Formation Control of Multirobot Based on I/O Feedback Linearization and Potential Function

Abstract

Standard techniques of I/O linearization are widely applied to leader-follower approach for multirobot formation control. However general leader-follower approach cannot adapt to the environment with obstacles. Concerning that issue, a formation control method of multirobot system based on potential function is proposed in this paper, and a new control law is designed by choosing a proper potential function and employing Lyapunov stability theory, which stabilizes the formation of the multirobot system. We combine the method with a leader-follower approach to solve the problem that the latter cannot avoid obstacles. Simulation results are given to validate the method.

1. Introduction

Multirobot formation control is a common cooperation problem of multirobot system. In recent years, as one of the basic coordination and cooperation issues, it has been gradually applied to military reconnaissance, search, demining, aircraft formation flying, space exploration, and many other fields and attracted a large number of researchers. It has become a very active research direction of multirobot system [1]. At present, formation control methods mainly include leader-follower approach [2–5], artificial potential field approach [6–9], behavior-based approach [10], and virtual structural approach [11]. To achieve a reasonable and efficient formation, the key is to use a proper control method that meets the system requirements. No matter what kind of control method is used, the stability of the system must be analyzed. Lyapunov stability theory [12–14] and graph theory [15, 16] are usually used in stability analysis, and these control theory researches contribute to the improvements of multirobot system. Lyapunov method plays a very important role in a multirobot system’s stability analysis and controller design. In [17], the idea of relative-position-based formation stability was proposed and the Lyapunov method was also used to design the decentralized controllers, along with an extended linear matrix inequality (LMI) to analyze the conditions required for formation stability. In [18], Lyapunov method was used to analyze the stability of multiagent coordination control. For each individual robot, there is a control Lyapunov function, and the Lyapunov function of the system is the weighted sum of all the agents’ Lyapunov functions. However, that paper presents a method based on Lyapunov stability theory, which gives a stable formation control law of the multirobot system by selecting an appropriate artificial potential field function. We combine this method with 𝑙−𝜑 control based on input-output feedback linearization [19] to solve the problem that 𝑙−𝜑 control cannot adapt to obstacles environment.

2. Mathematical Model

In this section, the kinematics and dynamics of mobile robots are described in Figure 1. The robot consists of two driving rear wheels and a supporting roller, and the nonholonomic constraint that the driving wheels allow only pure rolling without slipping is satisfied. The kinematic and dynamic equations of motion of the 𝑖th individual robot are given as 𝑥̇_𝑖=𝑣_𝑖cos𝜃_𝑖,𝑦̇_𝑖=𝑣_𝑖sin𝜃_𝑖,𝜃̇_𝑖=𝜔_𝑖,𝑣̇_𝑖=𝛼_𝑖,𝜔̇_𝑖=𝛽_𝑖, (1) where the position of the robot is given by (𝑥_𝑖,𝑦_𝑖)^𝑇. 𝜃_𝑖 is the heading direction of the robot in the global coordinates. 𝑞_𝑖=(𝑥_𝑖,𝑦_𝑖,𝜃_𝑖)^𝑇 is the pose vector of robot 𝑖. 𝑣_𝑖 and 𝜔_𝑖 are the linear and angular velocities at the center of the axle of each robot. 𝑢_𝑖=(𝛼_𝑖,𝛽_𝑖)^𝑇 is control input (acceleration).

Figure 1 

Mathematical model of the robot.

3. Control of Formations

3.1. 𝑙−𝜑 Control Based on Input-Output Feedback Linearization

𝑙−𝜑 control is a kind of leader-follower approach. A group of two nonholonomic mobile robots is shown in Figure 2. The aim of the controller is to make the follower (Robot2) follow the leader (Robot1) by keeping the relative distance 𝑙₁₂ and the relative orientation 𝜑₁₂ constant. As a result, both the values can be controlled to reach the desired value (𝑙^𝑑₁₂,𝜑^𝑑₁₂) and maintain the desired formation.

Figure 2 

Notation for 𝑙−𝜑 control.

