366 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. VOL. 44, NO. 2, MAY 1995

Path-Tracking for Car-Like Robots with Single and Double Steering

R. M. DeSantis

Absfrucf- A procedure for the design of path-tracking con- trollers for car-like robots is proposed which is based on a formalization of the notion of path-tracking that is traditional in automotive applications. The advantage of this procedure over current metbods is that it is applicable to a larger class of vehicles (car-like robots with single or double steering), to more general paths (in particular, circular paths), and to a tracking velocity that is not necessarily constant. When the path is a straight line or a circular arc, the tracking offsets are small and the assigned velocity is constant, then the proposed procedure leads to a controller with a linear, time-invariant, and decoupled PID (proportional, integral, and derivative) structure.

I. NOMENCLATURE

0

q := [.r y 01‘ 6 1 . 6 2 s

F w m

Work-space coordinates of vehicle center of mass (COM). Vehicle heading in work-space. Configuration vector. Front and rear steering angles. Steering angle in a single steering vechicle. Linear velocity in vehicle-frame coordinates. Angular velocity. Velocity vector. Orientation of COM velocity with respect to vehicle’s frame. Vehicle’s state vector. .- ) ’ I , ,

.- 1 1 ? ( . := (2.

Propulsion. Steering controls. Longitudinal forces exerted by front and rear tires. := [F,,i. Fu2]‘ Lateral forces exerted by front and rear tires. := [F,, 1. Fu?2]’

Vehicle mass.

._

. _

:= [a,, a,,, ne]/

Manuscript received December 5, 1992, revised March 21, 1994. R. DeSantis is with Ecole Polytechnique de Montrcial. Montrcial. PQ H3C

IEEE Log Number 9408538. 3A7. Canada.

a

b

l : = a + b

J

i31! i32

c1, c2 R

Vehicle yaw moment of inertia with respect to COM. Distance from COM to front axle. Distance from COM to rear axle. Distance between rear and front axles. Direction of permissible velocities in vehicle-frame coordinates. Path in configuration space.

Profile of required velocities and accelerations.

Vehicle’s desired state.

Heading offset. Lateral offset. Velocity offset. Front-steering offset. Rear-steering offset. Matrices of linearized path-tracking offsets dynamics. Feedforward propulsion. Feedback propulsion. Poles characterizing velocity offset dynamics. Speed-controller feedback gains. Steering-controller feedback gains. Slip angles. Tire’s stiffness factors. Distance from center of rotation to vehicle’s longitudinal axis. := Rtan&.

11. INTRODUCTION OST OF THE available procedures for the design of M path-tracking controllers for wheeled mobile robots,

while appropriate for a variety of industrial and commercial applications, are of an ad hoc nature and are limited to straight line paths and a constant tracking velocity (see, for example,

0018-9545/95$04.00 0 1995 IEEE

DESANTIS: PATH-TRACKING FOR CAR-LIKE ROBOTS 367

[l], [ l l ] , [4]). References 191, [21], [18], [14], [12], and [241 are representative of the state of the art. The common approach is to develop a dynamic model of the vehicle by considering the lateral forces exerted by the tires to be a function of the wheels’ sideslip angles. This function is assumed to be linear by Matsumoto and Tomizuka [18], polynomial by Lee [14], and of a more complex nature by Nisonger and Wormley [21]. Based on the selection of such a function, a heuristic path- tracking controller is proposed, the action of which is typically proportional to the lateral path-tracking offset and to the orientation offset. The properties of the controller are usually investigated by means of a combination of experimental tests and numerical simulations.

The objective of the present paper is to propose a more systematic approach where the analysis may be carried out in a more formal setting, and where more general path assignments may be considered. In particular, we seek to develop a design procedure for path-tracking controllers that is applicable to car- like robots with front-wheel steering as well as to robots with front- and rear-wheel steering. We would like this procedure to remain applicable when the path is not necessarily a straight line and the required velocity is not necessarily a constant. Whenever possible, we would like the procedure to lead to the realization of controllers with such practical features as linearity, decoupling, time invariance and a PID (proportional, integral, and derivative) structure. Moreover, we would like the procedure to clearly relate path-tracking assignment parameters (such as tracking speed and radius of curvature of the path) to the controller’s structure and gains.

