ARTICLE IN PRESS

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doi:10.1016/j.ne

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Neurocomputing 69 (2006) 2140–2151

www.elsevier.com/locate/neucom

Simple neuron-based adaptive controller for a nonholonomicmobile robot including actuator dynamics

Tamoghna Das, I.N. Kar�, S. Chaudhury

Department of Electrical Engineering, Indian Institute of Technology, Delhi, New Delhi 110016, India

Received 19 February 2005; received in revised form 7 September 2005; accepted 16 September 2005

Available online 15 February 2006

Abstract

In this paper, a simple neuron-based adaptive controller for trajectory tracking is developed for nonholonomic mobile robots without

velocity measurements. The controller is based on structural knowledge of the dynamics of the robot and the odometric calculation of

robot position only. The wheel actuator dynamics is also taken into account. An approximation network approximates a nonlinear

function involving robot dynamic parameters so that no knowledge of those parameters is required. The proposed controller is robust

not only to structured uncertainty such as mass variation but also to unstructured one such as disturbances. The real-time control of

mobile robot is achieved through the online learning of the approximation network. The system stability and the boundness of tracking

errors are proved using Lyapunov stability theory. Computer simulations with circular as well as square reference trajectories are

presented that confirm the simplicity and effectiveness of the proposed tracking control law. The efficacy of the proposed control law is

tested experimentally on a differentially driven mobile robot. Both simulation and experimental results are described in detail.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Adaptive control; Actuators; Mobile robots; Stability; Tracking; Uncertainty

1. Introduction

Wheeled mobile robots (WMRs) have been usedextensively in various industrial and service applications.The applications include security, transportation, inspec-tion and planetary exploration etc. These applicationsdemand WMRs with various mobility configurations(wheel number and type, their location and actuation,single or multibody vehicle structure). A detailed analyticalstudy of the structure of the kinematic and dynamic modelsof WMRs can be found in [4]. WMRs constitute a class ofmechanical systems called nonholonomic mechanicalsystems [13,3] characterized by kinematic constraints thatare not integrable and cannot therefore be eliminated fromthe model equations. Nonholonomic behavior in roboticsystems is particularly interesting, because it implies thatthe mechanism can be completely controlled with reducednumber of actuators. The tools for analyzing and control-

e front matter r 2006 Elsevier B.V. All rights reserved.

ucom.2005.09.013

ing author. Tel.: +9111 26591093; fax: +91 11 26581264.

ess: ink@ee.iitd.ernet.in (I.N. Kar).

ling nonholonomic mechanical systems based on knownmathematical model are presented in [5]. Using Lagrangeformalism and differential geometry, a general dynamicalmodel can be derived for mobile robots with nonholonomicconstraints [1,5].In the trajectory tracking problem, the WMR is to

follow a prespecified trajectory. Using the kinematic modelof WMRs the trajectory tracking problem was solved in[11]. Both the local and global tracking problem withexponential convergence have been solved theoreticallyusing time varying state feedback based on the back-stepping technique by Jiang et al. [15]. Using a kinematicmodel of mobile robots in polar coordinates, Weiguo et al.[18] have developed a stabilization scheme using back-stepping technique. Dynamic feedback linearization hasbeen used in [14] for trajectory tracking and posturestabilization of mobile robot system in chained form.The trajectory tracking algorithm presented in the

above-mentioned papers share a common idea of definingvelocity control inputs which stabilize the closed loopsystem. All these controllers consider only the kinematic

ARTICLE IN PRESST. Das et al. / Neurocomputing 69 (2006) 2140–2151 2141

model (e.g. steering system) of the mobile robot, and‘perfect velocity tracking’ is assumed to generate the actualvelocity control inputs. From real-time point of view, thenonholonomic kinematic controller is to be integrated withdynamics of mobile robot as ‘perfect velocity tracking’assumption may not hold good in practice [7]. In Ref. [6]Fierro and Lewis presented a dynamical extension thatmakes possible the integration of a kinematic controllerand a torque controller. First, feedback velocity controlinputs are calculated for the kinematic steering system tomake the position error asymptotically stable. Then acomputed torque controller is designed such that themobile robot’s velocities converge asymptotically to thecalculated velocity inputs. Various nonlinear controltechniques have also been used by many researchersconsidering the system disturbances and unknown dynamicparameters. Sliding mode motion control technique [19],robust adaptive control technique [12], adaptive controltechnique [8] have been used to solve the tracking controlproblem for WMRs. In [16] adaptive controls are derivedfor mobile robots, using backstepping technique, for thetracking of a reference trajectory and stabilization to afixed posture.

Neural networks are considered as an effective tool fornonlinear controller design. Neural networks offer excitingadvantages such as adaptive learning, fault tolerance andgeneralization. Fierro and Lewis [7] developed an artificialneural network-based controller by combining the feed-back velocity control technique and torque controller,using a multi-layer feed forward neural network. But thecontroller structure and the neural network-learningalgorithm are very complicated and it is computationallyexpensive. In [10] and [9] single layer neural network-basedcontroller is proposed for real-time fine control of mobilerobots. The above approaches can learn the dynamics ofthe mobile robot by its on-line learning algorithm. But theeffectiveness of these approaches is shown only bysimulation results without any experimental verification.

