Abstract— In this paper a nonholonomic mobile robot with

completely unknown dynamics is considered. An efficient

single layered neural network controller is assumed for the

real time path tracking control of the mobile robot. The

controller takes advantage of the robot regressor dynamics

that express the highly nonlinear robot dynamics in a linear

form in terms of the known and unknown robot dynamic

parameters. The influence of time delays in the input control

torque on the stability of the mobile robot motion has been

studied. The present work enables to estimate the maximum

admissible time delay in the input control torque with out the

loss of stability in robot motion and guaranteed tracking

performance.

I. INTRODUCTION

he objective of path tracking control is to guide the

mobile robot to follow the desired trajectory by

adjusting the forward and angular velocities [26]. Many

control algorithms have been proposed for the precise real

time motion control of mobile robots. Adaptive

controllers can produce fine motion control with partially

unknown dynamics [18]. In Kanayama et al. [12], a stable

control method under the assumption of perfect velocity

tracking is proposed. Jiang and Nijmeijer [11]

investigated local tracking motivated by practical issues.

Sarkar et al. [17] proposed a smooth nonlinear feedback

control algorithm to asymptotically track a desired

trajectory. Neural networks and fuzzy logic have also

been applied to mobile robot control in recent years.

Boquete et al. [3] presented a three-layer network plus

filters to learn the reference velocity. However, most

neural network based tracking controllers require offline

learning. One important tool for online control has been

the regressor formulation [5]. Lu and Meng [14] proposed

a methodology for the regressor formulation of robotic

manipulators. As a next step towards complete autonomy

of a mobile robot, control algorithms have been designed

in [7,20,26], where the robot dynamics is uncertain and

subjected to unmodeled and unstructured disturbances.

Fierro and Lewis [7] proposed a neural network based

V. Sree Krishna Chaitanya is with the Department of Mechanical

Engineering, Indian School of Mines, Dhanbad, India. (phone: +91-

9835718832; e-mail: vskc1986@yahoo.co.in).

P. Dwarikanath Patro is with the Department of Electronics &

Instrumentation Engineering, Indian School of Mines, Dhanbad, India. (e-

mail: dwarikanath@gmail.com).

Prabir Kumar Sarkar is Professor in the Department of Mechanical

Engineering, Indian School of Mines, Dhanbad, India (e-mail:

sarkarpk1955@yahoo.co.in).

model by combining the backstepping technique and

torque controller. But this algorithm is computationally

complex and expensive, which arises due to employment

of multi-layered neural networks. In Simon Yang et al.

[25, 26] a nonholonomic mobile robot with unknown

dynamics has been considered adopting the backstepping

velocity controller [12], which can provide the desired

linear and angular velocities. Yang et al. [26] proposed a

computationally efficient single layered neural network

controller, where the robot dynamics is completely

unknown and subjected to significant disturbances. Fuzzy

rules attracted the interest of many researchers in the

design of robot tracking control algorithms. Neuro-fuzzy

controllers for tracking are proposed in [10]. The

proposed controller in this paper has two components i.e.

a component proportional to the feedback error signal to

guarantee the global stability of the robot system and a

single layered neural network to achieve real-time fine

motion adopting the backstepping velocity controller [12].

In this paper the influence of time delays in the input

control torque on the stability of the mobile robot motion

and smoothness and continuity of the control signals has

been studied. The maximum admissible time delay in the

input control torque with out the loss of stability in the

robot motion has been estimated using an algorithm

proposed in [23]. The salient features are highlighted by

simulations.

II. MODEL DESCRIPTION

In this section we present the mathematical model of the

nonholonomic mobile robot. Later the model has been

derived in local coordinates attached to the centre of mass of

the mobile robot.