A critical step in pursuing this objective is to formulate the path-tracking problem so as to enable a development that is both physically meaningful and analytically transparent. In parallel research efforts by other authors, this step has been carried out by interpreting path-tracking as equivalent to having the vehicle’s state follow a desired state trajectory (a prescribed function of time; see, for example, [15], [25], and [6]). However, while rigorous and mathematically elegant, this formulation is not necessarily the most appropriate from a physical point of view. For example, when this result is applied to industrial AGV’s, the result is intercoupled and time- variant controllers even in cases where standard decoupled PID controllers are quite adequate for the job (see, for example, [ 2 ] , [7], [24], [22]). The pragmatic approach that is routinely adopted in automotive applications appears to be more natural and to lead to controllers that are more intuitive and easier to implement. This approach still entails the convergence of the vehicle’s state to a desired state. Rather than being a preassigned function of time, however, this prescribed state is now a function of the configuration of the vehicle with respect to the path to be tracked (see [91, 181, [121).

Based on this observation, we proceed by formulating a path-tracking concept that retains the geometric connotation mentioned above. In regard to the vehicle’s model, we use the standard assumptions that the motion of the vehicle is planar, that the vehicle’s properties are symmetrical with respect to the longitudinal axis and that the contact between wheels and surface of motion is pointwise. In addition, and in contrast with automotive applications, we will also assume a motion

median front tire

I

L - - -

Fig. 1. Vehicle geometry.

of the wheels exempt from sideslippage. Although perhaps questionable in highway applications [21], [19], [201, [14], this latter assumption has been proven to be quite realistic in robotic applications where the maximum speed of the vehicle is usually less than 3-4 m/s (see, for example, [2], [7]).

We proceed by first considering a car-like robot equipped with front- and rear-wheel steering and by then particularizing the results for the more familiar case of a single steering.

111. VEHICLE’S DYNAMIC MODEL Consider a car-like robot equipped with a pair of rear

wheels and a pair of front wheels, each pair of which is steerable (Fig. 1). The difference between the orientations of the front and rear wheels and the orientation of the vehicle (steering angles) is assumed to be sufficiently small that the vehicle’s dynamics can be analyzed by replacing these wheels with “median” wheels located at the center of the front and rear axles. The position of the vehicle is described by the coordinates of its center of mass, (2, y); its heading by the orientation of the vehicle’s longitudinal axis, 0 (Fig. 2). The steering angles are denoted by the symbols 61 and 62, respectively; the vehicle’s linear velocity in the vehicle’s frame coordinates is denoted by (vu, v,) and its angular velocity by 0. The lateral and longitudinal forces exerted by the tires (see Fig. 3) are denoted by F,, and F,, (z = 1 refers to the front tires, i = 2 refers to the rear tires). The propulsion, Fp, is applied through the rear tires and has the direction of the longitudinal axis of these tires. Finally, m denotes the mass of the vehicle and j its yaw moment of inertia; a and b denote the distance from the center of mass to the front and rear axles; 1 is the distance between front and rear axles.

With these notations and by applying the Newton-Euler ap- proach, a mobile robot with double steering may be modeled as follows (an outline of this proof and those of the propositions to follow are given in the Appendix).

Proposition 1: The dynamics of a mobile robot with double steering is given by

(2.1) i = qL cos0 - vu) sin0

y = ti,, sin 8 + v , cos 8 (2.2)

368 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 44, NO. 2, MAY 1995

‘ Work-space path r - / I

I

/

Work-space Frame

* X

Fig. 2. Vehicle configuration.

6, = sz

Ful cos S1 Ft,.l sin 61 FU1 cos 62 _ _ _ _ + m

ilu = w w Q + m m

/ Fig. 3. Longitudinal and lateral forces exerted by the tires.

For convenience, we rewrite (2.1)-(2.6) in the more compact form:

F u := [ F u l FU21’ (2.10)

(2.3)

F w 2 sin 62 F cos 62 +-- (2.4) ______ m m

(2.1 1) M := diag (m, m, j )

Go := [wWR - v,R 01’ (2.12)

and

+ m m -sin 61 -sin 62

Ful sin 61 Fwl cos 61 F,z sin 62 +- il, = -v,R + ___ m

Fw2 cos 62 +- F, sin 62 (2.5) G, := [cos& cosS2 ]

acos61 - ~ C O S ~ ; ? + m m (2.13)

. Fulasin61 F,lacosh1 - FU2bsin62 R = +

j .i j cos61 cos& - F,2 b cos 62 _ _ _ _ _ Fp b sin 62 . (2.6) G, := [sin& sin62 ] (2.14)

j j a sin 61 - b sin S2

The vectors q := [z, y, SI’, v := [w, uw R]’, X := [z y 8 vu ww Q]’ and fi := [;lu ilw R]’ are referred to as the configuration, velocity, state and acceleration of the vehicle, respectively. The vehicle model is completed by adding to (2.1)-(2.6) the dynamics of the steering angles. This dynamics will be described by

(2.7)

where Fsi is the steering control ( i = 1 denotes the steering control applied to the front wheels, i. = 2 that applied to the rear wheels).

