IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 5, OCTOBER 2000 609

Adaptive Tracking Control of a NonholonomicMobile Robot

Takanori Fukao, Hiroshi Nakagawa, and Norihiko Adachi

Abstract—A mobile robot is one of the well-known nonholonomic sys-tems. The integration of a kinematic controller and a torque controller forthe dynamic model of a nonholonomic mobile robot has been presented. Inthis paper, an adaptive extension of the controller is proposed. If an adap-tive tracking controller for the kinematic model with unknown parame-ters exists, an adaptive tracking controller for the dynamic model with un-known parameters can be designed by using an adaptive backstepping ap-proach. A design example for a mobile robot with two actuated wheels isprovided. In this design, a new kinematic adaptive controller is proposed,then a torque adaptive controller is derived by using the kinematic con-troller.

Index Terms—Adaptive backstepping, adaptive tracking control,dynamic model, nonholonomic mobile robot.

I. INTRODUCTION

A mobile robot is one of the well-known systems with nonholonomicconstraints, and there are many works on its tracking control [1]–[4].Their objects are mostly kinematic models, but recently one methodfor dynamic models has been proposed [5]. This method integrates akinematic controller and a torque controller for the dynamic model ofa nonholonomic mobile robot by using backstepping [6].

The control input of the controller for the kinematic model is gen-erally velocity, but it is more realistic that the input is torque. In [5], akinematic controller is designed first so that the tracking error betweena real robot and a reference robot converges to zero, and secondly atorque controller is designed by using backstepping so that the veloc-ities of a mobile robot converge to the desired velocities, which aregiven by the kinematic controller designed at the first step.

In this paper, we present a method to design an adaptive trackingcontroller for the dynamic model of a nonholonomic mobile robot withunknown parameters by adaptive backstepping. The adaptive controlmethods [7], [8] proposed so far for nonholonomic mobile robots donot consider the model with unknown parameters in its kinematic part,but our method considers the case. We show that there exists an adap-tive tracking controller for the dynamic model with unknown param-eters if it is possible to design an adaptive tracking controller for thekinematic model with unknown parameters. For an example, we de-sign an adaptive tracking controller of a mobile robot with two actuatedwheels. First, we present an adaptive tracking controller for the kine-matic model modifying the existing method [9]. Secondly, our maintheorem is applied to the dynamic model by using the kinematic adap-tive controller and we get a torque controller.

Manuscript received May 28, 1999; revised November 15, 1999. This paperwas recommended for publication by Associate Editor J. Laumond and EditorA. De Luca upon evaluation of the reviewers’ comments. This paper was pre-sented in part at the International Symposium on Intelligent Robotic Systems,Bangalore, India, January 1998.

T. Fukao and N. Adachi are with the Department of Systems Science, Grad-uate School of Informatics, Kyoto University, Kyoto 606-8501, Japan (e-mail:fukao@sys.i.kyoto-u.ac.jp).

H. Nakagawa is with Sumitomo Electric Industries, Osaka 554-0024, Japan.Publisher Item Identifier S 1042-296X(00)08352-X.

II. K INEMATICS AND DYNAMICS OF A NONHOLONOMIC

MOBILE ROBOT

Consider the following nonholonomic mobile robot that is subject tom constraints

M(q)�q + V (q; _q) _q +G(q) = B(q)� +AT (q)� (1)

whereq 2 Rn is generalized coordinates,� 2 Rr is the input vector,� 2 Rm is the vector of constraint forces,M(q) 2 Rn�n is a sym-metric and positive-definite inertia matrix,V (q; _q) 2 Rn�n is thecentripetal and coriolis matrix,G(q) 2 Rn is the gravitational vector,B(q) 2 Rn�r is the input transformation matrix, andA(q) 2 Rm�n

is the matrix associated with the constraints. In the following, we con-sider ther = n �m case.

The kinematic constraints are assumed to be expressed as

A(q) _q = 0: (2)

With respect to the dynamics of mobile robot (1), the following prop-erties are known [10].

