Sliding Mode Formation Control of Nonholonomic Robots Xiang Gao, Qiuju Huang, Ming Wan, Chunxiang Liu

Test Measurement Technology and Instruments, College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China

E-mail: y081426@njupt.edu.cn

Abstract: This paper discusses a method of nonholonomic

robot formation control based on leader-follower. The leader

moves along a designated trajectory, the follower keeps a

specified distance and angle when following the leader so as

to keep a certain formation. Sliding-mode control strategy is

introduced in, the sliding mode surface functions are designed

to make robots asymptotically stabilize to a time-varying

expected formation. Thus a more stable robot formation is

made.

Keyword: Formation Control, Leader-Follower, Kinematic

Equation, l ϕ− Control, Sliding-Mode Control

I. INTRODUCTION

Multi-robot formation control means making a number of robots to maintain formations and avoid obstacles or collisions with other robots in the process of reaching a target location [1], [2]. Multi-robot formation control methods are mainly behavior-based approach, follow the navigator approach and generated approach. The basic idea of Behavior-based approach is: Firstly, defining some expected behaviors for the robots, including collision avoidance, obstacle avoidance, walking along the wall, towards the goals and maintaining formations and so on, for the robots in normal circumstances. When the robots' sensors receive external environment's stimulus, they make reactions according to the sensors' input information, and output response vector as the expected reactions of the behavior (for example, directions and movement speeds). However, this method does not define group behavior explicitly, so it is difficult to make mathematical analysis and can not guarantee the stability of formations. Robustness of the traditional leader-follower formation control method [3], [4] is poor: the control of the whole formation totally depends on the navigator, slight deviation of navigator will affect the stability of the entire formation. Thus in this method mistakes of the navigator would not be allowed. In other words, the system has a bad tolerance for interference. Generated approach is rarely used, so there is no need to describe in detail.

For the shortcomings of above formation methods, a developed formation method, with the sliding mode

control strategy [5], [6] based on leader-follower introduced in, is proposed in this paper. Sliding mode control (SMC) is also known as variable structure control, which is essentially one type of the special nonlinear control, and the non-linear of SMC is mainly embodied in the discontinuity of control. The difference of this control strategy to other control systems is that the "structure" is not required fixed. It can be in a dynamic process according to the current state of the system, and can force the system to move in accordance with the expected "sliding mode" trajectory. For the sliding mode can be designed having nothing to do with the object parameters and disturbances, the sliding mode control has the advantages of fast response, insensitivity to parameter changes and disturbances and no need for system identification, the physical realization simple and so on. As the sliding mode control system can overcome the uncertainty of the disturbance and has a strong robustness for unmodeled dynamics, especially has good control effect for the nonlinear systems, it has a wide range of applications in the field of robot control .

II. PROBLEM DESCRIPTION AND

DEFINITION

Assuming the leader robot's trajectory has been pre-configured, the follower adjusts its own movement according to the leader's movement, and a desired formation is maintained. It is requested that the robots in the multiple mobile robot system are all the same type of a car model robot. It means that the parameters of the robots are the same. Two robots are used to form an example to explain the method of formation as follows.

As show in figure 1, the robot queue is given as 1 2{ , ,..., }nR R R , recording ijl as the relative distance

between iR and jR ,where iR represents the leader robot,

jR represents the follower robot; [0,2 )iθ π∈ as the direction angle of iR ; ( ) [ , )ij j iβ θ θ π π= − ∈ − is the

difference of direction angle of iR and jR , describing the direction of relative motion of the two robots;

[ , )ijϕ π π∈ − is the angle between the movement direction of iR and the connection of jR , describing the

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.21

67

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.21

67

direction of relative position of the two robots, where , {1, 2,..., },i j n i j∈ ≠ ; d is the length of the connection between the robot rotation center and the off-axis reference point jP on the robot. In the control strategy of l ϕ− between two robots [7], [8], the aim is to keep expected distance d

ijl and expected

declination dijϕ between two robots; the kinematic

equations of the two robots shown in Fig.1 are as follows:

Fig. 1 Two robots' leader-follower system

The kinematic equations of the leader robot iR are:

cossin

i i i

i i i

i i

x vy v

θθ

θ

==

= ω (1) The kinematic equations of the follower robot jR

are: cos cos sin

1 ( sin sin cos )

ij j ij i ij j ij

ij i ij j ij j ij ij iij

j j

l v v d

v v d ll

ϕ

ϕ ϕ

θ

= γ − + ω γ

= − γ + ω γ − ω

= ω (2)

Where ij i ij jθ θγ = + ϕ − , iv and iω , jv and jω are corresponding to the linear speed and angular velocity of the two robots, it is requested that ijl d> .