In most of the research works for controlling WMRs,wheel torques are control inputs though in reality wheelsare driven by actuators (e.g. DC motors) and thereforeusing actuator input voltages as control inputs is morerealistic. To this effect, actuator dynamics is combined withthe mobile robot dynamics. The contribution of the presentpaper is the utilization of a simple approximation networkfor estimating the nonlinear robot functions involvingunknown robot parameters. In most robust controlalgorithms, it is assumed that wheel velocities are available.However, it may not be always possible to measurevelocities, or it may not even be desirable to do it. Froma practical viewpoint, the velocity measurements, suppliedby the tachometers, are easily perturbed by noises and arenot directly available for the design of controller. Alsocosts are increased when more sensors are used. However,if velocity measurements are not available, they arereplaced by numerical derivatives of the measured position.That may lead to chattering of the control inputs due to

combined effect of noisy measurements and unmodelledphenomena, such as friction and elasticity. To avoid thisundesirable behavior, we have designed a control algorithmwith only wheel angular position as the required input.Also, in case of faulty behavior of real velocity sensorsattached to the wheels, a fault tolerant control of themobile robot can be achieved with the control algorithmdesigned here. The novelty of the approach presented herecan be understood by considering the following points: (i)the wheel actuator (e.g. DC motor) dynamics is integratedwith mobile robot dynamics and kinematics so that theactuator input voltages are the control inputs thus makingthe system more realistic, (ii) the control structure assumesno wheel velocity measurements, and (iii) an approxima-tion network approximates a nonlinear function of robotparameters so that no knowledge of those parameters isrequired. The proposed controller is robust to structureduncertainty as well as bounded unknown disturbancesincluding unstructured unmodelled dynamics. Also, thecontrol law proposed here is less computationally expen-sive than the MLP-based control algorithm presented in[7]. Simulation results are to this effect. The convergence oferror signals and other closed loop signals is shown to bebounded using Lyapunov Stability Theory [17]. Simulationand experimental results are described in detail that showsthe effectiveness of the proposed control law. Detailedprocedure of how to implement the control law in real timehas been demonstrated.The paper is organized as follows. Section 2 presents

the mobile robot model as well as the actuator dynamicmodel and its main properties. Tracking problem definitionand controller design details are described in Section 3.Section 4 presents the stability analysis of the proposedcontroller. Simulation and experimental results of theproposed controller with a mobile robot system are givenin Sections 5 and 6, respectively, followed by conclusion inSection 7.

2. Preliminaries

2.1. Model of a nonholonomic mobile robot

The mobile robot shown in Fig. 1 consists of a vehiclewith two driving wheels mounted on the same axis and apassive self-adjusted supporting wheel, which carries themechanical structure. The two driving wheels are indepen-dently driven two actuators (e.g. DC motors) for themotion and orientation. Both wheels have the same radiusdenoted by r. Two driving wheels are separated by 2R. It isassumed that the mobile robot under study is made up of arigid frame equipped with nondeformable wheels and thatthey are moving in a horizontal plane. Center of mass ofthe mobile robot is located at C; point P is located in theintersection of straight line passing through the middle ofthe vehicle and a section which is an axis of the two wheels.Distance between points P and C is denoted by d. The poseof the robot in the global coordinate frame fO;X ;Y g is

ARTICLE IN PRESS

Y

O 2r

CDrivingwheels

d

X C

Yc

2r

2R

C

Passive self adjusted supportingWheel

d

X

Fig. 1. Nonholonomic mobile robot.

T. Das et al. / Neurocomputing 69 (2006) 2140–21512142

completely specified by the vector P ¼ ½xc yc y�T, wherexc; yc are the coordinates of the point C in the globalcoordinate frame and y is the orientation of the local framefC;X c;Y cg attached on the robot platform measured fromX-axis.

Three generalized coordinates can describe the config-uration of the robot

q ¼ ½xc yc y�T. (1)

Generally, a nonholonomic mobile robot system having ann-dimensional configuration space with n generalizedconfiguration variables ðq1; . . . ; qnÞ and subject to m

constraints can be described by [7]

MðqÞ€qþ Vmðq; _qÞ_qþ sd ¼ BðqÞs� AðqÞk, (2)

where MðqÞ 2 Rnxn is a symmetric positive definite inertiamatrix, Vmðq; _qÞ 2 Rnxn is the centripetal and coriolismatrix, sd 2 Rn�1 denotes bounded unknown disturbancesincluding unstructured unmodelled dynamics, BðqÞ 2 Rn�r

is the input transformation matrix, s 2 Rr�1 is the inputvector, AðqÞ 2 Rn�m is the matrix associated with theconstraints and k 2 Rm�1 is the vector of constraint forces.Several properties of the dynamic equation (Eq. (2)) can beobtained as follows [6]:

P1. The inertia matrix MðqÞ is symmetric and positivedefinite.

P2. Skew symmetry: The matrix ð _M � 2VmÞ is skewsymmetric resulting in the following characteristic, xTð _M �2VmÞx ¼ 0 for all x 2 Rn.

All kinematic constraints are independent of time, andcan be expressed as

ATðqÞ_q ¼ 0. (3)

Let, SðqÞ be a full rank matrix ðn�mÞ formed by a set ofsmooth and linearly independent vector fields spanning thenull space of AT

ðqÞ i.e.

STðqÞAðqÞ ¼ 0. (4)

From (3) and (4), it is possible to find an auxiliary velocityvector time function vðtÞ 2 Rn�k such that for all t [7,5]

_q ¼ SðqÞvðtÞ. (5)

2.2. Kinematics and dynamics of a mobile platform

For the mobile robot system considered here, the purerolling and nonslipping nonholonomic condition statesthat the robot can only move in the direction normal to theaxis of the driving wheels [20,13]. Therefore, the compo-nent of the velocity of the contact point with the ground,orthogonal to the plane of the wheel is zero [2], i.e.

_yc cos y� _xc sin y� d_y ¼ 0. (6)

Thus the constraint matrix in (3) for a mobile robot is givenby

ATðqÞ ¼ ½� sin y cos y � d�. (7)

Therefore, matrix SðqÞ is given by

SðqÞ ¼

cos y �d sin y

sin y d cos y

0 1

264

375. (8)

Therefore, the kinematic equation in (5) can be describedas

_q ¼

_xc

_yc

_y

264

375 ¼

cos y �d sin y

sin y d cos y

0 1

264

375 v1

o

� �, (9)

where v ¼ ½v1 o�T; v1 and o are the linear velocity of thepoint P along the robot axis and angular velocity,respectively. System (9) is called the steering system ofthe vehicle.The Euler–Lagrange equations of motion is used to

derive the dynamics of the mobile robot. The dynamicalequations of the mobile robot can be expressed as (2),where

MðqÞ ¼

m 0 md sin y

0 m �md cos y

md sin y �md cos y I

2664

3775,

Vmðq; _qÞ ¼

0 0 md _y cos y

0 0 md _y sin y

0 0 0

2664

3775,

BðqÞ ¼1

r

cos y sin y

sin y sin y

R �R

2664

3775; s ¼

tr

tl

" #, ð10Þ

where tr and tl represents right and left wheel torques,respectively.

