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2019 16th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)Mexico City, Mexico. September 11-13, 2019
Output Adaptive Control of a Skid SteeringAutonomous Vehicle
1st Ruben Fuentes-AlvarezUMI LAFMIA 3175 CNRS
CINVESTAV-IPNMexico City, Mexico
2nd Isaac ChairezDept. of Bioprocesses
UPIBI-IPNMexico City, Mexico
3rd Kim AdamsFaculty of Rehabilitation Medicine
University of AlbertaEdmonton, Canada
4th Sergio SalazarUMI LAFMIA 3175 CNRS
CINVESTAV-IPNMexico City, Mexico
5th Ricardo LopezUMI LAFMIA 3175 CNRS
CINVESTAV-IPNMexico City, Mexico
Abstract—This work presents the design as well as theevaluation of an output adaptive controller which must inducethe stabilization of the tracking error for a class of skid steeredautonomous vehicle (SSAV). The control design includes anonlinear transformation (diffeomorphism) using a simplifiedSSAV mathematical model. This diffeomorphism justifies thetransformation of the control problem in the original coordi-nates to a suitable chain of integrator system (third-order).In this study, the available measurements are the positionand orientation of the SSAV. The aforementioned condition,encourages the implementation of a modified super twistingalgorithm (STA) which is applied as a recursive differentiatorwhich may estimate the velocity and acceleration of the SSAVefficiently. Based on the estimated states, an adaptive controllerprovides the asymptotic stability of the tracking trajectoriesfor the SSAV. Numerical evaluation comparing the proposedadaptive controller with a state feedback controller confirmedthe design of the suggested control structure.
Index Terms—Skid steering autonomous vehicle, super twist-ing, adaptive control
I. INTRODUCTION
An increasing interest in the developing mobile roboticshas been observed in recent years [1]. The quantity ofcommercial wheeled mobile robot platforms available onthe market is growing rapidly, presenting more constructive,but complex structures than the ones usually considered andfor which modeling and control are still a relevant field ofstudy [2]. Skid steering autonomous vehicles (SSAV) arecommonly used to achieve different indoor and outdoor tasksdue to its all-terrain capabilities [3]. The exclusion of thesteering aspect makes the four wheeled differentially driven(4WDD) mobile vehicles robust in mechanics terminology,but also easily manoeuvrable when considering the problemof accurate trajectory tracking. As a consequence, the modi-fication of SSAV orientation produces lateral slippage in thepoint where the wheels touch the ground making the controlapproach slightly different from the common wheeled mobilerobotic devices [4]. When SSAV follows a non-straight path,the wheels required skidding laterally and they must not movetangently to the reference trajectories [5]. Also, following acircular trajectory may produce some instantaneous centerof rotation (ICR) of the SSAV to displace away of therobot wheelbase, causing some kind of instability. Lateral
friction on the wheels produces skidding forces, which mo-tivates a specific control law design which may considerthe mathematical description of the SSAV dynamics [6].This consideration represents a strong difference betweenSSAV and vehicles using active steering kinematics as aconsideration to design the controller. The skidding forcesdeveloped by wheels lateral friction motivate the design ofa control based on the mathematical description of a skidsteered mobile robot dynamics [6], which has embraced thedesign of new controllers. In [4], [7], a trajectory tracking andregulation closed-loop controller based on the back-steppingmethod was implemented, motivated by the unknown groundinteraction forces and the dynamic uncertainties presented inthe model of an SSAV. This solution shows to be robustagainst dynamic non modelled disturbances with an expo-nential time of convergence. The classical linear quadraticregulating control design (LQR) with a feed-forward part thatcompensates the non-linearities of the dynamic-drive SSAVmodel was developed in [8] In that study, it was observedthat this type of algorithms is able to overcome the effectsof non-linearities