A mobile robot system having an n - dimensional

configuration space C with generalized co-ordinates

( nqq ,.......,1 ) and subjected to m constraints may be

described by [7]

λτ

τ

)()(

)()(),()(

qAqB

qGqFqqqVqqM

Ta

dm

−

=++++ &&&&&

(1)

where )(qM is a n x n symmetric, positive definite inertia

matrix, ),( qqVm& is the n x n centripetal and coriolis matrix,

)(qF & is the n x 1 surface friction, )(qG is the n x 1

gravitational vector, dτ denotes unknown disturbances

Delay Dependent Stability in the Real Time Control of a Mobile

Robot Using Neural Networks

V. Sree Krishna Chaitanya, P. Dwarikanath Patro, and Prabir Kumar Sarkar

T

Proceedings of the 2007 IEEE International Symposium onComputational Intelligence in Robotics and AutomationJacksonville, FL, USA, June 20-23, 2007

ThBT1.2

1-4244-0790-7/07/$20.00 ©2007 IEEE. 95

including unstructured, unmodeled dynamics, )(qB is the n

x r input transformation matrix, aτ is the r x 1 input vector,

)(qA is the m x n matrix associated with the constraints, and

λ is the m x 1 vector of constraint forces. Denoting the right

hand side of (1) by T we get

TqGqFqqqVqqM dm =++++ τ)()(),()( &&&&& (1b)

Fig. 1 A nonholonomic mobile robot

The mobile robot shown in Figure 1 is a typical example of a

nonholonomic mechanical system. The nonholonomic

constraint states that the robot can only move in the direction

normal to axis of the driving wheels, i.e., the mobile base

satisfies the conditions of pure rolling and non slipping [7]

0sincos =−− θθθ &&& dxy cc . (2)

The dynamical equations of the mobile base in Fig. 1 can be

expressed in the matrix form (1) where

,

−

−=

Imdmd

mdm

mdm

qM

θθ

θ

θ

cossin

cos0

sin0

)( (3)

=

000

sin00

cos00

),( θθ

θθ&

&

& md

md

qqVm

, 0)( =qG ,

−

=

RRr

qB θθ

θθ

sinsin

coscos1

)( ,

=

2

1

d

d

d τ

ττ ,

−

−

=

d

qAT θ

θ

cos

sin

)(,

θθθλ &&& )sincos( cc yxm +−=

in which m is the mass of the robot, I is the mass moment of

inertia about the mass centre C, R is the distance between the

driving wheels and r is the radius of the wheel of the mobile

robot. (refer Fig. 1).

Assuming the trajectory of the mobile robot is constrained on

the horizontal plane and there is no friction, model (1b) can

be transformed into a more appropriate form for control

purpose as described below

TvVvM dm =++ τ& (4)

in

which MSSMT= , )( SVSMSV m

T

m += & , d

T

d S ττ = ,

TSTT= , Svq =& and

−

=

10

cossin

sincos

)( θθ

θθ

d

d

qS

Equation (4) presents the nonholonomic mobile robot in

local coordinates attached to its centre of mass and S(q) is a

jacobian matrix that transforms independent velocities v in

the local coordinates to the constrained velocities q& in the

global coordinates.

In the local coordinate system, we get

−=

20

0

mdI

mM ,

=

00

00mV (5)

III. CONTROLLER DESIGN

In this section we present the control algorithm. The

controller employs a single layered neural network reducing

computational complexity to a great extent [20,26]. The

control algorithm is capable of generating real-time smooth

and continuous velocity signals that drive the mobile robot to

follow the desired trajectories. Further it does not require

any offline training procedures. Later the learning algorithm

for the neural network is presented [20,26].

The nonlinear mobile robot dynamics can be rewritten into a

linear form similar to the regressor dynamics formulation of

robot manipulators in the following manner,

TY d =+ τφ (6)

where φ is a vector consisting of the known and unknown

robot dynamic parameters, such as geometric size, mass,

moment of inertia etc; Y is the regressor matrix obtained

from the equations of motion. For the typical nonholonomic

mobile robot shown in Fig. 1, we obtain the robot regressor

Y and φ as

−=

ww

vY

l

&&

&

0

00,

−

=2

md

I

m

φ (6b)

The tracking position error between the reference robot and

the actual robot can be expressed in the robot local

coordinates as

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96

−

−

−

−=

=

θθ

θθ

θθ

r

r

r

M yy

xx

e

e

e

e

100

0cossin

0sincos

3

2

1

. (7)

It can be proven that the time derivative of the above

position error is given by

−

+−

+−

=

ww

evwe

evvwe

e

r

r

rl

M 31

32

sin

cos

& . (8)

The backstepping velocity tracking control input is chosen as

[12]

=cν

++

+

3322

113

sin

cos

evkevkw

ekev

rrr

r (9)

where k1, k2 and k3 are the feedback gains of 1e , 2e and 3e

respectively.

