Adaptive control of four-wheel-steering off-road mobile robots by slappypappy122

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									   Adaptive control of four-wheel-steering off-road mobile robots:
Application to path tracking and heading control in presence of sliding
                        Christophe Cariou∗ , Roland Lenain∗ , Benoit Thuilot† , Philippe Martinet†
                                  ∗ Cemagref                           † LASMEA

                              24, av. des Landais                  24, av. des Landais
                                     e
                        63172 Aubi` re Cedex France                       e
                                                               63177 Aubi` re Cedex France
                        christophe.cariou@cemagref.fr benoit.thuilot@lasmea.univ-bpclermont.fr

   Abstract— In this paper, automatic path tracking of a four-          a consequence, some recent marketed farm tractors, such as
wheel-steering vehicle in presence of sliding is addressed. The         Claas Xerion or JCB Fastract, propose manually steered rear
attractive feature of such a steering system is that, despite of        wheels, in order to compensate for the crabway motion on
sliding phenomena, both lateral and angular deviations can
be explicitly controlled. Indeed, previous research has demon-          a slope. The automatic control of the crab angle for such
strated that high-precision path tracking on a low grip terrain         vehicles would nevertheless provide an increased accuracy
can be achieved with two-wheel-steering vehicles. However, in           and would offer extended possibilities in soil exploitation.
this case, only the lateral deviation is kept satisfactorily close to      In the literature, four-wheel-steering vehicles have been
zero, the angular deviation is non null in order to compensate          studied with the aim, on one hand to increase the ma-
for sliding effects. In this paper, previous adaptive control laws
are extended to the case of four-wheel-steering mobile robots           noeuvrability of vehicles operating in confined space (e.g.
with the aim to servo both lateral and angular deviations.              handlers, loaders, self-propelled sprayers [9]) and, on the
Relying on an extended kinematic model, a backstepping                  other hand to design active security devices for high speed
control approach, which considers successively front and rear           road vehicles (see for instance [6]). In the latter case, the
steering control, has been designed. Real world experiments             objective is to modify the vehicle lateral dynamics thanks
have been carried out on a low adherent terrain with a four-
wheel-steering vehicle equipped with a single RTK-GPS. This             to the active rear wheels, in order to reduce side slip angles
demonstrates the capabilities of the proposed control law and           and vehicle yaw rate. Several control approaches have been
its robustness in real all-terrain conditions.                          proposed to meet this aim. Generally, they are based on gain-
                                                                        scheduled feedforward control techniques: the rear steering
                       I. I NTRODUCTION                                 angle, whose range never exceeds one or two degrees, is
   Accurate automatic guidance of an off-road vehicle at                computed as a function of the front steering angle, the yaw
relatively high speed is still challenging, since it requires           rate, and predefined gains related to the vehicle speed. These
to take into account the numerous dynamic phenomena,                    techniques have already been implemented on marketed cars,
usually disregarded in classical approaches. As pointed out             but do not address the variability of grip conditions pointed
for example in [13], the low grip conditions usually met in             out in an off-road context. Path tracking algorithms proposed
such a context damage particularly the accuracy of classical            for such vehicles, mainly concerned with automatic parking
