Target Tracking and Obstacle Avoidance for Multi-agent Networks with Input Constraints

Transcription

1 International Journal of Automation and Computing 8(1), February 2011, DOI: /s Target Tracking and Obstacle Avoidance for Multi-agent Networks with Input Constraints Jing Yan 1 Xin-Ping Guan 2 Xiao-Yuan Luo 1 Fu-Xiao Tan 3 1 Department 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: In this paper, the problems of target tracking and obstacle avoidance for multi-agent networks with input constraints are investigated. When there is a moving obstacle, the control objectives are to make the agents track a moving target and to avoid collisions among agents. First, without considering the input constraints, a novel distributed controller can be obtained based on the potential function. Second, at each sampling time, the control algorithm is optimized. Furthermore, to solve the problem that agents cannot effectively avoid the obstacles in dynamic environment where the obstacles are moving, a new velocity repulsive potential is designed. One advantage of the designed control algorithm is that each agent only requires local knowledge of its neighboring agents. Finally, simulation results are provided to verify the effectiveness of the proposed approach. Keywords: Target tracking, obstacle avoidance, multi-agent networks, potential function, optimal control. 1 Introduction In recent years, there is a spurt of interest in the area of unmanned system networks (UMSNs) [1 4], such as sensor networks (SNs). These networks can potentially consist of a large number of agents, such as unmanned underwater vehicles (UUVs), unmanned aerial vehicles (UAVs), and unmanned ground vehicles (UGVs). UMSN provide numerous applications in various fields of research, such as building automation, intelligent transportation systems, surveillance, terrain data acquisition, and underwater exploration. The advantages of UMSN over single system include cost reduction, increased efficiency, and improved robustness. One of the prerequisites for these networked agents is team cooperation and coordination for accomplishing predefined goals and requirements. Cooperation in multi-agent networks, including target tracking and obstacle avoidance, has received lots of attentions in the past few years. Some approaches to this problem have been investigated within different frameworks. In [5], a behavior-based approach was presented, in which the basic behaviors were assigned to the independent systems to form a guidance algorithm. Then, the controllers for achieving different objectives were combined. Virtual structure method was investigated in [6], and the control method was developed to force agents to be kept consensus in a rigid formation. In [7], receding horizon control (RHC) was applied to cooperative control of multiagent networks. RHC is a feedback control scheme, in which an optimization problem is solved at each sampling time. Manuscript received September 7, 2009; revised February 8, 2010 This work was supported by National Basic Research Program of China (973 Program) (No. 2010CB731800), Key Project of National Science Foundation of China (No ), National Nature Science Foundation of China (No ), Key Project for Natural Science Research of Hebei Education Department, PRC (No. ZD200908), and Key Project for Shanghai Committee of Science and Technology (No ). Each optimization yields an open-loop control trajectory, and the initial portion of the trajectory is applied to the agents until the next sampling instant. Among these control methods, potential function method is the most popular control strategy. It has been widely used for multi-agent cooperation problem, such as consensus problem [8,9], tracking problem [10], and obstacle avoidance problem [11]. However, most of these strategies are well solved for oversimplified agent dynamics without explicitly taking input constraints into account. For example, obstacle avoidance control is usually translated into minimization of a function, i.e., the value of this function will become big enough when a direction which would lead to collision is chosen. Such an approach does not reveal the real-life issues, such as limited acceleration and actuation ability of the agents. On the other hand, conventional potential function method is not suitable for path planning of multiagent networks in a dynamic environment where both the target and the obstacles are moving [10]. Because only the relative positions between agents, target, and obstacles are included in the definition of potential functions. Then, the potential field cannot reflect the whole information of the dynamic environment, because of neglecting the target and obstacles velocity information. One result of this demerit is that the agents are possible to collide with the obstacles when the relative velocities between agents and obstacles are large to some extent. Based on the above consideration, this paper attempts to overcome these limitations by combining the potential function method with dynamic optimal control. The objective is to make the agents track a moving target, while avoiding collision with the moving obstacles and other agents. First, we define a new velocity potential function that includes the velocities of the target and obstacles. Based on the new potential function, a presumed solution can be ob-

