive Engineers, ISBN 1-56091-199-9.

Goland, Martin and Frederick Jindra. (1961). “Car Handling Characteristics”. Automobile
Engineer , 51(8):296-302, August 1961.

Gordon, D. A. (1966a). “Experimental Isolation of Drivers’ Visual Input”, Public Roads,
33(12):266-273.

Gordon, D. A. (1966b). “Perceptual Basis of Vehicular Guidance”, Public Roads,
34(3):53-68.

Gruening, James and James E. Bernard. (1996). “Verification of Vehicle Parameters for
Use in Computer Simulation”, SAE Technical Paper Series , Paper #960176.

Guo, K. and H. Guan. (1993). “Modelling of Driver/Vehicle Directional Control System”,
Vehicle System Dynamics , 22:141-184.



                                            185
Heydinger, Gary J., W. Riley Garrot, Jeffrey P. Chrstos, and Dennis A. Guenther. (1990).
“A Methodology for Validating Vehicle Dynamics Simulations”, SAE Technical Paper
Series, Paper #900128.

Heydinger, Gary J., Paul A. Grygier, and Seewoo Lee. (1993). “Pulse Testing Techniques
Applied to Vehicle Handling Dynamics”, SAE Technical Paper Series , Paper #930828.

Heydinger, Gary J., Nicholas J. Durisek, David A. Coovert Sr., Dennis A. Guenther and
S. Jay Novak. (1995). “The Design of a Vehicle Inertia Measurement Facility”, SAE
Technical Paper Series , Paper #950309.

Hickner, G. B., J.G. Elliot and G.A. Cornell. (1971). “Hybrid Computer Simulation of the
Dynamic Response of a Vehicle With Four-Wheel Adaptive Brakes”, SAE Technical
Paper Series, Paper #710225.

Hoffman, E. R., (1975). “Human Control of Road Vehicles”, Vehicle System Dynamics ,
5(1-2):105-126.

Huang, Feng, J. Roger Chen and Lung-Wen Tsai. (1993). “The Use of Random Steer Test
Data for Vehicle Parameter Estimation”, SAE Technical Paper Series , Paper #930830.

Jindra, Frederick. (1976). “Mathematical Model of Four-Wheeled Vehicle for Hybrid
Computer Vehicle Handling Program”, Deparment of Transportation - National Highway
Traffic Safety Administration, DOT HS-801-800, January 1976.

Kim, Dojoong. (1990). “Dynamics and Optimal Design of High Speed Automotive Valve
Trains”, Doctoral Thesis, North Carolina State University, Raleigh, NC.

Kondo, M. and A. Ajimine. (1968). “Drivers Sight Point and Dynamics of the Driver-
Vehicle System Related to it.”, SAE Technical Paper Series , Paper #680104.

Kortüm W. and W. Schiehlen. (1985). “General Purpose Vehicle System Dynamics
Software Based on Multibody Formalisms”, Vehicle System Dynamics , 14:229-263.

Kortüm W. and R. S. Sharp. (1993). “Multibody Computer Codes in Vehicle System
Dynamics”, Supplement to Vehicle System Dynamics , Volume 22.

Lin, Y. and W. Kortüm. (1991). “Identification of System Physical Parameters for Vehicle
Systems with Nonlinear Components”, Vehicle System Dynamics , 20:354-365.

Lu, Zhengyu, Andrzej G. Nalecz, Kenneth L. d’   Entremont. (1993). “Development of
Vehicle-Terrain Impact Model for Vehicle Dynamics Simulation”, SAE Technical Paper
Series, Paper #930833.

McHenry, R. and N. Deleys. (1968). “Vehicle Dynamics in Single Vehicle Accidents”,
Technical Report CAL No. VJ-2251-V-3, Cornell Aeronautical Laboratory Inc.,
December 1968.


                                          186
McRuer, Duane T., R. Wade Allen, David H. Weir and Richard H. Klein. (1977). “New
Results in Driver Steering Control Models”, Human Factors, 19(4):381-397.

Milliken, William F. and David W. Whitcomb. (1956). “General introduction to a
programme of dynamic research”. Proceedings of the Automobile Division; The
Institution of Mechanical Engineers , (7):287-309.

