Target Tracking and Obstacle Avoidance for Multi-agent Systems

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1 International Journal of Automation and Computing 7(4), November 2010, DOI: /s z Target Tracking and Obstacle Avoidance for Multi-agent Systems Jing Yan 1 Xin-Ping Guan 1,2 Fu-Xiao Tan 3 1 Institute of Electrical Engineering, Yanshan University, Qinhuangdao , PRC 2 School of Electronic and Electric Engineering, Shanghai Jiao Tong University, Shanghai , PRC 3 School of Computer and Information, Fuyang Teachers College, Fuyang , PRC Abstract: This paper considers the problems of target tracking and obstacle avoidance for multi-agent systems. To solve the problem that multiple agents cannot effectively track the target while avoiding obstacle in dynamic environment, a novel control algorithm based on potential function and behavior rules is proposed. Meanwhile, the interactions among agents are also considered. According to the state whether an agent is within the area of its neighbors influence, two kinds of potential functions are presented. Meanwhile, the distributed control input of each agent is determined by relative velocities as well as relative positions among agents, target and obstacle. The maximum linear speed of the agents is also discussed. Finally, simulation studies are given to demonstrate the performance of the proposed algorithm. Keywords: Target tracking, obstacle avoidance, potential function, multi-agent systems. 1 Introduction In recent years, there is an increasing amount of research on the subject of unmanned system networks (UMSN) [1 4]. These networks can potentially consist of a large number of agents, such as unmanned underwater vehicles (UUV), unmanned aerial vehicles (UAV), and unmanned ground vehicles (UGV). UMSN provide numerous applications in various research fields, such as building automation, mail delivery, surveillance, and underwater exploration [5]. The advantages of UMSN over a single agent system include reducing cost and increasing robustness. Various military and civil applications often require agents to move autonomously in dynamic environment the target and obstacle are moving, rather than to simply follow a pre-planned path designated by an offline mission-level planning algorithm. Because the potential function method is simple and intuitional, it has been widely used in target tracking and obstacle avoidance for a multi-agent system [6 9]. This method is based on a simple and powerful principle, first proposed by Khatib in [10]. The agents are considered as particles that move in a potential field generated by the target and obstacle present in the environment. The target generates an attractive potential field while any obstacle generates a repulsive potential field. The agents immersed in the potential field are subject to the action of a force, which drives them to the target and keeps them away from the obstacle. In [4], a theoretical framework for design and analysis of distributed flocking algorithms for multi-agent dynamic systems was provided. The cases of flocking in free-space and presence of obstacle were both considered. According to behavior rules, three algorithms were proposed. But the Manuscript received August 3, 2009; revised November 30, 2009 This work was supported by National Basic Research Program of China (973 Program) (No. 2010CB731800), Key Program of National Natural Science Foundation of China (No ), and Key Project for Natural Science Research of Hebei Education Department (No. ZD200908). obstacle is assumed to be stationary, and then the proposed algorithms would not be suitable for path planning of multiagent systems in dynamic environment the obstacle is moving. Meanwhile, the conventional potential function method also exhibits shortages of local minima and goals unreachable with an obstacle nearby. To solve these shortages, some improvements were proposed [7, 8, 11]. In [7], the velocity of a target was included in the definition of the potential function. But the obstacle was still assumed to be stationary. In [8], the relative positions and velocities of the agents with respect to target and obstacle were included in the definition of the potential function. With