Remark 1. In this dynamic model of mobile robot, thepassive self-adjusted supporting wheel influence is not

ARTICLE IN PRESST. Das et al. / Neurocomputing 69 (2006) 2140–2151 2143

taken into consideration, as it is a free wheel. Thissignificantly reduces the complexity of the model for thefeedback controller design. However, the free wheel may bea source of substantial distortion, particularly in the case ofchanges of movement direction. This effect is reduced if weconsider small velocity of the robot.

+

-

u

Ra

+

-

emfe JL

Jm

r1

r2

, ,m m m

, ,L L L

�� �

�� �

Fig. 2. Simplified circuit model for a DC motor.

2.3. Structural properties of mobile platform

The system (2) can be transformed into a moreappropriate representation for control purposes. Using(5) and multiplying both sides of Eq. (2) by ST, thecomplete equations of motion of the nonholonomic mobileplatform are given by

_q ¼ SðqÞvðtÞ, (11)

M _vþ Vmvþ sd ¼ Bs, (12)

where M ¼ STMS, Vm ¼ STðM _S þ VmSÞ, sd ¼ STsd,

B ¼ STB.

Remark 2. Eq. (12) describes the behavior of the non-holonomic system in a new set of local coordinates, i.e.SðqÞ is a Jacobian matrix that transforms velocities v inmobile base coordinates to velocities in Cartesian coordi-nates _q. Therefore, the properties of the original dynamicshold for the new set of coordinates [7,10].

P3. The matrix MðqÞ is uniformly positive definite.P4. Skew symmetry: The matrix ð _M � 2VmÞ is skew

symmetric.P5. The disturbances are bounded so that ksdkpdB.Using (10), the matrices in Eq. (12) for the mobile robot

can be described as

M ¼m 0

0 I �md2

" #; B ¼

1

r

1

r

R

r�

R

r

2664

3775,

Vm ¼0 0

0 0

" #. ð13Þ

The nonlinear robot dynamics in (12) can be written in alinear combiner form as

M _vþ Vmv ¼ YU, (14)

where U is a vector consisting of the known and unknownrobot dynamic parameters, such as geometric size, mass,moment of inertia etc.; and Y is a coefficient matrixconsisting of the known functions of the robot position,velocity and acceleration, which is referred as robotregressor. For the mobile robot system consideredhere, the robot regressor Y and the coefficient vector U

are given as

Y ¼_v1 0 0

0 _o � _o

� �; U ¼ ½m I md2

�T. (15)

2.4. Actuator dynamics

The wheels of the robot are assumed to be driven by DCmotors.Consider the following notations used to model a DC

motor (see Fig. 2):u is the input voltage, i the motor current, Ra the motor

resistance, eemf the back emf due to the spinning motorshaft, N ¼ r2=r1 the gear ratio, tm and tL are the torqueexerted by the motor and the torque exerted on the load,respectively, om and oL are the angular velocities at themotor (before gears) and the load (after gears), respec-tively, ym and yL are the angles of the motor (before gears)and the load (after gears). The relationship between tm andtL is given by

tL ¼ Ntm, (16)

whereas the relationship between om and oL is given by

oL ¼1

Nom. (17)

The torque exerted by the motor is proportional to themotor current:

tm ¼ KTi, (18)

where KT is the torque constant.The back-emf is proportional to the angular velocity of

the motor shaft:

eemf ¼ Kbom, (19)

where Kb is the velocity constant.Applying Kirchoff voltage law to the circuit shown in

Fig. 2 we get

u ¼ iRa þ eemf . (20)

Using Eqs. (16)–(20), yields,

tL ¼NKT

Rau�

N2KT

RaKboL. (21)

ARTICLE IN PRESST. Das et al. / Neurocomputing 69 (2006) 2140–21512144

If xH ¼ ½or ol �T denotes the wheel velocity vector, where

or is the right wheel angular velocity and ol is the leftwheel angular velocity, then xH is related to the vector v asfollows

xH ¼

1

r

R

r1

r�

R

r

2664

3775 v1

o

� �¼ Xv. (22)

Using Eqs. (12) and (21), the mobile robot dynamics can bewritten as

M _vþ Vmvþ sd ¼NKT

RaBu�

N2KTKb

RaBXv. (23)

3. Control problem formulation and controller design

The tracking problem can be stated as follows:The trajectory tracking problem can be stated as follows:

Given a reference trajectory qrðtÞ ¼ ½xrðtÞ yrðtÞ yrðtÞ�T

generated by a reference vehicle/robot whose equations ofmotion are

_xr ¼ vr cos yr,

_yr ¼ vr sin yr,_yr ¼ or; vr ¼ ½vr or�

T, ð24Þ

with vr40 for all t, determine a smooth velocity controllaw vc ¼ f cðEP; vr;KÞ for (11) such that limt!1 qðtÞ ¼ qrðtÞ

where EP; vr and K are the tracking position error, thereference velocity and control gain, respectively. Then findthe actuator voltage input uðtÞ for (23), that can becomputed without measuring velocities such that vðtÞ !vcðtÞ as t!1.