when tracking a reference trajectory. Thesolution presented in [9] provides a state feedback controlconsidering the application of super-twisting algorithm whichmay stabilize the tracking error of the SSAV using a step-by-step sliding mode time derivator. This algorithm servedto estimate the first and second derivatives. A path followingalgorithm based on adaptive discontinuous posture controlwas designed in [10]. Hybrid control strategies proposed by[11] took advantage of the concept of extended transversefunctions to improve the performance of the controllerswhen reaching admissible reference motions. In addition,robust sliding mode fuzzy logic control was implementedas a trajectory tracking algorithm by [12] using a sequenceof way points, trying to improve the mechanical systemperformance in the presence of uncertainties and externaldisturbances with minimum reaching time, distance error andsmooth control actions. This study provides the followingmain contributions: a) a step by step high order differentiatorto reconstruct the unknown states of the transformed SSAVdynamics, b) an adaptive output feedback controller whichuses the estimated states provided by the output differentiator
978-1-7281-4840-3/19/$31.00 ©2019 IEEE
and c) the numerical evaluation of the proposed method-ology. Notice that the designed output adaptive controllerimplementing the third order step-by-step differentiator isnot exhibiting the overshoot effect (which is common inthis type of controllers). This may appear as an additionalmain contribution to the SSAV field. Section II describesthe problem of controlling the SSAV skidding while turning.Section III introduces the mathematical description of theSSAV following the results from [6]. This section presentsthe complete methodology to clarify the transformation of themathematical model of a SSAV into a chain of integrator thatlet implement the proposed controller. Section IV presents thedescription of a suggested step-by-step time differentiator,which can obtain the consecutive derivatives implementedin the control structure. In Section V the output adaptivecontrol strategy is developed, presenting the mathematicalproof, as well as the transformations that makes it feasible.The numerical evaluations comparing the proposed outputadaptive controller against a linear controller with constantgain are presented in Section VI. Finally, Section VII closesthe study with some final remarks on the obtained results.
II. PROBLEM STATEMENT
Let consider the SSAV must exert a path trajectory trackingdespite the lateral skidding produced when the SSAV turns.If the dynamics of the SSAV is defined by the states ζ andthe reference trajectories are defined by ζ∗, then the problemis to design an output feedback controller such that
lim supt
∞−→‖ζ(t)− ζ∗(t)‖ = 0 (1)
In particular, in this study, the SSAV must track a circulartrajectory defined as follows:
ζ∗ =
[rcos(θct)rsin(θct)
](2)
where r is the radius of the circle at constant speed θc ata time t. This available information z∗ will be selected insuch a way that the initial point D at a distance d0 overthe x-axis from the local inertial frame of the robot thatcorresponds to the non-holonomic constrain for the SSAV.The well suited problem statement requires that the SSAVexerts the circular movement following a tangential path tothe reference trajectories.
III. MATHEMATICAL MODEL OF THE SSAVThe dynamical model of the robot was inspired by the
study given in [6]. The model is obtained under the followingassumptions:a) Slow vehicle speed (for supporting the forward complete
characteristic).b) The longitudinal wheel slippage is neglected.c) The tire lateral force is a function of the vertical load
only.d) There is a neglecting of the suspension and tire defor-
mations.e) No side-way movement (no sudden change of the frontal
section of the mobile section).f) Both, SSAV mass and inertia are constant and known.
Assuming that no vertical movement of the SSAV is consid-ered, the free-body scheme of the mobile section in x-y planeappears in Figure 1(a).
xy
1
2
3
4
3
4
2
1 3x
4x
2x
1x
1y
4y
3y
2y
(a) Free body diagram of a SSAV.
c
xy
(b) Forces and velocities actuating on a SSAV.