the new potential function, agents were planned to track a moving target while avoiding the moving obstacle. However, this method is infeasible in practice. First, the measurement of an agent s velocity is very prone to noises which would directly affect the performance of the path planning. Second, position and velocity are different physical terms with different units, so it is hard to estimate the weighing of the position and velocity. Huang [9] proposed a velocity planning method for a single agent to track a moving target while avoiding collision with moving obstacle. Although this method is effective, it is only suitable for a single agent. In some applications, such as surveillance, rescue, and underwater exploration, it often requires multiple agents to perform a cooperative task. Therefore, the applications of this method fall under prodigious restriction. Based on the above considerations, a novel control scheme for target tracking and obstacle avoidance of multiagent systems is proposed in this paper. The objective is to make the agents track a moving target, while avoiding collision with a moving obstacle and other agents. According to the state whether an agent is within the area of its neighbors influence, two kinds of potential functions are presented. They are extern-potential function and interpotential function, respectively. Based on the information

2 J. Yan et al. / Target Tracking and Obstacle Avoidance for Multi-agent Systems 551 of the target and obstacle, an extern-potential function is designed to deal with target tracking and obstacle avoidance problems. Then, we implement the velocity planning strategy based on the result in [9]. Meanwhile, the interpotential function, which is used to deal with the interactions among agents, is determined by the relative positions between agents. Through the behavior rules, we dexterously link the extern-potential function and inter-potential function together. Under the total potential function, a distributed control algorithm can be gotten, while considering the maximum linear speeds of the agents. An outline of the paper is as follows. In Section 2, the preliminaries and problem statement of multiple agents are established. In Section 3, the distributed control algorithm is presented. Some simulation examples are provided to verify the effectiveness of the proposed approach in Section 3. Conclusion is done in Section 4. θ ao(i) : Angle of q ao(i). q ot = q tar q obs : Relative position vector from obstacle to target. θ ot: Angle of q ot. q ij = q j q i: Relative position vector from agent i to agent j. 2 Preliminaries and problem formulation A graph G [11] is a pair that consists of a set of vertices v = {1, 2,, n} and edges ε {(i, j) : i, j v, j i} (i.e., the graph is, in general, directed and has no self-loop). The graph is said to be undirected if (i, j) ε (j, i) ε. The adjacency matrix [11] A = [a ij] of a graph is a matrix with nonzero elements satisfying the property a ij 0 (i, j) ε. For an undirected graph, the adjacency matrix A is symmetric (or A T = A ). The set of neighbors of node i is defined by N i = {j v : a ij 0} = {j v : (i, j) ε}. (1) In multi-agent systems, the set of spatial neighbors of agent i is defined as N i = {j v : q j q i < r, r > 0} (2) r is the so-called distance of the neighbors influence. In this paper, we consider the problems of moving a group of agents toward a moving target, while avoiding collision with the moving obstacle and other agents. Therefore, there are two objectives. The first objective is to move the agents to their final destination, and to avoid collision with obstacle in the dynamic environment. The second objective is to avoid collision among agents. It is assumed that the agents and target are mass points. Specifically, the obstacle is treated as a sphere shown in Fig. 1. Suppose that the position and velocity of agents, obstacle, and target are all measurable. The following notations are used to describe the system: q tar R 2 : Position of the target. v tar R 2 : Velocity of the target. q obs R 2 : Position of the obstacle. v obs R 2 : Velocity of the obstacle. q (i) R 2 : Position of agent i. v (i) R 2 : Velocity of agent i. θ (i) : Angle of v (i). q at(i) = q tar q i : Relative position vector from agent i to the target. ψ (i) : Angle of q at(i). q ao(i) : Relative position vector from agent i to the obstacle. Fig. 1 The position and velocity vectors of the target, obstacle, and agents Consider a group of dynamic agents (or particles) moving in 2D Euclidean space. The motion of agent i is described as q i = u i, i = 1, 2,, n (3) q i R 2 denotes the position vector of agent i, and u i R 2 is its control input vector. 