3.1. Controller design

Many approaches exist in the literature to find thevelocity control vðtÞ for steering system (11). This controlinput is denoted as vcðtÞ. Then either ‘‘perfect velocitytracking’’ is assumed so that v ¼ vc, or a suitable sðtÞ for (2)is derived from a specific vcðtÞ that controls the steeringsystem. The approach assuming ‘‘perfect velocity tracking’’is unrealistic [7]. Also, keeping in mind the fact that thewheel torques are generated by actuators, we should findactuator input voltage u instead of sðtÞ, so that vðtÞconverge asymptotically to vcðtÞ. The contribution of thispaper lies in deriving a suitable uðtÞ, that does not requiremeasurement of vðtÞ and knowledge of dynamic parametersof robot, from a specific vcðtÞ that controls the steeringsystem (11).

First, smooth velocity control inputs vcðtÞ are calculatedfor the kinematic steering system (11) to make the positionerror asymptotically stable. Second, a suitable uðtÞ isderived to achieve that velocity control input such thatthe mobile robot’s velocities converge asymptotically to thecalculated velocity inputs.

The tracking error is expressed relative to the localcoordinate frame fixed on the mobile robot as Ep ¼

Tðqr � qÞ or

Ep ¼

e1

e2

e3

264

375 ¼

cos y sin y 0

� sin y cos y 0

0 0 1

264

375

xr � x

yr � y

yr � y

264

375. (25)

An auxiliary velocity control input that achieves trackingfor kinematic model (11), i.e. with which Ep ¼ 0 isuniformly asymptotically stable under the assumptionvr40 is given by [11]

vc ¼vc

oc

" #¼

vr cos e3 þ k1e1

or þ k2vre2 þ k3vr sin e3

" #, (26)

where k1; k2; k340 are design parameters. If only thekinematic model of the mobile robot (11) with velocityinput (26) is considered and assuming ‘‘perfect velocitytracking’’, then the kinematic model is asymptoticallystable with respect to the reference trajectory (i.e. Ep! 0as t!1) [11].To design the actuator voltage input and generate the

desired velocities vc, the auxiliary velocity tracking error isdefined as

ec ¼ vc � v. (27)

Differentiating (27) and using the result in (23), the robotdynamics using the velocity tracking error can be rewrittenas

M _ec ¼ � Vmec �N2KTKb

RaBXec þ ðM _vc þ VmvcÞ þ sd

þN2KTKb

RaBXvc �

NKT

RaBu. ð28Þ

The function in braces in Eq. (28) can be approximated byan approximation network, such that

M _vc þ Vmvc ¼ Yð_vc; vcÞU. (29)

Remark 3. To calculate _vc, the values of v1 and o arerequired, as is from the derivative of Eq. (25), vc willreplace v ¼ ½v1 o�T in this calculation.

3.2. Proposed control law

To make the error dynamics stable the input term in (28)is chosen as

NKT

RaBu ¼ Yð_vc; vcÞUþ

N2KTKb

RaBXvc þ K v

tanhðdc1Þ

tanhðdc2Þ

" #.

ð30Þ

So that, a single neuron model-based robot controller isproposed as

u ¼Ra

NKTB�1

Yð_vc; vcÞUþN2KTKb

RaBXvc þ Kv

tanhðdc1Þ

tanhðdc2Þ

" # !.

ð31Þ

ARTICLE IN PRESST. Das et al. / Neurocomputing 69 (2006) 2140–2151 2145

In the above equation U is a 3� 1 vector, whichapproximates the vector U. The term Yð_vc; vcÞU representsa linear combiner approximation network; the input to thenetwork is regressor Y . The variables for computing Y arethe vector ½_vTc vTc �

T, which can be obtained from (24). Also,½c1 c2�

T is a high-filtered version of ec ¼ vc � v synthesizedby

_W ¼ �aKWþ Kec, (32)

with WT ¼ ½c1 c2�T. K v ¼ diagfkvg; D ¼ diagfdg; K ¼

diagfkg and kv, d, k are positive scalar constants. The term

K v

tanhðdc1Þ

tanhðdc2Þ

" #

represents the output of a simple neuron controller asshown in Fig. 3, c1 and c2 are the inputs to two simpleneuron controllers with single neurons which processes theinput signals through a nonlinear activation function. Theabove used function is popular Tanh activation function.

Substituting (30) into (28), the closed loop errordynamics can be expressed as

M _ec ¼ � Vmec �N2KTKb

RaBXec � Kv

tanhðdc1Þ

tanhðdc2Þ

" #

þ Y ~Uþ sd, ð33Þ

where

~U ¼ Un � U (34)

is the estimation error of U.

Tanh Activation function

Output ��

vK

Fig. 3. Simple neuron controller.

T

Updation

of �

Reference cart

r

r r

r

x

q y

=

rv

x

q y

θ

θ

=

PE

Kinematic

Controller

cv

2T b

a

N K K

RBX

cv

_+

S

A

d

dt

Fig. 4. Block diagram of the

Remark 4. Eq. (32) is solved (for a sampling interval) toget the updation equation of W, given by

W ¼

c1ð0Þe�akt þ

1

aðe�aktÞðvr cos e3 þ k1e1Þðe

akt � 1Þ

c2ð0Þe�akt þ

1

aðe�aktÞðor þ k2vre2 þ k3vr sin e3Þðe

akt � 1Þ

2664

3775.(35)

To implement the control law given by Eq. (31), only thevalues of c1 and c2 are required, and these are calculatedfrom Eq. (35), without calculating the value of ec, thusavoiding the use of v. Thus wheel velocity information isnot required to implement this control law.The block diagram of the controller structure is given in

Fig. 4. Using appropriate parameter tuning laws forparameter vector U as given in (29) in Proposition 1, theclosed loop stability analysis can be done.

4. Learning algorithm and stability analysis

In this section the closed loop stability of the system isproved in the form of a proposition.

Proposition 1. Given the Nonholonomic system (9), (23), let,the feedback control input u given by (30) be used. Then the

closed loop error dynamics given by Eq. (33), (34) is locally

asymptotically stable with the approximation network

parameter tuning laws given by

_U ¼ CYTec � kCkeckU,

where C is a positive constant design matrix, for simplicity

we assume C ¼ diagfGg, and k is a small positive constant.

