A Safe Longitudinal Control for Adaptive Cruise Control and Stop

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```					SUBMITTED TO IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY, JANUARY 2005                                                                                                              1

A Safe Longitudinal Control for Adaptive Cruise
Control and Stop-and-Go Scenarios
John-Jairo Martinez and Carlos Canudas-de-Wit

Abstract— In this paper, we propose a novel reference model-                                                     60                                      Reference model
based control approach for automotive longitudinal control. An

Inter−vehicular distances (m)
Inter−distance rule (2 seconds)
important property of this proposed structure concerns the                                                          50
fact that the control design could be meet independent of the                                                            Average of the distances
peripheral Parisian−freeway
model design, permitting the additional control loop only be                                                        40
responsible of the model-matching between the actual system and
30
the desired reference dynamics. The reference model is non-linear
and provides dynamic solutions consistent with some deﬁned
20
safety and comfort constraints. Some model simulations together
with some experimental results are presented and discussed.                                                         10
Index Terms— Automotive, longitudinal control, reference
0
model, adaptive cruise control, stop-and-go.                                                                         0            20          40        60               80   100
Vehicle speed (km/h)

I. I NTRODUCTION                                 Fig. 1. Comparison of different distance policies: Constant time-headway
rule (2 seconds), Average distances in a parisian freeway (source [17]), and
DAPTIVE cruise control (ACC), and stop-and-go
A     scenarios are examples of problems related with
longitudinal control. The former concerns the inter-distance
the proposed reference distance policy.

control in highways where the vehicle velocity mainly                    equation, permitting to obtain the necessary distance to full
remains constant, whereas the latter deals with the vehicle              stop without collision, some examples are founded in [1], [12],
circulating in towns with frequent stops and accelerations.              [13], where the safe distance is calculated as
In both situations, goals of safety and comfort most often
2    2
oppose each other [16].                                                                                              dsaf e = λ1 (vf − vl ) + λ2 vf + λ3 ,                          (1)
for some constants λ1 , λ2 , and λ3 . The terms vf and vl
In most of the reported works, these two categories of
correspond to the follower and leader velocities, respectively.
problems are treated separately with little regard to the
The ﬁrst term is related to the relative braking distance
comfort speciﬁcations. Indeed, the behavior of the inter-
between two vehicles; the second term take into account the
distance dynamics often results from a particular feedback
system reaction time λ2 , the third term λ3 corresponds to the
loop, which makes difﬁcult to ensure a priori computable
minimal constant distance to respect.
bounds on the inter-distance and the vehicle acceleration and
Part of the attractiveness of this model is that it may
its time-derivatives. It is also suited that external factors such
be calibrated using common sense assumptions about driver
behavior, needing (in the most part) only the maximal braking
(among others), must be considered while defying the safety
rates that a driver will wish to use, and predicts other drivers
and the comfort metrics. This last point is naturally reinforced
will use, to allow it to fully function. However, this model
by the new safety programs including vehicles/infrastructure
correspond to a stationary solution of a motion equation,
communication [18],[19].
taking a non-exogen input (i.e. it depends of the own speed),
and assuming constant and similar decelerations for all the
Next we attempts to explain the main ideas about safety and
implied vehicles.
comfort criteria used in automotive longitudinal control. More
Although it produces acceptable results, for example, if
details and models could be found in [12] and [14].
one examines the “safety distance” concept, we see that this
model is not a totally valid starting point, as in practice,
A. Safe inter-distance policies.                                         during a urgence maneuver, the vehicles could present so large
During the last decades the well known “safe inter-distance”          transitory relative velocity, and then the actual inter-distance
has been calculated as a minimal distance to avoid a collision if        tends to decrease abruptly. This is opposed to the reference (1),
the preceding vehicle were to act “unpredictably”. In fact, the          which indiques that the safe distance should be increased, and
safe inter-distance is calculated from the Newtonian motion              by consequence, this safe distance is always violated during a
hard stop scenario. Thus, this model could be useful to dictate
Manuscript received January 10, 2005; revised November 7, 2005. This   at what moment the braking maneuver could be started, but it
work was supported by the ARCOS French Program.
Authors are with the Laboratoire d’Automatique de Grenoble, CNRS UMR   does not supply any braking strategy.
5528 ENSIEG, INPG, BP.46, St. Martin D’H` res 38440, France.
e                                Figure 1 illustrates three different distance policies. One
SUBMITTED TO IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY, JANUARY 2005                                                                                        2

..
^
xl
of these corresponds to the well known two seconds rule,
ur                                                          dr
that drives are forced to respect. In fact, this rule attempts                                                               Reference
Model

to respect a distance proportional to the human reaction time                                                                                               +
..
(approx. 1.5-2 seconds). Thus, starting from (1), this distance                                   xl                         ηd                             -
+                                                             ^
is calculated as                                                                              u        Inter-distance    d                             d
Sensor   Estimator
Dynamics
+

dsaf e = λ2 vf + λ3 .                        (2)
v                                 Feedback
controller
This rule is often called the Constant Time-headway rule,
where the constant λ2 stands for the Time-headway.                           Fig. 2.   The inter-distance control scheme.
Many works use this policy as a safe distance, with a little
regard in the original safe distance concept. Nevertheless,
some variations of this model try to perform other require-                  control, especially in Stop-and-go scenarios. This aspect will
ments, for example [2] proposes a control strategy where the                 be taken into account during the reference-model design.
safety inter-distance is computed as a non-linear function of
the speed, (i.e. the Time-headway is a function of the speed),               C. Paper contribution
in order to guarantee string stability in the platoon problem.
In this paper, we propose a novel reference model-based
Returning to the ﬁgure 1, we can notice that the current
control approach for automotive longitudinal control. The
driver behavior (almost in a parisian freeway) is very close to
proposed structure is intended to allow the controller and
a Constant Time-headway rule. Here, the drivers keep a Time-
the reference model be deﬁned independently. The proposed
headway inferior of the usual human reaction time (i.e. less
reference model is nonlinear and provides dynamic solutions
than one second) which is potentially dangerous.
which a priori verify safety constraints. The model is based
On the other hand, ﬁgure 1 also illustrates the proposed
on physical laws of compliant contact and has the particularity
reference distance policy, that contrarily to the precedent
that its solutions can be described by explicit integral curves.
models, it is obtained from an exogen dynamical motion
This allows to explicitly characterize the set of initial
equation. This fact allows to calculated explicitly the bounds
condition for which the safety constraints can be met. An
of the model solutions which are obtained through suitable
additional control loop is performed in order to compensates
integral curves. Thus, the proposed reference model does not
not modelling dynamics and external disturbances. In special,
suffer the problems discussed above, that is, we can guarantee
the control is intended to guarantee a good tracking of the
safety and comfort requirements in an explicitly way. All this
desired distance policy (i.e. tracking or model-matching
will be discussed in the Section III.
problem).

