Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 10–13___________________________________________________________________________________________________________Leader-follower Formation Control of Mobile Robots with Sliding Mode
Wenhao Zheng , Yingmin Jia

The Seventh Research Division and the Center for Information and Control,
School of Automation Science and Electrical Engineering,
Beihang University (BUAA), 37 Xueyuan Road, Haidian District,
Beijing 100191, China
E-mail: zhengwenhao152@163.com, ymjia@buaa.edu.cn
www.buaa.edu.cn

Abstract
This paper considers the formation control of nonholonomic mobile robots. The formation problem is converted tothe error model based on the leader-follower structure. A sliding mode controller, which is proved to be globally
finite-time stable by Lyapunov stability theory, is presented in this study. In addition, a continuous reaching law isdesigned to reduce the chattering which caused by the computation time delays and limitations of control.
Simulation results verify the feasibility and effectiveness of the control strategy.
Keywords: nonholonomic mobile robot, leader-follower, formation control, sliding mode, continuous reaching law.

designed to limited control inputs, was presented by
Lyapunov theory, and geometric analysis contributed tothe designment of optimal feedback functions which
made the controller more effective.
This paper mainly focuses on the formation control
of nonholonomic mobile robots with sliding mode. The
formation problem is proposed with leader-follower
setup, which is used to deduce the error model. Then, asimple sliding mode surface is chose to design control
law. And a continuous reaching law is presented toreduce the chattering. Moreover, the formation error
model is proved to be globally finite-time stable.
Comparing with existing results, this paper primarily
contributes to the novel solution of the formation
control with sliding mode.
The structure of the rest paper is organized asfollows. The section 2 introduces leader-follower
system and the formation error model. In the section 3,
the sliding mode controller is designed with acontinuous reaching law. Simulation results are

1. Introduction
Formation control of differential-drive mobile robots
have been widely researched in recent years. The
mobile robots are supposed to maintain the desiredgeometric configuration in lots of different situations
including personnel rescue, logistics transportation,
military affairs and environmental exploration. Various
formation control strategies have been proposed, such as
feedback linearization1, backstepping2, sliding-mode
control3 and sorts of intelligent control method. In Ref.3,
the kinematic model and the dynamic model were bothconsidered, then a sliding mode controller was proposed
to deal with model uncertainties and disturbances,
furthermore, the voltage inputs of driving motors weretaken as control inputs, which are more realistic than thevelocity. Basic controller based on backstepping method
was presented in Ref.4, a bioinspired neurodynamics
based approach was further developed to solve theimpractical velocity jumps problem. In Ref.5, acontroller with time-varying parameters which wereCopyright © 2017, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 10–13___________________________________________________________________________________________________________

that the distance between the point O f ( xof , yof , θ f ) andCl is d . The relation can be presented as (4)presented to show the validity of the control law in thesection 4. In the end, the section 5 summarizes thewhole paper and draws the conclusion.
2. Problem Statement

xof=x f + d cos θ f

yof=y f + d sin θ f(4)

Differentiating the above equations leads to

2.1. Kinematic model of nonholonomic mobilerobotxof v f cos θ f − d ω f sin θ f=y of v f sin θ f + d ω f cos θ f=

In this paper, the differential-drive mobile robot is
considered as the research object, which is subject tononholonomic constraint (1).
As shown in figure 1, let vL and vR be thevelocities of the left and right driving wheel. The
control inputs are usually taken as the linear speed v
and the rotational angular velocity ω . The relation(5)

The desired geometric shape can be denoted byFig. 2. Leader-follower formation structure

coordinate representation. The expected position of thefollower should be expressed as ( xd , yd ) in thecoordinate system of the leader. And the real position
Cd ( xc , yc ) in world frame can be deduced as equation(6).

Fig. 1. The configuration of a two-wheeled mobile robot.

between control inputs and driving wheels can beexpressed as (2). In practical systems, control inputs ofmobile robots are bounded.
y cos θ − x sin θ =0(1)v=(vL + vR ) / 2, ω =( vR − vL ) / 2 ρ(2)x =v =cos θ , y v=
sin θ, θ ω(3) xc   xl   cos θ l
=  + yc   yl   sin θ l− sin θ l   xd  cos θ l   yd (6)( xd , yd ) are considered as constants, and differentiating
both side of (6) will refer towhere ( x, y ) are the robot position and θ is the robot
orientation. Moreover, the kinematic of the mobile robot
can be expressed as equation (3).

 xc   cos θ l  =
 yc   sin θ l− sin θ l   vl − yd ωl cos θ l   xd ωl (7)We can define the formation errors as the Eq. (8).