G; := [cos& sinS2 - bsin621. (2.15)

Iv. ABSENCE OF SIDESLIPPAGE

The assumption of a slippage-free motion implies that the longitudinal, lateral and angular velocities of the vehicle are interdependent and that the lateral forces exerted by the tires are a well-defined function of all the other forces applied to the vehicle (propulsion and steering forces, inertia and perturbation forces). These two properties simplify the vehicle’s dynamic model as follows.

DESANTIS: PATH-TRACKING FOR CAK-LIKE ROBOTS

Proposirion 2: Under a slippage-free motion, the vehicle’s velocity is submitted to the constraint

v = JVU

where 1

Moreover,

s1 := tan-’{ y} v, + U R

and

369

ACCELERATOR

DYNAMICS

SYSTEM Angles

Steering Control I Prooulsion Control

Path to be tracked

Proposition 3: Under a slippage-free motion, the dynamics

(3.5)

of the vehicle is described by

= guFu + . g p F p + gs Fs

t i , ( b tan 61 + a tan 62) 1 (3.6) vu> =

(3.7) vu( tan 61 - tan 62)

I (1 =

j. = cos B v , , - sin Bv,. (3.10)

(3‘4) Fig. 4. Principle of operation of a path-tracking controller.

V. PATH-TRACKING From an intuitive point of view, the task of a path-tracking

controller may be stated in quite simple terms. A path-tracking controller receives as input the actual and desired values of the vehicle’s speed, heading and position relative to the path, and it provides as output the propulsion and steering required to bring the difference between the actual and desired values to zero (see Fig. 4). The description of the path-tracking task that is needed for an analytical development is slightly more complex, however, and requires careful definition of the concepts of admissible path-tracking assignment, the vehicle’s desired state and path-traclung offsets. A path-tracking assignment is the combination of a path in the configuration space and a profile of the linear and angular velocities and accelerations with which this path is to be tracked. A path (see Latombe 1991) is described by a smooth vector function,

where

q P ( s ) := b P ( 4 Y P ( S ) OP(S) l ’ (4.1) $ = sin Bv,, + cos Bv,,,

8 = C l

(3.1 1)

(3.12) where s E [0, m) is a parameter defining a point of the path, and qp(s ) is the vehicle’s configuration required at the point of the path defined by s. A velocity and acceleration profile along the path is described by a set of smooth functions

(3.13) (4.2)

(3.14) := [au,(s) % p ( S ) nep(s)l’, s E [O. m) (4.3) 1 1

cos61 cos62 where v,(s) and u,(s) are the velocity and acceleration that should characterize the motion of the vehicle at point s. A path-tracking assignment is admissible if (4.2), (4.3) are compatible with (3.17)-(3.24).

Given a state of the vehicle, X := [x y O vu v, G]‘, the vehicle’s desired state corresponding to a path-tracking

(3.15) 1

gr> := p-1- cos 62

(3.16) 1 assignment is defined by mbv, + j Q mav, - jfl

1 cos2 61 1 cos2 62

btanhl +atan62

+ j { tan61 12. (3.17) tan62

I L := m + m

370 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. VOL. 44. NO. 2. MAY 1995

Remark I : The above formulation relies on the assumption that the projection of the vehicle state onto the planar zy path is well defined and that a unique state from the path-tracking assignment can be associated with it. While this assumption is satisfied in a great many practical applications, applications may be envisioned where this is not necessarily the case. Background material on the existence, uniqueness, continuity and computational complexity of the projection may be found in Schartz and Yap [23] or Latombe [ 131.

, ,Instantaneous

_ -

VI. CONTROLLER DESIGN Using Proposition 3 and (4.17), the design of a path-tracking

controller may be viewed as equivalent to the following stabilization problem.

Proposirion 4: Under a slippage-free motion, path-tracking is equivalent to stabilizing the dynamic system:

Fig. 5. Constraints imposed by the absence of sideslippage.

with

[xd Y d f ld] := [.p(o) '&(a) @p(u)]' (4.7) ~ 0 s = W(t) (5.1) and

2rwd a d ] := [ V u p ( O ) V u p ( U ) Op(U)] (4.8) 8 - ('Ud + v O s ) { t a n ( S d l + &osl ) -tan(Sd2 + h o s Z ) } 1 os -

(5.2) (v,d + vos){tan

1 - tan b d 2 } where u E [0, 00) is selected so that (z,(a), yp(.)) is the

point of the work-space path closest (in Euclidean norm) to

The path-tracking offsets (velocity (vas), heading (des),

(4.9)

-

(2, Y ) . . lo, = (v,d + vos){cos 6, +tan8,: sin&,} lateral ( l o s ) and steering (bo,9l, S O S 2 ) offsets) are defined by

. sin ( O O S + 8 , - 6,d) (5.3) vos(t) := v u ( t ) - O U d ( t )

(4.10)

where 6,d is such that b tan 6d1 + a tan 6d2

1 tanflvd =

and

(5.4)

(4.15)

(4.16)

While the physical interpretation of hOs1(t), 6 0 s 2 ( t ) , vas, and OOs is rather obvious, it may be helpful to note that the lateral offset, l o s , represents the (signed) distance from the center of mass of the vehicle to the projection of the path in the work-space (Fig. 3).