Velocity tracking error is defined as

vve

ee cc −=

=

5

4 (10)

where v =

w

vlis the actual velocities of the mobile robot.

By differentiating (10) with respect to time and substituting

the result in (6), the mobile robot dynamics may be rewritten

as

dcmc YTeVeM τφ ++−−=& (11)

where cmc vVvMY += &φ , is the regressor dynamics

introduced earlier.

The input control torque is defined as

γ−+= cNN eKTT 4 (12)

where 4K =

87

65

kk

kkis a definite design matrix and γ is

a robustifying term to compensate the unmodeled

unstructured disturbances. NNT is the output torque of the

single layered neural network given by

YWTNN = (13)

in which W is a 3 x1 vector that represents the connection

weights of the neural network, which is also an

approximation ofφ .

By substituting (12) into (11), the closed-loop error

dynamics can be expressed as

γτφ ++++−= dcmc YeVKeM~

)( 4&

(14)

where W−= φφ~

is the estimation error ofφ .

The robustifying term can be defined as

cd

r

d eK

vkk

ek

e

K ′−−=

−−=

32

51

4

γ

(15)

where dK is a diagonal positive matrix.

The convergence and stability of the above control law have

been proved in [20]. The learning law for the neural network

controller is obtained from the stability analysis [20],

which assumes the form c

TeYW Γ−=& . (16)

where Γ : is a 2 x 2 positive constant design matrix

This learning law is much simpler than those mostly used

such as the multi-layer back-propagation learning.

Fig. 1. The structure of the proposed neural network controller

IV. DELAY DEPENDENT STABILITY

Considering the model (4) and with the help of (5), (6), (9),

(10), (12) we obtain the following equations governing the

motion of the mobile robot.

)sin(

)cos(

33226

113511

wevkevkwk

vekevkTvm

rrr

lrNNd

−++

+−++=+τ& (17)

)sin()

cos()(

33228

11372

2

2

wevkevkwkv

ekevkTwmdI

rrrl

rNNd

−+++−

++=+− τ& (18)

Theorem 4.1

Consider a stable mobile robot motion described by (4), (5)

with nonlinear feedback control described in (14). Suppose

that there exists a time delay τ in control input torque T (t-

τ ) given by (12), (13). Then the closed loop system remains

stable if

1. 08 >k , 08 >+ ak , 05 >+ ck in which

rdr

rrlrNN

mdIk

evkwkekvvkTa

θτθ &&& )(

)()(

2

8

2281172

2−−−−

+++−+=

and

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97

133

22611351

)sin

()cos(

drr

rrrrNN

vmevk

evkwwkekvevkTc

τ−−+

+−++−+=

&.

*0 ττ ≤≤ = min { }, 21 ττ

in which 1τ = )(tan2

11

1

1

Tww

−

where ck

mT

−=

5

1 and

1

52

1

)(

mT

ckw

+−=

and 2τ = )(tan2

22

1

2

Tww

−

where

)(2

)()(4))(2())(2(

388

3828

2228

28

2avkkk

avkkkmdIamdIkamdIkT

r

r

−

−−−−−±−−=

and))((

)(

28

2

382

2TkmdI

avkkw r

−−

+= .

( 1τ and 2τ are the assumed time delays in the input control

torque of (17) and (18))

Proof

Now we present the methodology in the derivation of 2τ [23].

A similar procedure is followed in the estimation of 1τ .

It is easy to observe the following relations

3er −= θθ , 3er&&& −= θθ and 3er

&&&&&& −= θθ .

Using the above mentioned relations and (18), we obtain

0)( 338383

2 =+++− aekkekemdI &&& . (19)

Substituting)(

3 )( τλτ −=− tete in (19), we obtain

0)( )(

83

)(

8

)(22 =+++− −−−aevkkekemdI

t

r

tt τλτλτλ λλ

We set

aevkk

ekemdIF

r +

++−=−

−−

λτ

λτλτ λλτλ

83

8

22 )(),( (20)

So that

)(

)()0,()(

83

8

22

1

avkk

kmdIFF

r ++

+−== λλλλ (21)

From Routh-Hurwitz criterion, it is easy to see that all roots

of 0)(1 =λF have negative real parts. Thus, 0)(1 ≠λF

for 0)Re( ≥λ .