Property 1: M(q) is a symmetric and positive-definite matrix.Property 2: There is a parameter vectorp0 2 Rl on dynamics that

satisfies the following equation [11]:

M(q) _v + V (q; _q)v +G(q) = Y0(q; _q; v; _v)p0 (3)

wherev 2 Rn andY0 is ann � l0 matrix whose elements consist ofknown functions.

Property 3: The matrix _M � 2V is skew-symmetric [12], that is,xT ( _M � 2V )x = 0, 8x 2 Rn.

The nonholonomic mobile robot (1) is transformed to and dividedinto the following two equations [5]:

_q =S(q)�(t) (4)

M(q) _� + V (q; _q)� +G(q) =B(q)� (5)

whereS(q) 2 Rn�(n�m) spans the null space ofA(q) and a full-rankmatrix formed by a set of smooth and linearly independent vector fields,� 2 Rn�m, M = STMS, V = ST (M _S + V S), G = STG, andB = STB. The system (4) represents the kinematics of a mobile robot.

The following properties are derived from the previously describedProperties 1–3 [5], [13].

Property 1′: M(q) is a symmetric and positive-definite matrix.Property 2′: There is a parametric vectorp1 2 Rl on dynamics

that satisfies

M(q) _� + V (q; _q)� +G(q) = Y1(q; _q; �; _�)p1 (6)

whereY1 is (n � m) � l1 matrix whose elements consist of knownfunctions.

Property 3′: The matrix _M � 2V is skew-symmetric.

In (6),p1 includes only the parameters on dynamics, not kinematics.The parameters on kinematics are included inY1. Now, we assume thestructure of the parameters on kinematics.

Assumption II.1: Some parameters in the kinematic part (4) of amobile robot appear as following:

_q =S(q; �)� =

n�m

i=1

si(q; �i)�i =

n�m

i=1

(�i0(q) +

l

j=1

�ij�ij(q))�i

(7)

1042–296X/00$10.00 © 2000 IEEE

610 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 5, OCTOBER 2000

where we let� = [�1; . . . ; �n�m], �i = [�i1; . . . ; �il ]T , 1 � i �

n�m are parametric vectors, and�ij(q), 1 � i � n�m, 0 � j � liare vectors whose elements consist of known functions.

Furthermore, the following Property 2″ is satisfied.Property 2″: There is a parametric vectorp 2 Rl on kinematics and

dynamics which satisfies

M(q) _� + V (q; _q)� +G(q) = Y (q; _q; �; _�)p (8)

whereY is (n�m)� l matrix whose elements consist of known func-tions andp is a parametric vector which is composed of the elementsof p1 and�i.

III. A DAPTIVE TRACKING CONTROL OF A NONHOLONOMIC

MOBILE ROBOT

In [5], �(t) is considered as a control input for the kinematic part(4), and an ideal control input�c(t) is designed to track a referencetrajectory. Since the real input of the mobile robot (1) is� , � is designedto make�(t)� �c(t)! 0 ast!1 by using backstepping [6]. But ifthere exist some unknown parameters in a mobile robot, that is,S(q)has unknown parameters orp is unknown in (8), we cannot design atracking controller according to [5].

In this paper, it is shown that an adaptive tracking controller can bedesigned for the dynamic model with unknown parameters if it is pos-sible to design an adaptive tracking controller for the kinematic modelwith unknown parameters.

Control Objective:Design an adaptive tracking controller for a non-holonomic mobile robot (1), in order that

limt!1

(q(t)� qr(t)) = 0 (9)

whereq(t) = Cq(t),C 2 Rs�n andqr(t) 2 Rs is its desired outputand differentiable.

Assumption III.1: An adaptive tracking controller

� = �c(q; qr; a) (10)_ai =Ti(q; qr; a); 1 � i � k (11)

exists for the kinematic model (7), that is, with this controllerq ! qrast ! 1.