The task of a path-tracking controller may now be stated as that of generating the propulsion and steering controls required to have

(5.9)

and gu, g p , and gB are as in Proposition 3. With this result, a path-tracking controller may now be

designed by bringing to bear well-known control theory tech- niques.

Proposition 5: Under a slippage-free motion and small offsets, path-tracking is equivalent to stabilizing

X = A x + B u (5.10)

where

1 0 0 0 0 B':= 0 0 0 1 0

[0 0 0 0 (5.1 1)

DESANTIS: PATH-TRACKING FOR CAR-LIKE ROBOTS

and

0 0 0 0 o 1 ’ ’ ’ 1CO$d6dl e A := vd 1Cz’;dl lCOS’d6d2

av

0 0 0 0 0 0 0 0 0 O d I

VII. THE CASE OF SINGLE STEERING

By blocking the orientation of the rear wheels (that is, by setting 62 = 0), Propositions 1-7 enable a rediscovery and an extension of results that are currently available for car- like robots with single steering. This extension means that our procedure for designing path-tracking controllers is applicable

(5.12)

37 1

where

Proposirion 3‘: Under a slippage-free motion, the dynamics Proposition 6: I f the path in the workspace is a straight

line or a circular arc, the desired tracking velocity is constant of a car-like robot with single steering is described by

and the path-tracking offsets are kept sufficiently small, then path-tracking may be ensured by the combination of two time- invariant controllers. First, a steering controller providing the action

where K, is a matrix of constant gains. Second, a speed controller providing the propulsion

Fp(t) = F p l ( t ) + F p 2 ( t ) (5.14)

where

with F, given by (5.13) and

Fp2 = - g p l { Kpl u,, + KPz/ v,, d t } (5.16) where

and where Kpl and Kp2 are two constant gains.

has the following properties: Proposition 7: The controller described by (5.13)-(5.16)

a) the dynamics of v,, is given by

where poles p l l and p12 are such that

X = ( A - BK,)x

where

v,6 tan S v, = ~

1

vu tans R = - 1

S = F,

X = cos Ov, - sin Ov,

y = sin Ov, + cos Ow,

O = C l

g p := { + ( y ) 2 ( m 6 2 + j ) } -1

mb2 -t j g, := -~ 1 COS2 6 R g p .

-(PII + P I Z ) = Kpl and ~ 1 1 ~ 1 2 = Kp2. (5.18) Proposirion 6‘: If the path in the workspace is a straight line or a circular arc, the desired tracking velocity is constant and the path-tracking offsets are kept sufficiently small, then path-tracking for a car-like robot with single steering may be ensured by the combination of two time-invariant controllers. First, a steering controller providing the steering action

b) ~~i~~ K, may be chosen so as to stabilize the system:

(5.19)

and

A :=

0 0 1 0 13‘:= [o 0 0 11

where K, is a row-vector of constant gains. Second, a speed controller providing the propulsion

F ~ l = - S F ’ { S ~ F ~ + g8F,} (6.13)

312 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 44. NO. 2. MAY 1995

with Fs given by (6.1 l), and Remark 3: In the spirit of Remark 2, the steering controller proposed by Proposition 6' may be given forms that differ from (6.11) and that are more amenable to a practical im- plementation. For example, the steering control described by (6.11) could be replaced with

Fp2 = -gpl Kplti,s + KP2/ uos d t ) (6.14) { and where K,1 and K,z are two constant gains.

Proposition 7': The- controller described by (6.1 1)-(6.14) has the following properties.

a) The dynamics of TI,, is given by Indeed, in the case of single steering, the stabilization of the system described by (5.10) is equivalent to the stabilization of

(6.15) the following system: cos - (Pll + P12)&8 + PllP12~0, = 0

where poles pll and y12 are such that

- ( ~ l l +PIZ) = Kpl and ~ 1 1 ~ 1 2 = Kp2. (6.16)

b) Gains Ks may be chosen so as to stabilize the system

j. = (.4 - B&)z (6.17)

where

(6.29)

(6.30)

(6.31)

so, = u. (6.32)

(6.18) This stabilization may be obtained by applying a control of the form described by (6.28).