3 Distributed control algorithm In this section, a novel distributed control algorithm is proposed for solving the problem described in Section 2. Based on whether an agent is within the area of its neighbors influence, two kinds of potential functions are presented. They can be denoted as extern-potential function and inter-potential function, respectively. The motion planning is first studied for agent i outside the area of its neighbors influence ( q j q i r, j N i). Second, the other case, when agent i is within the area of its neighbors influence ( q j q i < r, j N i), is investigated. Finally, the total control inputs can be gotten by summarizing the discussions above. 3.1 Distributed control algorithm when agent i is outside the area of its neighbors influence ( q j q i r, j N i ) When agent i is outside the area of its neighbors influence, there is no interaction between agent i and its neighbors. In other words, agent i is only under the target and

3 552 International Journal of Automation and Computing 7(4), November 2010 obstacle s influence. The target generates an attractive potential field, while the obstacle generates a repulsive potential field. Then, the potential function can be defined as [12 14] : U att(i) 1 2 ξ1qt at(i)q at(i) (4) U rep(i) { 1 2 ξ2(ρ 1 (i) ρ 1 0 ) 2, if ρ (i) ρ 0 0, else (5) From (6) and (7), we have (ξ 1x at(i) µx ao(i) )( v tar sin θ tar v(i) sin θ(i) ) = (ξ 1y at(i) µy ao(i) )( v tar cos θ tar v(i) cos θ(i) ) (8) µ = ξ 2ρ 1 (i) qao(i) 1 (ρ 1 (i) ρ 1 0 ) U att(i) is the attractive potential function, and U rep(i) is the repulsive potential function. ρ (i) denotes the minimum distance from agent i to the obstacle. The obstacle can be assumed as a circle, and ρ obs denotes the radius of the circle; then we can get ρ (i) = q ao(i) ρ obs. ρ 0 > 0 is the obstacle influence threshold. ξ 1 > 0 and ξ 2 > 0 are scaling factors for attractive and repulsive potentials, respectively. Then, the total potential function can be defined as U (i) U att(i) + U rep(i). (6) Let q at(i) = [x at(i), y at(i) ] T and q ao(i) =[x ao(i), y ao(i) ] T. The relative motion between agent i and the target is described by q at(i) = [ẋ at(i), ẏ at(i) ] T, and the relative motion between agent i and the obstacle can be described by q ao(i) = [ẋ ao(i), ẏ ao(i) ] T with ẋ at(i) = v tar cos θ tar v (i) cos θ (i) x at(i) = qat(i) cos ψ(i) y at(i) = q at(i) sin ψ (i) x ao(i) = qao(i) cos θao(i) y ao(i) = q ao(i) sin θ ao(i). Rearranging (8) and assuming v(i) 0, we have v tar (sin(θ tar ψ (i) ) λ sin(θ tar θ ao(i) )) = v (i) (sin(θ (i) ψ (i) ) λ sin(θ (i) θ ao(i) )) Define λ = µ qao(i) ξ 1 qat(i). (9) ẏ at(i) = v tar sin θ tar v (i) sin θ (i) ẋ ao(i) = v obs cos θ obs v (i) cos θ (i) ẏ ao(i) = v obs sin θ obs v (i) sin θ (i). Theorem 1. To make agent i track a moving target while avoiding the moving obstacle as quickly as possible, the control input for agent i should be planned such that q at(i) points to the negative gradient of U (i) with respect to q at(i). Proof. From [3], it is known to us that if an agent requires to track a static target as quickly as possible, q (i) should point to the negative gradient of U (i) with respect to q (i). For a moving target, we can assume that the target is static with relative to agent i. Then, the dynamic environment can be translated into quasi static environment, and the new state information can be expressed as follows: q (i) is the new position of agent i, q (i) is its velocity, and U (i) denotes the new potential function of agent i. Under the condition in static environment, the control input agent i should be planned so that q (i) points to the negative of gradient of U (i) with respect to q(i), if the agent i requires to track the quasi static