And there exists a positive-definite and radially unbounded functionV1 which satisfies

_V1(q; qr; ~a) =@V1@q

S�c +@V1@qr

_qr +

k

i=1

@V1@ai

Ti � 0 (12)

and the signals included in this function are bounded, wherea is theestimate of an unknown parametric vectora = [a1; . . . ; ak]

T , whichis composed of�ij , and~a = a� a is the estimated error.

The general design method of these adaptive tracking controllerswhich satisfy Assumption III.1 has not been established so far.

Assumption III.2: B(q) in (5) does not include unknown parametersand is nonsingular.

Assumption III.3: @V1=@q does not include unknown parameters.

Assumption III.2 is easily relaxed by the existing adaptive controltechnique, ifB is constant and the sign of each elements is known.Assumption III.3 can be always satisfied by the appropriate selectionof V1.

Now, we provide the following theorem to design an adaptivetracking controller of a mobile robot using adaptive backstepping

technique [6]. The adaptive control technique for the dynamic part (5)is based on [10].

Theorem III.1: If Assumptions II.1 and Assumptions III.1–III.3 aresatisfied for a nonholonomic mobile robot (1), the following adap-tive tracking controller (13)–(16) achieves the control objective:q !qr(t! 1) and the boundedness of the signals included inV2, whichis defined in (17).

� =B�1

�Kd~� + Ycp�@V1@q

ST

(13)

_ai =Ti(q; qr; a); 1 � i � k (14)

_�i =�i

@V1@q

�i

T

~�i; 1 � i � n�m (15)

_p =��Y Tc ~� (16)

where� = [�1; . . . ; �n�m] is the estimate of�,Yc � Y (q; _q; �c; _�c),S � S(q; �), ~� = � � �c = [~�1; . . . ; ~�n�m], �i = [�i1; . . . ; �il ],1 � i � n �m, andKd, �, �i, 1 � i � n �m are symmetric andpositive-definite matrices with appropriate dimensions.V2 is defined as

V2 = V1 +1

2~�TM ~� +

1

2~pT��1~p+

n�m

i=1

1

2~�Ti �

�1

i~�i (17)

with ~p = p � p, ~� = � � �.Proof: The derivative ofV2 is

_V2 =@V1@q

S(�c + ~�) +@V1@qr

_qr +

k

i=1

@V1@ai

Ti

+ ~�T1

2_M � V ~� + ~�T (B� � V �c �G�M _�c)

+ ~pT��1 _~p+

n�m

i=1

~�Ti ��1

i_~�i

=@V1@q

S�c +@V1@qr

_qr +

k

i=1

@V1@ai

Ti + ~pT��1 _~p

+ ~�T B� � Ycp+@V1@q

ST

+

n�m

i=1

~�Ti ��1

i_~�i

=@V1@q

S�c +@V1@qr

_qr +

k

i=1

@V1@ai

Ti + ~pT��1 _p+ �Y Tc ~�

+

n�m

i=1

~�Ti ��1

i_�i � �i

@V1@q

�i

T

~�i � ~�TKd~�

=@V1@q

S�c +@V1@qr

_qr +

k

i=1

@V1@ai

Ti � ~�TKd~� (18)

where we used the following equation:

M _~� =M( _� � _�c)

=B� � V � �G�M _�c

=B� � V (~� + �c)�G�M _�c: (19)

From Assumption III.1, (17), and (18), the signals included inV2are bounded. Because_~� is proved to be bounded,�V2 2 L1. FromBarbalat’s lemma [14], [6], we can show�(t) ! �c(t) ast ! 1.Therefore, the equation_q = S� = S(~� + �c) = S�c + S~� showsq(t) ! qr(t) ast ! 1, since~� ! 0 ast ! 1 and the kinematicmodel of a mobile robot satisfies Assumption III.1

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 5, OCTOBER 2000 611

Fig. 1. Mobile robot with two actuated wheels.

Remark III.1: Because adaptive control is applied to treat unknownparameters in the kinematic part, it is more important to consider thedynamic part properly, that is, our proposed model-based controlleris better than a high-gain feedback controller to treat the dynamics.As is generally known, the adaptive control system designed for thekinematics may be unstable if there exists the error~� [15].