Fig. 1. Caption place holder
The interaction forces between the surface and wheelsare the lateral skidding and the friction forces. As wheelsdevelop reactive forces Fjx
(dqdt
), they are restricted to the
longitudinal resistance forces described by Rjx
(dqdt
), for
j = 1, . . . , 4. It is assumed that wheel actuation is equalon each side, reducing longitudinal slip and causing lateralforces Fjy
(dqdt
)to act on the wheels because of the presence
of the lateral skidding. This means that:
F1x
(dqdt
)= F4x
(dqdt
), F2x
(dqdt
)= F3x
(dqdt
)(3)
Notice that lateral skidding occurs when ddty = 0. Also,
lateral skidding velocity ddtyj and longitudinal velocity d
dtxj(for j = 1, . . . , 4) of each wheel are defined in (4):
d
dty + c
d
dtθc =
d
dty2 =
d
dty1 (front)
d
dty − d d
dtθc =
d
dty4 =
d
dty3 (rear)
d
dtx+ l
d
dtθc =
d
dtx3 =
d
dtx2 (right)
d
dtx− l d
dtθc =
d
dtx4 =
d
dtx1 (left)
(4)
The resistive moment Mr
(dqdt
)is generated by Rjx
(dqdt
)and Fjy
(dqdt
)forces as follows:
Mr
(dqdt
)= c
(F1y
(dqdt
)+ F2y
(dqdt
))−d(F3y
(dqdt
)+ F4y
(dqdt
))+ l(R2x
(dqdt
)+R3x
(dqdt
))−l(R1x
(dqdt
)+R4x
(dqdt
))(5)
where the total longitudinal resistive forces and lateral forcesare:
Fy
(dqdt
)=µmg
c+ d
(bsign
(dy1
dt
)+ dsign
(dy3
dt
))Rx
(dqdt
)=frmg
2
(sign
(dx1
dt
)+ sign
(dx2dt
)) (6)
assuming both the lateral friction coefficient µ and the co-efficient of rolling resistance fr constants independent fromvelocity. Also, the rotational movement is produced by therotor effect forced by the motor and the rotational frictionbetween the wheel and the shaft. The complications on themovement of the mobile section arise due to the presence ofthe lateral skidding [3], [5], [13].
The dynamics for the SSAV can be represented by thefollowing nominal model:
M (q)d2
dt2q +Q
(dq
dt
)+ f (q, t) = B (q) τ (7)
Here, q> =[X Y θc
]defines the vector of state
variables. The states X and Y describe the planar coordinatesof the center of mass in the x−axis and y−axis respectively.The angle θc corresponds to the SSAV orientation as shownin Figure 1(b). The elements included in the model (7) are:
M =
m 0 00 m 00 0 I
, B (q) = 1R
cos(θc) cos(θc)sin(θc) sin(θc)−c c
,Q(
dqdt
)=
Frx
(dqdt
)Fry
(dqdt
)Mr
(dqdt
)
(8)The functions FRx
(dqdt
)and FRy
(dqdt
)represent the
forces affecting the wheels, which are generated by the lateralskidding as:
FRy
(dqdt
)= Rx
(dqdt
)sin(θc) + Fy
(dqdt
)cos(θc)
FRx
(dqdt
)= Rx
(dqdt
)cos(θc)− Fy
(dqdt
)sin(θc)
(9)
The term f (q, t) is the uncertain function representing thenon-modeled section of the mathematical model (uncertain-ties) as well as the external perturbations. This work assumesthat:
‖f (q, t)‖2 ≤ f+, f+ ∈ R+ (10)
Considering that the angular velocities are identical for thefrontal and rear wheels (which are also assumed alike) at eachside of the SSAV, the torques regulating its movement are:
τ =
[τLτR
]=
[τL1 + τL2
τR3 + τR4
](11)
where τL and τR are the torques for the left and right sides,respectively.