Mimuro, Tetsushi, Takahiro Maemura and Hiroshi Fujii. (1993). “Development and
Application of the Road Profile Measuring System”, SAE Technical Paper Series , Paper
#930257.

Modjtahedzadeh, A. and R. A. Hess. (1993). “A model of driver steering control behavior
for use in assessing vehicle handling qualities”. Transactions of the ASME: Journal of
Dynamic Systems, Measurement, and Control, 115:456-464, September 1993.

Mori, Yoshinori, Hironobu Matsushita, et al. (1991). “A Simulation System for Vehicle
Dynamics Control”, SAE Technical Paper Series , Paper #910240.

Morman, Kenneth N. (1977). “Non-Linear Model Formulation for the Static and Dynamic
Analyses of Front Suspensions”, SAE Technical Paper Series , Paper #770052.

Mousseau, C. W., M. W. Sayers and D. J. Fagan. (1991). “Symbolic Quasi-Static and
Dynamic Analyses of Complex Automobile Models.”, Proceedings, 12 th International
Association for Vehicle System Dynamics Symposium on the Dynamics of Vehicles on
Roads and Tracks, Lyon, France, Aug. 1991.

Murphy, Ray W. (1970). “A Hybrid Computer System for the Simulation of Vehicle
Dynamics”, SAE Technical Paper Series , Paper #700154.

Nalecz, Andrzej G. (1987). “Investigation into the Effects of Suspension Design on
Stability of Light Vehicles”, SAE Technical Paper Series , Paper #870497.

Nalecz, Andrzej G. et al. (1988). “Effects of Light Truck and Roadside Characteristics on
Rollover”, Final Report for First Period, NHTSA - US DOT Contract No. DTNH22-89-
C-07005, August 23, 1988.

Nalecz, Andrzej G. (1992). “Development and Validation of Light Vehicle Dynamics
Simulation (LVDS)”, SAE Technical Paper Series , Paper #920056.

Nikravesh, Parviz E. (1988). “Computer-Aided Analysis of Mechanical Systems”, ISBN
0-13-164220-0 025, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.

Oberdieck W., B. Richter and P. Zimmerman. (1979). “Adaptation of Mathematical
Vehicles Models to Experimental Results”, Proceedings of the 6 th IAVSD Symposium on
the Dynamics of Vehicles on Roads and Tracks, Berlin, Germany, pp. 367-378, September
3-7, 1979.



                                          187
Okada, T., T. Takiguchi, M. Nishioka, and G. Utsumomiya. (1973). “Evaluation of
Vehicle Handling and Stability by Computer Simulation at the First Stage of Vehicle
Planning”, SAE Technical Paper Series , Paper #730525.

Olley, M. (1934). “Stable and Unstable Steering”, Unpublished Report, General Motors,
1934.

Olley, M. (1937). “Suspension and Handling”, Engineering Report, General Motors
Corp., May 1937.

Pacejka, H. B. and I. J. M Besselink. (1997). “Magic Formulate Tyre Model with
Transient Properties”, Vehicle System Dynamics Supplement , Vol. 27, pp. 234-249.

Pacejka, H. B. and E. Bakker. (1992). “The Magic Formula Tyre Model”, Tyre Models
for Vehicle Dynamics Analysis , 1st International Colloquium on Tyre Models for Vehicle
Dynamics Analysis, Supplement to Vehicle System Dynamics , pp. 1-18. Vol. 21.

Petersen, Michael R. and John M. Starkey. (1996). “Nonlinear Vehicle Performance
Simulation with Test Correlation and Sensitivity Analysis”, SAE Technical Paper Series ,
Paper #960521.

Radt, H. S. and W. F. Milliken Jr. (1960a). “Motions of Skidding Automobiles”, SAE
Summer Meeting, Paper #205A, June 1960.

Radt, H. S. and W. F. Milliken Jr. (1960b). “Exactly What Happens When an Automobile
Skids?”, SAE Journal, pp. 27-33, December 1960.