target as quickly as possible. Finally, translating the quasi static environment into dynamic environment, we have q (i) = q at(i), q (i) = q at(i), and U (i) = U (i). Based on Theorem 1, it is necessary to make U (i) x at(i) ẏ at(i) U (i) y at(i) ẋ at(i) = 0 (7) then (9) is rearranged as ψ (i) = arctan sin ψ (i) λ sin θ ao(i) cos ψ (i) λ cos θ ao(i) θ (i) = ψ (i) + arcsin vtar sin(θtar ψ (i) ) v (i) (10) ψ (i) = arctan sin ψ (i) λ sin θ ao(i) cos ψ (i) λ cos θ ao(i). Meanwhile, we should also consider another case v(i) = 0. Thus, we can assume θ(i) = θ tar, if v(i) = 0. From (10), the direction of an agent i can be gotten. Then, we can plan the velocity for the i-th agent. With the planned velocity, potential function U (i) will be large enough but not infinite when agent i is within the area of the obstacle s influence. Meanwhile, the potential function will also be very small when q at(i) 0, and it reaches 0 when q at(i) = 0. Differentiating U (i) in (6) with respect to time t, we have U (i) = U att(i) + U rep(i) = ξ 1q T at(i) q at(i) ξ 2q T ao(i) q ao(i). (11) With q at(i), q ao(i), q at(i), and q ao(i) defined above, can be rewritten as U (i) = ξ 1 qat(i) ( vtar cos(θ tar ψ (i) ) v (i) cos(θ (i) ψ (i) ) λ v obs cos(θ obs θ ao(i) )) = ξ 1 q at(i) ( v tar cos(θ tar ψ (i) ) ( v(i) 2 v tar 2 sin(θ tar ψ (i) )) 1 2 U (i) λ v obs cos(θ obs θ ao(i) )). (12)

4 J. Yan et al. / Target Tracking and Obstacle Avoidance for Multi-agent Systems 553 To make U (i) < 0, we define v(i) =[( vtar cos(θ tar ψ (i) ) λ v obs cos(θ obs θ ao(i) ) + ξ 1 q at(i) ) 2 + v tar 2 sin 2 (θ tar ψ (i) )] 1 2. (13) Substituting v(i) into U (i), we have U (i) = ξ1 2 q at(i) 2 = 2ξ 1U att(i). Then, we can find U (i) > 0 and U (i) < 0, so the system is stable. It means the agent can track the moving target in the dynamic environment. Based on the above discussion, the following corollary can be gotten. Corollary 1. For multiple agents to track a moving target while avoiding collision with the obstacle under the situation that there is no interaction between agents, the disturbed control input for agent i can be planned as u (i) = ( v (i) cos θ (i), v (i) sin θ (i) ) T (14) (j N i). The parameters in (17) are the same as those in (5). Remark 1. Although q dij denotes the desired distance between agent i and agent j, the distance needs not strictly converge to q dij in fact. The reason is that the total potential function is a combination of some different potential functions, which are defined in Sections 3.1 and 3.2. Even though, this result does not influence the final aim of this paper because the objective of this paper is to accomplish collision avoidance task but not the consensus problem, and q dij > 0 for all agents means that the objective has already finished. The potential function U ret(ij) between agent i and agent j can be illustrated in Fig. 2, and the repelling and attractive forces U ret(ij) are illustrated in Fig. 3. As q i > q dij, U ret(ij) > 0 means agent i will attract agent j; as q i = q dij, U ret(ij) = 0 means agent i and agent j can be balanced; as q i < q dij, U ret(ij) < 0 means agent i will repel agent j. v (i) = [( v tar cos(θ tar ψ (i) ) λ v obs cos(θ obs θ ao(i) ) + ξ qat(i) 1 ) 2 + v tar 2 sin 2 (θ tar ψ (i) )] 1 2 ψ (i) + arcsin vtar sin(θtar ψ (i) ) θ (i) = v (i), if v (i) 0 θ tar, if v(i) = Distributed control algorithm when agent i is under the area of its neighbors influence ( q j q i < r, j N i ) The distributed control algorithm in Corollary 1 does not take the neighbors influence into consideration. Once the distance between agent i and its neighbors is smaller than r, there are interactions between agents. Therefore, the neighbors influence must be taken into consideration. Based on the above thought, another potential function Ǔ (i) for the interactions between agents is defined. In this potential field, agent i should be repulsed from its neighbors if they are nearly-spaced. Meanwhile, agent i should also avoid the obstacle during this process. By this virtual force, agent i can escape from its neighbors influence quickly. After escaping from its neighbors influence, agent i will autonomously track the moving target under the algorithm in Corollary 