B. Comfort criteria.                                                            The remainder of the paper is organized as follows. Section
Studies on comfort criteria are scarce. However, we can                   II presents the problem statement. Section III explore the
ﬁnd some works that try to ensure comfort by imitating the                   proposed inter-distance reference model. Section IV presents
human behavior. For example, [3] presents an ACC system for                  some experimental results. And ﬁnally in Section V, we
low speed motion, where the desired acceleration was obtained                present some conclusions and future directions.
from an estimated model using data of a real driver’s behavior.
On the other hand, [10] uses human perception theory in order                                          II. PROBLEM STATEMENT
to obtain an acceptable inter-distance reference. The problem                   The ﬁgure 2 shows the control scheme for which the inter-
here is that reproducing such as behavior may not necessarily                distance reference model is designed. The longitudinal control
lead to safe operation. Therefore, human-based methods may                   problem could be understood as a tracking problem of the
perform over heuristic approaches [19].                                      inter-distance reference signal dr (t). With this structure, the
In general, passenger comfort in public ground transporta-                controller and the reference model can be deﬁned indepen-
tion is determined by the changes in motion felt in all direc-               dently. Thereby, the reference model will include the safety
tions, as well as by the other environmental effects. Typically,             and the comfort constraints, and it could be seen as an exogen
acceleration magnitude is taken as a comfort metric, however                 system describing a reference vehicle dynamics. In that way
in [7] comfort due to the motion changes in a vehicle’s                      the controller can be designed to optimally reject other system
longitudinal direction (the “jerk”) has been used instead1 . So,             disturbances speciﬁc to the sensors characteristics as well as
the jerk is important when evaluating the discomfort caused                  other disturbance input torques such a side wind, road slopes,
to the passengers in a vehicle. For example, when designing                  and vehicle internal actuator dynamics. We next describe each
a train and elevators, engineers will typically be required to               of the elements of this control scheme.
keep the jerk less than 2 m/s3 for passenger comfort. Then, an
accepted criteria is that bounded longitudinal accelerations and             A. The inter-distance dynamics - The plant
jerks can guarantee a certain degree of comfort in longitudinal
The automotive longitudinal control is generally composed
1 theacceleration’s time-derivative is the best metric to reﬂect a human   by two loops: an internal or inner control loop which compen-
comfort criteria                                                             sates the nonlinear vehicle dynamics (acceleration and brake
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..
dr          ..
^
TABLE I
xr
f                                     xl                                      S AFETY C ONSTRAINTS

Collision avoidance     :       dr ≥ dc
dc                                  Maximum velocity        :     ˙f
xr ≤ Vmax
do
Maximum deceleration      :    ¨f
xr ≥ −Bmax
Fig. 3.   The inter-distance reference model.

It is assumed that the velocity and the acceleration of the
systems), and an outer control loop which is responsible for        leader vehicle can be estimated from suitable sensors2 .
guaranteeing a good tracking of the desired inter-distance             On the other hand, the constraints imposed by safety can
reference. In this work, we assume that the inner control loop      be set as bounds on the reference vehicle states and its
has already been designed to compensate the internal vehicle        time-derivatives. These constraints are summarized in Table
dynamics (acceleration loop), and we are only interested here       I, where dc , Vmax , Bmax are positive constants. Bounds dc
in the outer control loop, i.e., the inter-distance control loop.   and Vmax could be imposed by the driver or by the infras-
The inter-distance dynamics can be represented as a double       tructure manager, while Bmax is imposed by the dynamics
integrator driven by the difference between the leader vehicle      characteristics of the vehicle. Nevertheless, these bounds may
¨                                          ¨
acceleration xl and the follower vehicle acceleration xf , i.e.,    be dependent on the other road external factors as well. In this
study, we assume that they are invariant.
¨ ¨
d = xl − xf ,
¨                      (3)
C. The reference model-design problem
where d is the distance between the two vehicles.                                ˜
Introducing d = d0 −dr , as being the inter-distance error with
respect to the (constant) nominal inter-distance magnitude d0 .
The dynamics of this error coordinate will be
B. The reference model
¨
˜       ˜ ˜ ˙   ˆ
The inter-distance reference model is taken as an exo-system                               d = ur (d, d) − xl .
¨                            (5)
describing a virtual vehicle dynamics which is positioned at           The model-design problem is then to ﬁnd a suitable ur =
a distance dr (the reference distance) from the leader vehicle,          ˜ ˜˙
ur (d, d) such that all the solutions of (5), for a given set
as is illustrated in Figure 3. The reference model dynamics is
of initial conditions (at the moment when orange zone is
given by
reached), are consistent with the constraints indicated in Table
I. To this aim, we search for suitable nonlinear functions of
¨    ˆ
¨    ¨f
dr = xl − xr ,                  (4)   ur (·). This is investigated in the Section III.
ˆ
¨
where xl is an estimation of the leader vehicle acceleration
and xf¨ r              ˙
ur (dr , dr ) is a nonlinear function of the inter-
D. The inter-distance control objective
˙
distance reference dr and its time-derivative dr . This function       The control objective is for the inter-distance d, described
can be designed to meet safety and comfort requirements,            by the dynamics (3), to track an inter-distance reference signal
and is related to the safe nominal constant inter-distance,         dr that satisﬁes (4). This is illustrated in the Section IV, where
do , and the minimal constant inter-distance, dc , as it will be    it is employed a simple control feedback to solve the model-
discussed later.                                                    matching problem for a preliminary experimental benchmark.
Not a lot attention is reserved for the feedback design, and
In order to characterize different safety levels, three zones    this could be more elaborated in according to each designer.
are deﬁned as follow:                                               Thus, the proposed control structure and the new reference
model design/setting represent the main contribution of this
•   Green Zone : dr > do ,                                       paper.
where the inter-distance d is larger than the safe nominal
inter-distance do (do is a constant design parameter).
III. INTER-DISTANCE REFERENCE MODEL.
This is a safe operation region,
The inter-distance model-design problem can be studied by
•   Orange Zone : do ≥ dr > dc ,                                 making a parallel with the problem of compliant contacts.
where do − dc is the necessary inter-distance to avoid       In particular, nonlinear models resulting from the theory of
collision if a possible hard braking is produced by the      elasticity and mechanic of the contacts (i.e. Hertz contact
leader vehicle.                                              model) are a good source of inspiration.
Take for example the following case which considers two
•   Red Zone : dr ≤ dc ,                                         different laws for ur , i.e.
where dc is a constant minimal inter-distance to be             2 Actually, commercial inter-distance sensors give information about the
respected. This is a collision-free zone.                    inter-distance and the relative speed between two cars. Thus, leader speed
and/or acceleration should be estimated from these measurements.
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50

u1 (·)      ˜
d<0                                                        45                     Human driver

ur =             ˙                         (6)
˜ ˜
u2 (d, d)    ˜≥ 0
d
40

35

Inter−distance [m]
Model
where we assume C 1 continuity between these two struc-                                         30

tures, i.e. ∂u˜ |d=0 = ∂u˜ |d=0 . The particular proposed struc-
1
∂d ˜
2
∂d ˜
25

r
ture for u allows the equation (5) to be re-interpreted as an                                      20

equation describing the physics of a point mass moving in the
˜
15

free space if d < 0, and in contact with a compliant surface
˜ ≥ 0.
10

if d
˜
5

In this work we assume that in d < 0 (i.e. into the green
0
zone), u1 is dictated by the type of mode selected by the driver                                        60     80       100         120   140       160