We observe that by continuity, all the characteristic roots of

the above equation for ),( τλF will continue to have

negative real parts for sufficiently small 0>τ .

Substituting

2

2

1

1

T

Te

λ

λλτ

+

−=−

for some 02 >T in ),( τλF ,

we obtain

0)()(

))(()()(

8382238

2

28

2

2

23

2

=+−−−+

−−+−=

avkkkaTTvkk

mdITkTmdIF

rrλ

λλλ(22)

Now, )(2 λF has pure imaginary roots 2iw=λ if

[ ]πτ kTww

−= − )(tan2

22

1

2

2 , ..3,2,1,0 ±±±=k

In order to obtain the pure imaginary roots for the expression

(22), we set the Routh-Hurwitz determinant to zero, i.e.

0)(

))(2()(

2

8

2

2

8

22

2838

=−

+−−+−

mdIk

TkmdITkavkk r (23)

and we obtain from (23),

)38(82

])38(28)

2(4

2))

2(28(

))2

(28[(

2arvkkk

arvkkkmdIamdIk

amdIk

T−

−−−−−

±−−

= (24)

Differentiating the right hand side of ),( τλF with respect

toτ and equating it to zero and simplifying, we get

τλλ

λ

λλλ

τ

λ

++−−+−

++−=

)83

8)2()

8)2(2(

)883

2)2((

rvkk

kmdIkmdI

krvkkmdI

d

d

(25)

Evaluating τ

λ

d

d at 2iw=λ , we obtain

in which

[ ]2832

22

)( rvkkwmdI −−=δ [ ]2282

2 )(2 τwkmdIw −−+

We shall now discuss the possible cases in

which 0)( 22 =iwF . Substituting 2iw=λ in (22) and

simplifying it, we obtain

))((

)(

28

2

382

2TkmdI

avkkw r

−−

+= (26)

Now, substituting the value of 2T from (24) in (26) and

simplifying further, we obtain 2w .

Further at, 2τ = )(tan2

22

1

2

Tww

− we note that

22 ,

Reττλτ

λ

==iwd

d> 0

This completes the proof of the theorem.

Remark

δ

τ

τ

λ

λ

−+−

−−−++−

==

))(2)((

))(()(2

Re

228

2

28

22

8

23

288

2

22

2

mdIkmdI

kwkmdIvkkkmdI

w

d

d

r

iw

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98

After obtaining the plots versus time of the expressions for

the time delays, the minimum value in the entire range will

be the maximum admissible time delay in the control torque.

Any time delay above this value would lead to instabilities

and the kind of resulting instabilities are not discussed in this

paper.

V. SIMULATION RESULTS

This section illustrates the performance of the proposed

control scheme (in the presence of time delays in the input

control torque) using MATLAB. The objective is to track a

path 1)1( 22 =−+ yx such that the errors in position and

velocity tend to zero. The following parameters were

adopted: m = 10.0 Kg, I = 5.0 Kg m2, R = 0.5 m, r = 0.05 m,

rv = 1.0m/s, rw = 1 rad/s. The control parameters were

selected as k1 = 0.5, k2 = 2.1, k3 = 3, k5 = 1.5, k6 = 1.5, k7 =0, k8

= 6, Γ = I2. The initial position of the robot has been chosen

as (0,0) as shown in the Figure. 2. For the chosen control

parameters, the maximum admissible time delay in the input

control torque has been obtained as 0.3s such that the

condition1 in theorem 4.1 holds. The simulations have been

conducted with this time delay in the input control torque. It

can be seen from the following figure that the robot perfectly

tracks the circular path. Moreover the control signals are

smooth and continuous. We observe that as the delay in the

control torque increases to a value of 0.5s, the stability of the

mobile robot motion is lost.

Fig. 2.

VI. CONCLUSION

In this paper a nonholonomic mobile robot with unknown

dynamics is considered. A novel single layered neural

network controller is assumed for the stable path tracking

control of the mobile robot. The influence of time delays on

the stability and smoothness of control signals has been

studied. An analytical estimate for the maximum acceptable

time delay in the input control torque such that there is no

loss of stability in the robot motion has been derived. The

salient features are demonstrated by appropriate simulations.

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