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Microsoft Word document doc International Journal of Advanced Robotic Systems, Vol 5, No 4 (2008) ISSN 1729 8806,pp 403 410 403 BacksteppingBacksteppingBacksteppingBackstepping based trajectory tracki[.]

Backstepping Backstepping based trajectory tracking based trajectory tracking control of a four wheeled mobile robot control of a four wheeled mobile robot

Umesh Kumar & Nagarajan Sukavanam Umesh Kumar & Nagarajan Sukavanam Department of Mathematics, Indian Institute of Technology, Roorkee , India Corresponding author E-mail :(umeshdma@iitr.ernet.in)

Abstract : For a four wheeled mobile robot a trajectory tracking concept is developed based on its kinematics A trajectory is a time–indexed path in the plane consisting of position and orientation The mobile robot is modeled

as a non holonomic system subject to pure rolling , no slip constraints.To facilitate the controller design the kinematic equation can be converted into chained form using some change of co-ordinates.From the kinematic model of the robot a backstepping based tracking controller is derived Simulation results demonstrate such trajectory tracking strategy for the kinematics indeed gives rise to an effective methodology to follow the desired trajectory asymptotically

Key words :wheeled mobile robot, chained form systems, nonholonomic systems, trajectory tracking

1 Introduction

A mobile robot is one of the well known system with

nonholonomic constraints and there are many works on

its tracking control Their objective are mostly the

kinematics model For non holonomic systems such as

mobile robots their kinematics constraints make time

derivative of some configuration variables nonintegrable

(Xiaoping,Y & Yamamoto,Y.,1996) Due to the

appearance of the nonholonomic constraints the motion

planning and the tracking control of mobile robots are

difficult to be managed In the phase of motion planning

(Wilson ,D.E.,and Luciano,E.C.,2002) a suitable trajectory

is designed to connect the initial posture (i.e the position

and orientation of the robot) and the final one such that

no collisions with obstacles would occur and kinematics

constraints are satisfied Once the feasible path is

obtained the navigation and control process enters the

tracking phase In this paper we restrict our attention to

the kinematics of the nonholonomic systems such that

every path can be followed efficiently In (Kanayama, Y

,kimura, Y ,Miyazaki,F & Noguchi,T.,1990) a

linearization based tracking control scheme was

introduced for a two degree of freedom mobile robot A

similar idea was independently examined by Walsh et al

in (Walsh, G., Tilbury, D , Sastry, S., Murray, R &

linearization was further explored by (Oelen,W &

Amerongen ,J.,1994) for a two degree of freedom mobile

robot All these papers solve the local tracking problem

for some classes of nonholonomic systems The class of

nonholonomic system in chained form was introduced by

(Murray, R M., & Sastry , S.S.,1993) and has been studied

as a bench mark example by several authors It is well known that many mechanical system with nonholonomic constraints can be locally or globally, converted to chained form under coordinate change and state feedback Interesting examples of such mechanical systems include tricycle-type mobile robots ,cars towing several trailers, the knife edge(Murray, R M., & Sastry , S.S.,1993), (Kolmanovsky ,I & McClamroch, N H ,1995) Trajectory planning algorithm for a four-wheel-steering (4WS) vehicle based on vehicle kinematics was introduced by (Danwei, W & Feng, Q.,2001).Trajectory tracking control of tri-wheeled mobile robots in skew chained form system was introduced by(Tsai, P.S.,Wang, L.S.,& Chang, F.R.,2006)

In this paper we develop the kinematic model of a four wheel nonholonomic mobile robot subject to pure rolling and no side slipping condition and we propose a systematic control design procedure for chained form obtained from the kinematic model.The stepwise control design procedure used in this paper is based on the backstepping approach The problem to be solved here is the tracking of a desired reference trajectory for a four wheeled mobile robot moving on a horizontal plane and simulation results were presented to illustrate the approach

2 Model of the mobile robot 2.1 Introduction

In this section kinematic model of the four wheel nonholonomic mobile robot is presented In order to simplify the mathematical model of the four wheel nonholonomic wheeled mobile robot we assume that