path tracking control laws designed from rolling without                applications, also disregard grip conditions: for instance,
sliding assumptions (as developed in [10] or in [12]). This             in [5], a horizontal plane and rolling without sliding are
can be seriously penalizing from the application point of               assumed, in order for the vehicle models to exhibit the
view, especially in agriculture, where autofarming emerges              flatness property. This feature is then used to plan a suitable
as a promising solution [2], but where high-precision path              parking trajectory and servo the four-wheel-steering vehicle
tracking is required whatever the grip conditions encoun-               along it.
tered.                                                                     On the contrary, the objective of this paper is to take
   In previous work [8], high accuracy guidance of a farm               advantage of the rear steering actuation to compensate for
two-wheel-steering vehicle has been achieved, despite sliding           sliding effects during path tracking on a natural environment.
phenomena. Nevertheless, a crab angle between the vehicle               Based on a backstepping approach and adaptive techniques,
heading and the tangent to the reference path is systemati-             the proposed control law on one hand ensures accurate path
cally observed, since such a behavior is necessary to com-              following, despite sliding phenomena, and on the other hand
pensate for the sliding forces acting on the uncontrolled rear          explicitly controls the angular deviation whatever grip condi-
wheels. The actuation of these latter theoretically enables the         tions. This paper is organized as follows. First, an extended
explicit control of both lateral and angular deviations, and            kinematic model accounting for sliding effects is developed,
appears to be very interesting from an agricultural point of            and on-line estimation of grip conditions is addressed. Then,
view: an angular error, which is moreover varying according             path tracking is considered using a backstepping approach. In
to the sliding conditions, leads to an unsatisfactory heading           a first step, the rear steering angle is assumed to be measured
of the mounted implement, and may cause indiscriminate                  and the extended model is then turned into chained form,
placement of field inputs (seeds, fertilizers, pesticides). As           so that a front steering control law can be designed with
the sole aim to servo vehicle lateral deviation. Then, a rear
steering control law is designed in order to ensure also the
convergence of the vehicle angular deviation to some set
point. Next, the experimental mobile robot is presented, as
well as the available measurements. Finally, the capabilities
of the proposed control law are investigated through full-
scale experiments on a slippery ground.
             II. E XTENDED KINEMATIC MODEL
A. Model formalism
   Since vehicles considered in this paper are expected to
move on a natural terrain, the low grip conditions must be
                                                                             Fig. 1.   Path tracking parameters and variables
accounted in the modeling, in order to enable, in next section,
the design of accurate path following control laws. Dyna-           As the control objective is to follow reference path Γ, the
mical models incorporating tire/ground interaction forces, as     equations of motion have to be derived with respect to this
described in [4] or in [1], do not appear tractable when          path. It can be established see [10], that:
vehicles are moving off-road: numerous parameters have                                        ˜
then to be known. This is a concern, especially when grip                     s = Vr cos(θ+δR −βR )
                                                                              ˙              1−c(s) y
conditions are variable. Alternatively, it is here proposed to               
                                                                             