B' ._ 0 0 1 0

0 ° *

' -0 0 0 1

and VIII. APPLICATION EXAMPLE

A : = [;; ; +] (6.19) Given a car-like robot with double steering and with

a = 2 m 6 = l m where

j = 100 kg * m2 m = 300 kg (7.1)

consider the task of having the center of mass track a circular arc of radius R = 10 m3 with a constant longitudinal velocity vud = 2 s 5 ~ S V and with a null lateral velocity vwd = 0. Propositions and suggest that this task may be Carried Out by a path-trackng controller made UP of a steering Controller and a speed controller.

The steering controller receives as input the heading, lateral and steering offsets. It gives as output

vd := U,Ld ( 1 + -p-- ( I1tan6)~) 1'2. (6.20)

Remark 2: It should be noted that while the speed controller described by (6.13)-(6.14) has the structure of a PI (propor- tional + integral) controller, the steering controller described by (6.1 1) can also be interpreted as a controller of the PID family. Indeed, from (5.12) we obtain

1 cos2 & . (6.21) S,,s = ___ dm

t ' l l d

and

= v & ~ ~ + 6 Qo5. (6.22) Fs2 := -Ks21Qos - Ks221os - Ks236osl - K~246os2 (7.3)

By rewriting (6.11) as where Kst1 are selected so as to stabilize the system described by (5.19)-(5.21). Since v,d = 0, using (3.3), (3.4) and (3.19),

I?q = -K,ioos - KsiLs - K i S o s (6.23) we have

and using (6.21), (6.22), it follows that

and

- b o d I cos2 S,] 62d := tan-' { =}. (7.6)

Kd := _____ Ks3 (6.26) Moreover, (from (A.7) in the Appendix), t 'ud

(6.27)

DESANTIS: PATH-TRACKING FOR CAR-LIKE ROBOTS 313

2.5 0 0.85 +1.7 0 0 0 0 ’ A = 1

The speed controller receives as input the desired veloc- ity, v,,j and the velocity offset, v ~ , ~ . It gives as output the propulsion

Fp = Fpl + Fp2 (7.12)

where Fpl is given by

~ _ _ __ Fpi = -gi1{.9.uif‘ui + gu2FtL2 + g,iFsi + gszFs2} (7.13)

where FUl and Fu2 represent the estimated values of FUl and Fu2 (potential perturbations), gpgul, gu2, g s l r and gs2 are as established in Proposition 4 and F, is descnbed by (2), (3) and (1 1); Fp2 is given by