IV. M OBILE ROBOT WITH TWO ACTUATED WHEELS

In this section, we consider a mobile robot with two actuatedwheels as an example which the theorem can be applied to. Anadaptive tracking controller is designed for the kinematic model andthe dynamic model, and some simulation results are provided.

A. Model of a Mobile Robot with Two Actuated Wheels

We consider the mobile robot with two actuated wheels, which isshown in Fig. 1 [16].

With regard to the mobile robot shown in Fig. 1,2b is the width ofthe mobile robot andr is the radius of the wheel.O � xy is the worldcoordinate system andP0 �XY is the coordinate system fixed to themobile robot.P0 is the origin of the coordinate systemP0�XY and themiddle between the right and left driving wheels. The center of mass ofthe mobile robot isPc, which is on theX-axis, and the distance fromP0 to Pc is d. For the later description,mc andmw are the mass ofthe body and wheel with a motor,Ic, Iw, andIm are the moment ofinertia of the body about the vertical axis throughPc, the wheel witha motor about the wheel axis, and the wheel with a motor about thewheel diameter, respectively.

The configuration of the mobile robot can be described by five gen-eralized coordinates

q = [x; y; �; �r; �l]T (20)

where(x; y) are the coordinates ofP0, � is the heading angle of themobile robot, and�r; �l are the angles of the right and left drivingwheels.

We assume the wheels roll and do not slip. Then, there exist threeconstraints; the velocity ofP0 must be in the direction of the axis ofsymmetry and the wheels must not slip

_y cos�� _x sin� =0 (21)

_x cos�+ _y sin�+ b _� = r _�r (22)

_x cos�+ _y sin�� b _� = r _�l: (23)

These constraints can be rewritten in the form

A(q) _q = 0 (24)

where

A(q) =

sin� � cos� 0 0 0

cos� sin� b �r 0

cos� sin� �b 0 �r

: (25)

Equations (4) and (5) can be written as the following:

_q =S(q)�(t) (26)

M(q) _� + V (q; _q)� =B(q)� (27)

whereS(q) is selected as

S(q) =

r

2cos�

r

2cos�

r

2sin�

r

2sin�

r

2b�

r

2b1 0

0 1

(28)

andM , V , B are expressed as

M =

r2

4b2(mb2 + I) + Iw

r2

4b2(mb2 � I)

r2

4b2(mb2 � I)

r2

4b2(mb2 + I) + Iw

V =0

r2

2bmcd _�

�

r2

2bmcd _� 0

B =1 0

0 1(29)

and� = [�r; �l]T consists of motors’ torques�r and�l, which act

on the right and left wheels, respectively, and letm = mc + 2mw,I = mcd

2 + 2mwb2 +Ic + 2Im.

B. Adaptive Control of the Kinematic Model

We design an adaptive tracking controller for the kinematic part (26)modifying the method proposed by Kanayamaet al. [9].

First, we consider� as a control input and construct the adaptivecontrol system for the following kinematic model:

d

dt

x

y

�

�r

�l

=

r

2cos�

r

2cos�

r

2sin�

r

2sin�

r

2b�

r

2b1 0

0 1

�1

�2(30)

where �1 and �2 represent the angular velocities of right and leftwheels.

We focus on only three statesx, y, �, except�r and�l. The relation-ship betweenv, w, and�1, �2 is the following:

�1

�2=

1

r

b

r1

r�

b

r

v

w(31)

wherev is the straight line velocity andw is the angular velocity of themobile robot at the pointP0.

Substituting (31) for (30), we get the ordinary form of a mobile robotwith two actuated wheels

d

dt

x

y

�

=

cos� 0

sin� 0

0 1

v

w: (32)

The various design methods for this system (32) have already beenproposed. Our method is based on the method [9] whose objective istracking on a reference robot shown in Fig. 2.