extend classical four-wheel-steering kinematic models, such                      ˙              ˜
                                                                                y = Vr sin(θ + δR − βR )                    (1)
                                                                             
as [10], since they propose a suitable structure to address                  
                                                                              ˙
                                                                              ˜
control design.                                                                 θ = Vr [cos(δR − βR )λ1 − λ2 ]
   Consequently, each two front and rear wheels are con-                                                                        ˜
                                                                     with: λ1 = tan(δF −βF )−tan(δR −βR ) , λ2 = c cos(θ+δR −βR )
                                                                                             L                        1−c(s) y
sidered equivalent to two virtual wheels located at mid-
distance between the actual ones, as depicted on figure 1.            It can be noticed that this model becomes singular when
                                                                         1
In order to account for sliding phenomena, two additional         y = c(s) , i.e. when points A and R (depicted on figure 1)
parameters - homogeneous with sideslip angles in a dynamic        are superposed. This problem is not encountered in practice
model - are added to the classical representation. In the same    since, on one hand actual path curvatures are quite small,
way than in [8], these two angles denoted respectively βF         and on the other hand, the vehicle remains close to Γ when
and βR for the front and rear axle, represent the difference      properly initialized. The lateral deviation is thereby always
between the theoretical direction of the linear velocity vector   smaller than the radius of curvature of Γ. As a result, the
at wheel centers, described by the wheel plane, and their         assumption (2) can be made and will be used in the sequel.
actual direction. These angles are assumed to be entirely                               1
                                                                                |y| <          ⇒ 1 − c(s) y > 0                (2)
representative of the sliding influence on vehicle dynamics.                           |c(s)|
   The notations used in the paper are listed below and
                                                                  B. Known data and grip estimation
depicted on figure 1.
                                                                     Model (1) accurately describes the vehicle motion in
   • F and R are respectively the center of the front and
                                                                  presence of sliding as soon as the two additional parameters
     rear virtual wheels. R is the point to be controlled.
                                                                  βF and βR are known. Therefore, the estimation of these two
   • L is the vehicle wheelbase.
                                                                  variables appears to be of crucial importance. As pointed out
   • θv is the orientation of vehicle centerline with respect
                                                                  for example in [11] in the case of dynamical representations,
     to an absolute frame [O, XO , YO ).
                                                                  the direct evaluation of side slip angles appears to be hardly
   • Vr is the vehicle linear velocity at point R, assumed to
                                                                  feasible at a reasonable cost. Their estimation classically
     be strictly positive and manually controlled.
                                                                  requires the use of huge measurement systems (such as
   • δF and δR are the front and rear steering angles. They
                                                                  expensive inertial measurement units...), and needs some pre-
     constitute the two control variables.
                                                                  liminary assumptions with respect to adherence conditions.
   • βF and βR are the front and rear side slip angles.
                                                                  The variability of the soil conditions encountered in natural
   • M is the point on the reference path Γ which is
                                                                  environment, as well as their on-line modifications, do not
     the closest to R. M is assumed to be unique, see
                                                                  permit to apply directly observer algorithms used in on-road
     hypothesis (2) below.
                                                                  context.
   • s is the curvilinear abscissa of point M along Γ.
                                                                     An observer is here proposed to achieve sideslip an-
   • c(s) is the curvature of path Γ at point M .
                                                                  gles indirect estimation, relying on the sole exteroceptive
   • θΓ (s) is the orientation of the tangent to Γ at point M                                      T
     with respect to the absolute frame [O, XO , YO ).            measurements X = y ¯
                                                                                  ¯        ¯ θ ˜     (respectively the measured
      ˜
   • θ = θv −θΓ is the vehicle angular deviation with respect     lateral and angular deviations). This observer is based on
     to Γ.                                                        the duality between observation and control. As proposed
   • y is the vehicle lateral deviation at point R with respect   in [7], βF and βR are considered as control variables to be
     to Γ.                                                        designed in order to ensure the convergence of the extended
model outputs to the measured variables. More precisely,            and rear steering angles (δF and δR ). The expression of
model (1) without curvilinear abscissa equation, is rewritten       extended kinematic model (1) accounting for sliding effects
as a non-linear state representation:                               is still consistent with classical models of wheeled mobile
                       ˙
                       ˆ           ˆ                      (3)       robots. As a consequence, according to [10], it can be
                       X = f (X, u)
                                                                    turned into a linear model named chained form without
  where f is derived from (1):                                      any approximation. Nevertheless, in the case of two steering
                         ˆ
               
                         ˜
                Vr sin(θ + δR + u2 )
                                                                   axles, such a transformation requires the integration of δR
               
                                       +u
                  Vr cos(δR +u2 )[tan(δFL 1 )−tan(δR +u2 )] −       into the state vector, rewriting model (1) as model (9):
    ˆ
 f (X, u) =
                         ˆ
                c(s) cos(θ+δR +u2 )
                          ˜                                                     ˙
                                                                                 x = f (x) + g1 (x)w1 + g2 (x)w2                 (9)
                             1−c(s)ˆ
                                  y                                with
                                                              (4)          
                                                                                 y
                                                                                                
                                                                                                           0
                                                                                                                            
                                                                                                                                0
                                                                                                                                    
X = [ y ˆ ]T is the observed state and u =
 ˆ           ˆ θ ˜                                                    x=        θ˜     , g1 =  Vr cos(δR −βR )  , g2 =  0 
                                                                                                           L
[ u1 u2 ]T = −[ βF βR ]T are the side slip angles to be                          −
                                                                             δR  βR                       0                  1
estimated, considered as the control variables of system (3).                                       ˜
                                                                                            Vr sin(θ + δR − βR )
As side slip angles do not exceed few degrees in practice,                                                        ˜
                                                                           f =  −Vr sin(δR −βR ) − Vr c(s) cos(θ+δR −βR ) 
                                                                                                                            