Time [s]
(e.g. cruise control or speed regulation). Our interest here is
restricted in the “constrained” zone (the orange zone), hence       Fig. 4. Inter-distance produced by a human driver vs. that produced by
the design of u2 .                                                  the proposed model. With dc = 5m, d0 = 22m, Bmax = 10m/s2 and
Hertz, for example, has proposed a model of the form u2 =        Vmax = 15m/s.
˜    ˜
−k dn , ∀d ≥ 0, where n = 1, 2 · · · accounts for contact surface
topology. However, the model has the major inconvenient of
the constant d0 (the design parameters of (8)), such that the
being non-dissipative, producing a oscillatory effect that may
restrictions in Table I will be satisﬁed for all possible solutions
induce a non feasible negative vehicle velocity. To cope with
of (8) starting in Ωorange .
0
this problem, Hunt and Crosseley [8], and then Marhefka and
Orin [9] have introduced a non-linear damper/spring model
˜ ˜˙   ˜      ˜                A. Setting the Model
of the general form u2 = −c|d|n d − k dn , ∀d ≥ 0. Then,
the forces are proportional to the penetration of the object          Note that Equation (8) can be solved analytically, i.e.,
into the surface. One of the advantages of this model is that
˙
˜       c˜        ˆ
in connection with (5), it is possible to compute the integral                        d(t) = − d(t)2 − xl (t) + β,
˙                       (9)
2
curves associated to the autonomous nonlinear differential
c ˜                              ˜
equation of the form:                                                  with β xr (0) + 2 d2 (0). Note that by deﬁnition d(0) = 0,
˙f
¨         ˙                                     then β = xf (0). Upon substitution of the relation xr (t) =
˙ r
˙f
˜    ˜ ˜       ˜
d + c|d|n d + k dn = 0,   ∀n.                     ˙
˜     ˆ
d(t) + xl (t) in (9) one can obtain an explicit relation between
˙
However, with k = 0, this equation has a“bouncing” effect;       the reference vehicle velocity and the “penetration” distance,
solution of this equation may produce motion with velocity          i.e.
reversal. It is clear that in our framework, we may want that
c˜
the vehicle velocity behaves monotonically in the forward                               xr (t) = − d(t)2 + β.
˙f                                      (10)
direction. For this, we can remove the storage-term in the                                          2
damper/spring model discusses previously and let u2 be only            From this expression, we can ﬁnd a c such that for all β =
deﬁned by a dissipation term as (for n = 1):                        xr (0) = Vmax , the critical distance dc is not attained. From:
˙f

˜˜ ˙         ˜                                                                                       ˙f
2(β − xr (t))
u2 = −c|d|d,        ∀d ≥ 0,               (7)                                             ˜
d(t) =                             ,         (11)
which leads to the following equation                                                                                              c
˜
the maximum penetration distance dmax can be computed as
¨
˜         ˙
˜˜ ˆ  ¨
d = −c|d|d − xl .                     (8)    ˜             ¯
¯                                    ˜
dmax = 2c ; (β = max∀t {β − xr (t)} = β). Making dmax ≤
β
˙f
Due to the necessity of eliminating the excess in kinetic        do − dc , (do − dc is the orange zone length), we have,
energy that the vehicle has once it enters in the orange
zone, it is then natural to only use a dissipation term to                                   2β              ˜
≤ do − d c , d(t) ≤       (12)
avoid collisions. Note that the goal of this structure is not                                 c
˜
to regulate back the reference vehicle to d = 0, but to stop          which imposes us a ﬁrst constraint, C1 , on the possible
the vehicle before it reaches the critical distance dc , while      values of c, i.e.
respecting the imposed comfort constraints as it is illustrated
in ﬁgure 4.                                                                                                                        2β
C1 :     c≥                   .              (13)
(do − dc )2
Consider for simplicity t = 0 the time at which the orange           Figure 5 displays the integral curves (10) for different initial
zone is reached. Let Ωorange be deﬁned as
0                                             reference vehicle velocities. The constant c is computed to
˜      ˙f              ˜
Ωorange = xr (0), d(0) : xr (0) = Vmax , d(0) = 0 ,
˙f                                                    ensure that the vehicle inter-distance dr is larger than dc for
0                                                                                                      ˜
different initial velocities xr (0) and d(0) = 0.
˙f
the set of all admissible initial state values at the crossing         By taking the time-derivatives on (10), and using (9), we
˜
point d = 0. Now, the problem is to ﬁnd a value for c and for       get
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30                                                                                                                          25
d
c

25                                                                                                                                                            c=0.0375
20
c=0.0250
Reference Car Speed [m/s]

Reference Car Speed [m/s]
20
c=0.0125
15

15

10
10

5
5
xr                                                                                                      x
1                                                                                                       2

o o                                0
0     10        20           30       40       50   60   70       o o
Penetration Distance [m]                                                                             0
0       10    20         30           40              50         60
Penetration Distance [m]
Fig. 5. Speed vs. Penetration Distance for different initial velocities. (c =                                                                                                       0

0.0125, do = 75m and dc = 5m).
−1

−2

Reference Car Acceleration [m/s2]
−3

˜ c˜
xr = −c|d|[− d2 + β − xl (t)],
¨f                    ˙                                             (14)
−4

2                                                                                                             −5
c=0.0125

proceeding in the same way as before, and introducing the                                                                                                                     −6
c=0.0250

¨f
deceleration constraint, xr (t) ≥ −Bmax we have:                                                                                                                                −7

−8                                 c=0.0375

2                     2cβ                                                                                       −9

¨f
xr (t) ≥ − β                        ≥ −Bmax .                       (15)
3                      3                                                                                        −10
0    10    20         30          40
Penetration Distance [m]
50         60

Appendix I presents more details on the derivation of (15).                                                                                                                   10
c=0.0375

c=0.0250
5
Figure 6 shows solutions of (8) by different values of c.                                                                                                                                                                              c=0.0125

Notice for example that high values of c yield high values
Reference Car Jerk [m/s3]

0

for deceleration and jerk magnitudes, while small values for
c are required to get large stoping distances. This relation                                                                                                                    −5

demonstrates clearly the tradeoff between safety (that require
large c), and comfort (that associate small c).                                                                                                                                 −10

−15

Relation (15) yields an upper bound for c, i.e.
−20
0    10    20         30          40              50         60
Penetration Distance [m]
27 B 2
C2        :     c ≤ ( ) max .                                (16)
8  β3                                         Fig. 6. Speed, Acceleration and Jerk vs. Penetration Distance for the same
˜
initial conditions (xr (0) = 20m/s; d(0) = 0m), and different c values.
˙f
The problem can thus be formulated as ﬁnding a value of
c , subject to the set of constraints C1 and C2 . Therefore, a
sufﬁcient condition so that c exists is that C1 and C2 holds,
i.e.                                                                                                                                                                                                        2
27Bmax
c=       3
.                                          (20)
2β         27 B 2                                                                                                                            8Vmax
2
≤ ( ) max ,                                           (17)
(do − dc )      8  β3                                                       Note that the design parameters could be obtained from (18)
which together with β = Vmax , implies that the design                                                                         and (20) as functions of the imposed bounds dc , Vmax and
parameter do should meet the following relation                                                                                  Bmax . If (18) and (20) hold, the reference inter-distance model
provides an inter-distance reference dr that avoids collision
2
16 Vmax                                           respecting the maximum braking capacity. All this is true for
do ≥                           + dc ,                                                                                 c ˜
(18)   all initial conditions that satisfy xr (0) + 2 d2 (0) = β = Vmax .
˙f
27 Bmax
Notice also that the equation (18) gives an important
If do is selected according by taking the smaller value                                                                        relationship between the maximal vehicle velocity and the
compiling with (18), i.e.                                                                                                        safe inter-distance do for a given braking capacity Bmax .
In fact, equation (18) corresponds to the braking distance
2
16 Vmax                                           dictated by the model. This braking distance is quite similar
do =                           + dc ,                             (19)   to the Newtonian braking distance, equation (1) (i.e. both are
27 Bmax
quadratic functions of the speed). The ﬁgure 7 illustrates this
then we can calculate c from C2 , as:                                                                                         relationship.
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100                                                                                                                               1.4
to an infinite value

90                                                                                                                               1.3
Normalized

Safe distance do and max Jerk
distance d
o
80                                                                                                                               1.2
B       =5m/s2
70                               max                                                                                             1.1
Safe Distance [m]

2
Bmax=7m/s                                                                                                   1
60
2
B       =10m/s
50       max                                                                                                                     0.9
Normalized
max. Jerk
40                                                                                                                               0.8