3 Orientation offset - -1 0

Speed offset ; -:5ryr-Jf77r4

~~~ _ _ -1 5 -

-2 0 0 2 4 6 8 10 12

distance travelled along the path (in meters) 4 2 := -..I{ K p l V o s + K p 2 / vo, d t } (7.14)

where gains Kpl and Kp2 are selected so as to have a desired dynamics for vas. By requiring this dynamics to be characterized by the poles p l l = -6 and p12 = -0.1, we obtain Kpl = -pll - p12 = 6.1, and Kp2 = p11p12 = 0.6.

The behavior of the overall path-tracking controller has been

eqns (2.1H2.7). In implementing the simulations, we have not assumed a slippage-free motion (this assumption is only used in the design of the controller). Rather, the lateral forces exerted by the tires have been described as a function of the orientations of the velocity of the tires’ points of contact with

axes (slip angles). In agreement with [12] and [18] this function has been described by

0 05

0

0 05

-0 10

-0 15 D simulated by considering the vehicle’s dynamic described by p -020

-0 25

0 30

-0 35

-0 40

-0 45 0 2 4 6 8 12 the surface of motion with respect to the tires’ longitudinal 10

Fig 6 Path-tracking off\ets under nominal operating conditions

Assuming a clockwise tracking of the circle, it follows that (7.15)

f l d = -0.2 rads Fur2 := -C2P2 (7.16)

6d1 = -0.16 rad

S d 2 = 0.08 rad (7.8)

and, therefore, the A matrix of (5.21) becomes

r 0 0 0.85 -0.841

(7.9)

L o 0 0 0 1 Using linear quadratic optimization (e.g., Chen [3]), the con- troller’s gains may be computed by the formula K , = R-’ B’P, where P is the solution of the Riccati equation:

A’P + PA’ - PBR-lB’P + Q = 0 (7.10)

and C,? and R are positive definite matrices. With the selection Q = 1, and R = 12, we obtain:

Ksll = 3.2 Ks12 = 0.93 Kr13 = 2.8 Ks14 = -0.2

Ks21 = -0.2 Ks22 = 0.37 Kn23 = -0.2 Ks24 = 1.6. (7.1 1)

where the stiffness factors, C1 and (22, have been taken as equal to 30000 m * kg/s * rad, and the slip angles, /31 and p2, have been computed by means of the following equations:

(7.17)

and I ) , - bR

~2 = tan-’{ y} - 6 2 . (7.18)

To illustrate, Fig. 6 shows path-tracking offsets in cor- respondence with a simulation implemented under nominal operating conditions. Observe that, in spite of a reasonably large initial lateral offset (equal to 2 m), the vehicle’s con- figuration converges quite rapidly to that required by the path-tracking assignment. It should also be noted that the vehicle’s overall dynamics is essentially identical to that which would be obtained under the slippage-free motion hypothesis. Fig. 7 shows results obtained when the controller’s gains and the desired steering angles are determined under the (incorrect) expectation that the path’s radius of curvature will be equal to 12 m (whereas it is in fact equal to 10 m). Observe that while the tracking remains satisfactory, steady-state lateral and orientation offsets are now present. These offsets could be

374 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. VOL. 44. NO. 2, MAY 1995

PATH-TRACKING OF A CIRCLE WITH CONSTANT SPEED

p 1 0 . - T

o 2 4 6 8 12 IO

distance travelled along the path (in meters)

0

-0 05

-0 10

-0 15

-0 20 v1 p -025

-0 30

-0 35

I I 8 10 12

-050- ’ Fig 7. diua of curvature of the path.

Path-tracking offsets in correspondence to bigger-than-expected ra-

readily eliminated by introducing in the steering controller a term proportional to the integral of the lateral offset.

In simulations reported elsewhere, the influence of unaccounted-for vehicle mass and geometric parameter variations, the presence of drag forces and of path assignment uncertainty was also considered. The results have confirmed that under nominal operating conditions and small path- tracking offsets, the controller’s behavior is as predicted by the theoretical development. They also suggest that performance remains acceptable even when the path-tracking offsets are relatively large, when realistic perturbations are present and when the vehicle’s and the path-assignment’s parameters differ from their nominal values.

IX. CONCLUSIONS The hypothesis of a slippage-free motion makes it possible

to simplify considerably the model of the kinematic and dynamic behavior of a car-like robot with double steering (Propositions 1-3). This model leads, in turn, to a simplified description of the dynamics of lateral, heading and velocity path-tracking offsets (Proposition 4). It follows that a con-

troller capable of tracking a planned path may be designed using linear, albeit time-variant, techniques (Proposition 5). If the assigned path is a straight line or a circular arc and the tracking velocity is constant, then the controller may be implemented using a simple, linear and decoupled structure, the gains of which may be computed using classical time- invariant PID techniques (Propositions 6 and 7). By bloclung the orientation of the rear wheels, these results enable a rediscovery and an extension of results currently available for car-like robots with single steering (Propositions 3’, 6’, and 7’).