In the following paragraphs, for two robots' control interconnection, we use the sliding mode control strategy based on leader-follower. The main idea of this method is to design sliding mode surfaces to achieve the formation of multi-robot control [9], [10]. Sliding mode variable structure control means that designing system switching hyper-plane by the sliding mode controller, based on the dynamic characteristics of the expected system, making the system state converging to

the switching hyper-plane from outside of the hyper-plane. Once the system reaches the switching hyper-plane, control effect will take the system reach the origin along the switching hyper-plane. The process of moving to the origin along the switching hyper-plane is known as sliding mode control. Because the characteristics and parameters of the system depend only on the designed switching hyper-plane and have nothing to do with the outside interferences, sliding mode control has a strong robustness. At present, in the leader-follower model, the leader's movement determines the whole team's movement. The single robot's movement is described referring to the main members' trajectory. The design concept of the sliding mode control strategy will be described in detail later, which is an important part of the whole control framework. Corresponding simulation results will be given as well.

III. FORMATION CONTROL METHOD DESIGN

In this section we design a sliding mode controller making the follower robot jR with desired angle ijϕ and desired distance ijl tracking the leader robot iR , the leader robot moves along a designated trajectory, and the vector ( , )T

i iv ω is already known.

Fig.2 Schematic diagram of two robots' formation control

Two robots are used to form an example for analysis. The robots here used are abstraction of the physical robots. Assuming the mid-point of the connection between two front-wheels of the robots are M, the intersections of robots' axis with the front and rear ends of the robots are N, P. XOY is two-dimensional global coordinate system, the position of iR in the plane is

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determined by the point of ( , )i iM x y , the definitions of

, ,ij ij il ϕ θ are the same as defined in section II, ijα is the

angle between the movement direction of iR and the connection between the M-point of the two robots. i i iX O Y is the local coordinate system based on iR , the origin of coordinates is the M-point of iR , where the abscissa's direction is the direction of vector MN , the abscissa according to counter-clockwise rotation / 2π to be the vertical axis.

,N Pl l are the connections between the M-point of

iR and the N、P-point of jR respectively, ,NX NYl l are the

projections of Nl in the axis of i X and iY respectively, the three edges form a right triangle triangular, the length of MN is 1d .

The * *,ij ijl α in the target formation are the distance between the two M-points of the robots and the relative position angle, through the relationship of triangular edges and angles, it is easy to transform parameters to be * * *, ( )N N ijl ϕ ϕ ,which are the parameters between M -point of leader robot and N-point of follower robot, so the formation control objectives can be translated into:

* * *, ,N N N N ij ijl l ϕ ϕ β β→ → → To establish kinematic equations of the leader robot

and the follower robot in the Cartesian coordinate system, the control of * *,N N N Nl l ϕ ϕ→ → is equivalent to control * *,NX NX NY NYl l l l→ → . The specific derivation of the process is not elaborated, only to give the final results:

1 1

1 1

( cos )cos ( sin )sin

( cos )sin ( sin )cosNX i j j i i j j i

NY i j j i i j j i

l x x d y y d

l x x d y y d

θ θ θ θθ θ θ θ

= − − − − − −

= − − − − −

Define the error variable * *1 2,NX NX NY NYe l l e l l= − = − ,

thus

*1 2 1

*2 1 1

cos sin

sin cosi j ij j ij i NY i

i j ij j ij i NX

ij j i

e e v d l v

e e v d l

ω β ω β ω

ω β ω β ω

β ω ω

= − + − +

= − − − +

= −

(3)

The goal of robot system control is making a simple, common sliding surface function s under a condition that the actual trajectory tracking on the desired trajectory. The error functions of formation shape

1 2,e e are used to define the sliding mode surface vector s , where 1 2( , )Ts s s= . 1 2,e e would converge to zero as long as the sliding surface functions converge to zero.

Theorem: Define sliding surfaces

1 1 1 1

2 2 2 2

s e k es e k e

= += + (4)

Here 1 2,k k are positive constants. If the sliding

surfaces 1 2,s s asymptotically stable, then 1 2,e e asymptotically converge to zero. The control law can be defined as follows.

[ ][ ] [ ] [ ]

1

1 2 1 2

(

sgn( ))

T T Tj j j j i i

T T Ti i

v A B v C v

D v E e e F e e G s

ω ω ω

ω

−⎡ ⎤ ⎡ ⎤= − −⎣ ⎦ ⎣ ⎦

− − − − (5)

Matrix , , , , ,A B C D E F are given in the following proof, where the function of sgn( )s is defined

as [ ]1 2sgn( ) sgn( ) Ts s , and for x R∀ ∈ , when 0,sgn( ) 1x x> = , when 0,sgn( ) 0x x= = , when 0,sgn( ) 1x x< = − , this control law makes the sliding mode surfaces globally asymptotically stabilizes to zero.