30
0.7

0.6
20

0.5
10                                                                                                                                  0           0.5                   1                1.5                2
Parametre n
0
0       5            10        15           20    25      30   35
Velocity [m/s]
Fig. 8.    Safety distance do and maximum jerk w.r.t. the parameter n.
Fig. 7. The distance d0 as a function of the maximum velocity (See equation
(18)), for dc = 5m and different braking capacities.
scenario according to the acceleration/deceleration capabilities
B. The comfort behavior
A comfortable braking is here understood as the ability to
decelerate with “low” jerk while respecting the safe stopping                                                    C. Inﬂuence of different values of n
distance. One of the principal advantages of the proposed
We have, until now, analyzed the model with n = 1 in
reference model, concerns the possibility of evaluate the
(8). Proceeding in the same way, as in Section III-A, and
expected comfort behavior using the solutions of the equation
˜                                                        considering different values of the parameter n, we can obtain
(9). Taking the times-derivatives of (9) in function of d, we
a more general expression of (18):
have the reference acceleration given by
1
˜    c˜    ˆ
xr = −c|d| − d2 + β − xl (t) ,
¨f                   ˙                                                                      (21)                                nn (n + 1)2(n+1)                                                    n+1     2
Vmax
2                                                                                                           do ≥                                                                                 + dc .           (27)
(2n + 1)2n+1                                                              Bmax
and the reference jerk given by
Similarly, if condition (27) is satisﬁed, then there exists c
...r
˜   ˜   c˜          ˆ     ˆ                                                                     such that the maximum braking value Bmax is respected and
xf     = − c d −cd − d2 + β − xl (t) − xl (t)
˙     ¨
2                                                                                       the inter-distance is always larger or equal than minimal inter-
c˜             2                                                                        (22)
ˆ                                                                                     distance dc , (for all initial speed smaller or equal to Vmax ),
+ − d2 + β − xl (t)
˙       .
2                                                                                              i.e. the parameter c could be calculated as follow:
Thus, assuming that the estimated leader vehicle accelera-                                                                                                                                                 2n+1
tion/deceleration is bounded as:                                                                                                                                               1 2n + 1                                    n+1
Bmax
c=                                                    .               (28)
nn n + 1                                  β 2n+1
ˆ
¨
−γ ≤ xl (t) ≤ α,                                   (23)
In addition, (27) suggests the existence of a minimum value
where γ and α are positive constants, with γ >> α, we can                                                     for do as a function of n. Figure 8 illustrates this. Although
re-write (21) and (22) as                                                                                        reducing n gives a smaller safe distance do , the comfort may
be affected. Figure 8 also shows a numerical plot of the
¨f
xr (t) ≤ α,                               (24)   maximum possible jerk values with respect to n, assuming
˙f       ˆ
˙                           ˆ
¨
max∀t {xr (0) − xl (t)} = Vmax , and −γ ≤ xl (t) ≤ α.
...r
| x f (t)| ≤ max(cβ 2 ,                      2cβγ).             (25)
...r                      ˜
˜ 2 d2n+1      ˜ ˙          ˆ     ˆ
These equations (see appendix I and II for derivations), sug-                                                           xf                                             ˙     ¨
= −cdn [ c n+1 − cdn (xr (0) − xl ) − xl ]
f
˜n+1                                                                                   (29)
gest that the maximum positive reference vehicle acceleration                                                                                  ˜
−cndn−1 [− cd                ˆ
+ xr (0) − xl ]2 .
˙        ˙                 n+1                 f
and the jerk depend of the chosen design parameter c and the
constant β (β = xr (0)), but also depend of the maximum
˙f                                                                                               Note that the maximum possible jerk explose to the inﬁnity
leader vehicle acceleration α and deceleration γ.                                                                values for n < 1, and it decrease for larger values of n.
Taking, for example, the parameter c as it is calculated                                                      However the distance do (that determines the length of the
in (20) we can, under assumption (23), bound the reference                                                       orange zone), becomes larger for larger n. That means that
vehicle jerk as:                                                                                                 we can set the model in order to obtain more comfort, tourist
2
mode , requiring more distances, or we can set the model for
27 Bmax
...r    Bmax                                                                   smaller safe distances, sporting mode, demanding more jerk.
, 2.6
| x f (t)| ≤ max(γ).      (26)
8 Vmax       Vmax                                                                      In the rest of the paper we continuous to use n = 1, which
The vehicle safety is then guarantee for all operation con-                                                   concerns a raisonnable value of inter-distance with bounded
ditions, while the vehicle comfort level is adapted to each                                                      jerk.
SUBMITTED TO IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY, JANUARY 2005                                                                                        7

D. Model simulations
1) Case 1: Simulations without noise: To illustrate the be-
havior of the proposed inter-distance model, we have designed
a test proﬁle including cruise control, hard-stop, and stop-and-                                     90

go scenarios. The simulations have been done considering                                             80
ACC
Vmax = 30m/s, Bmax = 10m/s2 , and dc = 5m. These                                                     70

Inter−distance [m]
values are used to compute d0 , and c as shown in previous
section. This results in d0 = 75m and c = 0.0125. Initial
60

˙1
conditions are xr (0) = 0m, x2 (0) = 85m, xr (0) = 30m/s,
1
50

and xl (0) = 20m/s. The dotted lines in the ﬁgure 9 shows
˙                                                                                              40                                Stop−and−go

the curves produced by the simulated leader vehicle.                                                 30

Hard
When the reference vehicle comes near to the leader vehicle,                                      20                   stop
the velocity is adapted with comfortable deceleration and the                                        10

reference vehicle is positioned to a safe distance. Then, at                                         0
t = 25s the leader vehicle is stopped with elevate braking                                            0         10         20    30
Time [s]
40     50   60   70

value (approximately 10m/s2 ), while the reference vehicle                                           30

obtains completed stop before critical distance dc = 5m                                                       ACC

with a braking smaller than 6m/s2 . Thereafter, the leader                                           25

vehicle is accelerated and decelerated (stop-and-go) with usual
Stop−and−go

Speed [m/s]
acceleration values but elevate jerk; however, the reference                                         20

vehicle is maintained to a safe distance, and a bounded jerk
(< 3m/s3 ).
15

Figure 9 shown the resulting inter-distance evolution                                             10                  Hard
stop
predicted by the virtual vehicle along the complete test-
proﬁle. As expected, the red zone is never reached while the                                         5

acceleration and jerk are keep within the predicted limits.
0
0         10         20    30       40     50   60   70
Time [s]
2) Case 2: Simulations with noise or bias: Here we
4
simulate the reference model driving by a leader vehicle                                                      ACC

acceleration affected by noise or a bias, i.e.                                                            2
Acceleration [m/s2]

0
ˆ
xl = xl + ηl ,
¨    ¨                            (30)
−2

where ηl could be a zero mean, gaussian noise signal, with                                         −4                                Stop−and−go
variance equal to 0.1 (Figure 10), or a bias equal to 0.1m/s2                                                            Hard
stop
(Figure 11).                                                                                          −6

The minimal inter-distance and the maximal braking is                                              −8

always respected as a consequence of assuring the leader dy-
namics hypothesis dictated in Section III-A, i.e. the estimated                                      −10
0           10      20    30      40     50   60   70
Time [s]
leader dynamics (30) is subject to:
15

0     ˆ
≤ xl (t)
˙         ≤ Vmax , ∀t.               (31)                                       10
Stop−and−go

Figure 10 illustrates the response of the reference model                                              5
ACC
Jerk [m/s3]

when it is driven by an noisy leader acceleration. Note that
the reference model always respect the minimal inter-distance                                             0

with bounded jerk. Note also that the effect of a noisy
−5
measurement only is observed in the jerk response that                                                                   Hard
stop
becomes noisy too.                                                                                   −10