The free-body dynamics appears in equation (7). Suchrepresentation does not consider the non-holonomic restric-tions inherent to the SSAV. Considering the non-inertialconfiguration of the mobile robot (Figure 1(b)), the x−axisprojection of the center of rotation cannot be bigger thand. If such condition is violated, the vehicle skids along they−axis, and the SSAV is not longer controllable. The propervehicle movement requires the satisfaction of the followingmovement condition: ∣∣∣∣∣∣∣−
d
dty
d
dtθc
∣∣∣∣∣∣∣ < d (12)
Hence, the operative constraint [6] relating the y-linear andangular velocities, must be considered in the SSAV controldesign:
d
dty + d0
d
dtθc = 0, 0 < d0 < d (13)
where d0 is the x-coordinate of the ICR, that is xICR. Thenon-holonomic restriction (13) is equivalent to (in general-ized variables):
A (q)d
dtq = 0 (14)
Here, A (q) =[− sin(θc) cos(θc) d0
]and
d
dtq> =
[d
dtX
d
dtY
d
dtθc
]. These non-holonomic
constraints can be included in the dynamic model (7) usingthe Lagrange-multiplier technique:
M (q)d2
dt2q +Q
(dqdt
)+ f (q, t)
= B (q) τ+A> (q)λi(15)
Here λi defines the vector of Lagrange multipliers as-sociated to the restriction equations (14). Also, admissiblegeneralized velocities d
dtq can be represented as:
d
dtq = S(q)η (16)
where η ∈ IR2 refers to a pseudo-velocity and S(q) is a3× 2 full range matrix, whose columns are in the null spaceof A(q) as:
S
(dq
dt
)=
cos(θc) −sin(θc)sin(θc) cos(θc)
0 −d−10
(17)
Differentiating ddtq from equation (16) and eliminating λ
from equation (15), the following dynamic system is ob-tained:
d
dtq = Sη
d
dtη =
(S>MS)−1S>(Eτ −M d
dtSη − f(q, t)
) (18)
Applying a nonlinear static state feedback law to (15), onemay take the explicit control action τ from (18) as:
τ = (S>E)−1(S>MSu+ S>M
d
dtSη + S>c
)(19)
where u =[u1 u2
]>contains the new control variables,
letting the reduced dynamic model becomes a pure secondorder kinematic model as follows:
d
dtq = Sη
d
dtη = u
(20)
This transformation is possible considering the followingnew set of variables:
d
dtX = cos(θc)η1 − sin(θc)η2
d
dtY = sin(θc)η1 + cos(θc)η2
d
dtθc = −d−10 η2 (21)
d
dtη1 = u1
d
dtη2 = u2
By choosing a particular output, the equation systempresented in (21) can be linearized completely and decoupledby means of a dynamic state feedback. The position of a pointD placed on the x-axis at a distance d0 from the vehicleframe origin must be chosen as a the selected linear outputform:
ζ =
[X + d0 cos(θc)Y + d0 sin(θc)
](22)
by adding a dynamic extension corresponding to the integra-tion of the input u1 :
u1 = ξd
dtξ = v1 (23)u2 = v2
where ξ is the controller state and v1 and v2 are the new con-trol inputs. By decoupling the standard input-output, equation(22) is differentiated until the input v appears explicitly,obtaining:
d3
dt3ζ =
[cos(θc) d0
−1η1 sin(θc)
sin(θc) −d0−1η1 cos(θc)
]v
+
[d0−2ξη2 sin(θc)− 2d0
−2η1η22 cos(θc)
−d0−2ξη2 cos(θc)− 2d0−2η1η
22 sin(θc)
]= α(q, η)v + β(q, η) (24)
Sincedet[α(q, η)] = −d0−1η1 (25)
the decoupling matrix α is non-singular if η1 6= 0. Wheneverdefined, the law control (under a certain relationship betweenX , Y and θc)
v = α−1(q, η) [r − β(ζ, η)] (26)
Here r is the trajectory jerk reference, yielding to
d3
dt3ζ = r (27)
The system (27) can be represented as
d
dtZ = AZ +Br
Z =
[ζ,d
dtζ,d2
dt2ζ
]> (28)
where A and B are controllable companion matrices ofappropriate dimensions. Notice that the SSAV starts movingtoward the corresponding trajectory by the application of anoutput adaptive controller τ . The goal of this controller isbringing the center of mass of the SSAV into the desiredcircular trajectory as the SSAV has the acceptable orientation(tangential movement).
The control design consider two steps: first the next sectiondescribes the application of a step by step differentiator. Oncethe no measurable states are estimated, the output feedbackcontroller is developed.