let us linearize this state equation with respect to “control                                L                 1−c(s)y
vector” u in the vicinity of zero (i.e. no sliding). It leads to:                                      0
                                                                                                         ˙     ˙                (10)
                 ˙
                 ˆ         ˆ         ˆ
                                                                                  T
                                                                    and [w1 w2 ] = [tan(δF − βF ) δR − βR ]T .
                 X    = f (X, 0) + B(X)u                     (5)
                                                                    Using this representation, a model with two chains can be
with B denoting the derivative of f with respect to u,              obtained and control laws can be designed for δF and δR       ˙
evaluated at u = 0:                                                 thanks to linear control theory. Unfortunately, this imposes to
                                                                         ˙
                                                                    use δR as the control variable for the rear axle. An integration
                                    ˆ
                                     ˜
                                                     
                  0        Vr cos(θ + δR )                          is then required, which may generate instability in practice,
    ˆ                           ˆ˜
 B(X) =  Vr cos δR Vr c(s) sin(θ+δR ) − Vr           (6)
                                                    
              L cos2 δF    1−c(s)ˆ y        L cos δR
                                                                    due to the inevitable delays present on steering actuators.
                                    tan δF −tan δR
                        −Vr sin δR         L                        B. Backstepping approach
                                  ˆ˜                                   An alternative based on backstepping method is then
The matrix B is invertible when θ + δR = π [π], Vr = 0
                                              2
           π
and δr = 2 [π]. These conditions are met in practical path          proposed, allowing the design of control laws for steering
following conditions. Using this formalism and hypothesis,          angles instead of their derivative. In a first step, let us
the observation of sliding parameters is achieved thanks to         consider the rear steering angle as a measured parameter in
the following observer equation:                                    model (1). From this hypothesis, it results that δR − βR can
                                                                    be considered as a unique rear side slip angle denoted βR2 .
        u       ˆ               ˆ       ˙M
                                        ¯
            = B(X)−1 G · e − f (X, 0) + X                    (7)    Model (1) is then consistent with a front steering mobile
                                                                    robot in presence of sliding such as considered in [8],
            ˆ ¯                                                     with a unique control variable δF . Following the same
where e = X − X, G is an Hurwitz matrix, which constitutes
                                                                    methodology than in this reference, the state and control
                       ¯˙M
the observer gain, and X is the numeric derivative of the           transformations (11) can be introduced.
measured state. This expression of control vector u leads to
                                                                             ˜                                           ˜
                                                                      [s, y, θ] → [a1 , a2 , a3 ] = [s, y, (1 − c y) tan(θ + βR2 )]
the following error dynamics:                                                                              ˜
                                                                     [Vr , δF ] →    [m1 , m2 ] = [ Vr cos(θ+βR2 ) , da3 ]
                                                                                                       1−c(s) y       dt
                         ˙
                         e = G·e                           (8)                                                                  (11)
which ensures the convergence of the observed state to              It leads to the following chained system:
the measured one. The gain matrix G allows to decrease                                          a
                                                                                       a1 = ddt1 = m1
                                                                                      
the impact of sensor noise. Observer equation (7) provides                             ˙
                                                                                      
then an estimation of the side slip angles introduced into                                      a
                                                                                         a = ddt2 = a3 m1
                                                                                          ˙                                     (12)
the bicycle model depicted on figure 1 and ensures that                                 2
                                                                                      
                                                                                                a
                                                                                         a3 = ddt3 = m2
                                                                                          ˙
                                                                                      