On the other hand, ﬁgure 11 illustrates the response of the                                       −15
0           10      20    30      40     50   60   70

reference model when it is driven by a leader acceleration                                                                            Time [s]
affected by a constant bias. Notice that the reference inter-      Fig. 9.   Inter-distance, velocities, acceleration and jerk for a given leader
distance response changes a little with respect to the above       proﬁle.
case. However, the minimal inter-distance is always respected
with bounded jerk. During braking, the positive jerk becomes
bigger than the above case. Notice too that due to the bias
the maximal inter-distance decrease a little. These aspects
SUBMITTED TO IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY, JANUARY 2005                                                                                                                                                                                                                                  8

30                                                                                                                   18
16

25                                                                                                                   14
Inter−distance [m]
12

Speed [m/s]
20                                                                                                                   10

8

15
6

4

10
2

0

5
0   10   20      30        40         50   60   70                                                              −2
0    10
Reference vehicle speed
20       30          40       50   60   70                                    (a)                                (b)
Time [s]                                                                                                                     Time [s]

5
Measured
4
Fig. 12.   (a) The LOLA car in the track and (b) its brake pedal.
4            Estimated                                                                                               3

Reference Acceleration [m/s2] and Jerk [m/s3]

3
2
2

1
1

0

−1
0
obtained some preliminary experimental results. The different
−1

−2
−2
Jerk                                           algorithms have been integrated on the “LOLA” test car (see
−3

−4
−3
Acceleration
ﬁgure 12).
−5
0   10   20      30        40

Time [s]
50   60   70
−4
0    10         20       30          40

Time [s]
50   60   70
During the test the inter-distance value is computed as
the difference of the absolute position of each vehicle. The
Fig. 10. Behavior of the model affected by noise in the leader acceleration
measurement.                                                                                                                                                                                                                       absolute position and the speed are obtained from an odometer
30
available in each vehicle. The measures are transmitted by
18

16
radio-frequency to the central computer-station located into the
25
14
follower car. These values are both used for control (reference
Inter−distance [m]

12

model and control feedback), and for recording in real-time.
Speed [m/s]

20
10

8

15
6                                                               Leader and follower vehicle accelerations are obtained from
10
4

2
their gyros (inertial sensors) with the purpose of appreciate its
5
0   10   20      30        40         50   60   70                                                              −2
0
Reference vehicle speed                            behavior.
0   10         20       30          40       50   60   70

Time [s]                                                                                                                     Time [s]

4                                                                                                                    4
Reference Acceleration [m/s ] and Jerk [m/s ]
3

3

3

2                                                                                                                    2                                                                B. Implemented longitudinal controller
1                                                                                                                    1
2

0                                                                                                                    0                                                                   The automotive longitudinal control is generally composed
Measured
−1

−2
Estimated
−1

Jerk
by two loops: an internal or inner control loop which compen-
−2

−3
−3
Acceleration                sates the nonlinear vehicle dynamics (acceleration and brake
−4
0   10   20      30        40         50   60   70                                                              −4
0    10         20       30          40       50   60   70
systems), and an outer control loop which is responsible for
Time [s]
guaranteeing a good tracking of the desired reference inter-
Fig. 11. Behavior of the model affected by a bias in the leader acceleration                                                                                                                                                       distance (given by the reference model).
measurement.
The inner control loop, i.e. the throtle/brake control loop,
is a non-trivial control problem. The difﬁculty is due to
concern the main disadvantage of the model, which reﬂects an                                                                                                                                                                       the complexity, and lack of symmetry of the throttle and
important sensibility to the quality of the leader acceleration                                                                                                                                                                    brake sub-systems that control the vehicle acceleration and
measurement. In fact, during implementation we use the leader                                                                                                                                                                      deceleration. In addition, the vehicle dynamics is highly non
velocity measurements instead of the acceleration ones. This                                                                                                                                                                       linear and behaves differently than our idealized point mass.
was possible thanks to the integrability property of the model                                                                                                                                                                        This important topic has been tackled elsewhere (the inter-
which permits to calculate the inter-distance reference and the                                                                                                                                                                    ested readers can be refereed to [3] and [13] ). In this paper,
reference acceleration in terms of the leader speed. This will                                                                                                                                                                     we assume a perfect inner controller performance, yielding
be illustrated next in the Section IV-B.1.

IV. E XPERIMENTAL RESULTS                                                                                                                                                                                                                          ¨
xf     u,                          (32)
In this Section, we describes some experimental results of
where u stands for the outer controller output signal. Figure
an implemented longitudinal control. The structure of the total
13 illustrates an actuator test (i.e. the inner control loop) in
implemented control law is depicted in ﬁgure 2. We start this
the LOLA car for an arbitrary reference acceleration. Thus,
Section describing the used equipment, then we describes the
we can accept the above assumption, equation (32) in the rest
implemented longitudinal controller and ﬁnally we discuss the
of the section.
main experimental results both for a stop-and-go scenario and
In this work we are only interested in the design of an outer
for a car-following scenario.
control loop. Here the outer controller is composed for both
the reference model dynamics and an additional regulation
A. Equipment description
feedback. The latter will be responsible to guarantee a
Within the framework of the ARCOS French program                                                                                                                                                                                 compensation of non-modelled and neglected dynamics in
and in collaboration with the LIVIC3 Laboratory, we have                                                                                                                                                                           assumption (32). Next, we describe brieﬂy every element of
3 LIVIC is a French laboratory about the interaction between driver, vehicle                                                                                                                                                     the implemented longitudinal control.
and infrastructure; see http://www.inrest.fr/ur/livic
SUBMITTED TO IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY, JANUARY 2005                                                                                                           9

2

Vehicle acceleration
1.5
ˆ
u = ur − H(s) dr − d ,                (37)
1

where ur is given by (34), and H(s) corresponds to a
Acceleration [m/s2]    0.5

0
linear feedback operator.
−0.5

−1
Substituting (35) in (37) and using (4) we have:
−1.5

−2     Arbitrary reference
acceleration
¨ ¨      ¨
ˆ    ¨         ˜
d = xl − xl + dr + H(s)(de − ηd (t)) − ηu ,       (38)
−2.5

−3
ˆ                   ¨
where d = d + ηd (t), and xl = xl + ηy (t). This yields the
ˆ    ¨
300     400     500     600    700     800    900   1000   1100   1200