Radt, H. S. (1964). “Analysis of Nonlinear Problems in Vehicle Handling”, Advanced
Dynamic Institute, Wayne State University, September 1964.

Roland, R. D. and T. B. Sheridan. (1966). “A Normative Model for Control of Vehicle
Trajectory in an Emergency Manoeuvre”, Paper presented at Annual Meeting of the
Highway Research Board, Jan. 1966.

                                                                          s
Roland, R. D. and T. B. Sheridan. (1967). “Simulation Study of the Driver’ Control of
Sudden Changes in Previewed Path”, Mass. Inst. Tech., Department of Mechanical
Engineering, Report DSR 74920-1, 1967.

Sayers, Michael W. and Paul S. Fancher. (1993). “Hierarchy of Symbolic Computer-
Generated Real-Time Vehicle Dynamics Models”, Transportation Research Record,
1403:88-97.

Segel, Leonard. (1956a). “Research in the Fundamentals of Automobile Control and
Stability”, SAE National Summer Meeting, Atlantic City, June 5, 1956.




                                          188
Segel, Leonard. (1956b). “Theoretical prediction and experimental substantiation of the
response of the automobile to steering control”. Proceedings of the Automobile Division;
The Institution of Mechanical Engineers , (7):310-330, 1956.

Segel, Leonard. (1966). “On the Lateral Stability and Control of the Automobile as
Influences by the Dynamics of the Steering System”, Transactions of the ASME: Journal
of Engineering for Industry , August 1966.

Sharp, R. S. (1994). “The Application of Multi-body Computer Codes to Road Vehicle
Dynamics Modeling Problems”, Proceedings of the Institution of Mechanical Engineers,
Part D, Journal of Automobile Engineering , 208(D1):55-61.

Shigley, Joseph Edward and Charles R. Mischke. (1989). “Mechanical Engineering Design
- 5th Edition”, McGraw-Hill, Inc.

Smiley, R. F. and W. B. Horne. (1958). “Mechanical Properties of Pneumatic Tires with
Special Reference to Modern Aircraft Tires”, NACA, TN 4110, 1958 or NASA, TR R-64,
1960.

Speckhart, Frank H. (1973). “A Computer Simulation for Three-Dimensional Vehicle
Dynamics”, SAE Technical Paper Series , Paper #730526.

Society of Automotive Engineers. (1965). “Vehicle Dynamics Terminology SAE J670a”.
An updated version is available (SAE J670e).

Tiffany, N. O., G. A. Cornell and R. L. Code. (1970). “A Hybrid Simulation of Vehicle
Dynamics and Subsystems”, SAE Technical Paper Series , Paper #700155.

Walker, G. E. L. (1950). “Directional stability, a study of factors involved in private car
design”. Automobile Engineer , 40(530,533):281-370.

Weir, David H., C. P. Shortwell, and W. A. Johnson. (1968a). “Dynamics of the
automobile related to driver control”. Transactions of the Society of Automotive
Engineers , SAE 680194.

Weir, David H. and Duane T. McRuer. (1968b). “A Theory for Driver Steering Control of
Motor Vehicles”, Highway Research Record, 247(2):7-27.

Weir, David H. and Richard J. DiMarco. (1978). “Correlation and Evaluation of
Driver/Vehicle Directional Handling Data”, SAE Technical Paper Series , Paper #780010.

Whitcomb, David W. and William F. Milliken. (1956). “Design implications of a general
theory of automobile stability and control”. Proceeding of the Automobile Division; The
Institution of Mechanical Engineers , (7):367-391.

                                                                          s
Yoshimoto, K. (1969). “Simulation of Man-Automobile Systems by the Driver’ Steering
Model with Predictability”, Bulletin of the J.S.M.E. , 12(51):495-500.


                                            189
Appendix A Useful Derivatives
       In this section the derivatives of commonly appearing quantities are calculated.

Derivatives which are zero are not listed explicitly. Derivatives are taken with respect to

each generalized coordinate, with respect to each generalized velocity and with respect to

time for most quantities.