Proof. Consider the Lyapunov function candidate

V ¼ V 1 þ V 2,

cv.

q

+

+

uimple neuron

Controller

pproximation

Network

-1a

T

R

NKB

Mobile robot

System

mobile robot controller.

ARTICLE IN PRESST. Das et al. / Neurocomputing 69 (2006) 2140–21512146

where

V1 ¼1

2ðe21 þ e22Þ þ

1

k2ð1� cos e3Þ½1�, (36)

V2 ¼1

2ðeTc MecÞ þ

1

2~UTC�1 ~Uþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLN coshðdc1Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLN coshðdc2Þ

p24

35T

�KvK�1D�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLN coshðdc1Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLN coshðdc2Þ

p24

35. ð37Þ

Obviously VX0.Differentiating (36), we get

_V ¼ _V1 þ _V 2.

Now,

_V1 ¼ �k1e21 �

k3

k2vr sin

2 e3. (38)

Since the reference linear velocity vr40, therefore _V 1p0.Differentiating (37) and substituting the error dynamics

from Eq. (33), we get

_V2 ¼1

2eTc ð

_M � 2VmÞec �N2KTKb

RaeTc BXec

þ ~UTðYTec þ C�1

_~UÞ � eTc K v

tanhðdc1Þ

tanhðdc2Þ

" #

þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLN coshðdc1Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLN coshðdc2Þ

p24

35T

K vK�1D�1

�

tanhðdc1Þd _c1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLN coshðdc1Þ

ptanhðdc2Þd _c2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLN coshðdc2Þ

p

2666664

3777775þ eTc sd. ð39Þ

Owing to the property ð _M � 2VmÞ is skew symmetric,

eTc ð_M � 2VmÞec ¼ 0.

And choosing, _~U ¼ �CYTec þ kCkeckU, Eq. (39) yields

_V2 ¼ �N2KTKb

RaeTc BXeþ k ~U

TkeckU

� aWTKv

tanhðdc1Þ

tanhðdc2Þ

" #þ eTc sd

¼ � V3 þ k ~UTkeckðU

n � ~UÞ þ eTc sd, ð40Þ

where

V3 ¼N2KTKb

RaeTc BXeþ aWTK v

tanhðdc1Þ

tanhðdc2Þ

" #.

Now since, ~UTðUn � ~UÞpk ~UkFðkFnkF � k

~FkFÞ with k � kFthe Frobenius norm, there results

_V2p� V3 þ keck½kc1 þ dB�; where c1 ¼ k ~UkFðkUnkF � k

~UkFÞ

¼ � V3 þ keckc2 where c2 ¼ kc1 þ dB.

By using the fact that x tanhðxÞ is positive definitefunction in x 2 R, and BX is positive definite matrix, as isevident from Eqs. (13), (22), we conclude that _V is negativesemidefinite function provided

keckpV3=c2.

This proves the local asymptotic stability of the system.The system has a spherical domain of attraction of radiusV3=c2. The domain of attaraction can be made large bychoosing a large value of kv and small value of k.Now since, ~U ¼ Un � U, then _~U ¼ � _U, therefore, the

learning law for the approximation network is obtained as

_U ¼ CYTec � kCkeckU. (41)

This completes the proof of the proposition. &

5. Simulation results

The proposed control law (31), (32) is first verified withcomputer simulation using MATLAB. For simulationpurposes, two types of reference trajectories have beenchosen. The first simulation is carried out with a circularreference trajectory with an initial error of 10 cm in theposition of the robot and the other simulation is carried outwith a square reference trajectory with an initial error of20 cm in the position of the robot.For the first simulation, the circular reference trajectory

is given by

xrðtÞ ¼ cost

20

� �; yrðtÞ ¼ sin

t

20

� �; yrðtÞ ¼

p2þ

t

20.

The reference velocity vector is vr ¼ ½0:05 0:05�T. Thereference trajectory starts from

qrð0Þ ¼ 1 0p2

h iTand the robot initial posture is taken as

qð0Þ ¼ ½xð0Þ yð0Þ yð0Þ�T ¼ 1:1 0p2

h iT.

The parameters of the controller are chosen as k1 ¼ 4;k2 ¼ 10; k3 ¼ 6; kv ¼ 15; C ¼ diagf100; 100; 100g; a ¼ 3:0;d ¼ 0:01; k ¼ 20; k ¼ 0:01. No prior information ofthe robot parameters such as mass, moment of inertiaetc. are required, the controller learns the system dynamicson-line. However, for simulation purposes the parametervalues are taken as m ¼ 9 kg; I ¼ 5 kgm2; 2R ¼ 0:306m;r ¼ 0:052m. These values are selected based on approx-imate measurements or calculations made from theexperimental mobile robot system. A disturbance termsd ¼ ½0:1 sinð2tÞ; 0:1 sinð2tÞ�T is added to the robot system

ARTICLE IN PRESS

-1-1

-0.5

0

0.5

Robot actual trajectory and reference trajectory

Actual trajectory

Desired trajectory

Position error plot of the robot

Error along abscissaError along ordinate

Linear and angular velocity profile of the robot

Linear velocity profileAngular velocity profile

1

1.5

-0.12

0.1

0.08

0.06

0.04

0.02

0

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

-0.5 0 0.5X (meters)

Y (

met

ers)

Pos

ition

al e

rror

s (m

eter

s)lin

ear

velo

city

(m

eter

s/s

ec)

angu

lar

velo

city

(ra

d/s

ec)

1 1.5

0 20 40 60time (sec)

time (sec)

80 100 120 140

0 20 40 60 80 100 120 140

(A)

(B)

(C)

Fig. 5. (A) Tracking performance. (B) Tracking errors. (C) Velocity

profile.