Time [s]                                         following tracking error dynamics
¨
˜        ˜
Fig. 13.   An actuator test (inner control loop) in the LOLA car.                                                             de + H(s)de = ηy (t) + H(s)ηd (t) + ηu ,         (39)
with ηd , and ηy being the measurement noise associated
1) The reference model - A feedforward term: The refer-                                                          to their respectively measures. We can re-write (39) in its
ence model acts as a feedforward term into the longitudinal                                                         equivalent Laplace representation as:
control law. This control action is described by the following
˜            1                    H(s)
nominal inter-distance dynamics given by equation (9). In                                                                    de =               (ηy + ηu ) + 2       ηd .      (40)
terms of dr we have:                                                                                                                s2   + H(s)             s + H(s)
ˆ
This means that increasing accuracy in signal d, we can
˙  c              ˆ
dr = (do − dr )2 + xl − β,
˙                                               (33)   effectively compensate both the leader estimation uncertainties
2
ηy and the inner control loop inaccuracy ηu for increasing the
ˆ
˙
where β = Vmax . Notice that we have an input, xl , and two                                                       H(s) gain. Hence, we could chose H(s) to account for the
outputs ur and dr . The output ur is obtained from                                                                  speciﬁc frequency properties of ηd , and ηy .
Note also that the term H(s) is taken here as a linear
˙
ur = c|do − dr |dr .                                      (34)                                                    ˆ
operator of the measured tracking error (dr − d). However,
this feedback compensation could be obtained from more
The parameters c and do are calculated from (20) and (19)                                                        elaborated control designs, as for example H∞ /H2 control,
respectively. The model parameters are summarized in Table                                                          optimal control, state feedback, etc.
ˆ
II. The initial condition dr (0) is calculated as dr (0) = d(0),
ˆ
where d(0) stands for the initial inter-distance estimation or
C. Results discussion
measurement.
Notice that this part of the control uses directly the                                                              1) Case 1: A Stop-and-go scenario: The Figures 14a, 14b,
solution of (9) instead of (8) itself. Thanks of this property                                                      14c and 14d, correspond to the inter-distances, velocities,
(integrability of the model), the input required to drive the                                                       acceleration/deceleration, and jerks respectively during a stop-
reference model will be the estimated leader velocity instead                                                       and-go scenario. Notice that we have different initial con-
of the estimated leader acceleration. This aspect is well                                                           ditions (i.e. the reference inter-distance and the actual inter-
appreciated during the control implementation due to the fact                                                       distance at t = 40s). However the controller was charged of
that the leader velocity estimation is, in general, more easy                                                       the attractiveness of the system states to the reference model
to estimate, and often presents better signal-noise ratio.                                                          states.
During the experimental tests we have used a Proportional-
2) Control feedback - A model matching: Take the inter-                                                          Derivative (PD) controller as the function H(s). The Table III
distance dynamics (3), i.e.                                                                                         summarizes the PD-gains which give goods results (models of
sensors noise was not available, so, a trial and error PD-gains
¨ ¨
d = xl − xf .
¨                                         (35)   adjustment was used). Due to the elevated noise in the inter-
distance measurement, the controller bandwidth (dictated from
Assuming that the follower vehicle acceleration (32) could                                                        the PD-gains) has been quite limited, and then, the tracking
be described as follow:                                                                                             error becomes appreciable.
The reference model was adjusted using the parameters
xf = u + ηu ,
¨                                                  (36)   values depicted in Table II. The maximal deceleration was
decreased to 7m/s2 in order to compensates the low band-
where ηu stands for an inner control loop inaccuracy.                                                            width of the control feedback (i.e. smaller deceleration and
˜
And deﬁning de = dr − d as the tracking error signal, with                                                       jerk magnitudes require less controller bandwidth).
dr subject to (33) and d subject to (35). Then, the problem                                                            Notice that the jerk is so smaller and consequently better
is to design a control feedback u that minimizes the tracking                                                       in terms of comfort. The jerk is not obtained directly from
˜
error de . During experiments, the chosen control structure has                                                     measurements, actually the jerk is calculated from derivation
the following form:                                                                                                 of acceleration measurements with suitable ﬁltering. In fact,
SUBMITTED TO IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY, JANUARY 2005                                                                                                                                                                                                                                                          10

18
Inter−distance
8
16                                                                        7
Reference                                                              Follower
14                                                                        6                                                                                                     C ONTROL FEEDBACK PARAMETERS
Inter−distance [m]

5

Speed [m/s]
12

0.3
10                                                                        4

8                                                                        3
Proportional action gain                                   Kp          :
6                                                                        2                                                                                                  Derivative action gain                                     Kd          :             1.0
4                                                                        1

2                                                                        0

0                                                                        −1                                                                                                                                                 25
40           50         60      70   80     90                            40     50         60      70   80   90                                 40

Time [s]                (a)                                         Time [s]              (b)                           35
20
4                                                                         4

Inter−distance [m]
30
3                                                                         3                                                                                                                                                 15

Speed [m/s]
25
2                                                                         2
Acceleration [m/s ]
2

20                                                                         10
1                                                                         1

Jerk [m/s3]
0                                                                         0                                                                      15
5                                   Follower
−1                                                                        −1                                                                      10
Inter−distance
0

−3            Follower                                                    −3      Follower
0                                                                         −5
Leader                                                          25   30   35    40    45    50    55   60   65   70                        25        30    35    40       45       50    55      60   65   70
−4

−5

−5
Time [s]                         (a)                                                 Time [s]                              (b)
40           50         60      70   80     90                            40     50         60      70   80   90                                  4                                                                                8

Time [s]                (c)                                         Time [s]              (d)                                                                                                             6
2
Follower

Acceleration [m/s2]
Fig. 14. Inter-distance, velocities, acceleration and jerk from experiment. A                                                                                                    0
4

Jerk [m/s ]
3
stop-and-go scenario.                                                                                                                                                                           Follower
2

−2
0

TABLE II                                                                                                                                                                      −2

−6
R EFERENCE MODEL PARAMETERS USED DURING EXPERIMENTS                                                                                                                                                                        −4

−8                                                                            −6
25   30   35    40    45    50   55    60   65   70                            25     30    35       40    45       50      55   60   65   70

Sample time                      Ts                         :         0.1 s
Time [s]                         (c)                                                 Time [s]                              (d)
Maximal velocity                 Vmax                       :         30 m/s                       Fig. 15. Inter-distance, velocities, acceleration and jerk from experiment. A
car-following scenario.
Max. braking capacity            Bmax                       :         7    m/s2
Minimal distance                 dc                         :         5m
Max. distance                    d0                         :         104 m
Parameter                        c                          :         0.006125
The proposed structure combines an exogen reference model
with an additional control loop. The former is charged of
verify some safety and comfort constraints, while the latter
is charged of the model-matching between the model and
the Figure 14d is depicted in order to illustrate the magnitude
the actual system, assuring a good tracking of the desired
of the obtained comfort.
reference inter-distance.
2) Case 2: A Car-following with Hard stop scenario:
The ﬁgures 15a, 15b, 15c and 15d, correspond to the inter-                                                                                                The model has few parameters that can be also set in
distances, velocities, acceleration/deceleration, and jerks re-                                                                                         accord to other external factors, such as the road conditions
spectively, during a car-following with hard stop scenario.                                                                                             and the trafﬁc load. In this work, the main assumption
During car-following (i.e. between 25s and 57s), the inter-                                                                                          concern the fact that the parameters model are invariant. As
distance tracking error behavior is very acceptable, with ac-                                                                                           a future work, adaptability of the model with respect to the
celerations and jerks so smaller. However, during the hard                                                                                              external information should be studied.
stop scenario (after 57s), the leader speed decreases abruptly
with a deceleration near to 8m/s2 . The speeds were almost                                                                                                 The proposed model is described by a nonlinear set of
20m/s just before to start the braking maneuver. The follower                                                                                           equations that are driven by the vehicle leader acceleration.
vehicle makes use of its maximal braking capacity, exceeding                                                                                            This last aspect corresponds to the main disadvantage of
the maximal braking performed by the model. This behavior                                                                                               the model. In fact the model requires a good estimation of
is attributed to a large time delay in the inner control loop                                                                                           the leader acceleration that could be relatively difﬁcult to
(about 300ms inherent to the used brake actuator/controller),                                                                                           obtain directly. Nevertheless, thanks of the integrability of
which becomes appreciable during a hard stop scenario. As                                                                                               the model, this problem could be solved by expressing the
a consequence of this time delay, a large deceleration and                                                                                              reference inter-distance and the reference acceleration in
an elevated positive jerk are reached. However, this jerk                                                                                               terms of the leader speed.
magnitude is raisonnable according to the scenario. In fact, this
scenario is quite extreme, permitting to test the effectiveness of                                                                                         Although this approach seems quite similar to some early
the proposed approach, specially the fact to avoid a collision.                                                                                         works, for example [6], [4], and [20], where an impedance
In addition, this positive jerk occurs near to zero speed and                                                                                           control is proposed, the distance policy presented in this
therefore the related uncomfort is not so perceptible in practice                                                                                       paper is obtained from an exogen dynamical motion equation,
(see for example [7]).                                                                                                                                  instead of non-exogen stationary ones based on the classi-
cal safe distance, equation(1). This fact allows to calculated
explicitly the bounds of the model solutions which are ob-
V. C ONCLUSION
tained through suitable integral curves. Thus, the proposed
In this paper we have presented a novel reference model-                                                                                              reference model does not suffer the problems discussed in
based control approach for automotive longitudinal control.                                                                                             Section I-A. In addition, the proposed distance policy gives
SUBMITTED TO IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY, JANUARY 2005                                                                                                                             11