Angular Velocity Derivatives
       Derivatives are calculated in this section for the angular velocity       R
                                                                                     ω R/E of a

rotating coordinate system R with respect to the inertial coordinate system E. The angular

velocity of a rotating coordinate system with respect to the inertial coordinate system is

                                                                        &
                                                                       β 
                                          − β1    β0     β3    − β2   &0 
                                    r
                                    &                                β 
                R
                    ω R /E   = 2 [L]β = 2 − β2   − β3    β0     β1  &1 
                                                                        β
                                          − β3
                                                  β2    − β1    β0  &2 
                                                                      β 
                                                                        3               (A.1)
                                     &     &      &      &
                                 β β − β β − β2β3 + β3β2 
                                  0 &1 1 &0      &      &
                             = 2 β0β2 + β1β3 − β2β0 − β3β1 
                                 β β − β β + β β − β β 
                                  0 &3    &
                                          1 2
                                                  &
                                                 2 1
                                                         &
                                                        3 0

                                                                            &
The only nonzero derivatives are those taken with respect to the βi and the βi .


                                                   &
                                                  β 
                                        0 1 0 0  &0       &
                                                            β1 
                       ∂ R ω R /E               β       & 
                                    = 2 0 0 1 0 &1  = 2 β2                          (A.2)
                         ∂0β                       β        β 
                                        0 0 0 1  2 
                                                &        &3 
                                                  β3 




                                                  190
                                                             &
                                                            β 
                                 − 1    0     0        0   &0        &
                                                                      − β0 
                ∂ R ω R /E                                β        & 
                             = 2 0      0     0        1  &1  = 2  β3               (A.3)
                  ∂1β                                        β        − β 
                                 0
                                        0     −1       0  &2 
                                                                    &2 
                                                            β3 

                                                             &
                                                            β 
                                 0      0     0        −1  &0        &
                                                                      − β3 
                ∂ R ω R /E                                β        &
                             = 2 − 1    0     0        0  &1  = 2 − β0              (A.4)
                  ∂2β                                        β        β 
                                 0
                                        1     0        0  &2 
                                                                    &1 
                                                            β3 

                                                              &
                                                             β 
                                 0   0        1         0   &0      &
                                                                       β2 
                ∂ R ω R /E                                 β1      &
                             = 2 0 − 1        0         0  &  = 2 − β1              (A.5)
                  ∂3β                                         β       − β 
                                 − 1 0
                                              0         0   &2 
                                                                    &0 
                                                             β3 

                                                                 1
                              − β1     β0      β3        − β2       − β1 
             ∂ ω R /E                                          0         
                R
                          = 2 − β2     − β3    β0         β1    = 2 − β2            (A.6)
              ∂&0
               β                                                  0          
                              − β3
                                       β2     − β1        β0   
                                                                0    − β3 


                                                                 0
                              − β1     β0      β3        − β2        β0 
             ∂ ω R /E                                          1         
               R
                          = 2 − β2     − β3    β0         β1    = 2 − β3            (A.7)
              ∂&1
               β                                                  0          
                              − β3
                                       β2     − β1        β0   
                                                                0     β2 


                                                                 0
                              − β1      β0        β3     − β2        β3 
             ∂ R ω R /E                                        0         
                          = 2 − β2     − β3       β0      β1    = 2  β0             (A.8)
               ∂&2
                 β                                                1          
                              − β3
                                        β2    − β1        β0   
                                                                0    − β1 


                                                                 0
                              − β1      β0     β3        − β2       − β2 
             ∂R ω R /E                                         0         
                          = 2 − β2     − β3    β0         β1    = 2  β1             (A.9)
               ∂&3
                β                                                 0          
                              − β3
                                        β2    − β1        β0   
                                                                1     β0 

Note that the time derivative, taken with respect to the inertial coordinate system, is




                                                    191
                       E
                           d ∂R ω R /E  R d ∂R ω R /E  R         ∂R ω R /E
                                       =              + ω R /E ×
                           dt  ∂&i  dt  ∂&i 
                                 β                β                   ∂&i
                                                                        β
                                                                                                  (A.10)
                                                 ∂R ω R /E  R         ∂R ω R /E 
                                              =−          +  ω R /E ×           
                                                  ∂iβ                  ∂&i 
                                                                          β
where