612 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 5, OCTOBER 2000

Fig. 2. Reference robot and real robot.

The kinematics of the reference robot is given as

d

dt

xryr�r

=

cos�r 0

sin�r 0

0 1

vrwr

(33)

wherexr , yr, and�r are the configure of the reference robot, andvr,wr are its reference inputs.

We definee1, e2, e3 as following:

e1e2e3

=

cos� sin� 0

� sin� cos� 0

0 0 1

xr � x

yr � y

�r � �

: (34)

e1, e2, e3 describe the difference of position and direction of thereference robot from the real robot. The inputsv, w, which makee1,e2, e3 converge to zero, are given by the following [9], [5]:

vf = vr cos e3 +K1e1

wf =wr + vrK2e2 +K3 sin e3 (35)

whereK1; K2; K3 are positive constants.We can easily confirm thate1, e2, e3 satisfy

d

dt

e1e2e3

= v

�1

0

0

+ w

e2�e1�1

+

vr cos e3vr sin e3

wr

: (36)

We defineV0 as

V0 =1

2(e21 + e22) +

1� cos e3K2

(37)

then, the derivative ofV0 satisfies the following inequality:

_V0 = e1 _e1 + e2 _e2 + _e3sin e3K2

= �K1e2

1 �

K3 sin2 e3

K2

� 0:

(38)

If the parameters in kinematics (30),r andb, are unknown, we cannotchoose the inputs as (35) because of the relationship (31) betweenv,wand�1, �2. Hence, we design an adaptive controller to attain the controlobjective by using the estimates ofr andb.

By using�1 and�2, (36) is transformed to

d

dt

e1e2e3

= �1

�r

2+

r

2be2

�r

2be1

�r

2b

+ �2

�r

2�

r

2be2

r

2be1

r

2b

+

vr cos e3vr sin e3

wr

(39)

where we set

a1 =1

rand a2 =

b

r(40)

and it is assumed that we know a positive constant� which satisfiesa2 � � noticinga2 > 0.

Then,�1 and�2 are chosen as the following:

�1�2

=a1 a2a1 �a2

vfwf

(41)

=a1 + ~a1 a2 + ~a2a1 + ~a1 �a2 � ~a2

vfwf

: (42)

Therefore

d

dt

e1e2e3

= 1 +~a1a1

vf

�1

0

0

+ 1 +~a2a2

wf

e2�e1�1

+

vr cos e3vr sin e3

wr

: (43)

We defineV1 as

V1 = V0 +1

2 1a1~a21 +

1

2 2a2~a22 (44)

with positive constants 1, 2.The derivative ofV1 is

_V1 = e1 _e1 + e2 _e2 +_e3 sin e3K2

+~a1 1a1

_a1 +~a2 2a2

_a2

= e1 �K1e1 �~a1a1

vf + e2vr sin e3

+ �K2e2vr �K3 sin e3 �~a2a2

wf

sin e3K2

+~a1 1a1

_a1 +~a2 2a2

_a2

= _V0 +~a1 1a1

_a1 � 1e1vf +~a2 2a2

_a2 � 2wf sin e3

K2

:

(45)

Now, the parameter update rules are chosen as

_a1 = 1e1vf

_a2 = 2wf sin e3

K2

+ f(a2) (46)

where

f(a2) =

0; a2 > �

1�a2�

2

(f20 + 1); a2 � �(47)

with f0 = ( 2wf sin e3)=K2.Then

_V1 = _V0 +~a2 2a2

_a2 � 2wf sin e3

K2

= _V0 +~a2 2a2

f(a2)

= _V0 +a2 � a2 2a2

f(a2): (48)

Whena2 > �, the second term of the right-hand side of (48) is zero.Whena2 � �, the second term is less than zero becausef(a2) � 0,a2� a2 � �� a2 � 0. Therefore, it is shown that_V1 � 0. Also, from(43), _e1 and _e3 are bounded since~a1 and~a2 are bounded. After all,�V1is bounded. Barbalat’s lemma shows that_V1 ! 0 ast ! 1, that is,e1 ! 0 andsin e3 ! 0. Furthermore, (35) and (41) show that�1 and

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 5, OCTOBER 2000 613

�2 are bounded. If we want to avoide3 ! ��, one sufficient conditionis that the initial value satisfiesV1(0) < (2=K2).