Table 1

Number of Proposed controller MLP-based controller

Weights 3 46

Sigmoidal — 5

Tanh 2 —

Iðxr � xÞ 0.0239 0.0376

Iðyr � yÞ 0.0099 0.0290

T. Das et al. / Neurocomputing 69 (2006) 2140–2151 2147

as unmodelled disturbances. To demonstrate the on-linelearning ability of the proposed controller the mass of themobile is taken as m ¼ 9 kg for tp60 s and m ¼ 10:5 kg fort460 s. The simulation results are shown in Figs. 5A–C.Fig. 5A shows the mobile robot tracks the virtual referencerobot moving on the circular path, and the tracking errorssharply drop to very small values as shown in Fig. 5B.Fig. 5C shows that the mobile robot linear and angularvelocities converge to the corresponding reference velo-cities. The proposed controller is capable of compensatingany sudden change of the robot dynamics because of itslearning ability as is evident from simulation results. Theproposed controller is robust not only to structureduncertainty such as mass variation but also to unstructuredone such as disturbances.The performance of the proposed controller is compared

to that of the multilayer perceptron (MLP) controllerproposed by Lewis [7] for a circular reference trajectory,same as above. The controller parameters and initialconditions are taken same in both the cases. Thedisturbance vector is taken as sd ¼ ½0:1 sinð2tÞ;0:1 sinð2tÞ�T. The MLP controller has a single hidden layerwith 5 neurons in the hidden layer as given in [7]. For thiscontroller the measurements of both position and velocityare required. To get an objective idea of the performance of

the system, the index Ið�Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=NÞ

PN1 k � k

2q

is used,

where N is the total number of samples. Table 1 gives acomparison of the computational complexity and theperformance of the proposed controller and the MLPcontroller [7]. It is observed that the number of tunableparameters of the proposed controller is less than theMLP-based NN controller.Simulation is also carried out with a square reference

trajectory. It is a trajectory with explicitly noncontinuousgradient. The reference trajectory starts from

qrð0Þ ¼ 3 0p2

h iTand the robot initial posture is

qð0Þ ¼ 3:2 0p2

h iT.

The controller parameters are taken same as in thesimulation with circular reference trajectory. Also, samemobile robot parameters and the disturbance are used inthis simulation. However the mass of the robot is notvaried. Figs. 6A and B show the simulation results. Fig. 6A

ARTICLE IN PRESS

Position plot with respect to reference trajectory

Position error plot with respect to reference trajectory

3.5

3

2.5

2

1.5

1

0.5

0

-0.5

0.2

0.15

0.1

0.05

0

-0.05

-0.1

-0.15

-0.2

-0.250 50 100 150

time (sec)

Error along abscissaError along ordinate

200 250

-0.5 0 0.5 1 1.5X (meters)

Y (

met

ers)

Pos

ition

err

ors

(met

ers)

Actual trajectoryDesired trajectory

2 2.5 3 3.5(A)

(B)

Fig. 6. (A) Tracking performance (square trajectory). (B) Tracking errors

(square trajectory).

Fig. 7. Mobile robot, Multimedia lab, IIT Delhi.

T. Das et al. / Neurocomputing 69 (2006) 2140–21512148

shows that the mobile robot tracks the square referencetrajectory quite accurately. Starting from the initial posture

qð0Þ ¼ 3:2 0p2

h iTthe robot tracks the right side of the square referencetrajectory and the tracking errors sharply drop to verysmall values as shown in Fig. 6B. Note that, along any oneside of the square, the reference orientation angle isconstant. At the end of one side of the square, thereference orientation angle changes suddenly. Conse-quently the mobile robot takes a slow turn (see Fig. 6A)at each corner of the square and ultimately tracks anotherside of the square path. The sudden increase in the positionerrors of the robot against the reference trajectory at thecorners of the square are shown in Fig. 6B.

The simulation results of tracking a circular and a squarereference trajectory by the mobile robot demonstrates that,with the proposed control algorithm the robot is capable oftracking any trajectories with continuous or noncontin-uous gradients.

6. Implementation results

The proposed control algorithm is implemented on atwo-wheel differentially driven mobile robot system asshown in Fig. 7. The wheels have a radius of r ¼ 5:2 cm andare mounted on an axle of length 2R ¼ 30:6 cm. The wheelradius includes the o-ring used to prevent slippage; therubber is stiff enough that point contact with the groundcan be assumed. A small passive self adjusted supportingwheel is placed in the front of the vehicle to carry themechanical structure. The wheels are driven by DC motorshaving rated torque of 20mNm at 3000 r.p.m. and at 24Vrated voltage. Each motor is equipped with incrementalencoder counting 600 pulses/turn and a gear and beltdriven assembly, which reduces the speed by a factor ofN ¼ 147.The mobile robot described here is a low-cost prototype

and presents, therefore, the typical nonidealities ofelectromechanical systems, namely, gear backlash, wheelslippage, actuator dead zone, and saturation. Theselimitations clearly affect the control performance. In viewof the bounded velocity of the wheel actuators, each wheelcan achieve a maximum angular velocity. That’s why toprevent wheel slippage as much as possible, the referencelinear and angular velocities are always limited by jvrjmax ¼

0:05m=s and jorjmax ¼ 0:05 rad=s, respectively. Also, thevehicle is supplied with very thin wheels making trajectorytracking in real time easier. However, for potentialapplication areas such wheels cannot be used. This is alimitation of our experimental mobile robot.

6.1. Control system architecture

As shown in Fig. 8 it has a two level control architecture.High level control algorithms (including reference motiongeneration) are written in VC++ and run with a samplingtime of 16ms on a remote server (a Pentium III processor).The PC communicates through serial port with the

ARTICLE IN PRESS

Control

Algorithm running on PC

Wheel

motor

Wheel

motor

L R�

Right wheel Left wheel

ULUR

Power

Electronics

Power

Electronics

Encod

∆ �∆

er Encoder

TxRx

PIC Microcontroller

Serial Duplex communication

Fig. 8. Control architecture of the mobile robot.