2.2. Leader-follower system xe   cos θ f  =  − sin θf ye  

In the figure 2, leader-follower formation structure ispresented in Cartesian coordinates. The nonholonomic
mobile robot Rl is treated as the leader. The center ofthe leader Cl ( xl , yl , θl ) describes the position and theorientation of the leader in the world frame. And (vl , ωl )are control inputs. The follower is the robot R f . The
center of the robot C f ( x f , y f , θ f ) and inputs (v f , ω f )have the analogous definitions like the leader. Assumesin θ f  xc − xof cos θ f  yc − yof (8)Moreover, the angular difference between the leader
and the follower is remarked as (9).

β=θl − θ f , β =ωl − ω f11(9)Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 10–13___________________________________________________________________________________________________________xe =−v f + ω f ye + (vl − yd ωl ) cos β − xd ωl sin βy e =−(d + xe )ω f + (vl − yd ωl )sin β + xd ωl cos βHere, f ( s ) is given as (16) and sgn( s ) is a sign function.
TTBesides,=ε (ε=(k=(α1 , α 2 )T , ϕ > 01,ε2 ) , k1 , k2 ) , αand ε1 , ε 2 , k1 , k2 > 0, 0<α1 , α 2 < 1 .

(10)

 si αi sgn( si )fi ( s) = αi siϕ / si

The destination of control law design is to find theappropriate control inputs (v f , ω f ) to meet the desiredoutcome:lim xe → 0, lim ye → 0, β < σ (σ > 0)
t →∞

t →∞

(11)

(e1 s1 + k1 f1 ( x)) cos β − (e 2 s2 + k2 f 2 ( x))sin βωf =The most significant process of sliding mode control isto choose the suitable sliding surface. Here, we design
the sliding surface which ensures that the formation
error will converge to zero when the system trajectorylies on the sliding surface (12).
(12)

Proof. Choose the Lyapunov function as Vi = si2 / 2(i = 1, 2) . It is easy to refer toω f =[(vl − yd ωl )sin β + xd ωl cos β − xeωl − s1 sin β − s2 cos β ] / d(18)Vi =si si =−ε i si2 − ki f i ( si ) si

Case1: if si < ϕ , the equation (18) will be converted toVi =−2ε iVi − 2(1+αi )/ 2 kiVi (1+αi )/ 2 . Define V = Vi (0) at t = 0 .
By solving differential equation, we can deduce formula1−α i

α i −1

Vi 2 =−2 2k

α i −1+ (2 2kee

1−α i+ Vi 2 (0))e − (1−αi )e t (19)Let Vi = 0 ; the reaching time can be expressed as(13)

α i −1Furthermore, the formation controller can be simplywritten as:v f =(vl − yd ωl )cos β − xd ωl sin β + yeωl − s1 cos β + s2 sin β(17)1[(vl − yd ωl )sin β + xd ωl cos β − xeωl +d(e1 s1 + k1 f1 ( x))sin β + (e 2 s2 + k2 f 2 ( x)) cos β ]Theorem 1. Consider the formation error system (10).
Using the control law (17), the sliding surface (12) will
converge to zero in finite time Tr , and xe → 0, ye → 0
as t = Tr .

Let V = 0.5sT s , thus we can deduce=V 0.5( s12 + s22 )22= 0.5( xe + ye ) from equation (12), which indicates that
if s1 → 0, s2 → 0 , the xe → 0, ye → 0 will hold. Hence,
when the sliding surface converges to zero, thegeometric shapes of the robot formation will beestablished.
Differentiating the formula (12) will conclude (13)s=v f sin β − d ω f cos β + xd ωl − s1ωl2(16)if si > ϕv f =(vl − yd ωl ) cos β − xd ωl sin β + yeωl +3.1. Sliding surface designs1 =−v f cos β − d ω f sin β + vl − yd ωl + s2ωlif si ≤ ϕThus, we can get the sliding mode controller (17)from equation (12), (14) and (15).