this extended model fits with the measured behavior of the
vehicle. Therefore, it constitutes a relevant basis for mobile      which can be also expressed with derivatives with respect to
robot control design.                                               the curvilinear abscissa:
                                                                                              da2
                 III. C ONTROL LAW DESIGN                                            a′ = da1 = a3
                                                                                      2
                                                                                                                           (13)
                                                                                              da3             m2
A. Motivations                                                                       a′ = da1 = m3 = m1
                                                                                      3
   The control objective is to perform an accurate path                In order to control the lateral deviation, a judicious choice
tracking with respect to lateral and angular deviations (re-        for m3 is (14), since it leads to a second order differential
                  ˜
spectively y and θ), compensating for the effects of low            equation, ensuring the convergence of a2 = y to zero.
grip conditions. As the velocity is viewed as a measured                         m2
parameter (manually controlled), system inputs are the front              m3 =       = −Kd a3 − Kp a2 (Kd , Kp > 0)             (14)
                                                                                 m1
Injecting (14) into (11) and considering βR2 as slow-varying               As α is assumed to be always strictly positive, see hy-
with respect to the dynamic imposed by the two gains Kp                  pothesis (2), the condition ∆ > 0 leads to:
and Kd , the control law for the front axle can finally be                    
                                                                                   ˜ ˜                Kd2
                                                                              (θ − θref ) < 4 c(s) K α if c(s) > 0
derived as:                                                                                               d2
                                                                                                                              (23)
                                                                                                        2
                                                                              ˜ ˜                    Kd
    δF   =    arctan     tan(δR − βR )                                           (θ − θref ) > 4 c(s) Kd2 α if c(s) < 0
                                     ˜
                                    cos             ˜
                                             A cos3 θ2
              + cos(δR −βR ) ( c(s) α θ2 +
                     L
                                                α2     )   + βF            The choice for (Kd , Kd2 ) and the limit values of c(s) and
                                                                                                                            ˜ ˜
                                                                         y lead, in the worse case, to a ±30◦ bound on (θ − θref ),
  with:                                                           (15)
   ˜                                                                    which is always satisfied in practice.
          ˜
   θ2 = θ + δR − βR                                                       Since ∆ has been shown to be strictly positive, two
      α = 1 − c(s)y                                               (16)   solutions can be derived. Considering the actuators range
              2
            Kd y        ˜               ˜
      A = − 4 − Kd α tanθ2 + c(s)α tan2 θ2                               of variation, only one of the solutions can be applied.
  
                                                                         As a result, the rear control law achieving the expected
The gains (Kp , Kd ) allow to specify a settling distance
                                                              K2
                                                                         convergence can be written as following:
instead of a settling time. In the sequel, it is chosen Kp = 4d
                                                                                                           2
                                                                                                          Kd
in order to obtain a critical damping ξ = 1. With control law                                     Kd −                       ˜ ˜
                                                                                                               − 4 c(s) Kd2 (θ − θref )
(15), the lateral deviation is satisfactorily servoed to zero, as                 ˜
                                                                         δR = βR −θ+arctan
                                                                                                          α