30

both the reference inter-distance and the necessary accelera-                                                                                   2   ~3
(c /2)d
3 1/2
(2cβ )

tion/deceleration, leaving an additional control loop to com-                                                 20

Reference acceleration [m/s ]
2
~
pensate other no modelling dynamics of the car and external                                                               cβd

perturbations. The design of the control is independent of the                                                10

reference model design. Therefore, more elaborated control
0
strategies could be used for this task.                                                                                                                                             ~
d
Contrary to the other methods, the proposed control does                                                  −10
max

not divise each scenario, and does not need to build different
~
references and controllers for each one. The proposed model                                                  −20
−cβd
~3   ~
verify safety and comfort for all the range of speed, and in                                                              2
(c /2)d −cβd

this way the proposed longitudinal control could be useful                                                   −30
0      10        20       30     40        50        60     70             80

Penetration distance [m]
into highways and suburban areas, in particular in stop-and-
go scenarios.                                                    Fig. 16. The reference acceleration decomposed in two different functions,
The string stability problem is not analyzed here. This                         ˆ ˜
where max{c(β − xl )d} = cβ d.
˙          ˜
problem together with the acceptability of this approach in
commercial cars could be the object of future works.
¨f
∂ xr
A PPENDIX I                                                                                                        ˙
= 0, for xl = 0,                                               (46)
∂d˜
D ERIVATION OF THE MAXIMUM REFERENCE
ACCELERATION AND B RAKING                            that is
Take the equation describing the dynamics of a reference                                                                ¨f
∂ xr  c2 ˜         ˜
= (d∗ )3 − cβ d∗ = 0.                                                 (47)
vehicle, equation (14). The reference vehicle acceleration                                                                 ˜  2
˜                                                                                         ∂d
could be described in terms of d as follow:
Where the index “∗” stands for an extremal of the function
c2 ˜         ˆ ˜
(41). Solving (47) we obtain
xr = d3 − c(β − xl )d,
¨f                  ˙                   (41)
2
˜               2β
where the constant β is deﬁned in terms of the reference-                                                                              d∗ =                                                      (48)
c ˜                                                                                                       3c
model initial conditions , i.e. β     xr (0) + 2 d2 (0). See
˙f
2    r
¨
∂ x              ˜            ˜
Section III.                                                        which veriﬁes ∂ d2f |d=d∗ = 3c2 d∗ > 0, i.e. d∗ minimizes
˜ ˜ ˜
the function (41), and the maximum braking could be calcu-
˜
The maximum penetration distance, denoted as dmax             lated as
˜
max∀t {d(t)}, have been calculated from equation (12) as
follow:                                                                                        2                                                               2cβ
min{¨r (t)} = − β
xf                                                                             , ∀t                            (49)
3                                                                3
˜          2β
dmax          ,                (42)      or, in other words,
c
˙f
for all 0 ≤ xr (t) ≤ β, ∀t.                                                                 2     2cβ
¨f
xr (t) ≥ − β         , ∀t              (50)
3      3
In addition, from equations (11) and (14) we can obtain the     Notice that the equation (50) determines the maximum
following boundary conditions:                                   value of the reference vehicle braking. This value depends of
both the parameter c and the constant β. Remember that β is
xr = 0,
¨f          ˙f
xr = β        ˜
at d = 0            (43)
calculated from the model initial conditions.

¨f
xr   = 0,   ˙f
xr = 0        ˜ ˜
at d = dmax .       (44)

Figure 16 illustrates the reference model braking and/or       B. Maximum Acceleration
ˆ
acceleration for xl = 0.
˙                                                 Proceeding in the same way that in Section I-A, we can
obtain a maximal bound of the reference positive acceleration.
A. Maximum Braking                                               That is
3
Taking the equations (41)-(44), and based in the ﬁgure 16,                                                            c2 ˜      c2                     2β        2
1
we have that                                                                 ¨f
xr (t)                                    ≤ d3 max =                                      = (2cβ 3 ) 2 ,              (51)
2         2                       c
xf ˜ ˙
min{¨r (t)} ≡ min{¨r (d)|xl =0 },
xf                                     (45)          ˜ ˜                            ˆ
˙
at d = dmax and assuming that xl instantaneously reaches
t                 ˜                                                     ˆ
d
the maximal value xl = β, i.e. assuming inﬁnite leader
˙
i.e. the maximum value of the reference braking could be                                      ˆ ˜
acceleration, and then −c(β − xl )d = 0. Notice from the ﬁgure
˙
calculate from                                                   16, that this bound could be so large, and then so conservative.
SUBMITTED TO IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY, JANUARY 2005                                                                                                                 12

4
~
J(d*)=(1/3)cβ2
~
~
cdmaxα                     Take the right hand side of the above equation. Notice that
cdα
2
the ﬁrst term is always positive, the second and the third one
Reference Jerk [m/s3]
0
~
d
are always negative, while the last one depends of the sign of
max

−2
~
cdγ
−4
Proceeding similarly as in Appendix I, we ﬁrst separate the
−6
~
J(d)
equation (58) in two different functions, that is:
−8
...r
˜     ˜ˆ
~
−cdmaxγ
xf                ¨
= J(d) + cdxl                  (58)
−10
~*
J(d )=−cβ  2
˜
where J(d)             2˜2 ˆ
˙           ˜          ˆ
2c d (β − xl ) − 3 c3 d4 − c(β − xl )2 .
˙
−12
0   10       20        30     40     50       60   70         80
4
Penetration distance [m]
So, a simple way to calculate the maximum values of jerk
Fig. 17.    The reference jerk decomposed in two different functions.                                                 is described as follow: First we calculate the extremals of the
˜
nonlinear function J(d). Then, we calculate the maximum
˜ˆ
¨
values (positive and/or negative) of the linear term cdxl , and
Another way to calculate the positive reference acceleration                                                        ﬁnally, we add this value to the previous calculated extremals
could be taking into account a bound in the estimated leader                                                                ˜                                                     ...r
of J(d). This procedure gives a bound of the total function x f .
acceleration, that is, assuming that:
˜
Thus, the extremals of J(d) can be calculated from
ˆ
xl (t) ≤ α.
¨                                               (52)
˜
∂J(d)        ˜      ˆ          ˜
= 4c2 d∗ (β − xl ) − 3c3 d∗3 = 0.
˙
¨
dt xf (t)   = 0,
d r                                                                                (59)
Using the above assumption, we will calculate                                                                                   ˜
∂d
i.e.
The above equation has two solutions:
d r              ˙
˜       ˜        ˙
˜ ˜     ˆ                                                                                                               ˆ
x (t) = −c(d∗ )2 − cd∗ (−cd∗ d∗ − xl (t)) = 0
¨                                    ¨               (53)                                                                     ˜        ˜     4 (β − xl )
˙
dt f                                                                                                                              d∗ = 0 ; d∗ =              .           (60)
3     c
Notice that into this equation it appears explicitly the leader
ˆ                                                                                                Therefore, we have two extremals to be taken into account:
¨
vehicle acceleration xl (t). Then, the following equation gives
a condition for existence of an extremal of xr (t) as a function
¨f                                                                                           ˜ ∗                 ˆ
ˆl (t). So, after simpliﬁcations, we have                                                                                             J(d∗ )|d˜ =0 = −c(β − xl )2
˙
¨
of x

˜˙                                                                            ˜            1       ˆ
˜ ˜ ˙            (d∗ )2    ˜                                                                  J(d∗ )| ˜        c(β − xl )2
˙ =
ˆ
−cd∗ d∗ = xl (t) −
¨               ; ∀d∗ > 0                                      (54)                           ∗ d =  3
ˆ
β−xl
˙