1. In view of the transitory repulsive process, the target s influence will not be considered during the repulsive process. The potential function can be defined as Ǔ (i) = j N i U ret(ij) + U rep(i) (15) U rep(i) U ret(ij) = In(q ij) + q dij (16) q ij { 1 2 ξ2(ρ 1 (i) ρ 1 0 ) 2, if ρ (i) ρ 0 (17) 0, else q ij = q i q j and q dij > 0 are, respectively, the actual and desired distance between agent i and agent j Fig. 2 Potential function between agent i and agent j Fig. 3 The forces produced by U ret(ij) Remark 2. In conventional definition, the desired distance q dij < r. But in this paper, we define q dij > r. There are several advantages associated with this definition. First, the objective is to make agent i escape from its neighbors influence, and then it can autonomously move under the algorithm in Corollary 1. In other words, the attractive forces are not our expectation. Second, we define a large constant q dij, and then agent i can also move toward the desired position by the repulsive force and inertia. After converging to the desired position, agent i can also escape from the neighbors influence (q dij > r). Agent i s control input is along the negative gradient of

5 554 International Journal of Automation and Computing 7(4), November 2010 Ǔ (i) with respect to q (i), and u (i) = q(i) Ǔ (i) = ( 1 q dij µq ao(i) (18) q ij q ij q i µ = { j Ni ξ 2ρ 1 (i) qao(i) 1 (ρ 1 (i) ρ 1 0 )2, if ρ (i) ρ 0 0, else. The obstacle can be assumed as a circle, and ρ obs denotes the radius of the circle, and then we can get ρ (i) = qao(i) ρ obs. ρ0 > 0 is the obstacle influence threshold. Theorem 2. With the control input in (18), agent i can escape its neighbors influence when ρ (i) > ρ 0. Proof. The potential function U ret(ij) is a sub-function. ρ (i) > ρ 0 denotes that agent i has already escaped from the obstacle s influence. Thus, we can choose Lyapunov function as U = j Ni Ǔ(i). Differentiating U with respect to time t, we have U = n Ǔ (i) = i=1 n ( 1 q dij q ij i=1 j Ni q 2 ij q i u (i). (19) Substitute (18) into (19) when ρ (i) > ρ 0, we have U = n [( 1 q dij q ij i=1 j Ni q 2 ij q i ] 2 < 0. Based on Lyapunov stability theory, we can get the following conclusion: if the desired position is known, the agent i will be repulsed by its neighbors and reach the desired position. When ρ (i) > ρ 0, it means that agent i should avoid collision with its neighbors and the obstacle. By contradictions, we can also get there is no collision. The proof is similar to the result in [2], and hence omitted. Remark 3. The maximum linear speed of the agents should also be considered. First, we consider the situation in Section 3.1. Similar to the analysis in [9], the convergence is still guaranteed if the agent s maximum linear speed is v max. Then, we consider the situation in Section 3.2. Because, in this section, we ignore the target s influence, the convergence of q ao(i) to zero is still guaranteed regardless of the limitation of the maximum linear speed. The results are summarized by the following theorem. Theorem 3. For multiple agents to track a target while avoiding collision with moving obstacle and between agents, the distributed control input for agent i can be planned such that v(i) (cos θ(i), sin θ(i)) T, if q j q i r u(i) vmax u (i) = ( 1 q dij µq ao(i), j Ni q ij q ij q i if q j q i < r u(i) vmax v max(cos θ (i), sin θ (i) ) T, if u(i) > vmax. θ (i) = ψ (i) + arcsin vtar sin(θtar ψ (i) ) v (i), if v (i) 0 θ tar, if v (i) = 0 µ = { ξ 2ρ 1 (i) qao(i) 1 (ρ 1 (i) ρ 1 0 ), if ρ (i) ρ 0 0, else. 4 Simulation results To test the performance of the proposed approach, numerical simulations are performed. The simulations are done for three agents to track a moving target. The route of the target is specified by: ( cos t, sin t) T, 0 t π q tar = 2 ( cos t, 3.2) T π, < t (cos( π v tar = 2 t), sin( π 2 t))t, 0 t π 2 (1.2, 0) T π, < t The initial states of the three agents are given as follows (when t = 0) q 1 = (0, 0) T q 2 = (0.2792, ) T q 3 = (0.2666, ) T v 1 = v 2 = v 3 = (0, 0) T. The simulation results are done for two cases. First, we study the situation that there is no obstacle in the dynamic environment. Then, we consider the situation that there is an obstacle in the dynamic environment. 4.1 Three agents track a moving target In this subsection, agents, which are affected by the target and neighbors, will track a moving target. Some parameters in the simulation are given as follows: ξ 1 = 2, ξ 2 = 1, of ρ obs = 0.2, ρ o = 0.3, q dij = 0.5, and r = 0.2. The position trajectories of the three agents and target are given in Fig. 4. with v (i) = [( v tar cos(θ tar ψ (i) ) λ v obs cos(θ obs θ ao(i) ) + ξ 1 qat(i) ) 2 + v tar 2 sin 2 (θ tar ψ (i) )] 0.5 Fig. 4 Trajectories of the agents 1 3 and the target