We introduce 𝛾₁=𝜃₁+𝜑₁₂−𝜃₂; according to the geometrical relationship, the differential equation of 𝑙₁₂ and 𝜑₁₂ is given by 𝑙̇₁₂=𝑣₂cos𝛾₁−𝑣₁cos𝜑₁₂+𝑑𝜔₂sin𝛾₁,𝜑̇₁₂=1𝑙₁₂𝑣₁sin𝜑₁₂−𝑣₂sin𝛾₁+𝑑𝜔₂cos𝛾₁−𝑙₁₂𝜔₁,𝜃̇₁₂=𝜔₂, (2) where 𝑣_𝑖 and 𝜔_𝑖(𝑖=1,2) are the linear and angular velocities at the center of the axle of each robot. The distance between the supporting roller and the center of the axis of the driving wheels of each robot is denoted by 𝑑. In order to avoid collisions between robots, we will require that 𝑙₁₂>𝑑.

We use standard techniques of I/O linearization to generate a control law [12], which is given by 𝜔₂=cos𝛾₁𝑑𝑎₂𝑙₁₂𝜑^𝑑₁₂−𝜑₁₂−𝑣₁sin𝜑₁₂+𝑙₁₂𝜔₁+𝜌₁₂sin𝛾₁,𝑣₂=𝜌₁₂−𝑑𝜔₂tan𝛾₁, (3) where 𝜌₁₂=𝑎₁𝑙^𝑑₁₂−𝑙₁₂+𝑣₁cos𝜑₁₂cos𝛾₁. (4) That leads to dynamics in the 𝑙−𝜑 variables of the form 𝑙̇₁₂=𝑎₁𝑙^𝑑₁₂−𝑙₁₂,𝜑̇₁₂=𝑎₂𝜑^𝑑₁₂−𝜑₁₂, (5) where 𝑎₁ and 𝑎₂ are constants. By selecting appropriate values, the relative distance and angle between the robots can converge to (𝑙^𝑑₁₂,𝜑^𝑑₁₂), and the system becomes stable.

3.2. Control Based on Artificial Potential Field Function

An artificial potential field approach is a virtual force method proposed by Khatib. Its basic idea is that the robots move in a virtual force field. Obstacles are surrounded by repulsive potential field, which produces a repulsive force that decreases rapidly with the closer distance between the robot and the obstacle. The target is surrounded by attractive potential field, which produces an attractive force that decreases as the robot is close to the target. Under the role of the resultant force, the robots move along the direction of minimizing potential energy. For multirobot formation control, we need to design a suitable potential function to maintain the formation while reaching the target and avoiding obstacles. Potential function is defined as follows.

Definition 1 (potential function). Potential function 𝑉_𝑖𝑗 is a differentiable, nonnegative, and unbounded function of the distance ‖𝑟_𝑖𝑗‖ between robot 𝑖 and 𝑗, and the following conditions are satisfied.(1)When ‖𝑟_𝑖𝑗‖→∞, 𝑉_𝑖𝑗(‖𝑟_𝑖𝑗‖)→∞.(2)When the distance ‖𝑟_𝑖𝑗‖ achieves the desired distance, that is, ‖𝑟_𝑖𝑗‖=𝑑_𝑖𝑗, 𝑉_𝑖𝑗 gets the only minimum value.The potential function of neighbor robot 𝑖 and 𝑗 is chosen as follows: 𝑉_𝑖𝑗𝑞_𝑖𝑗=𝐾_𝑖𝑗𝑞_𝑖𝑗ln𝑞_𝑖𝑗+𝑞²_𝑖𝑗+1−𝑞²_𝑖𝑗+1+1, (6) where 𝑞_𝑖𝑗=‖𝑟_𝑖𝑗‖−𝑑_𝑖𝑗, 𝐾_𝑖𝑗 is a positive constant, and then the force is the negative gradient of the potential energy: 𝑓_𝑖𝑗𝑟_𝑖,𝑟_𝑗=−grad𝑉_𝑖𝑗=𝐾_𝑖𝑗arsh𝑞_𝑖𝑗𝑟_𝑗−𝑟_𝑖𝑟_𝑖𝑗, (7) where 𝑟_𝑖, 𝑟_𝑗 are the position vectors of robots 𝑖 and 𝑗.