From (43), the derivative of_e3 is

�e3 =� 1 +~a2a2

_wf �_a2a2

wr + _wr

=� 1 +~a2a2

( _wr + _vrK2e2 + vrK2 _e2 +K3 _e3 cos e3)

�_a2a2

wf + _wr: (49)

Sincesin e3 ! 0 ast!1, e3 goes to some finite number. Since~a2,e2, _e2, _e3,wf are bounded,�e3 is bounded if we choosevr ,wr, _vr, _wrto be bounded. Barbalat’s lemma shows_e3 ! 0 ast ! 1. From theequation

_e3 = � 1 +~a2a2

(wr + vrK2e2 +K3 sin e3) + wr (50)

we can show�(a2=a2)vrKre2 ! 0 if we choosewr ! 0, becausee1 ! 0, sin e3 ! 0 ast ! 1.

Finally, we showa2 > 0.If a2 � ((2�p2)=2)� < �, the following is satisfied:

_a2 = f0 + 1� a2�

2

(f20 + 1) � f0 +1

2(f20 + 1)

=1

2(f0 + 1)2 � 0: (51)

From this inequality, we can seea2 � ((2�p2)=2)� > 0.Therefore, we can obtain the following theorem.Theorem IV.1: If we choose the control inputs as (41) and the pa-

rameter update rules as (46) for the kinematic model (30) of a mobilerobot with unknown parametersr andb, the closed-loop signals arebounded. If we choose the reference inputs such thatvr does not go tozero andwr goes to zero, that is, the reference path is a straight line,thenx ! xr , y ! yr, � ! �r .

Remark IV.1: In Theorem IV.1, we assume thatwr goes to zero, thatis, if the reference trajectory is not a straight line, the tracking errorsdo not converge to zero. Recently, we got some results [17], [18] thatresolve this difficulty.

C. Adaptive Control of the Dynamic Model

From above sections, the mobile robot with two actuated wheels sat-isfies Assumptions II.1 and Assumptions III.1–III.3. Therefore, we candesign an adaptive tracking controller for the dynamic model (26) and(27) from Theorem III.1. According to Theorem III.1, we design anadaptive controller and perform some simulations.

From Theorem III.1, the adaptive tracking controller for the dynamicmodel is

� =�kd ~�1~�2

+_�1c _�2c _��2c_�2c _�1c � _��1c

p1p2p3

��11

@V1@x

cos�+@V1@y

sin� + �12@V1@�

�21@V1@x

cos�+@V1@y

sin� � �22@V1@�

(52)

d

dt

p1p2p3

=� _�1c _�2c_�2c _�1c_��2c � _��1c

~�1~�2

(53)

d

dt

�11�12

=�

@V1@x

cos�+@V1@y

sin�

@V1@�

~�1 (54)

Fig. 3. Reference inputsv ; w .

d

dt

�21�22

=�

@V1@x

cos�+@V1@y

sin�

�@V1@�

~�2 (55)

wherep1 = (r2=4b2)(mb2 + I) + Iw, p2 = (r2=4b2)(mb2 � I),p3 = (r2=2b)mcd, andp1, p2, p3 are the estimates, and

@V1@x

=�e1 cos�+ e2 sin�

@V1@y

=�e1 sin�� e2 cos�

@V1@�

=� sin e3K2

:

Moreover, _� is defined by (46) and�c is given as (41).

D. Simulation Results

In this section, we perform a computer simulation on the dynamicmodel of a mobile robot by using the adaptive tracking controller whichwas designed in the previous section. In this simulation, physical pa-rameters and design parameters area = 2, b = 0:75, d = 0:3, r =0:15, mc = 30, mw = 1, Ic = 15:625, Iw = 0:005, Im = 0:0025,K1 = K2 = K3 = kd = 5, = � = 5. The initial values of theestimated parameters are about 1/10—four times the real values.