6 J. Yan et al. / Target Tracking and Obstacle Avoidance for Multi-agent Systems 555 As shown in Fig. 4, the agents track the target from point A to C. The 2nd agent catches up with the target at point C. The other agents also track the target and keep a pimping distance with the target; the reason of this phenomenon is that each agent should avoid collision with the other agents. 4.2 Three agents track a moving target while avoiding a moving obstacle We consider the situation that the agents track a moving target while avoiding the obstacle in a dynamic environment. The obstacle is assumed to be circular with the radius ρ obs = 0.2 and the influence: ρ o = 0.3. On the way to the target, if sensors installed on agents detect the obstacle, the agents regenerate a local trajectory to avoid the obstacle by using the algorithm in Section 3. First, an obstacle that flatly moves on the plane is chosen, and the initial states of the obstacle are given as follows: q obs = (2.5, 1.3) T, v obs = ( 1.1, 0) T. As shown in Fig. 5, the agents move from point A to D. At point B, an agent meets a moving obstacle, then the agent moves quickly to escape the obstacle s influence. Finally, the agents catch up with the target at point D. The distances between agents are shown in Fig. 6 (a), and there is no collision between agents (d ij > 0). Because the objective of this paper is to accomplish collision avoidance task but not consensus, q ij > 0 for any agent i means that the objective has already been achieved. Fig. 6 (b) shows the distances between agents and obstacle. It is obvious to find that there is no collision between agents and obstacle because the relative distances are all greater than zero. To test the feasibility of the proposed approach, another obstacle is chosen which vertically moves on the plane, and the initial states of the obstacle are given as follows: q obs = (2.5, 4.25) T, v obs = (0, 0.363) T. Meanwhile, the target is virtual in this paper, and so it does not need to avoid the obstacle. Fig. 7 shows the trajectories of the agents and target. The distances between the agents are shown in Fig. 8 (a), and Fig. 8 (b) shows the distances between agents and obstacle. Similarly to the results in Figs. 6 and 7, there is no collision between agents and obstacle. Fig. 5 Trajectories of the three agents and target when there is an obstacle which flatly moves on the plane Fig. 6 The distances. (a) The distances between agents (d 12 denotes the distance between agent 1 and agent 2, d 13 is for agent 1 and agent 3, and d 23 is for agent 2 and agent 3; (b) The distances between agents and the obstacle (d 1 denotes the distance between agent 1 and the obstacle, d 2 is for agent 2, and d 3 is for agent 3) 5 Conclusion Based on the potential function and behavior rules, a novel control approach is proposed for target tracking and obstacle avoidance of multiple agents in dynamic environment. Proper potential functions concerning with target, obstacle and agents are chosen to design the new distributed control algorithm. To avoid collision between the agents, we also consider the interactions between agents. The simulation results are provided to verify the effectiveness of the approach. Under the proposed method, multiple agents can track a moving target while avoiding collision with moving obstacle and between agents. In this paper, the proposed approach is based on cooperative strategy. But, in real-life, some information cannot be acquired, such as a competition game. Then, in the future, we will use non-cooperative game theory to analyze the target tracking and obstacle avoidance problems for multi-agent systems. Fig. 7 Trajectories of the three agents and target when there is an obstacle which vertically moves on the plane