The potential energy 𝑉_𝑖𝑗 and force𝑓_𝑖𝑗 described by (6) and (7) are shown in Figure 3. It indicates that when 𝑞_𝑖𝑗>0, that is, ‖𝑟_𝑖𝑗‖>𝑑_𝑖𝑗, 𝑓_𝑖𝑗>0, the robots are attractive, when 𝑞_𝑖𝑗<0, that is, ‖𝑟_𝑖𝑗‖<𝑑_𝑖𝑗, 𝑓_𝑖𝑗<0, the robots are repulsive, and when 𝑞_𝑖𝑗=0, that is, ‖𝑟_𝑖𝑗‖=𝑑_𝑖𝑗, 𝑓_𝑖𝑗=0, the force between the robots achieves balance, and 𝑉_𝑖𝑗 is minimized.

(a)

(b)

Figure 3 

Potential energy and force.

4. Control Based on I/O Feedback Linearization and Potential Function

4.1. Design of a Stable Formation Control Law

Considering a system of n robots, the dynamic of each robot is described by (1). The total potential energy and force acting on robot 𝑖 contributed from all the other robots are given as 𝑉_𝑖𝑟_𝑖=_𝑗≠𝑖𝑉_𝑖𝑗𝑞_𝑖𝑗,𝐹_𝑖𝑟_𝑖=_𝑗≠𝑖𝑓_𝑖𝑗𝑟_𝑖,𝑟_𝑗. (8)(9) When the formation moves toward the target in an environment with obstacles, 𝐹_𝑖(𝑟_𝑖) also includes the attractive force of the target and the repulsive force of the obstacles. The magnitude of 𝐹_𝑖(𝑟_𝑖) projected in the heading direction of the robot is used to control the translational acceleration of the robot. The angle is used as the desired orientation for the heading direction of the robot. Let 𝜙_𝑖 be the direction of the resultant force of robot 𝑖, and the orientation error can be described as 𝑒_𝑖=𝜃_𝑖−𝜙_𝑖,𝑒_𝑖̇=𝜃_𝑖̇−𝜙_𝑖̇,𝑒_𝑖̈=𝜃_𝑖̈−𝜙_𝑖̈. (10)

Theorem 2. For a multirobot system described by (1), the potential force shown by (9) acts on each robot, and the angle of the resultant force is defined as the desired orientation of them. Then, the following control law stabilizes the formation of the system: 𝛼_𝑖=−𝑓²_𝑖𝑥+𝑓²_𝑖𝑦−𝑘₁𝑣_𝑖,𝛽_𝑖=−𝑘₂𝜃_𝑖−𝜙_𝑖−𝑘₃𝜃_𝑖̇−𝜙_𝑖̇+𝜙_𝑖̈, (11) where 𝑘₁, 𝑘₂, 𝑘₃ are positive constants and 𝑓_𝑖𝑥, 𝑓_𝑖𝑦 are the components of 𝐹_𝑖(𝑟_𝑖) in the 𝑥 and 𝑦 directions, respectively.

Proof. (1) 𝑣_𝑖≥0. Consider the following Lyapunov candidate: 𝑊=^𝑛_𝑖=1𝑉_𝑖+12^𝑛_𝑖=1𝑘₂𝑒²_𝑖+12^𝑛_𝑖=1𝑒²_𝑖̇+12^𝑛_𝑖=1𝑣²_𝑖. (12) The time derivative of 𝑊 is computed as 𝑊̇=^𝑛_𝑖=1∇_𝑟𝑖𝑉_𝑖^𝑇𝑟_𝑖̇+𝑘₂𝑒_𝑖𝑒_𝑖̇+𝑒_𝑖̇𝑒_𝑖̈+𝑣_𝑖𝑣_𝑖̇. (13) Insert (11) into (13): 𝑊̇=^𝑛_𝑖=1𝑣_𝑖𝑓_𝑖𝑥cos𝜃_𝑖+𝑣_𝑖𝑓_𝑖𝑦sin𝜃_𝑖−𝑘₃𝑒²_𝑖̇−𝑣_𝑖𝑓²_𝑖𝑥+𝑓²_𝑖𝑦−𝑘₁𝑣²_𝑖=^𝑛_𝑖=1𝑣_𝑖𝐹_𝑖cos𝜃_𝑖−𝜙_𝑖−1−𝑘₃𝑒²_𝑖̇−𝑘₁𝑣²_𝑖, (14) where |𝐹_𝑖|=𝑓²_𝑖𝑥+𝑓²_𝑖𝑦, due to cos(𝜃_𝑖−𝜙_𝑖)≤1, so 𝑊̇≤0.
(2) 𝑣_𝑖<0. Consider the following Lyapunov candidate: 𝑊=2^𝑛_𝑖=1𝑉_𝑖+12^𝑛_𝑖=1𝑘₂𝑒²_𝑖+12^𝑛_𝑖=1𝑒²_𝑖̇+12^𝑛_𝑖=1𝑣²_𝑖. (15) Insert (11) into (15), and then calculate the derivative of 𝑊: 𝑊̇=^𝑛_𝑖=1𝑣_𝑖𝐹_𝑖2cos𝜃_𝑖−𝜙_𝑖−1−𝑘₃𝑒²_𝑖̇−𝑘₁𝑣²_𝑖. (16)
From (10), 𝑒_𝑖=𝑐₁cos4𝑘₂−𝑘²₃2𝑡+𝑐₂sin4𝑘₂−𝑘²₃2𝑡𝑒^{−(𝑘3/2)𝑡}. (17)
Let 𝑐₁=𝑐₂=0.5, and then |𝑒_𝑖|<1, that is, 2cos(𝜃_𝑖−𝜙_𝑖)−1=2cos𝑒_𝑖−1>0, so 𝑊̇≤0.
Consequently, the stability of the formation is proved with Lyapunov stability theory.