a The wheeled mobile robot is built from rigid

mechanism

b There is zero or one steering link per wheel

c All steering axes are perpendicular to the surface of

motion

d The surface is a smooth plane

e No slip occurs between the wheel and the floor

2.2 Instantaneous centre of rotation (ICR)

It is an imaginary point around which a rigid body

appears to be rotating momentarily (for an instant) when

the body is rotating and translating For a four wheel

mobile robot the instantaneous centre of rotation is the

cross point of all axes of the wheels Its importance lies in

the fact that wheel axes must intersect at a point if there is

no slipping

2.3 Kinematic model of the four wheel mobile robot

In this section the kinematic model is developed.A

wheeled mobile robot is a wheeled vehicle which is

capable of an autonomous motion (without external

human driver) because it is equipped ,for its motion ,with

motors that are driven by an embarked computer The

four wheel mobile robot considered in this paper is

shown in figure 1, its front wheels are steering wheels,

and its rear wheels are driving wheels The distance

between the front wheel axle and platform centre of

gravity is c and distance between the rear wheel axle and

platform centre of gravity is d and 2b is the wheel span

The trajectory planning will be done for the platform

centre of gravity Let the generalized co-ordinates be

0 0 0

[ , , , ]T

q= x y φ φ ,where ( , )x y0 0 are the cartesian

coordinates of the centre of gravity of the mobile platform

with respect to co-ordinate frame { }U The four wheels

are located at p1,p p2, 3 and p4on the mobile platform

respectively pc is the centre of the mobile platform Six

co-ordinate frames are defined for describing position

and orientation of the mobile robot {1} is the frame fixed

on wheel 1.x1 - axis is choosen to be along the horizontal

radial direction and y1 axis in the lateral direction

Likewise {2},{3} and {4} are the frame defined for the

wheel 2, 3 and 4 respectively {0} is the frame defined at

point pc.The orientation of the vehicle body is

characterized by φ0 which is the angle from xU to x0

1

φ and φ2 are the front two steering angle and φ is the

angle at which the whole platform changes the

orientation due to steering angles φ1 and φ2 of the front

two steering wheels With these notations we establish

homogenious transformations describing one frame

relative to another abT denotes the homogenious

transformation of frame { }b relative to frame { }a

Because the motion of the mobile robot is restricted to

2-dimentional plane homogenious transformation

Fig.1:Four wheel mobile robot and co-ordinate frame matrices are 3 3× rather than 4 4× and are given by

U

x

−

0

c

−

0

c

−

= − 

=  = − 

2.4 Velocities

Fig.2:Velocities of wheels With the help of homogenious transformation given above the velocities of point p1 ,p p2, 3 and p4 can be easily computed

The homogenious position vector of point p1 in

frame{0} is

0 1

1

c

 

 

= 

  The same point is represented in frame { }U by

0

1

φ φ

φ φ

=

− +

= + + 

The velocity of point p1relative to frame { }U expressed

in frame { }U is then

0 U

= − + 

& & &

& &

In order to derive the nonholonomic constraint equation

of wheel 1 The velocity of point p1 relative to frame

{ }U is expressed in frame {1} is

1

( )

cos sin cos sin

sin cos sin cos

cos( ) cos sin sin( )

sin( ) sin cos cos(

U

P T P

φ φ φ φ φ φ φ φ

φ φ φ φ φ φ φ φ

−

−

=

 −   − 

& &

&

& &

& &

0

y

&

Similarly the velocity of point 2, 3, 4 relative to frame

{ }U expressed in frame {2} , {3} and {4} as follows

2

cos( ) cos sin sin( )

sin( ) sin cos cos( )

0

φ φ φ φ φ φ φ φ

φ φ φ φ φ φ φ φ

& &

& &

& & &

3

4

cos sin sin cos

0 cos sin sin cos

0

φ φ φ

φ φ φ

φ φ φ

φ φ φ

&

& &

&

& & &

&

& &

&

& & &

2.5 Constraint equations

To develop the kinematic model of the wheeled mobile

robot ,the ith wheel is considered as rotating with angular

velocity θ& where i θ&i,i=1, 2,3, 4.denotes the angular

velocities for each wheel.For simplicity the thickness of

the wheel is neglected and is assumed to touch the plane

at Pias illustrated in fig.2.Further it is assumed that

during the motion the plane of each wheel remain vertical

and the wheel rotates around its (horizontal) axle whose

orientation with respect to the frame can be fixed or

varying The first four constraints in which two constraint

are identical are due to no slip condition i.e pure rolling

and the other four constraints obtained from the

condition that the wheels can not move in the lateral direction (i.e y-component of 1P&1,2P&2,3P&3 and 4P&4 are all zero).So finally we have seven constraints given by

sin( ) sin cos cos( ) 0 sin( ) sin cos cos( ) 0

φ φ φ φ φ φ φ φ

φ φ φ φ φ φ φ φ

φ φ φ

& &

& &

&

& &

cos( ) cos sin sin( ) cos( ) cos sin sin( ) cos sin

cos sin

φ φ φ φ φ φ φ φ θ

φ φ φ φ φ φ φ φ θ

φ φ φ θ

φ φ φ θ

& & &

& & &

& &

& &

& &

& &

Where r is the radius of each wheel and θ θ θ1, 2, 3and

4

θ are the angular displacement of the wheels

Choosing the following as generalised co-ordinate vector

0 0 1 2 3 4 0

q= x y θ θ θ θ φ φ

The seven constraint can be written as

A q q& =

where A is a 7 8× matrix given by

1

( )