IV. STATE ESTIMATION FOR THE SSAV
The realization of the output feedback controller requiresthe on-line measurements of ζ and its derivatives (at leats tothe second order) which are not available in principle. Insteadof using the actual values of such derivatives, a robust highorder differentiator based on a step-by-step super-twistingalgorithm form, given by:
d
dtζ̂1,j = ζ̃2,j + k11,jφ11 (e1,j)
d
dtζ̃2,j = k12,jφ12 (e1,j)
d
dtζ̂2,j = E2,j
[ζ̃3,j + k21,jφ21 (e2,j))
]d
dtζ̃3,j = E2,j [k22,jφ22 (e2,j)] + vi,j −
d3
dt3ζ∗i,j
(29)
with: ei,j = ζ̃i,j − ζ̂i,j . All the initial conditions for theobserver (29) are zero. Notice here that j = 1, 2 representsthe individual state included in ζ due to the this variable hastwo components.
The variables ζ̂i,j represent the corresponding estimatedtrajectories of ζi,j . Consider that ζ̃1,j = ζ1,j which is neededto complete the observer design. The indicator functionEi,j(t) fulfills the following definition
Ei,j(t) =
{0 t < T ∗i,j1 t ≥ T ∗i,j
(30)
The switching time T ∗i,j is found as result of the fixed-time converge obtained for the observer [14]. The nonlinearfunctions φ1k (ek) and φ2k (ek) (k = 1, 2) were designedin agreement to the proposal given in [15]–[18]. Then, thefollowing structures were considered for the observer design:
φ1k (ei) = |ei|1/2 sign(ei), φ2k (ei) =1
2sign(ei) (31)
The details to adjust the gains k11,j , k12,j , k21,j and k22,jas well as the times when each section of the observer turnson Ti,j appear in [14]. Notice that the observer (29) extendsthe dimensions of the state, which may introduce additionalimplementation complexity. However, the estimation qualityof the derivatives of ζi produced by the suggested observerjustifies such additional complexity, even if it should beimplemented on-board for the SSAV. This step-by-step dif-ferentiator is not the unique possible solution to recover thestates but it has a natural simplicity that may help to tune thegains easier than some other competitive options. Notice thathigh order sliding mode could offer interesting options butthey gain tuning may take long periods and their embeddedimplementations could be energetic demanding.
V. ADAPTIVE CONTROL DESIGN FOR THE SSAV
Once the estimation of the state Z is ready and consideringthe forward complete characteristics of the SSAV dynamics,it is feasible to design the output feedback adaptive controller.Considering that the reference trajectory admits is at leasttwice differentiable, then exists a state variable representationgiven by Z∗ ∈ R6 such that
d
dtZ∗(t) = AZ∗ +Bh(Z∗(t), t) (32)
Hence, the introduction of the tracking error ∆ = Z −Z∗yields the following dynamics for the tracking error:
d
dt∆(t) = A∆(t) +B (r(t)− h(Z∗(t), t)) (33)
Let propose the controller r as follows:
r(t) = h(Z∗(t), t) +K>(t)∆(t) (34)
where K satisfies
d
dtK>(t) = −αQ∆(t)∆>(t)P + K̃(t) (35)
with αQ = λminP−1/2QP−1/2 a positive scalar, and P ∈
R6×6, q ∈ R6×6 positive definite matrices which regulatesthe time variation of the controller gain. This structurecorresponds to the model reference adaptive control in theindirect form with the reference model given in (32). Thedeviation gain K̃(t) satisfies K̃(t) = K(t) − K0 with K0
any matrix such that AK = A − BK0 is Hurwitz. Themotivation to apply the adaptive control form is the necessityof compensating the potential perturbations affecting theSSAV as well as reducing the applied control energy duringthe tracking of the reference trajectory. The following lemmaprovides the main result of this study.
Lemma 1. If the positive definite matrices Q and P arerelated by the following matrix inequality
PAK +A>KP ≤ −Q (36)
then the origin is an exponential stable equilibrium point forthe tracking error ∆ with a rate of convergence given by αQ.