The reference inputsvr, wr are chosen as following:

0 � t < 5: vr =0:25 1� cos�t

5

wr =0

5 � t < 20: vr =0:5

wr =0

20 � t < 25: vr =0:25 1 + cos�t

5

wr =0

25 � t < 30: vr =0:15� 1� cos2�t

5

wr =�vr=1:530 � t < 35: vr =0:15� 1� cos

2�t

5

wr = vr=1:5

35 � t < 40: vr =0:25 1 + cos�t

5

wr =0

40 � t: vr =0:5

wr =0:

Fig. 3 shows the reference inputs.

614 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 5, OCTOBER 2000

Fig. 4. Simulation resultx � y.

Fig. 5. Tracking errorse , e , e .

Fig. 6. Errors between ideal and real value:~� , ~� .

The simulation results are shown in Figs. 4–8. From these simulationresults, we can confirm the usefulness of Theorem III.1 and the limi-tation of Theorem IV.1. The control performance is good whenwr isclose to zero, but the performance becomes bad aswr is far from zero.

V. CONCLUSION

In this paper, we proposed a design method of an adaptive trackingcontroller for a nonholonomic mobile robot with unknown parame-ters. It was proved that an adaptive tracking controller for the dynamicmodel can be designed by using adaptive backstepping if an adaptivetracking controller for the kinematic model exists. As one example,

Fig. 7. Estimated parametersa , a .

Fig. 8. Estimated parametersp , p , p .

we designed an adaptive controller of a mobile robot with two actu-ated wheels and provided some simulation results. In future works, theclass of systems which satisfy Assumption II.1 should be clarified andthe design method of an adaptive tracking controller for the kinematicmodel written in Assumption III.1 should be established.

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[2] C. Samson and K. Ait-Abderrahim, “Feedback control of a nonholo-nomic wheeled cart in cartesian space,” inProc. IEEE Int. Conf.Robotics and Automation, 1991, pp. 1136–1141.

[3] Y. Nakamura and S. Savant, “Nonholonomic motion control of an au-tonomous underwater vehicle,” inProc. IEEE/RSJ Int. Workshop Intel-ligent Robots and Systems, 1991, pp. 1254–1259.

[4] M. Sampei, T. Tamura, T. Itoh, and M. Nakamichi, “Path tracking controlof trailer-like mobile robot,” inProc. IEEE/RSJ Int. Workshop IntelligentRobots and Systems, 1991, pp. 193–198.

[5] R. Fierro and F. L. Lewis, “Control of a nonholonomic mobile robot:backstepping kinematics into dynamics,” inProc. 34th IEEE Conf. De-cision Control, 1995, pp. 3805–3810.

[6] M. Krstic, I. Kanellakopoulos, and P. Kokotovic,Nonlinear and Adap-tive Control Design. New York: Wiley, 1995.

[7] Y. Chang and B. Chen, “Adaptive tracking control design of nonholo-nomic mechanical systems,” inProc. 35th IEEE Conf. Decision Control,1996, pp. 4739–4744.

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[9] Y. Kanayama, Y. Kimura, F. Miyazaki, and T. Noguchi, “A stabletracking control method for an autonomous mobile robot,” inProc.IEEE Int. Conf. Robotics and Automation, 1990, pp. 384–389.

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New Potential Functions for Mobile Robot Path Planning

S. S. Ge and Y. J. Cui

Abstract—This paper first describes the problem of goals nonreachablewith obstacles nearby when using potential field methods for mobile robotpath planning. Then, new repulsive potential functions are presented bytaking the relative distance between the robot and the goal into considera-tion, which ensures that the goal position is the global minimum of the totalpotential.

Index Terms—GNRON problem, new repulsive potential function, po-tential field.