                                        R
                                            d ∂ R ω R /E     ∂ R ω R /E
                                                         = −                                    (A.11)
                                            dt  ∂&i 
                                                   β             ∂iβ

The time derivative of the angular velocity can be found using the vector differentiation

theorem

                             E                   R
                                 d R
                                 dt
                                    ( ω R /E ) = dt ( R ω R /E ) + R ω R /E ×R ω R /E
                                                   d

                                                 R
                                                                                                  (A.12)
                                               =
                                                   d R
                                                   dt
                                                      ( ω R /E )
R                        r        r        r
    d R
    dt
       ( ω R /E ) = 2 [L]β + 2 [L]β = 2 [L]β
                         &
                         &      & &        &
                                           &

                                                              &
                                                              &
                                                             β 
                         − β1         β0        β3         0
                                                       − β2 &           &
                                                                        &      &
                                                                               &      &&      &
                                                                                              &
                                                                     β0β1 − β1β0 − β2β3 + β3β2  (A.13)
                                                             &
                                                            β1                               
        R
            ω R /E
            &        = 2 − β2        − β3      β0                      &
                                                                        &      &&      &
                                                                                       &      &&
                                                        β1  &  = 2β0β2 + β1β3 − β2 β0 − β3β1 
                                                              &
                                                              β
                         − β3                          β0  & 2   β β − β β + β β − β β 
                                      β2       − β1        & 
                                                                        &
                                                                      0& 3    &&
                                                                              1 2
                                                                                       &
                                                                                       &
                                                                                     2 1
                                                                                              &
                                                                                              &
                                                                                             3 0
                                                             β  3



Transformation Matrix Derivatives
            Derivatives of the coordinate transformation matrix [E C R ] , which transforms a

vector from its representation in the rotating system to its representation in the inertial

system, are calculated in this section. The coordinate transformation matrix is expressed in

terms of Euler Parameters. The definition of the transformation matrix is given below in

terms of two linear transformation matrices. The resulting nonlinear transformation matrix

depends only on the βιand time (indirectly).




                                                            192
                                   β2 + β1 − 1 β1β2 − β0β3 β1β3 + β0β2 
                                      0
                                           2
                                              2
                                                                       
            [E C R ] = [G ][L] = 2  1β2 + β0β3 β2 + β2 − 1 β2β3 − β0β1 
                              T
                                    β             0    2  2                                           (A.14)
                                    1β3 − β0β2 β2β3 + β0β1 β2 + β2 − 1 
                                   β                         0    3  2 



              − β1     β0     − β3        β2           − β1         β0       β3      − β2 
                                                                                          
       [G ] = − β2     β3      β0        − β1  ; [L] = − β2         − β3      β0      β1          (A.15)
              − β3
                      − β2     β1         β0          − β3
                                                                       β2      − β1     β0 
The derivatives of the transformation matrix with respect to the βι are


                      2β0      − β3       β2                          2β1 β2         β3 
         ∂E C R ]
          [                                              ∂E C R ]
                                                            [                              
                  = 2  β3      2β0       − β1                     = 2 β2  0         − β0 
            β
           ∂0                                                 β
                                                             ∂1
                      − β2
                                β1       2β0                         β3 β0
                                                                                        0 
                                                                                                      (A.16)
                       0        β1       β0                          0     − β0      β1 
         ∂E C R ]
          [                                              ∂E CR ]
                                                            [                             
                  = 2  β1      2β2       β3                      = 2 β0      0      β2 
            β
           ∂2                                                 β
                                                             ∂3
                      − β0
                                β3       0                          β1
                                                                              β2      2β3 
                                                                                           