Proof. Consider the following Lyapunov candidate functionV (∆, K̃) = ∆>P∆ + tr
{K̃>K̃
}. Then the time derivative
of such candidate function is
d
dtV (t) = 2∆>(t)P
d
dt∆(t) + 2tr
{K̃>(t)
d
dtK̃(t)
}(37)
Taking the time derivative of ∆ on (37) and noticing thatd
dtK̃(t) =
d
dtK(t), one gets:
d
dtV (t) = 2∆>(t)P (A∆(t) +B (r(t)− h(Z∗(t), t))) +
2tr
{K̃>(t)
d
dtK(t)
}(38)
The substitution of (34) in (38) provides
d
dtV (t) = 2∆>(t)P
(A∆(t) +B
((K∗)>∆(t)
))+
2∆(t)PK̃>(t)∆(t) + 2tr
{K̃>(t)
d
dtK(t)
} (39)
The algebraic manipulation of (40) yields
d
dtV (t) = ∆>(t)
(PAK +A>KP
)+
2tr
{K̃>(t)
d
dtK(t) + K̃>(t)∆(t)∆>(t)P
} (40)
Based on the adjustment law for the gain K, one gets
d
dtV (t) ≤ −∆>(t)Q∆(t)− tr
{K̃>K̃
}(41)
Using the Raleigh’s inequality, the following inclusiontakes place
d
dtV (t) ≤ −αQV (t) (42)
This inclusion finalizes the proof based on the applicationof the comparison principle and the direct integration of (42).
VI. NUMERICAL EVALUATIONS
Both the differentiator and the controller gains were com-puted using the theoretical results presented in the previoussections. The differentiator was simulated with the followinggains:
kObs =
[3 32 2
], KCtrl =
[−1 −2.4142 −2.4142
]In Figure 2 the trajectories of the SSAV tracking a circulartrajectory are shown, beginning at initial condition (0, 0).No overshoot was obtained by the output adaptive controllerproposed in this paper. Also a comparison against a statefeedback control was carried out, showing a smoother move-ment when converging to the trajectory.
-1 0 1
-1
0
1
Fig. 2. SSMR trajectories in the xy plane. A second comparison is presentedinvolving a state feedback controller and an output adaptive controller.
The convergence and smoothness of the trajectory ap-proaching are depicted in Figure 3(a) for x-axis and Figure3(b) for y-axis. In these figures, both the state feedbackcontroller and the output adaptive controller are plotted,showing a smaller overshoot phenomenon in the output adap-tive controller, without compromising the convergence time.The simulation time was established in Matlab-Simulink at100 seconds using the Bogacki-Shampine (ode3) fixed-stepsolver.
Figure 4 shows the trajectory tracking error in logarithmicscale in each of the derivatives of the system. It can beobserved that as the simulation converge to the desiredtrajectory, the error remains in 0.1 for the position, 0.15for its primitive derivative and 0.2 for its second derivativerespectively. The evolution of φi,k gains through time arepresented in Figure 5, showing that the amount of energy
0 2000 4000 6000 8000 10000
-1.5
0
1.5
0 500 1000 1500
Time (s)
0
1.5OAC
Reference
SFC
(a) Trajectories in the x-axis for the SSAV. Comparison between a state-feedback controller and an output adaptive control.
0 2000 4000 6000 8000 10000-1.5
0
1.5
0 500 1000 15000
1.5OAC
Reference
SFC
(b) Trajectories in the y-axis for the SSAV. Comparison between a state-feedback controller and an output adaptive control.
Fig. 3. Trajectories in the xy-frame.
Fig. 4. System error.
consumed by the proposed step-by-step observer is nearlydespicable. Also the chattering effect that is commonlypresented by the STA can be observed as the signal convergesto the origin.
VII. CONCLUSIONS
The design of the proposed output adaptive controller fortrajectory tracking of an SSAV showed a similar time ofconvergence compared to the state-feedback controller inpresence of modelling uncertainties. Also the effect of over-shooting is observed to be reduced, generating a smootherconvergence of the desired circular trajectory. Future workswill describe the comparison of this scheme of control withdifferent control algorithms, but also its implementation inan SSAV prototype.
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