7 556 International Journal of Automation and Computing 7(4), November 2010 [9] L. Huang. Velocity planning for a mobile agent to track a moving target A potential field approach. Robotics and Autonomous Systems, vol. 57, no. 1, pp , [10] O. Khatib. Real-time obstacle avoidance for manipulators and mobile robots. International Journal of Robotics Research, vol. 5, no. 1, pp , [11] C. Godsil, G. Royle. Algebraic graph theory, Graduate Texts in Mathematics, New York, USA: Springer-Verlag, vol. 207, [12] D. Kolokotsa, A. Pouliezos, G. Stavrakakis, C. Lazos. Predictive control techniques for energy and indoor environmental quality management in buildings. Building and Environment, vol. 44, no. 9, pp , Fig. 8 The distances. (a) The distances between agents (d 12 denotes the distance between agent 1 and agent 2, d 13 is for agent 1 and agent 3, and d 23 is for agent 2 and agent 3; (b) The distances between agents and the obstacle (d 1 denotes the distance between agent 1 and the obstacle, d 2 is for agent 2, and d 3 is for agent 3) Acknowledgement The authors thank the anonymous reviewers for their constructive comments and suggestions that improved the quality of this paper. References [1] H. S. Su, X. F. Wang, Z. L. Lin. Flocking of multi-agents with a virtual leader. IEEE Transactions of Automatic Control, vol. 54. no. 2. pp , [2] W. Li. Stability analysis of swarms with general topology. IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics, vol. 38, no. 4, pp , [3] W. H. Huang, B. R. Fajen, J. R. Fink, W. H. Warren. Visual navigation and obstacle avoidance using a steering potential function. Robotics and Autonomous Systems, vol. 54, no. 4, pp , [4] R Olfati-Saber. Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Transactions of Automatic Control, vol. 51, no. 3, pp , [5] L. Guzzella. Automobiles of the future and the role of automatic control in those systems. Annual Reviews in Control, vol. 33, no. 1, pp. 1 10, [6] W. Kowalczyk, K. Kozlowski. Artificial potential based control for a large scale formation of mobile robots. In Proceedings of the 4th International Workshop on Robot Motion and Control, IEEE, pp , [7] G. V. Raffo, G. K. Gomes, J. E. Normey-Rico. A Predictive controller for autonomous vehicle path tracking. IEEE Transactions on Intelligent Transportation Systems, vol. 10, no. 1, pp , [8] T. T. Yang, Z. Y. Liu, H. Chen, R. Pei. Formation control and obstacle avoidance for multiple mobile robots. Acta Automatica Sinica, vol. 34, no. 5, pp , [13] Y. Yoon, J. Shin, H. J. Kim, Y. Park, S. Sastry. Modelpredictive active steering and obstacle avoidance for autonomous ground vehicles. Control Engineering Practice, vol. 17, no. 7, pp , [14] P. Ögren. Formations and Obstacle Avoidance in Mobile Robot Control, Ph. D. dissertation, Royal Institute of Technology, Sweden, Jing Yan received the B. Eng. degree in automation from Henan University, PRC in He is currently a Ph. D. candidate in control theory and control engineering at Yanshan University, PRC. His research interests include cooperative control of multi-agent systems and wireless networks. yanjing6663@163.com (Corresponding author) Xin-Ping Guan received the M. Sc. degree in applied mathematics in 1991, and the Ph. D. degree in electrical engineering in 1999, both from Harbin Institute of Technology, PRC. Since 1986, he has been at Yanshan University, PRC, he is currently a professor of control theory and control engineering. In 2007, he also joined Shanghai Jiao Tong University, PRC. His research interests include robust congestion control in communication networks, cooperative control of multi-agent systems, and networked control systems. xpguan@ysu.edu.cn; xpguan@sjtu.edu.cn Fu-Xiao Tan received the B. Eng. degree in automation from Hefei University of Technology, PRC in 1997, and the Ph. D. degree in control theory and control engineering from Yanshan University, PRC in He is currently an associate professor in Fuyang Teachers College, PRC. His research interests include robust congestion control in communication networks, cooperative control of multi-agent systems, and networked control systems. Tanfxme@163.com