The first time derivative of the transformation matrix is

d                 &        &              &         &               &
   [ E C R ] =[ E C R ] = [G ][L]T + [G ][L]T = 2[G ][L]T = 2[G ][L]T
dt
           &        &       &       &          &       &
      β0β0 + β1β1 − β2β2 − β3β3 − β0β3 + β1β2 + β2β1 − β3β0  &      &          &      &       &      &
                                                                              β0β2 + β1β3 + β2β0 + β3β1 
      &            &       &       &         &       &      &      &            &      &       &      & 
= 2 β0β3 + β1β2 + β2β1 + β3β0             β0β0 − β1β1 + β2β2 − β3β3         − β0β1 − β1β0 + β2 β3 + β3β2 
     − β β + β β − β β + β β
            &        &        &      &        &       &      &      &
                                           β0 β1 + β1β0 + β2 β3 + β3β2        β0β0 − β1β1 − β2 β2 + β3β3 
                                                                                &      &       &      &
      0 2          1 3     2 0     3 1                                                                   

                                                                                                      (A.17)
Note that

                                          &                 ~
                                      [ E C R ] =[E C R ][R ω R /E ]                                  (A.18)
         ~
where [R ω R/ E ] is the skew symmetric matrix associated with the angular velocity

vector R ω R/ E . R ω R/ E is the angular velocity of the non-inertial coordinate system with

respect to the inertial coordinate system expressed in terms of the unit vectors of the non-

inertial coordinate system. The result of the product of the skew symmetric matrix

    ~
[ R ω R/ E ] with a vector x is a vector containing the cross product of R ω R/ E and x.



                                                       193
The derivatives of [E CR ] with respect to the βi are
                      &


                      &     &
                     β0 − β3         &
                                      β2                    &
                                                           β1      &
                                                                    β2         &
                                                                               β3 
         [ &
        ∂E CR ]     &     &           &       [ &
                                               ∂E CR ]     &        &          & 
                = 2  β3   β0        − β1                  β
                                                       = 2 2      − β1       − β0 
           β
          ∂0                                      β
                                                 ∂1
                    − β2 β1
                       &   &          β0 
                                      &                    &3
                                                            β       &
                                                                    β0        − β1 
                                                                                &
                                                                                
                                                                                       (A.19)
                       & &
                    − β2 β1         &
                                     β0                       &
                                                           − β3    −    &
                                                                         β0    &
                                                                               β1 
         [ &
        ∂E CR ]     &    &          &         [ &
                                               ∂E CR ]     &            &     & 
                = 2  β1 β2          β3               = 2  β0     −    β3    β2 
           β
          ∂2                                      β
                                                 ∂3
                    − β0 β3
                       & &            &
                                    − β                    β1
                                                             &          &
                                                                        β      β3 
                                                                               &
                                      2                                2       
                                                 &
The derivatives of [ E C R ] with respect to the βi are
                       &


                  β0        − β3     β2                β1       β2          β3 
         [ &
        ∂E CR ]                                [ &
                                                ∂E CR ]                           
           & = 2  β3
           β
          ∂0
                              β0     − β1         & = 2 β2
                                                   β
                                                  ∂1
                                                                   − β1       − β0 
                 − β2
                             β1      β0               β3
                                                                  β0         − β1 
                                                                                   
                                                                                       (A.20)
                 − β2 β1            β0                − β3       − β0       β1 
         [ &
        ∂E CR ]                               [ &
                                               ∂E CR ]                           
           & = 2  β1 β2
          ∂2
           β
                                     β3          & = 2  β0
                                                  β
                                                 ∂3
                                                                    − β3       β2 
                 − β0 β3
                                   − β2 
                                                        β1
                                                                    β2        β3 
                                                                                  
Note that


                                    [ &          &
                                 d ∂ E C R ] ∂[E C R ]
                                            =                                          (A.21)
                                 dt ∂&i
                                      β         β
                                               ∂i




                                              194
The second time derivative of the transformation matrix is

         d2               & &         &&            & &             &&
          2 E
             [ C R ] = 2[G ][L]T + 2[G ][L]T = 2[G ][L]T + 2[G ][L]T
        d t
                      & &2 & &2                 & &
                     β2 + β1 − β2 − β3 − 2β0β3 + 2β1β2  &&        & &       &&
                                                                 2β0β2 + 2β1β3 
                       0          2
           &&         & &        &&       & &2 & &2                & &       & & 
        [E C R ] = 2 2β0β3 + 2β1β2        β2 − β1 + β2 − β3 − 2β0β1 + 2β2β3 
                                             0         2
                     − 2β0β2 + 2β1β3
                          & &       &&         & &     & &
                                             2β0β1 + 2β2β3      β2 − β1 − β2 + β2 
                                                                & &2 & &
                                                                0          2    3