Robot actual position and reference trajectory1.5

1

Actual robot trajectoryReference trajectory

0.5

0

-0.5

-1-1.5 -1

-0.06

-0.04

-0.02

0

0.02

0.04

-0.5X (meters)

Error in position of the robot

Error along X coordinateError along Y coordinate

Y (

met

ers)

sitio

n er

ror

(met

ers)

0 0.5 1 1.5(A)

T. Das et al. / Neurocomputing 69 (2006) 2140–2151 2149

microcontroller on the robot. Wheel PWM duty cyclecommands are sent to the robot and encoder measures DfL

and DfR are received for odometric computation. Thelower level control layer is in charge of the execution of thehigh-level velocity commands. It consists of a PIC 16F877micro controller.

The microcontroller performs three basic tasks, (1) Tocommunicate with the higher-level controller through RS232. (2) To read the encoder counts and (3) to generate ofPWM duty cycle commands, UR and UL (see Fig. 8).

0-0.12

-0.1

-0.08

20 40 60 80 100 120 140 160 180time (sec)

Po

(B)

Fig. 9. (A) Real time tracking performance. (B) Tracking errors.

6.2. Experimental results

To demonstrate the feasibility and effectiveness of thecontrol law (31), (32), the control laws are applied to theexperimental mobile robot. The current robot configura-tion is reconstructed from odometric data. If DfR and DfL

be the wheel angular displacements measured duringsampling time T s by the encoders, the robot linearand angular displacements are reconstructed asDs ¼ r

2ðDfR þ DfLÞ, Dy ¼ r

2RðDfR � DfLÞ. The posture

estimated at time tk ¼ kT s is

qk ¼

xk

yk

yk

0B@

1CA ¼ qk�1 þ

cos yk 0

sin yk 0

0 1

0B@

1CA Ds

Dy

� �,

where yk ¼ yk�1 þ Dy=2. Robot localization using theabove odometric prediction (commonly referred to as dead

reckoning) is quite accurate in the absence of wheel slippageand backlash. These effects are largely reduced when thevelocity is kept reasonably small and the number of backmaneuvers is limited.

For experimental purpose, the reference trajectory ischosen, given by

xrðtÞ ¼ cost

20

� �; yrðtÞ ¼ sin

t

20

� �; yrðtÞ ¼

p2þ

t

20,

so that the reference velocities are vrðtÞ ¼ 0:05m=s andorðtÞ ¼ 0:05 rad=s. The reference trajectory starts from

qrð0Þ ¼ ½1 0 p2�T and the robot initial posture is

qð0Þ ¼ ½1:1 0 p2�T, so that the initial error in the position

of the robot with respect to the reference trajectory is0.1m. The controller parameters are chosen same as usedin the simulation, k1 ¼ 4; k2 ¼ 10; k3 ¼ 6; kv ¼ 15; C ¼

diagf100; 100; 100g; a ¼ 3:0; d ¼ 0:01; k ¼ 20; k ¼ 0:01.Figs. 9A and B show the results obtained by applying theproposed controller (31), (32). Fig. 9A shows the trajectoryfollowed by the mobile robot with respect to the referencetrajectory. The tracking errors drop to small values asshown in Fig. 9B. From these figures, it is clear that, thetracking of the reference trajectory is very accurate. Thecontroller learns the function in Eq. (29) online; so that noprior information of the physical parameters such as mass,moment of inertia is required. The initial value of the co-efficient vector U defined in Eq. (15) is taken asUð0Þ ¼ ½0 0 0�T. To verify the robustness of the proposedcontrol scheme against the parameter variation, anadditional mass of 2 kg is added deliberately with themobile robot after the robot starts tracking the reference

ARTICLE IN PRESS

0-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Error along abscissa

Position error plot (with extra mass added)

Error along ordinate0.02

0.04

20 40 60 80 100

time (sec)

erro

r (m

eter

s)

120 140 160 180

Fig. 10. Tracking errors (with some extra mass added).

T. Das et al. / Neurocomputing 69 (2006) 2140–21512150

trajectory. The corresponding position error plot is shownin Fig. 10. Comparison of Fig. 9B with Fig. 10 reveals thatas the extra mass is added at about t ¼ 95 s, there is a smallincrease in the position error of the robot against thereference trajectory. But owing to the online learningcapability of the approximation network, the controllerlearns the function in Eq. (29) with the extra mass. Thusthe tracking errors remain bounded and the robot tracksthe reference trajectory. The tracking of the referencetrajectories is reasonably accurate; the residual errors in theexperimental result, which is not present in the simulationis caused mainly due to the inherent friction present in thereal system, the neglected wheel dynamics and thenonsmoothness of robot path in addition to the encodertruncation and quantization error.

7. Conclusion

In this paper a controller is proposed for solving thetracking control problem of nonholonomic mobile robotwith robot dynamics and wheel actuator dynamics takeninto account. For the controller, the stability and errorboundness is proved using Lyapunov stability theory.Simulation and experimental results validate the proposedcontroller. There are several features worth pointing out.

�

A simple neuron-based adaptive controller for trajectorytracking of a wheeled mobile robot has been proposed. � The wheel actuator input voltage is chosen as thecontrol input for the torque controller in the proposedcontroller. This approach is more realistic than mosttorque controller where wheel torques are chosen as thecontrol input. � The proposed controller design does not require wheelvelocity measurement. � The proposed controller is robust to structured un-certainty as well as bounded unknown disturbancesincluding unstructured unmodelled dynamics.

�

Detailed procedure of how to implement the control lawin real time has been demonstrated. For real-timeconstraints the conventional torque control output isreplaced by actuator input voltage control output.

References

[1] B. d’Andrea-Novel, G. Bastin, G. Campion, Modelling and control

of nonholonomic wheeled mobile robots, in: Proceedings of IEEE

International Conference on Robotics and Automation, April 1991,

pp. 1130–1135.