0

s i n

A q

r r r r

d

φ

=











−





+

−

−

0 0

b b

φ

















Now using the fact that wheel axes must intersect at a point when the mobile robot turns Thus we get

1

tan

φ φ

φ

+

= + −

and

2

tan

φ φ

φ

+

= + +

We introduce a vector of quasi velocities instead of generalized velocities because control runs in an abstract space Quasi-velocities are function of kinematic parameters Since the generalised velocities is always in the null space of ( )

A q ,according to (1) the generalized velocities

.

qcan be exp- ressed in terms of quasi-velocities ( )v t as follows

q&=s q v t where ( )s q is a 8 2× full rank matrix, whose columns are

in the null space of ( )A q and is given by ( using the

MATHEMATICA package)

11 21 31 41 51 61 71 81

0 0 0 0 ( )

0 0 0 1

S S S S

s q S S S S

= 

Where

2

( ) cos cot sin

cos ( ) cot sin

2 ( ) cot 2 cos 2 ( 2( ) )

S

( ) ( tan ) 1

( ( ) cot )

S d c d

b c d b ec b c d

c d

r c d b

b c d

φ

φ

=

+

− +

2

2

S

φ

φ

=

+ + + +

+ +

51

S

r

φ

− + +

=

61

S

r

φ

+ +

=

71 81

1 0

S S

=

=

Define

= −

= +

&

&

where vxand vy are the velocities of the centre of gravity

of the mobile platform along the xand y-axes respectively

Thus we get

2 0

2 0

2

3

4

0

cos sin tan 0 sin cos tan 0

2 ( ) cot 2 cos 2 ( 2( ) ) tan

0 ( )

( tan ) 1

( ( )cot ) ( ) cos sec (2( ) tan

d

c d d

c d

b c d b ec b c d

c d

r c d b y

b c d

c d ec b c d b

φ

φ θ

θ

θ

θ

φ

φ

− + + +

 

 =

 

 

 

 

 

 

&

&

&

&

&

&

&

&

2 2

) tan

0 ( )

( tan ) 1

( ( )cot ) tan ( )

0 ( )

tan ( )

0 ( )

tan

0

x

v

c d

c d

r c d b

b c d

b c d

r c d

b c d

r c d

c d

φ φ φ

φ φ

φ φ

+

+

+

+

&

Since the control objective for the robot is to ensure that ( )q t

tracks a reference position and orientation denoted by

q td( )=[x t y td( ), d( ),φd0( )]t so we consider only

0

1

2 0

(3) tan

0

d

x

d

v y

v

φ

−

&

&

&

&

Where v1=vx and v2=φ&

2.6 Chained form

In order to convert the kinematic model of the mobile robot in chained form following change of co-ordinates is used

0

c o s

t a n ( ) c o s

t a n

s i n

x x d x

c d x

φ φ φ φ

φ

= +

=

Together with two input transformations

1 1 0 cos

u V

φ

=

2

0

0

3 s i n s i n

( ) c o s c o s ( ) c o s

φ

−

+

Then from above transformation we get the following two input four state chained form

x u

x u

x x u

x x u

=

=

=

=

&

&

&

&

where x=( ,x x x x1 2, 3, 4) is the state and u1and u2are the two control inputs

3 Design of trajectory tracking controller Denote the tracking error as xe= −x xd.The error differential equation are

= −

= −

= + −

= + −

&

&

&

&

The goal is to find a , Lipschitz continuous time-varying state-feedback controller %u i.e

% 1

1 2 2

( ,e d, d )

u

u

 

= =

  such that the tracking error xeconverges to zero

t x x

conditions on the reference control functions u1and

2

u and initial tracking errors xe(0) ,with a good choice of

λ

We first introduce a change of coordinates and rearrange

system (5) into a triangular –like form so that the

integrator backstepping can be applied

Denote %xd=(xd2,xd3)and let η1(.;x%d) :R4→R4be the

mapping defined by

(4 1) ( ) ( ) 1

i e i e n i d n i e

e

e

x

x

ζ

ζ

ζ

=

=

In the new coordinates ζ=( ,ζ ζ ζ ζ1 2, 3, 4) system (5)is

transformed into

(6)

d

d

d

ζ

ζ

= − −

= −

= −

= −

&

&

&

&

The basic design idea for backstepping based control law

is to take for every lower dimension subsystem, some

state variables as virtual control inputs and at the

same time ,recursively select an appropriate

Lyapunov function candidate Thus each step results in a

new virtual control -er In the end of the overall

procedure ,the true control law results which achieves

the original design objective

Consider the ζ1−subsystem of (6)

1 ud1 2 x u2( 1 ud1) 4 (7)