                  &
                  &       &
                          &      &
                                 &       &
                                         &          &&     &&
             β0β0 + β1β1 − β2β2 − β3β3 − β0β3 + β1β2 + β2β1 − β3β0&
                                                                   &      &&                  (A.22)
             &   &       &
                          &      &&      &
                                         &         &&     &&     &&      &&
        + 2 β0β3 + β1β2 + β2β1 + β3β0           β0β0 − β1β1 + β2β2 − β3β3
            − β0β2 + β1β3 − β2β0 + β3β1 β0β1 + β1β0 + β2β3 + β3β2
                   &&      &&      &&     &&       &&     &
                                                          &      &&      &
                                                                         &
            
                &&     &&     &&     &&
              β0β2 + β1β3 + β2β0 + β3β1 
                 &&     &&     &&     & 
                                       &
             − β0β1 − β1β0 + β2β3 + β3β2 
              β0β0 − β1β1 − β2β2 + β3β3 
                &&     &&     &
                              &      &&
                                         
Alternatively, the second time derivative of the transformation matrix may be found by

differentiating equation A.18

                    d2
                     2 E
                    d t
                                   d &
                        [ C R ] = [E C R ] =
                                  dt
                                                 d
                                                dt
                                                       (            ~
                                                      [ E C R ][R ω R/ E ]  )
                                     &       ~                       ~&
                                =[ E C R ][R ω R/ E ] + [ E C R ][ R ω R / E ]                (A.23)
                                             ~            ~
                                =[ C ][R ω ][R ω ] + [ C ][R ω                 &
                                                                               ~          ]
                                    E   R       R/ E         R/ E   E   R          R/ E

                                                                &
                                                                ~
and substituting the time derivative of the angular velocity [R ω R/ E ]


                                          0
            [ R   &
                  ~
                       ]    β β − ββ + β β − β β
                  ω R /E = 2 0 3
                            
                                &&
                                 &&
                                       &
                                       &
                                      1 2
                                        &&
                                              &&
                                             2 1
                                               &&
                                                     &
                                                     &
                                                    3 0
                                                      &&
                            
                            − β0β2 − β1β3 + β2β0 + β3β1
                                                                                              (A.24)
                 &
                 &      &
                        &      &
                               &      &
                                      &
             − β0β3 + β1β2 − β2β1 + β3β0                  &&     &&     &&     &&
                                                        β0β2 + β1β3 − β2β0 − β3β1 
                         0                                 &&     &&     &&     & 
                                                                                 &
                                                       − β0β1 + β1β0 + β2β3 − β3β2 
                & − ββ − β β + β β
                &
              β0β1    &
                      &     &
                            &    &
                                 &                                   0             
                     1 0   2 3  3 2                                                
to obtain the same result as before.




                                                           195
Appendix B Wheel Inertia Estimate
           The inertia of the wheel brake and tire assemblies can be estimated by

approximating the assembly as a collection of cylindrical disks and cylindrical shells of

varying thicknesses and materials. The inertias for the basic shapes are computed below.


Thin Cylindrical Disk
           The general formula for computing the inertia of a body about a particular axis a

is


                                                                   I a = ∫ a dm
                                                                            2
                                                                          r                                                (B.1)
                                                                        V

where ra is the distance of the mass element dm from the axis of rotation. The inertias for

objects possessing cylindrical symmetry are most easily computed in cylindrical

coordinates. The mass element in this case is given by

                                                               dm = ρ r dr dθ dz                                           (B.2)

where ρ is the density of the material. For a thin disk, with thickness T, outside radius Ro

and inside radius Ri, there are only three unique inertias. They are as follows:

                                                T / 2 2 π Ro
                                    I xy =       ∫ ∫∫r
                                                               2
                                                                   cos(θ)sin (θ)ρ r drdθ dz = 0                            (B.3)
                                               − T / 2 0 Ri