While more realistic simulations and experimental tests are required before the practical potential of the proposed controller can be conveniently assessed, prospects for a suc- cessful practical implementation are nevertheless good. Simu- lation results suggest that the performance of the controller is reasonably robust not only with respect to variations in the vehicle’s mass and geometric parameters, but also with respect to a relaxation of the key hypothesis that led to its design (absence of sideslippage). As for the technological requirements, they are precisely the same as those found in current implementations of advanced automatically guided vehicles with single steering (AGV’s). These requirements may be adequately met by using standard components like optical encoders, gyros, computer vision, laser range-finders, sonar, and other similar components.

APPENDIX OUTLINES OF PROOFS

Proof (Proposition 1)

With reference to Fig. 2, the linear velocity and acceleration of the vehicle, in the vehicle’s frame coordinates, are given by

.? cos 0 + sin 6’ ir,, - v,.R [z] = [ -;i:sin6+ycosO ] = [.. + ?&(l]. (A.1)

The resultant of the forces applied to the vehicle (again in the vehicle’s frame coordinates) is The moment of these forces about the vehicle’s center of mass is

r = F,lasin& + F,lacos61 - F,absin& - F,,,26 cos 62 - F p b sin 152. (A.3)

By writing the Newton-Euler equations (in the vehicle’s frame coordinates; see, for example, Craig [5 ] , Kane and Levinson [17]) we have

F = M a ,

which, combined with (2.1 )-(2.3), lead to (2.1)-(2.6).

DESANTIS: PATH-TRACKING FOR CAR-LIKE ROBOTS 375

Proof (Proposition 2)

have

Proof (Proposition 4 ) With reference to Fig. 5, under a slippage-free motion, we From (3.17) and (3.19) we have

(A. 18) it,,, = ,quFu + gpFp + .93K - V f L d

R = 5 {I,, = (6 - 1 + L1)R (A.5) I2 . vu(tan61 -tan&) vud(tan6dl - tan6d2)

00, = - (A.19) 1 1

and, from (3.20), (3.21), 1 - 11 = -Rtanb2 (A.6)

where R is the distance between the vehicle’s instantaneous center of rotation and its longitudinal axis, and 11 := R tan 61. It follows that

(A.20) i o s i = 81 - i d 1 = -&dl + Fsl

. . 60~2 = (52 - 6d2 = -6d2 + Fs2. (A.21) 1

(tan61 - tan&) u,, (tan 61 - tan 52)

To prove (5.3), derive the two members of (4.1 1) to obtain R = (A.7)

R = I (A.8) ios(t) : = -{.i(t) - i d ( t ) ) sin ( ~ d ( t ) + ~ , , d )

vu, = (6 + R tan61)R (A.9)

- - ~ , , ( 6 t a n 5 ~ +atanS2) (A.lO) 1

Using (2.1), (2.2) it follows Proof (Proposition 3)

Following the Kane approach (Kane and Levinson [17]) we premultiply the two members of (2.8) by J’. Since J’G, = 0,

. l o s ( t ) = ( U T I CosO~l + V U Sine,.) sin (Oos + 8, - (A.24)

it follows that

J’Mu = J‘Go + .JIG,, F,,

By deriving (3.2) now, we obtain

u = Jit, + 74,AJ[&

where

0 0 (A.13)

Using (A.12) in (A.ll) gives . .

J‘MJil,, = -J’MuuVJ[61 & I 1 + J’G,F,, + J’GpFp. (A.14)

The desired result follows by now computing

gl, := (.J’M.J)-’J’G,, (A.15)

,9p := ( J W J - ~ J I G , (A.16)

gs := - ( J ’ M J ) - ’ v J . (A.17)

Proof (Proposition 5) Note that the system represented by (5.10)-(5.12) represents

the linear approximant (in the Lyapunov sense) of the nonlin- ear system described by (5.1)-(5.9). We prove stabilizability by applying the Lyapunov’s linearization approach (lemma 1 below) and, therefore by proving the stabilizability of the system given by (5.10)-(5.12). This stabilizability may be obtained by proving controllability. Controllability is a consequence of the fact that dim {B AB} = 4, and of Lemma 2. Observe that, by appropriatedly modifying a first segment of the path-tracking assignment, the assumption of small initial values of vos(t), rOs(t), los(t) , S o s 1 ( t ) , and 6,,2(t) may always be satisfied in practice. The required modification of the path may be carried out either by inspection or by applying the specialized techniques discussed by Latombe [ 131 or by Femandez et al. [lo]).

376 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. VOL. 44. NO. 2, MAY 1995

Lemma 1 [16, p . 1841 : Let .T = 0 be an equilibrium point where for the nonlinear system:

x = f ( t , .T) (A.27)

where f : [O. cc) z D + Rrl is continuously differentiable, D := {x E R”. x2 < T } , and the Jacobian matrix [Sf/Sz] is bounded and uniformly Lipschitz continuous on D. Then,

nonlinear system, if and only if it is an exponentially stable equilibrium point for the linear approximant

and

the origin is an exponentially stable equilibrium point for the A22 :=

0 0 1 0 ’ B 2 : = [ ] 0 0 0 1

0 0 0 0

(A.39)