         ˜                                                      ˜                                                2c(s)
well as θ2 . Convergence of this latter variable implies that θ                                                                    (24)
converges to βR − δR , and not to some desired set point, as             Expressions (20) and (24) constitute the rear steering law for
expected.                                                                respectively straight and curve line following. The continuity
                        ˜
   Actual control of θ can now be addressed using the rear
                                                                         of these expressions, when c(s) tends to zero, can be
steering variable. This constitutes the second step of the ap-
                                                                         established by standard but tedious computations.
proach. Reporting control law (15) into the third equation in
model (1) leads to the following angular deviation dynamic               C. Stability of the backstepping controller
with respect to curvilinear abscissa:                                      The stability of the whole non-linear control strategy,
              2
             Kd y                                                        composed of control law (15) for the front steering angle
   ˜′
   θ = (−                     ˜             ˜       ˜
                  − Kd tanθ2 + c(s) tan2 θ2 ) cos2 θ2 (17)
             4α                                                          and (20) if c(s) = 0 or (24) if c(s) = 0 for the rear steering
                                                                         angle, can be checked using Lyapunov theory. Consider
                                                      ˜
As above mentioned, control law (15) imposes that θ2 stays                                                        ˜       ˜
                                        2 ˜
                                                                         Lyapunov function candidate, with ǫ = θref − θ:
close to zero. As a result, the term cos θ2 can be considered
as equal to 1, so that:                                                              V    =   1                ˜
                                                                                                  y 2 + (α tan θ2 )2 + ǫ2          (25)
                                                                                              2
                     2
           ˜′     K y            ˜             ˜                            The derivative of the positive function V with respect to
          θ = − d − Kd tanθ2 + c(s) tan2 θ2            (18)
                   4α                                                    curvilinear abscissa (homogeneous with the time derivative
In view of (18), two cases must be distinguished, according              considering a non-null velocity) leads, after calculations, to
to the curvature value.                                                  the following expressions (whatever the curvature value):
  1) Straight line following: (c(s)=0): in that case, the                       dV                 ˜               ˜
                                                                                     = −Kd α2 tan2 θ2 − Kd2 (ǫ cos θ2 )2           (26)
                                                                                ds
angular deviation dynamics (18) can be simplified as:
                                                                         which is always negative. The stability of the mobile robot
                            2
                            y
                           Kd                                                                                                     ˜
                                                                         trajectory tracking and the convergence of both ǫ and θ2 to
                   ˜′
                   θ =−                 ˜
                               − Kd tan θ2           (19)
                          4                                              zero is then ensured. As a result, injecting the asymptotic
                                                                                    ˜
                                                                         value of θ2 into equation (17) establishes that the lateral
                           ˜′         ˜      ˜
   Then, the error dynamic θ = Kd2 (θref − θ) with Kd2 >                 deviation y also converges to zero. This finally demonstrates
0 can easily be imposed by proposing the following rear                  the stability of path tracking control in presence of sliding,
steering law:                                                            with respect to lateral and angular deviations, with front and
                                           ˜
                          −Kd y Kd2 (θref − θ)  ˜
              ˜
  δR = βR − θ + arctan            −                  (20)                rear control laws (15) and (20) if c(s) = 0 or (15) and (24)
                              4            Kd
                                                                         if c(s) = 0.
                                 ˜ ˜
This ensures the convergence of θ to θref
                                                                                         IV. E XPERIMENTAL RESULTS
 2) Curve line following (c(s) =0): using the notation
        ˜                                                                   The experimental platform is the all-terrain four-wheel
X = tan θ2 , equation (18) can be rewritten as:
                                                                         steering vehicle depicted on figure 2. The vehicle weight and
                     2
              ˜′  Kd y                                                   maximum speed are respectively 600 kg and 18 km/h, and
             −θ −       − Kd X + c(s) X 2 = 0         (21)
                   4α                                                    it can climb slopes up to 45◦ . The only exteroceptive sensor
                                      ˜′        ˜       ˜                on-boarded is a RTK-GPS receiver, whose antenna has been
Once more, the objective is to impose θ = Kd2 (θref − θ).
                                                                         located straight up the point R (see figure 1). It supplies an
If it was achieved, then the discriminant of equation (21)
                                                                         absolute position with a 2cm accuracy, at a 10Hz sampling
would be:
                  K2                                                     frequency, and allows to estimate the vehicle heading thanks
                                    ˜ ˜
             ∆ = d − 4 c(s) Kd2 (θ − θref )           (22)
                   α                                                     to a Kalman filter.
                                                                                    tracking error within ±10cm is obtained. Finally, path track-
                                                                                    ing results when using both front and rear steering control
                                                                                    laws, with a null desired angular deviation, are shown in
                                                                                    black dashed line. With this control strategy, both lateral and
                                                                                    angular deviations are able to reach null values.
                                                                                                              0.4                                                                                                   6

                                                                                                              0.3
                                                                                                                     δ controlled                                                                                            δF controlled
                                                                                                                         F
                                                                                                                     sliding accounted                                                                              4
                                                                                                                                                                                                                                                      δF and δR controlled




                                                                                     Lateral deviation (m)




                                                                                                                                                                                           Angular deviation (°)
                                                                                                              0.2                                δ and δ controlled                                                          sliding accounted
                                                                                                                                                  F        R
                                                                                                              0.1                                sliding accounted                                                  2
                                                                                                                                                                                                                                                      sliding accounted

                                                                                                               0
                                                                                                                                                                                                                    0
                                                                                                             −0.1

                                                                                                             −0.2                                                                                                  −2