                     T / 2 2π Ro
                                                                                π                  π
     I xx = I yy =    ∫ ∫∫r
                                      2
                                          cos2 (θ)ρ r drdθ dz =
                                                                                4
                                                                                     (         )
                                                                                  ρT Ro 4 − Ri 4 =
                                                                                                   64
                                                                                                         (
                                                                                                      ρT Do 4 − Di 4   )   (B.4)
                     − T / 2 0 Ri


                       T / 2 2π Ro
                                                                     π                 π
             I zz =      ∫ ∫∫r
                                          2
                                              ρ r drdθ dz =
                                                                     2
                                                                            (            )         (
                                                                       ρT Ro 4 − Ri 4 = ρT Do 4 − Di 4
                                                                                       32
                                                                                                             )             (B.5)
                       − T / 2 0 Ri



                                                                            196
The z-axis is the axis of rotational symmetry. The x and y axes are in the plane of the disk

and are orthogonal to each other and to the z-axis.


Thin Walled Cylindrical Shell

         The inertias for a thin-walled cylindrical shell are computed in the same manner as

for the disk. The shell is assumed to have a mean radius R, thickness T and a length L. The

z-axis is assumed to be coincident with the axis of rotational symmetry. The x and y axes

are constructed to be orthogonal to one another and to the z-axis. The volume of the

cylindrical shell is


                                    
                   L / 2 2π R + T / 2      2
                                         T      T 
                                                     2

            V = ∫ ∫ ∫r drdθ dz = π LR +  − R −   = 2π RLT                                    (B.6)
               − L / 2 0 R− T / 2   
                                        2       2 
and thus, the total mass is

                                                        m = 2πρ RLT                                (B.7)

The inertias are

                                              L / 2 2π R + T / 2
                         I xx = I yy =         ∫ ∫ ∫ρ( z                            )
                                                                       + r 2 sin 2 (θ) r drdθ dz
                                                                   2

                                              − L / 2 0 R− T / 2

                                 π
                               ≈ ρ L 3 RT + πρLR 3T                                                (B.8)
                                 6
                               =
                                 m
                                 12
                                        (
                                    6 R 2 + L2 =
                                                 m
                                                 24
                                                           )
                                                    3D 2 + 2 L2        (           )
                                         π              T 
                        L / 2 2π R + T / 2          4        4
                                                 T 
               I zz = ∫ ∫ ∫ ρ r drdθ dz = LρR +  − R −  
                                     r    2

                      − L / 2 0 R− T / 2
                                         2     2      2 
                      π
                       2
                                 [
                    = Lρ 4 R3T + RT 3 ≈2πρ LTR3          ]                                         (B.9)

                                   m
                    = mR 2 = D 2
                                   4




                                                                       197
Rotating Assembly Model
       The rotating assembly can be modeled as a collection of thin cylindrical shells and

thin disks. The rotating assembly includes the brake disk or brake drum, the wheel

mounting flange assembly, the wheel and the tire. The wheel center is modeled as a thin

disk and the wheel rim is modeled as a thin cylindrical shell. The portion of the brake disk

swept by the brake pads is modeled as a thin disk. The brake disk mounting flange is also

modeled as a thin disk and the material connecting the brake disk to the mounting flange is

modeled as a thin walled cylinder. The tire sidewalls are modeled as thin disks and the

tread surface is modeled as a thin walled cylinder. The wheel mounting flange is modeled

as a thin disk. Note that, when computing the inertia of the complete assembly, it is

necessary to apply the parallel axis theorem to obtain the proper result.

       The densities for the materials which make up the rotating assembly are easily

obtained with the exception of the tire density. The tire density can be estimated by

             s
measuring it’ weight and computing an approximate volume. This approach ignores the

uneven distribution of mass in the tire due to the presence of the steel belts but can

provide a reasonable estimate. If a tire is not available for measurement it is possible to

estimate the weight of the tire by measuring the weight of the wheel and tire together and

subtracting the weight of the wheel rim. The weight of the wheel rim is computed by

              s
estimating it’ volume and multiplying by the appropriate material density.




                                             198