Y. Tan, Y. Shi, and K.C. Tan (Eds.): ICSI 2010, Part II, LNCS 6146, pp. 291–298, 2010. © Springer-Verlag Berlin Heidelberg 2010

Leader-Follower Formation Control of Multi-robots by Using a Stable Tracking Control Method

Yanyan Dai, Viet-Hong Tran, Zhiguang Xu, and Suk-Gyu Lee

Department of Electrical Engineering, Yeungnam University 214-1 Daedong Gyeongsan Gyeongsbuk, Korea 712-749

guangyanyan1129@hotmail.com, t_v_hong@yahoo.com xuzhiguang629@hotmail.com, sglee@ynu.ac.kr

Abstract. In this paper, the leader-waypoint-follower robot formation is con-structed based on the relative motion states to form and maintain the formation of multi-robots by stable tracking control method. The main idea of this method is to find a reasonable target velocity and angular velocity to change the robot’s current state. The proposed Lyapunov functions prove that robots change cur-rent velocities to target velocities which we propose, in globally asymptotically stable mode. The simulation results based on the proposed approach show bet-ter performance in accuracy and efficiency comparing with EKF based ap-proach which is applied in multiple robots system in common.

Keywords: leader-waypoint-follower robots formation, stable tracking control method, EKF, Lyapunov function.

1 Introduction

Coordination of multiple mobile robots has received considerable attention over the past decade [1]. There are many advantages in coordination of multiple mobile robots comparing with independent robot. Multiple robots localization has been developed for tasks which can be accomplished more efficiently by a whole team of robots than just by single robot localization [2]. Multiple robots may also produce more accurate maps by using more information than single robot case as described in [3].

The formation problem for multiple robots system has been regarded as an impor-tant problem in multi robots system. When a team of mobile robots move along a desired trajectory, three aspects of requirements should be considered such as forma-tion forming, maintaining and switching. In formation forming, leader-follower me-thods [1], [2], [5] and [11], behavior-based methods [12] and virtual structure me-thods [13] have been proposed. This paper concerned with a leader-follower forma-tion control, where the follower robots keep track of the leader robot in the desired distance and bearing angle.

There are several methods to control robot in the multiple robots system to move, such as an Extended Kalman Filter[4], distributed Extended Kalman Filter[2], onboard sensor information[5], graph theory [6] and Lyapunov function control me-thod [1], [7], [8]. In [9], the EKF based approach approximates the posterior

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p X , Y|Z, U by a multivariate Gaussian which includes mean µ and cova-riance Σ . We linearize the nonlinear function g and h using standard Kalman filter equations in order to update this Gaussian. However, since a single vehicle measure-ment generally affects all parameters of the Gaussian, the update procedure requires long calculation time when applied in environments with many landmarks (or fea-tures). Moreover, in [3] and [5], since EKF requires long time to update when the follower robot calculates the desired state, the leader robot has moved. This case may cause hysteresis and reduce the accuracy of multiple robots formation. In addition, the extended Kalman filter is not good at minimizing the error of θ which is the most important aspect about the motion of the robot. To over these drawbacks, we propose a stable tracking control method which is not only stable as EKF but also produces smaller error than EKF with less time required.

2 Formation Control Framework

2.1 Modeling of the Robot

For the n mobile robots under consideration, the motion of each robot is described in terms of P x, y, θ , where x, y and θ are x coordinate y coordinate and bearing respectively. The trajectory of each robot has the form of x, y where input of each robot is velocity v and angular velocity ω. The model of robot R has the form of

x v cosθ .

y v sinθ . (1)

θ ω .

2.2 Problem Formulation

In the formation control which is considered in this paper, the n robots are con-trolled to move following the desired distance and desired angle in relative coordi-nate frame in a stable mode. In multiple robot system, one robot is assigned as the leader robot determining the follower robot’s motion by defining the desired dis-tance and desired bearing angle which the follower robot must follow. According to the desired distance and desired bearing angle, the follower robot calculates its waypoint and then moves toward it. In this system, leader and follower robots are linked by wireless connection. Therefore, the speed of the data connection and up-date is important since it produces direct effect on the overall formation in the mul-tiple robots system. There are two problems: the first one is how to define and keep the formation stable, the second one is how to accelerate the speed of connection and reduce calculation time.