˜
d∗
c

ˆ
¨
On the other hand, assuming −γ ≤ xl ≤ α, ∀t, we have
˙
˜ ˜
Hence, substituting xr∗ (t) = −cd∗ d∗ (by deﬁnition, see
¨f
equation 7), in the above equation, we have                                                                                               ˜         ˜ˆ     ˜
−cdmax γ ≤ cdxl ≤ cdmax α
¨                                (61)

˜˙                                                                                                                          5
Therefore, assuming 8β >> γ >> α, (i.e. negative jerk
ˆ       (d∗ )2   ˆ        ˜                                                                                                  9c
xr∗ (t)
= xl (t) −
¨f  ¨              ≤ xl (t); ∀d∗ > 0
¨                (55)                                                            always greater than positive one), the maximum jerk will be
˜
d∗
bounded as follow:
˜˙          ˜
This is true for any d∗ and any d∗ > 0. Thus, from (55)
and taking the assumption described by (52), the maximal
...r
˜
| x f (t)| = max{cβ 2 , cdmax γ}               (62)
reference acceleration is bounded as follow:                                                                            In others terms, using (42):
xr (t) ≤ α
¨f                                              (56)                     ...r
| x f (t)| = max{cβ 2 ,     2cβγ}              (63)
Notice that the maximum value of the reference vehicle                                                                 The maximum value of jerk depends of the parameter c,
acceleration depends of the maximum leader vehicle                                                                    the constant β (initial conditions) and also of the maximal
acceleration α.                                                                                                       acceleration/deceleration of the estimated leader vehicle γ.

ACKNOWLEDGMENT
A PPENDIX II                                                                                       The authors would like to express their gratitude to their
D ERIVATION OF THE MAXIMUM REFERENCE JERK                                                                                                                 e
colleagues Axel Von-Arnim and Cyril Roy` re from the LIVIC
Take the equation (53) which describes the reference vehicle                                                       laboratory who kindly gave suggestions and discussion during
˜
jerk. The jerk could be expressed in terms of d, as follow:                                                           algorithms integration and test. Thanks also to the ARCOS4
French Program by its contribution and ﬁnancial support.
...r
˜       ˆ     3 ˜            ˆ        ˜ˆ
4 ARCOS is a French program on safety vehicle and secure roads. For
xf     = 2c2 d2 (β − xl ) − c3 d4 − c(β − xl )2 + cdxl
˙                    ˙         ¨                                                      (57)
4                                                                                      details, see http://www.arcos2004.com
SUBMITTED TO IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY, JANUARY 2005                                                                                   13

R EFERENCES                                                                 John-Jairo Martinez was born in Cali, Colombia.
He received the B.S. degree in electrical engineering
[1] Chien C. and Ioannou P., “Automatic Vehicle-Following”. Proceeding of                                and the M.S. degree in automatic control from
American Control Conference 1992, Chicago, IL, pp.1748-1752.                                         the Universidad del Valle, Colombia, in 1997 and
[2] Yanakiev D. and Kanellakopoulos I., “Variable Time Headway for String                                2000 respectively. He has joint to the Universidad
Stability of Automated Heavy-Duty Vehicles”, Proc. of the 34th. IEEE                                 Nacional de Colombia as a teacher assistant during
Conference on Decision and Control, New Orleans, LA, December 1995.                                  the period 2001-2002. He received the Ph.D de-
pp. 4077- 4081.                                                                                      gree in automatic control from the Institut National
[3] Persson M., Botling F., Hesslow E., Johansson R., “Stop & Go Controller                              Polytechnique de Grenoble INPG, France, in March
for Adaptive Cruise Control”, Proceeding of the 1999 IEEE International                              2005. Dr. Martinez has been an invited visitor in the
Conference on Control Applications, Hawaii, USA.                                                     Centre for Complex Dynamic Systems and Control
[4] Gerdes J.C., Rossetter E.J., Saur U., “Combining Lanekeeping and            of the Newcastle University, Australia 2005. And currently He has a post-
Vehicle Following with Hazard Maps”. Vehicle System Dynamics, Vol.36,       doctoral position in the INPG, France. His main research interests include
No.4-5, pp.391-411, 2001.                                                   hybrid and nonlinear systems, applications of switching control theory and
[5] Alvarez L. and Horowitz R., “Hybrid controller design for safe maneu-       automotive control.
vering in the PATH AHS architecture”. Proceeding of American Control
Conference, Albuquerque, New Mexico, pp. 2454-2459, June 1997.
[6] Hennessey M.P., Shankwitz C., Donath M., Sensor Based “Virtual
Bumpers” for Collision Avoidance: Conﬁguration Issues. In: Proceeding
of the SPIE, Vol.2592, pp. 48-59, 1995.
[7] Hoberock L.L., “A Survey of Longitudinal Acceleration Comfort Studies
in Ground Transportation Vehicles”. Journal of Dynamic System, Mea-
surement; and Control, pp. 76-84, June 1977.
[8] Hunt K.H. and Crossley F.R.E., “Coefﬁcient of restitution Interpreted as
damping in Vibroimpact”, Journal of Applied Mechanics, pp. 440-445,
June 1975.
[9] Marhefka D.W. and Orin D.E., “Simulation of Contact Using a Nonlinear
Damping Model”, Proceeding of IEEE International Conference on
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1996.
[10] Fancher P., Bareket Z., Ervin R., “Human-Centered Design of an
Acc-With-Braking and Forward-Crash-Warning System”. Vehicle System
Dynamics, Vol. 36,No.2-3, pp. 203-223, 2001.
[11] Martinez J. and Canudas de Wit C., “Model reference control approach
for safe longitudinal control”. American Control Conference, Boston,
USA. 2004.
[12] Brackstone M. and McDonald M., “Car-Following: a historical review”,
Transportation Research Part F 2, Pergamon, pp.181-196, 2000.
[13] Seiler P., Song B., Hedrick J.K., “Developmet of a Collision Avoidance
System”, Society of Automotive Engineers (SAE), 98PC-417, 1998. 7p.
[14] Bengtsson J., “Adaptive Cruise Control and Driver Modeling”, Ph.D.
Thesis, Department of Automatic Control, Lund Institute of technology.
Sweden, 2001. 91p.
[15] Germann St., and Isermann R., “Nonlinear distance and cruise control                                Carlos Canudas-de-Wit Biography text here.
for passenger cars”, Proceeding of the American Control Conference,
Vol.5, pp.3081-3085, Seatle, Washington, June 1995.
[16] Goodrich M.A., and Boer E.R., “Designing Human-Centered Automa-
tion: Tradeoffs in Collision Avoidance System Design”, IEEE Transac-               PLACE
tions on Intelligent Transportation System, Vol.1, No.1 pp.40-54, March            PHOTO
2000.                                                                               HERE
e
[17] Nouveliere L. ”Commandes Robustes Appliqu´ es au Control Assist´       e
e      `          e
d’un V´ hicle a Basse Vit´ sse”. Ph.D. Thesis on Versailles-Saint Quentin
en Yvelines University. France, 2002. 296p.
[18] Jones W., Keeping Cars from Crashing. IEEE Spectrum, Vol. 38, No.9,
pp. 40-45, September 2001.
[19] Vahidi A. and Eskandarian A. “Research Advances in Intelligent Colli-
sion Avoidance and Adaptive Cruise Control”. IEEE Trans. on. Intelligent
Transportation Systems, Vol.4, No.3, pp. 143-153, September 2003.
[20] Gorjestani A., Shankwitz C. and Donath M. “Impedance Control for
Truck Collision Avoidance”. Proceeding of the American Control Con-
ference, Chicago, Illinois, June 2000.

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