. (A.40)

x = i l ( t ) z (A.28)

where

A ( t ) := 2) (A.29)

Lemma 2 [3, p . 1791 : The system

.i. = A ( t ) z + B(t )u (A.30)

with matrices A(t) , B( t ) continuously differentiable n - 1 times, is controllable if

dim{[Mo(t)(hfl(t)( ” . IMT1-l(t)]} = n (A.31)

where t l d t Mk+l( t ) = -A(t)Mk(t)+-hMk(t) . k = 0, l , . . . , n - l

(A.32) with

M ( ) ( f ) = B(t ) . (A.33)

Proof (Propositions 6, 7) From Proposition 5, if the path in the work-space is a

straight line or a circular arc, and if the required forward, lateral and angular velocities are constant, then the linearized model of the path tracking offsets dynamics is time-invariant. The application of the control represented by (5.13)-(5.16) gives a dynamics of these offsets that is described by (5.10)-(5. I 2) with

111 = -Kl.rl - K2 ~ - 1 d t (A.34) s and

u2 = -Ks[x2 2-3 zq z g ] ’ (A.35)

where

51 := 7f*s

and

(A.30)

It follows that

Since limt,, zl(t) = 0, and since K, is selected so that A22 - B2K,9 is stable, it follows that:

ACKNOWLEDGMENT Thanks are due to the IEEE TRANSACTIONS ON VEHICULAR

TECHNOLOGY reviewers for helpful editorial and technical suggestions.

REFERENCES

[I] S. E. Anderson, Ed., Proc. 3rd In!. Con$ on Automuted Vehicle Systems, Stockholm, Sweden, Oct. 1985.

121 J. Borenstein and Y. Koren, “Motion control analysis of a mobile robot,” J. Dynamic Systems Measurement and Contr., vol. 109, June 1987.

[3] C.-T. Chen, Linear System Theory und Design, HRW 1987. [4] I. Cox, G. T. J. Wilfong, Eds., Autonomous Robot Vehicles. Reading,

MA: Springer-Verlag, 1990. [ 5 ] J. J. Craig, Introduction to Robotics Mechanics and Control. 2nd

ed. Reading, MA: Addison-Wesley, 1989. [6] B. D’Andrea-Novel, G . Bastin, and G. Campion, “Dynamic feedback

linearization of nonholonomic wheeled mobile robots,” Proc. 1992 Int. Con$ on Robotics and Automation, Nice, France, May 1992.

[7] R. M. DeSantis. and R. Hurteau, “On the motion control of electric AGVs.” Automaz. Strument.. vol. 3 , Mar. 1990, pp. 137-150 (Special issue on AGV; also available as EPMmr-89/22).

[8] R. M. DeSantis. “Path-tracking for a carlike mobile robot,” in ACC Proc., pp. 64-68, 1993.

[9l R. E. Fenton, G. C. Melocik, and K. W. Olson. “On the steering of automated vehicles: Theory and experiments,” IEEE Trans. Automat. Contr., vol. AC-21, pp. 306314. June 1976.

1101 C. Fernandez. L. Gurvits, and 2. X. Li, “Foundations of motion planning,” 1991 IEEE Int. Cunj in Robotics and Automation.

[11] G . Giralt, Les Robots Mobiles Autonomes, LAAS, Toulouse, France, Publ. 87308. 1988.

112) A. Hemami, M. G. Mehrabi, and R. M. H. Cheng, “A synthesis of an optimal control law for path tracking in mobile robots,” Automalica, vol. 8, no. 2. pp. 383-387, 1992.

[I31 J. C. Latombe, Robot Motion Planning. Nonvell, MA: Kluwer 1991. [14] A. Y. Lee, “A preview steering autopilot control algorithm for four-

wheel steering passenger vehicles,” J . Dyn. Syst. Meas. Contr.. vol. 114, pp. 401-409, Sept. 1992.

[I51 Y. Kanayama, Y. Kimura, F. Miyazaki, and T. Noguchi, “A stable tracking control method for an autonomous mobile robot,” Proc. Int. Conj on Robotics and Automation. Cincinnati, OH, pp. 384-389, 1990.

[I61 H. K. Khalil. Nonlinear Sysrems. New York: MacMillan, 1992. [I71 T. R. Kane and D. A. Levinson. Dyamics: Theory and Applications.

New York: McGraw-Hill, i985. [IS] N. Matsumoto and M. Tomizuka, “Vehicle lateral velocity and yaw rate

control with two independent control inputs,” J . o f Dynamic Systems Meas. Contr.. vol. 114, pp. 606613. Dec. 1992.

[I91 M. G . Mehrabi, A. Hemami, and R. M. H. Cheng, “Analysis of steering control in vechicles with Two Independent left and right traction wheels.” Fifth Int. Con5 Advanced Robotics, Pisa, Italy, 1991,

[2Ol -, “Control of a wheeled mobile robot with double steering,” Proc. IROS Y l , Osaka, Japan.

[211 R. L. Nisonger and D. N. Wormley, “Dynamic performances of auto- mated guideway transit vehicles with dual-axle steering. IEEE Trans. Veh. Technol.. vol. VT-28, Feb. 1979.

DESANTIS: PATH-TRACKING FOR CAR-LIKE ROBOTS 377

[22] N. Sarkar, X. Yun. and V. Kumar, “Control of mechanical systems with rolling constraints: Application to dynamic control of mobile robots,’’

[23] J. T. Schwartz and C. K. Yap, Advances in Roborics I: Algorithmic Hillsdale, NJ: Lawrence Erlbaum

[24] D. H. Shin, S. Singh, and J. J. Lee, “Explicit path tracking by

[25] G. Walsh, D. Tilbury, S. Sastry. R. Murray, and J. P. Laumond, Pmc. 1992 IEEE In?. Con$ on Robotics and Automation, Nice, France, pp.

Inr. J . Robot. Res., vol. 13, no. 1, pp. 55-69, Feb. 1994. 1999-2004.

and Geometric Aspects of Robotics. Assoc., 1987.

autonomous vehicles,” Roborica, vol. 16, pp. 537-554. 1992. R M. DeSantis, photograph and biography not available at the time of publication.