Proof: In order to obtain the control law T

j jvμ ω⎡ ⎤= ⎣ ⎦ , we need to calculate 1 2,s s

1 1 1 1

2 2

* *1 1

*1 2 1

2 2 2 2

1 1

1 1

cos sin

sin cos

( cos sin )

sin cos

cos

i i j ij j ij ij

j ij j ij ij i NY i NY

i i j ij j ij i NY i

i i j ij j ij ij

j ij

s e k e

e e v v

d d l l

v k e v d l v

s e k e

e e v v

d d

ω ω β β β

ω β ω β β ω ω

ω β ω β ω

ω ω β β β

ω β

= +

= + − +

+ + − −

+ + − + − +

= +

= − − − −

− + * *

*2 1 1

sin

( sin cos )j ij ij i NX i NX

i j ij j ij i NX

l l

k e v d l

ω β β ω ω

ω β ω β ω

+ +

+ − − − +

(6)

Thus obtains

[ ][ ]

[ ] [ ] [ ]

1 2

1 2 1 2

T

T T Tj j j j i i

T T Ti i

s s s

A v B v C v

D v E e e F e e

ω ω ω

ω

=

⎡ ⎤ ⎡ ⎤= + +⎣ ⎦ ⎣ ⎦

+ + +

(7)

Where 1

1

1 1 1 1

2 1 2 1

*2

*1

* *1 2 1 2 1

*1 2 1

cos sinsin cos

sin cos cos sincos sin sin cos

10

0

ij ij

ij ij

ij ij ij ij ij ij

ij ij ij ij ij ij

NY

NX

NY NY

NX

dA

d

k d k dB

k d k d

e lC

e l

k e l k e k lD

e l k e

β ββ β

β β β β β ββ β β β β β

−⎛ ⎞= ⎜ ⎟− −⎝ ⎠⎛ ⎞− +

= ⎜ ⎟⎜ ⎟− − −⎝ ⎠⎛ ⎞−

= ⎜ ⎟− +⎝ ⎠

− + −=

− + − *2

1

2

00

00

NX

i

i

i i

i i

k l

E

kF

k

ωω

ω ωω ω

⎛ ⎞⎜ ⎟

+⎝ ⎠⎛ ⎞

= ⎜ ⎟−⎝ ⎠+⎛ ⎞

= ⎜ ⎟− −⎝ ⎠

Consider the Lyapunov function 12

TV s s= , it can

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be obtained

[ ][ ] [ ] [ ]1 2 1 2

(

)

T

T T TTj j j j i i

T T Ti i

V s s

s A v B v C v

D v E e e F e e

ω ω ω

ω

=

⎡ ⎤ ⎡ ⎤= + +⎣ ⎦ ⎣ ⎦

+ + +

(8)

11 2 1 1 2 2

2

0( , ) , ,

0 m m

fG diag f f f f f f

f⎛ ⎞

= = ≤ ≤⎜ ⎟⎝ ⎠

Under the condition of existing outside interference, 1mf and 2mf are the upper bounded of interference of the

robots' left and right wheels, thus 1 1 2 2V f s f s= − − (9)

For any of 0s ≠ , the function is strictly less than 0, so the control law makes the sliding mode surfaces globally asymptotically stabilizes to zero.

IV. SIMULATION RESULT

To verify the formation of sliding mode control strategy, two robots are used for simulations. In these two robots: iR is the leader robot, jR is the follower robot. In order to observe the robots' movement trajectory, assuming that the leader robot's linear speed and angular velocity are vi=1.5+t/(t+10) (m/s), wi=1rad/s respectively; select parameters k1=k2=12, f1=f2=0.5N, d1=0.1m; When the leader’s initial positions are xi(0)=0, yi(0)=-1.5m, (0) 0iθ = then the follower’s are xj(0)=2.2m, yj(0)=-1.8m, (0) / 2jθ π= , also could get e1(0)=0.6, e2(0)=0.8, the simulation results are as follows.

In Fig.4, the red line represents the leader's trajectory, and the blank line represents the follower's trajectory. From the trajectory it can be seen that there is a certain bias between the movement of the leader robot and the follower robot at the first time, but with the extended of time, the trajectories of the two robots meets strict consensus, it indicates that under the sliding mode control strategy, the formation can achieve strictly consistent, which also shows the effectiveness of the control strategy. From Figure 5 (a) - (c), it can be seen that the distance error variables and orientation error variable are all gradually approaching to zero, indicating that the proposed controller algorithm enables robots to maintain desired distance and angle to achieve a desired formation.

Fig.3 Control inputs jv and jω .

Fig.4 Robot trajectory

Fig.5 (a) Robot distance error 1e

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Fig.5 (b) Robot distance error 2e

Fig.5(c) Robot orientation error 3e

V. CONCLUSIONS

This paper discusses a method of robot formation control based on leader-follower, and makes an important discussion on two robots, as well as simulation. The simulation results show that in robot formation control based on leader-follower, the proposed sliding mode control strategy is robust to uncertainties such as measurement errors of the relative distance and estimation of the leader's orientation and linear and angular velocities, thus the robot formation can reach a steady state. The desired formation is mainly determined by two parameters (distance function and declination function), these two parameters are allowed to change over time. In this article, the specific simulation is made for two robots, future work will cover more than one leader and one follower in validation of the proposed control strategy, expansion of the simulation results.

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