4.2. Combination Control Method

𝑙−𝜑 control based on I/O feedback linearization is simple, that we just control the follower robot to follow the trajectory of the leader robot. In this way, the formation control can be simplified to an independent tracking problem. Each robot only needs to obtain the status information of its leader, which greatly simplifies the issue of cooperation among robots. However, this feedback control cannot avoid obstacles, while the artificial potential field approach is effective in obstacle avoidance. For this reason, if we combine those two methods, the problem will be solved.

The strategy is as follows. The leader adopts the artificial potential field approach for path planning to reach the target while avoiding obstacles. The follower uses I/O feedback linearization combination with potential function to maintain formation in the obstacles environment; that is, when the follower is in the region influenced by obstacles, the combination method is applied, and the control performance can be improved by adjusting the weight, and then when out of the influenced region, 𝑙−𝜑 control based on I/O feedback linearization is only used. The algorithm flow chart is shown in Figure 4.

Figure 4 

The algorithm flow chart of combination method.

5. Simulation

In this section, computer simulation is used to verify the previous conclusions. For a group of 3 robots, the robots maintain triangular formation to reach the target while avoiding obstacles, which were idealized as circular objects just as described in [20].

(1) Formation Control Based on Potential Function. The initial positions of the robots are −20, 03, and 00. The position of the target is 2520. The obstacle is a circle of radius 1 centered at 79. Under the role of the potential energy described by (6) and applying the control law (11), the robots form an equilateral triangle formation with the side length of 3, avoid the obstacle, keep the same speed and direction, and reach the target at last just as shown in Figure 5. Figure 6 illustrates the change of errors 𝑒_𝑖. It is observed that the formation is stable.

Figure 5 

Formation control based on potential function.

Figure 6 

The orientation error of robots.

(2) Control Based on 𝑙−𝜑 and Potential Function. The initial positions of the robots are 01, −3−2, and −20. The position of the target is 2016. The obstacle is a circle of radius 1 centered at 97. Robot1 is the leader, and Robot2, Robot3 are the followers keeping a distance of 2, angles of 𝜋/3, −𝜋/3, respectively, from Robot1. When the formation enters into the influenced region of the obstacle, use the approach of control based on 𝑙−𝜑 and potential function. Otherwise, 𝑙−𝜑 control based on I/O feedback linearization is applied. Figure 7 illustrates that single 𝑙−𝜑 control cannot avoid obstacles, while by using the proposed combination control method, the formation can successfully round the obstacle and move to the target keeping a stable formation, as shown in Figure 8.

Figure 7 

𝑙−𝜑 control cannot avoid obstacles.

Figure 8 

Control based on 𝑙−𝜑 and potential function.

6. Conclusions

Formation control of multirobot system was studied in the paper. A new artificial potential field function and corresponding formation controller were designed, and the stability analysis was given. In addition, we proposed a combination control method based on I/O feedback linearization and potential function, of which feature is choosing the appropriate control strategy according to obstacles environment, solving the problem that 𝑙−𝜑 control cannot avoid obstacles.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of China (NSFC) under Grant no. 61074085 and by Beijing Natural Science Foundation (4142035), China.