We consider the variable ζ2as virtual control input and

the variable u1 and ζ4 as time varying functions

Denote ζ1=ζ1

Differentiating the function 1 12

1 2

V = ζ along the solution

of (6)yields

V& =u ζ ζ −xζ u −u ζ

Since the variable ζ2is a virtual control input

Selecting ζ2= −k1 1ζ k1≥0

2

2

1 1 d 1 1

V& = −k u ζ whenever

ζ = From above equation we observe that functionα ζ1( )1 =0

is a stabilizing function i.e the desired value of virtual

control ζ2 for which V1 is negative semi definite for

subsystem (7) This desired value is called stabilizing

function and ζ2is called virtual control As in this case

2

1 21 d1 1

V& = − k u V Above implies that

1 1 1lim

0 0

as t

i e V

ξ

→ → ∞

→ → ∞

=

So the origin ζ2=0 is asymptotically stable Hence the

function α ζ1( )1 =0 is a stabilizing function

Define ζ2=ζ2−α ζ1( )1

Differentiating the function

along the solution of (6)yields

V& =u ζ ζ −xζ u −u ζ −uζ ζ

As

2 2 1 1

1

2

1

2 1 3 2 4

1 3 2 4 1 1 1 1

1 3 1 2 4 1 1

1 3 2 2 4 1 1

( )

d

α

= −

∂

= −

∂

= = −

= − + −

= + − −

= − − −

Where

2( ,1 2) 1

α ζ ζ = −ζ

Again define

3 3 2 1 2

3

2 2 1 2 2 1 1 4

= −

∂ ∂

= − +

∂ ∂

= − − = +

= − + − −

Consider the positive definite and proper function

Differentiating the function V3along the solution of (6)yields

3 3 1 2 2 2 1 2

2 1 2 3 1 1 4 2 2 4

d

= + − +

− + − −

&

In order to make V3negative definite we choose the following control input

where c3>0 Thus we have

2

3 3 3 ( 2 1 2 3)( 1 d1) 4 2 2 4

V& = −cζ − xζ +xζ u −u ζ −u ζ ζ

Finally consider the positive definite and proper function which serves as a candidate Lyapunov function for the whole system (6)

Where λ>0is a design parameter

Differentiating the function V4along the solution of (6)yields

4 3 3 2 1 2 3 1 1 4 2 2 4

4 1 1

2

d d

d

λζ

= − − + − − +

−

= − + − + − −

&

&

In order to make V4negative definite we choose the

following control input

From(9)and (10)we get the following control law

d

− +

= +

− +

= − + −

4 Simulation results:

To examine the effectiveness of the proposed trajectory

tracking control methodology, the simulation for a four

MATHEMATICA The system parameters of the four

wheel mobile robot were selected as c=1.3m, d=1.4 m,

2b=1.5m

4.1.Tracking a straight line

Tracking a straight line is a simple case for all robots.Here

for simulation we consider that the straight line.We

λ = = for tracking a straight line It is further

assumed that initially xe(0)=(2,0.8,0.8,0.8)

The straight line reference trajectory to be tracked is given

by x td( )=t y t, d( )=0

The desired trajectory and the norm of the tracking

error is shown in fig 3 and fig.4 respectively

4.2.Tracking the curve x td( )=t y t, d( )=asinωt

we consider the following desired sinusoidal

trajectory x td( )=t y t, d( )=asinωt as shown in fig 5

Fig.3:desired straight line trajectory

Fig.4: norm of tracking errorx te( )versus time

Fig.5:Desired sinusoidal trajectory

secs -0.1

-0.05 0 0.05 0.1

Fig.6 Desired steering angle versus time Fig.7 demonstrates the evolution of the norm of the tracking error x te( ) based on the following choice of design parameters and initial condition:

λ=3,c3=c4=5,xe(0)=(2,0.5,0.5,1.5)

secs 0

2

4

6

8

10

secs

0.5 1 1.5 2 2.5 3

-0.2 -0.1 0 0.1 0.2

with a=0.2 and =0.5ω

Fig.7: Norm of tracking errorx te( )versus time

5 Conclusion

In this paper the nonholonomic constraints and the

kinematics model of the four wheel (front steering and

rear

driving) mobile robot under pure rolling and no side

slipping condition is derived Using the change of

coordinates the system is transformed into chained form

and then a backstepping based tracking controller is

derived Simulation results are presented with two

examples to illustrate the approach

Acknowledgement:

The author Umesh Kumar gratefully acknowledges

the financial support of University grant commission

(UGC),Government of India through senior research

fellowship with grant No.6405-11-61

The author Nagarajan Sukavanam acknowledges

financial support by the Department of Science

and Technology (DST) ,Government of India under the

grant No DST-347-MTD

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