3. Sliding-Mode Controller Design

 s1   xe cos β + ye sin β =s =  s2   − xe sin β + ye cos β (15)s =−ε s − kf ( s )Combining the equations from (4) to (7) anddifferentiating both sides of (8), formation error model
can be expressed asTri =(14)1ln((1 − α i )ε2 2kε21−α i+ Vi 2 (0)α i −12)(20)

kt T=Therefore, si (i = 1, 2) will converge to zero at =rmax(Tri ) (i = 1, 2) and xe → 0, ye → 0 as t = Tr .
Case 2: if si > ϕ , the equation (18) can be presented as
the formula (21).

3.2. Controller design with the reaching law
In this paper, a continuous reaching law is adopted tomake each state reaching the sliding surface. Comparing
with the switching function of general reaching law, asaturation function is taken in order to reduce thechattering which caused by the computation time delays
and limitations of control. The reaching law is as follow:

Vi =−2ε i si2 − ki si ϕ αi < 0(21)Consequently, there must be a Trϕ which makes si → ϕ .
Thus, formation error model will be globally finite timestable.

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 10–13___________________________________________________________________________________________________________

Theorem 2. Consider the equation (9) with the control−1m, yd 1 =−0.5m . The movement ofxd 2 =0.5m, yd 3 =law (17). β will be bounded, if the velocity of thethe leader robot is set as the circular motion withleader
andthe =vl 1=m / s, ωl π / 12rad / s , and the initial poses of
0 < vl < const , β 0 < π at t =0
boundedness of the angular velocity ωl hold.
three robots are (0, 0, π/2), (−3, −2, 0) and (2.5, −3,−π/2). Moreover, Controller (16) is used withProof. Combining the formula (9) with the control law,
ε1 = ε 2 = 5,
we can obtain the equation (22).
k1 = k=7, α=α=
1 / 3, ϕ= 5, d=d=0.25m. The21223controlinputs
arelimitedasβ =−(vl sin β ) / d + f (t )v f ≤ 3m / s, ω f ≤ π / 2rad / s .
1The formation error is denoted as=
Exe2 + ye2 , . The
f (t ) = ωl − [− yd ωl sin β + xd ωl cos β − xeωl + (22)dsimulation results are presented in Fig.3 and Fig.4.
(e1 s1 + k1 f1 ( s1 )) cos β + (e 2 s2 + k2 f 2 ( s2 )α 2 )sin β ]5. Conclusion
The nominal system can be given as β = −(vl sin β ) / d
In this paper, the formation control problem of theat φ = 0 . Select the Lyapunov function as Vβ = 1 − cos β .
2nonholonomic mobile robot has been solved. ASo we can get Vβ =−(vl sin β ) / d ≤ 0 and Vβ is noncontroller with sliding mode is proposed using aincreasing. β 0 < π at t =0 , there must exist a positive
continuous reaching law. The simulation results show
constant δ which meets β 0 ∈ [−π + δ , π − δ ] . Then, Vβthat the control law can guarantee great formation∈ [0, 2 − δ ), β (t ) ∈ [−π + δ , π − δ ] and Vβ ≤ −(vl / d )d Vβ .
222
performance and the specified geometrical shapes ofBesides, Vβ = 2sin ( β / 2) , it is easy to obtain 2 β / π2
multi-mobile robots will be achieved in finite time.
< Vβ < β / 2 at β ∈ (−π , π ) . Thus, the nominal system
is exponentially stable using the Lyapunov method.
Acknowledgements
Obviously, ωl , xe , s1 , s2 ,sin β and cos β are bounded,
This work was supported by the NSFC (61327807,
which can be used to deduce the conclusion that φ (t ) is61521091, 61520106010, 61134005) and the Nationalbounded. Therefore, in the system (21), β is bounded
Basic Research Program of China (973 Program:by using the stability theory of perturbed systems.
2012CB821200, 2012CB821201).
4.

Simulation Results

References
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2. X. Li, J. Xiao, and Z. Cai, “Backstepping based multiple
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3. Park, Bong Seok, B. P. Jin, and Y. H. Choi. "Robustformation control of electrically driven nonholonomic
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Fig. 3. Formation process and formation errors.

Fig. 4. The sliding surface and inputs of followers.

Three robots are expected to keep a triangle formationand the expected formation shapes are designed as xd 1 =13