[2] B. d’Andrea-Novel, G. Bastin, G. Campion, Dynamic feedback

linearization of nonholonomic wheeled mobile robots, in: Proceed-

ings of IEEE International Conference on Robotics and Automation,

May 1992, pp. 2527–2532.

[3] G. Campion, B. d’Andrea-Novel, G. Bastin, Controllability and state

feedback of nonholonomic mechanical systems, in: C. Canudes, De

Wit (Eds.), Lecture Notes in Control and Information Science,

Springer, Berlin, Germany, 1991, pp. 106–124.

[4] G. Campion, G. Bastin, B. D’AndreaNovel, Structural properties

and classification of kinematic and dynamic models of wheeled

mobile robots, IEEE Trans. Robotics Autom. 12 (1) (1996) 47–62.

[5] A. De Luca, G. Oriolo, Modelling and Control of Nonholonomic

Mechanical Systems, pp. 277–342 (Chapter 7).

[6] R. Fierro, F.L. Lewis, Control of a nonholonomic mobile robot:

backstepping kinematics into dynamics, in: Proceedings of the

34th Conference on Decision & Control, December 1995,

pp. 3805–3810.

[7] R. Fierro, F.L. Lewis, Control of a nonholonomic mobile robot

using neural networks, IEEE Trans. Neural Networks 9 (4) (1998)

589–600.

[8] T. Fukao, H. Nakagawa, N. Adachi, Adaptive tracking control of a

nonholonomic mobile robot, IEEE Trans. Robotics Autom. 16 (5)

(2000) 609–615.

[9] T. Hu, S.X. Yang, An efficient neural controller for a nonholonomic

mobile robot, in: Proceedings of IEEE International Symposium on

Computational Intelligence in Robotics and Automation, July

29–August 1, 2001, pp. 461–466.

[10] T. Hu, S.X. Yang, F. Wang, G.S. Mittal, A neural network controller

for a nonholonomic mobile robot with unknown robot parameters,

in: Proceedings of IEEE Internatioal Conference on Robotics and

Automation, May 2002, pp. 3540–3545.

[11] Y. Kanayama, Y. Kimura, F. Miyazaki, T. Noquchi, A stable

tracking control method for an autonomous mobile robot, in:

Proceedings of the IEEE International Conference on Robotics and

Automation, Cincinnati, OH, USA, May 1990, vol. 1, pp. 384–389.

[12] M.S. Kim, J.H. Shin, J.J. Lee, Design of a robust adaptive controller

for a mobile robot, in: Proceedings of IEEE/RSJ International

Conference on Intelligent Robots and Systems, 2000, pp. 1816–1826.

[13] I. Kolmanovsky, N.H. McClamroch, Developments in nonholo-

nomic control problems, IEEE Control Syst. (1995) 20–36.

[14] G. Oriolo, A. De Luca, M. Vendittelli, WMR control via dynamic

feedback linearization: design, implementation and experimental

validation, IEEE Trans. Control Syst. Technol. 10 (6) (2002)

835–852.

[15] Z. Ping, H. Nijmeijer, Tracking control of mobile robots: a case study

in backstepping, Automatica 33 (7) (1997) 1393–1399.

[16] F. Pourboghrat, M.P. Karlsson, Adaptive control of dynamic mobile

robots with nonholonomic constraints, Comput. Electr. Eng. (2002)

241–253.

[17] J.-J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall,

Englewood Cliffs, NJ, 1991.

[18] W.U. Weiguo, C. Hutang, W. Yuejuan, Backstepping design for

path tracking of mobile robots, in: Proceedings of IEEE/RSJ

International Conference on Intelligent Robotics and Systems,

1999, pp. 1822–1827.

ARTICLE IN PRESST. Das et al. / Neurocomputing 69 (2006) 2140–2151 2151

[19] J.-M. Yang, J.-H. Kim, Sliding mode motion control of nonholo-

nomic mobile robots, in: Proceedings of IEEE International

Conference on Robotics and Automation, May 1998, pp. 15–23.

[20] Q. Zhang, J. Shippen, B. Jones, Robust backstepping and neural

network control of a low quality nonholonomic mobile robot, Int. J.

Mach. Tools Manu. 39 (1999) 1117–1134.

Tamoghna Das received the B.E. degree in

electrical engineering from B.E. College (Deemed

University), W.B., India, in 2002, and the M.S.

(Research) degree from Indian Institute of

Technology, New Delhi, India in 2005.

His research interests include nonlinear control

of nonlinear systems with parametric uncertainty,

adaptive visual servo control of mobile robots

and robot manipulators, neural network and

fuzzy logic-based adaptive control of robotic

systems.

Indra Narayan Kar received the Bachelor of

Engineering (1988) degree from Bengal Engineer-

ing College (Deemed University), India, and the

M.Tech. (1991) and Ph.D. (1997) degrees from

Indian Institute of Technology, Kanpur, India,

all in Electrical Engineering.

He worked as a research student at Nihon

University, Tokyo under the Japanese govern-

ment Monbusho scholarship program from 1996

to 1998. He joined as a lecturer in the department

of Electrical Engineering, Indian Institute of

Technology, Delhi where he became assistant professor in the year 1999.

At present, he is working as an associate professor at the Indian Institute

of Technology, Delhi. His research interests include robust and intelligent

control systems, embedded systems, nonlinear control and identification.

He has published more than 40 papers in international journals and

conference proceedings.

Santanu Chaudhury did his B.Tech. (1984) in

Electronics and Electrical Communication En-

gineering and Ph.D. (1989) in Computer Science

and Engineering from I.I.T, Kharagpur, India.

Currently, he is a professor in the department of

Electrical Engineering at I.I.T, Delhi. He was

awarded INSA medal for young scientists in

1993. His research interests are in the areas of

Computer Vision, Artificial Intelligence and

Multimedia Systems. He has published more

than 100 papers in international journals and

conference proceedings.