                                                                                                             −0.3            δ controlled
                                                                                                                              F
                                                                                                                                                                                                                   −4              δF controlled
                                                                                                             −0.4    sliding neglected
                                                                                                                                                                                                                              sliding neglected
                   Fig. 2.                      Experimental platform                                        −0.5
                                                                                                                 0   5        10     15     20        25       30   35   40
                                                                                                                                                                                                                   −6
                                                                                                                                                                                                                     0   5    10     15      20     25     30   35    40
                                                                                                                              Curvilinear abscissa (m)                                                                        Curvilinear abscissa (m)



                                                                                                                     (a) Lateral deviation                                                                               (b) Angular deviation
   The path to be followed is recorded by a preliminary run
achieved in manual driving. In this paper, two types of path                                                                           Fig. 4.             Validation of algorithm in slope
have been recorded: a straight line achieved on a 15% sloping                          These first results permit to point out the benefit of rear
ground (mobile robot running perpendicularly to the slope as                        steering control in order to achieve accurate path tracking in
depicted on figure 2) and a curved path depicted on figure 3                          sliding conditions. The proposed algorithm indeed permits to
achieved on a flat ground. In both cases, the terrain was an                         compensate for sliding effects in order to preserve an almost
irregular wet grass ground, where the vehicle is inevitably                         null tracking error, but also to ensure the convergence of the
prone to slide as it will be experimentally checked in the                          angular deviation to a set point. In slope, sliding phenomena
sequel. In the forthcoming experimental results, the vehicle                        can then be compensated without admitting a crab angle.
speed is 1.8 m/s (6.5 km/h). The control gains (Kp , Kd ) are                          Beyond this improvement, the proposed control laws for
set to (0.16, 0.8) in order to impose a 11m settling distance                       front and rear axles, allow to specify any desired value for the
for the convergence of the lateral deviation. Finally, Kd2 =                        robot heading with respect to the reference path orientation.
1.1 has been chosen in order to impose a 3m settling distance                       Using the same reference path than in the previous tests,
for the convergence of the angular deviation.                                       three path tracking have been performed with the proposed
                                                                                    algorithm using different values of desired angular deviation.
                                       −15
                                                                                    The tracking results are compared on figure 5: the references
                                       −20                                          ˜             ˜                      ˜
                                                                                    θref = 0◦ , θref = −10◦ and θref = −20◦ are shown
                                                                                    respectively in black plain line, gray plain line and black
                    Ycoordinates (m)




                                       −25



                                       −30
                                                                                    dashed line.
                                       −35



                                       −40
                                                                                                              0.2                                                                                                   5
                                                                                                                                    θref= −10°
                                       −45
                                          75   80   85     90      95   100   105                                                                      θref= 0°                                                     0
                                                                                                                                                                              Angular deviation (°)
                                                                                     Lateral deviation (m)




                                                                                                              0.1
                                                    X Coordinates (m)
                                                                                                                                                                                                                                                    θref= 0°
                                                                                                                                                                                                                   −5
    Fig. 3.   Path to be followed on a flat and slippery ground                                                 0

                                                                                                                                                                                                      −10
                                                                                                             −0.1                                                                                                                                 θref= −10°
                                                                                                                                                                                                      −15

A. Result during straight line on sloping ground                                                             −0.2                                     θref= −20°                                      −20
                                                                                                                                                                                                                                                  θref= −20°
                                                                                                                                                                                                      −25
   Several straight line following on a slope have been                                                         0    5        10     15     20
                                                                                                                              Curvilinear abscissa (m)
                                                                                                                                                      25       30   35   40                              0               5    10    15       20     25
                                                                                                                                                                                                                              Curvilinear abscissa (m)
                                                                                                                                                                                                                                                           30   35    40