2.3 Formation Control Framework

Let R and R be the leader and follower robot respectively. We denote d as the actual distance between R and R , d is the desired distance, and β as the actual

Leader-Follower Formation Control of Multi-robots 293

bearing angle from the orientation of follower robot to the axis connecting R and R , β is the desired bearing angle. The formation among leader robot R –waypoint R – follower robot R with desired distance and desired bearing angle is shown in Figure1. The waypoint of follower robot R is denoted by P x , y , θ which is calculated by Equation (2):

xyθ

x d cos β θy d sin β θθ . (2)

Fig. 1. Leader- waypoint- follower robots formation control framework

3 Stable Tracking Control Method for Each Robot

When controlling the motion of each robot, two points should be considered such as reducing the connection and calculation time, and keeping the robot motion stable. To achieve these goals, the proposed approach is motivated by the method introduced in [10]. In the proposed method, robot’s waypoint model is defined as Px , y , θ and robot current state model has the form of P x , y , θ . According to the waypoint and current state error posture, the robot calculates the target velocity and angular velocity v, ω and changes robot current state velocities to track it. Finally, based on the velocities of the robots, each robot produces the de-rivative of current state P to realize the goal of minimizing error and making the formation stable, especially the θ error which is the weak point of EKF. The deriva-tive of current state P is derived from the target velocity and angular velocity. Since this method does not require landmarks for measurement, the matrix is smaller than that in EKF. This contributes directly to accelerate the speed of calculation, solve the problem of hysteresis and improve accuracy and efficiency. The related equations are derived similarly as done in [10]. We define the waypoint and the current state error e x , y , θ P P and transfer state error e to state error posture E by the following equation.

E xyθ cosθ sinθ 0sinθ cosθ 00 0 1 e T e . (3)

0iβ

1itθ −

0id

itθ

1itθ −

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If P P , and the error e 0, then E 0. Using Equation (1) and an equal equa-tion x sinθ y cosθ 0 [10], the derivative matrix E can be derived.

E xyθ cosθsinθ0 v 001 ω 100 v yx1 ω . (4)

Based on the waypoint and state error posture, we propose target velocities as fol-lows:

vω v cosθ K xω y v K sinθ . (5)

where K and K are positive constants. To transfer the current state velocities to target velocities, the derivative matrix E is described by

E xyθ cosθsinθ0 v 001 ω 100 v yx1 ω cosθsinθ0 v 001 ω 100 v cosθ K x

yx1 ω y v K sinθ . (6)

Theorem. (Lyapunov stability of autonomous systems). Let be an equilibrium point for a system described by:

x f x . (7)

Where f:U R is a locally Lipschitz and U ⊂ R a domain that contains the origin. Let V: U R be a continuously differentiable, positive definite function in U.

1. If V x ∂V/ ∂x f is negative semidefinite, then x=0 is a stable equilibrium point.

2. If V x is negative definite, then x=0 is an asymptotically stable equilibrium point.

In both cases above V is called a Lyapunov function. Moreover, if the conditions hold for all x R and x ∞ implies that V x ∞, then x=0 is globally stable in case 1 and globally asymptotically stable in case 2.

Proposition. Assume we use the target velocities as the current state velocities. if waypoint velocities , and , , then as ∞, E=0 is an asymptotically stable equilibrium point.

Proof. We propose a natural Lyapunov function candidate for this system as follows:

V E x y 1 cosθ . (8)

Leader-Follower Formation Control of Multi-robots 295

Notice that V 0 0 and V E E 0 is positive definite. The derivative of V E along the trajectories of the system is given by

V E E x x y y θ sinθ

K x Kθ sinθ 0 . (9)

Therefore, by Theorem, as t ∞, the origin E=0 is globally asymptotically stable.