I. INTRODUCTION

The potential field method has been studied extensively for au-tonomous mobile robot path planning in the past decade [1]–[16].The basic concept of the potential field method is to fill the robot’sworkspace with an artificial potential field in which the robot isattracted to its goal position and is repulsed away from the obstacles[1]. This method is particularly attractive because of its mathematicalelegance and simplicity. However, it has some inherent limitations. Asystematic criticism of the inherent problems based on mathematicalanalysis was presented in [3], which includes the following: 1) trapsituations due to local minima; 2) no passage between closelyspaced obstacles; 3) oscillations in the presence of obstacles; and 4)oscillations in narrow passages. Besides the four problems mentionedabove, there exists an additional problem, goals nonreachable with

Manuscript received August 31, 1999; revised June 15, 2000. This paperwas recommended for publication by Associate Editor J. Ponce and EditorV. Lumelsky upon evaluation of the reviewers’ comments. This paper waspresented in part at the Third Asian Control Conference, Shanghai, China, July4–7, 2000.

The authors are with the Department of Electrical and Computer En-gineering, National University of Singapore, Singapore 117576 (e-mail:elegesz@nus.edu.sg).

Publisher Item Identifier S 1042-296X(00)09775-5.

obstacles nearby (GNRON). In most of the previous studies, the goalposition is set relatively far away from obstacles. In these cases, whenthe robot is near its goal position, the repulsive force due to obstaclesis negligible, and the robot will be attracted to the goal position by theattractive force. However, in many real-life implementations, the goalposition needs to be quite close to an obstacle. In such cases, when therobot approaches its goal, it also approaches the obstacle nearby. Ifthe attractive and repulsive potentials are defined as commonly used[2]–[4], the repulsive force will be much larger than the attractiveforce, and the goal position is not the global minimum of the totalpotential. Therefore, the robot cannot reach its goal due to the obstaclenearby.

To overcome this problem, the repulsive potential functions for pathplanning are modified by taking into account the relative distance be-tween the robot and the goal. The new repulsive potential function en-sures that the total potential has a global minimum at the goal position.Therefore, the robot will reach the goal finally. Note that we are nottrying to tackle the common local minima problems due to obstaclesbetween the robot and the goal. We shall restrict our attention to theformulation and solution of the GNRON problem only.

This paper is organized as follows. In Section II, the cause of theGNRON problem is analyzed after the introduction of the potentialfield methods. Section III presents the new repulsive potential func-tion and its properties. In Section IV, the relationship between scalingparameters of the potential functions is presented. In Section V, safetyissues of the new potential functions are discussed, and a control systemdirectly making use of the new potentials is also suggested. Simulationresults are presented in Section VI to show the problems of the conven-tional potential field method and the effectiveness of the new method.

II. POTENTIAL FIELD METHOD AND GNRON PROBLEM

For simplicity, we assume that the robot is of point mass and movesin a two-dimensional (2-D) workspace. Its position in the workspace isdenoted byq = [ x y ]T .

Different potential functions have been proposed in the literature.The most commonly used attractive potential takes the form [1]–[3]

Uatt(q) =1

2��m(q; qgoal) (1)

where� is a positive scaling factor,�(q; qgoal) = kqgoal � qk is thedistance between the robotq and the goalqgoal, andm = 1 or 2.Form = 1, the attractive potential is conic in shape and the resultingattractive force has constant amplitude except at the goal, whereUatt issingular. Form = 2, the attractive potential is parabolic in shape. Thecorresponding attractive force is then given by the negative gradient ofthe attractive potential

Fatt(q) = �rUatt(q) = �(qgoal � q) (2)

which converges linearly toward zero as the robot approaches the goal.One commonly used repulsive potential function takes the followingform [1]:

Urep(q) =1

2�

1

�(q; qobs)�

1

�0

2

; if �(q; qobs) � �0

0; if �(q; qobs) > �0

(3)

where� is a positive scaling factor,�(q; qobs) denotes the minimaldistance from the robotq to the obstacle,qobs denotes the point on the

1042 296X/00$10.00 © 2000 IEEE