performed using different control laws during a straight
line following on a slope. Firstly, a classical control law                                                          (a) Lateral deviation                                                                               (b) Angular deviation
neglecting for sliding effects has been applied only on the                                                                                                                            ˜
                                                                                                                              Fig. 5.            Path tracking in slope with different θref
front steering wheels (results are reported in black plain
line on figure 4). The expression of this one-axle control                              As it can be seen, after a settling time, the tracking error
law can be derived from (15) by setting sliding parameters                                                            ˜
                                                                                    is not affected by the value of θref since lateral deviations
to zero ((βF , βR ) = (0, 0)). It can be seen that, with                            reported on figure 5(a) present the same evolution whatever
this classical approach, the tracking error as well as the                          the reference. In the meanwhile, after a settling distance,
angular deviation cannot reach the desired zero value because                       the angular deviations (depicted on figure 5(b)) reach the
of sliding effects (the lateral deviation converges close to                                          ˜
                                                                                    desired value for θref . This demonstrates the capabilities of
−30cm, while the angular deviation reaches -2◦ ). A second                          the proposed algorithm to control both lateral and angular
test has been performed, still using only the front steering                        deviations almost independently, despite sliding phenomena.
wheels, but accounting for sliding (control law (15)) (results
are depicted in gray plain line). The same angular deviation                        B. Result during curved path following
than before can be observed (the asymptotic value -2◦ allows                          The last result proposed in this paper is devoted to curved
to compensate for rear side slip angle), but an acceptable                          path following, and in particular the transition between rear
control law expressions when c(s) becomes non null. Path                                                                                                          backstepping approach, using an incomplete linear form, is
tracking of the reference path depicted on figure 3 has                                                                                                            proposed. It permits to design two control laws for front and
been achieved at a velocity of 6.5 km/h on a wet grass                                                                                                            rear steering wheels.
ground. The front and rear steering control laws proposed in                                                                                                         As a result, lateral and angular dynamics with respect to a
this paper have been used and desired angular deviation of                                                                                                        reference path can be controlled almost independently with
˜
θref = −10◦ has been chosen to highlight angular deviation                                                                                                        a high accuracy (around 5cm for lateral deviation and one
control. Path following has first been run disregarding sliding                                                                                                    degree for vehicle orientation during full-scale experiments)
phenomena (βF and βR set to zero). The result is reported                                                                                                         whatever the ground (grip conditions and geometry) and
in gray plain line on figure 6. The second tracking has                                                                                                            whatever the shape of path to be followed.
been achieved with sliding accounted. Results are reported                                                                                                           Due to material limitations (mobile robot capabilities and
in black plain line.                                                                                                                                              sensor sample time), the velocity was limited to 8 km/h
                                                                                                                                                                  during experiments. Nevertheless, the theoretical validity of
                          0.1                                                                                 5                                                   the approach is preserved at faster evolutions (tested in simu-
                                 Straight line part            Curved part                                         Straight line part              Curved part
                                                                                                                                                                  lation). However, the increase in velocity can lead to tracking
                                                                                    Angular deviation (°)
 Lateral deviation (m)




                           0                                                                                  0
                                                                                                                                             Sliding neglected    error overshoots during curvature transient phases. This point
                                                              Sliding accounted
                         −0.1                                                                                −5                                                   can be addressed by predictive algorithms, considering future
                         −0.2                                                                               −10
                                                                                                                                                                  path curvature (as achieved on bigger vehicles). The increase
                                                           Sliding neglected                                                                 Sliding accounted
                                                                                                                                                                  in velocity with respect to both material and control points is
                                                                                                            −15
                         −0.3
                             0      5      10         15      20
                                         Curvilinear abscissa (m)
                                                                     25        30                              0      5       10        15    20
                                                                                                                            Curvilinear abscissa (m)
                                                                                                                                                      25     30   under development. The extension of this work to the case of
                                                                                                                                                                  a vehicle with a trailer is currently also investigated. Since we
                                  (a) Lateral deviation                                                             (b) Angular deviation                         are concerned with agricultural applications, the case of large
                                                                                                  ˜
                                                                                                                                                                  trailers or on-loaded implements constitutes a challenging
                                                Fig. 6.            Path tracking on a curved path θref
                                                                                                                                                                  problem.
   As it can be noticed, when sliding are accounted, the
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