4 Simulation Results

Figure 2 shows simulation result of the square formation by one leader robot and three follower robots. In the simulation, the first follower robot is designed to keep the desired distance d 0.2m, the desired angle β π/4, and the second follower robot is supposed to maintain the desired distance d 0.2m, the desired angle βgle β π/4. In addition, the third robot should hold the desired distance d√0.08m, the desired angle β 0. Based on the tracking control method, each robot is controlled. By using experimental trial, the constant K 0.1s , K 0.28s was chosen.

Fig. 2. One leader and three follower robots square formation

Figure 3, 4 and 5 compare the state error, (x, y, θ of followe robot based on the EKF control method and stable tracking control method with the same input and mea-surement noises. The simulation was performed during 240 seconds and it was run 100 times to take average values. The leader robot’s velocity is kept at 0.3 m/s, while the angular velocity is changed smoothly. The input noise is σv = 0.01 m/s and σω = 0.01 rad/s. The results present clearly that the stable tracking control method is supe-rior to EKF when controlling each robot with smaller error.

In order to see the merit of stable tracking control method over EKF under high noisy condition, we simulated the system with input noise of 10 times bigger than those used in the above simulation, i.e. σv = 0.1 m/s, σω = 0.1 rad/s. The root mean square(rms) of error values were calculated and shown in Table 1.

10 10.2 10.4 10.6 10.8 11 11.2 11.4-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x-axis (meter)

y-ax

is (

met

er)

the x-y-position of leader robot and follower robots

leader robotfollower robot1follower robot2follower robot3

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Fig. 3. Comparison of the error of EKF and stable tracking control method in x-axis

Fig. 4. Comparison of the error of EKF and stable tracking control method in y-axis

Fig. 5. Comparison of the error of EKF and stable tracking control method in θ-axis

0 40 80 120 160 200 240-3

-2

-1

0

1

2

3x 10

-3

Time (sec)

x E

rror

(m

eter

)

x Error

x error: EKFx error: stable tracking control method

0 40 80 120 160 200 240-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-3

Time (sec)

y E

rror

(m

eter

)

y Error

y error: EKFy error: stable tracking control method

0 40 80 120 160 200 240-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

θ E

rror

(D

egre

e)

θ Error

θ error: EKFθ error: stable tracking control method

Leader-Follower Formation Control of Multi-robots 297

Table 1. The rms errors in various noisy conditions

x-error y-error θ-error

stable tracking

control method

σv=0.01 m/s σω=0.01 rad/s

2.0224 × 10-4 4.9353 × 10-4 0.0609

σv=0.1 m/s σω=0.1 rad/s

8.6353 × 10-4 0.0014 0.0999

EKF

σv=0.01 m/s σω=0.01 rad/s

5.2112× 10-4 0.0011 0.3223

σv=0.1 m/s σω=0.1 rad/s

0.0039 0.0041 1.2222

As shown in Table 1, at each noisy level, stable tracking control method shows bet-

ter performance comparing with EKF. Moreover, when the noisy level increased 10 times, the x, y and θ error caused by using stable tracking control method increased by 4.2698, 2.8367 and 1.6404 times while EKF produces the error by 7.4839, 3.7273 and 3.7921 times higher. Therefore, the proposed method shows increased stability for bigger noise condition comparing with EKF based approach.

5 Conclusion

In this paper, we presented the leader-waypoint-follower formation framework using a stable tracking control method based on the target velocities which are less complicated than other paper’s equations. The two advantages of the proposed control method include stable formation through navigation and enhanced accuracy of the motion despite reduced calculation time. The simulation results shows that the pro-posed method has better performance in tracking accuracy comparing with EKF based approach which is the common, fundamental control method. In the future, in order to improve the safety and stability of multi-robot system, we will consider clusters of robots. Cooperation of leader robots in each cluster will be deeply considered to en-hance efficiency of the whole system.

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