2 J. Yan et al. / Target Tracking and Obstacle Avoidance for Multi-agent Networks with 47 tained without considering the input constraints. Optimizing the presumed solution with taking the input constraints into consideration, a novel distributed control algorithm is proposed. Meanwhile, the approach is cooperative since every neighboring agent exchanges its presumed solution prior each update. As each agent only exchanges information with its neighbors and the optimization is implemented at each update time, therefore, the optimal algorithm is local and dynamic. The outline of this paper is as follows. In Section 2, the preliminaries and problem statement of multiple agents are described. In Section 3, the control algorithm is presented. Some simulation examples are provided to verify the effectiveness of the approach in Section 4. Conclusion is given in Section 5. 2 Problem statement and preliminaries Multi-agent networks. Assume a set of agents = {i = 1,, N}, where N is the number of agents. Each agent can be described by its dynamics. For agent i with 2-dimensional (2D) coordinates, the state and control vectors are denoted by z i(t) = [q i(t) T, q i(t) T ] T and u i(t) R 2. The vectors q i(t) R 2 and q i(t) R 2 are the position and velocity of agent i, respectively. The dynamical representation of each agent is governed by ż i = az i + bu i (1) [ ] [ ] 0 I (2) 0 where a =, b =. The matrix I (2) 0 0 I (2) is a 2D identity matrix. The state and control vectors of agent i are defined as z i(t) Z and u i(t) C, respectively. C and Z are convex sets, which will be illustrated in Assumption 1. By concatenating the states of all agents in a team into a vector z = (z1 T,, zn T ) T, the team dynamics are ż(t) = Az(t) + B iu i(t), t 0 (2) where A = I (N) a and B i = [s 1,, s j,, s N ] T b, s j = 1, if i = j, 0, else, j N +. The operator is the Kronecker product. Graph [12]. A graph G is a pair that consists of a set of vertices V = {1, 2,, N} and edges E {(i, j) : i, j V, j i} (i.e., the graph in general is directed and has no self-loops). The graph is said to be undirected if (i, j) E (j, i) E. Information structure and neighboring sets [13]. To ensure cooperation among agents, each agent has to know the status of the other agents, and therefore, agents have to communicate with each other. For a given agent i, the set of agents from which it can receive information is called a neighboring set N i, that is i = 1,, N, N i = {j = 1,, N (i, j) E}. (3) It is assumed that the graph describing the information structure is connected. Laplacian matrix [13,14]. This matrix is used to describe the graph associated with information exchanges in a network of agents, and is defined as L = [L(i, j)] N N d(i), if i = j L(i, j) = 1, if (i, j) E and i j (4) 0, otherwise where d(i) is equal to the cardinality of the set N i, and N i is called the degree of vertex i. For an undirected graph, the degree of a vertex is the number of edges incident to that vertex (total number of links connected to that vertex). 3 Distributed control algorithm In this section, we design a term of agents that can track a moving target while avoiding collisions with obstacles and other agents. Based on potential function method, a presumed solution can be obtained without taking the input constraints into consideration. After optimizing the presumed solution considering the input constraints at each sampling instant, a distributed optimal control algorithm is proposed. To optimize control performance, the convexity assumption is necessary for optimization algorithms. Assumption 1. C is a compact and convex subset of R 2, which contains the origin in its interior. Z is a convex and connected subset of R 4, which contains zi c (zi c is the desired states vector for agent i) in its interior for every i. Following the general practice of the research in potential fields, the following assumptions are made. Assumption 2. The obstacles can be covered by convex circles. The k-th (k = 1, 2, ) obstacle in the environment is included in a 2D circle and denoted by B k (O k, ρ k ), where O k is its center, and ρ k (ρ k > 0) is its radius. Assumption 3. The agent s target can be treated as mass points. Definition 1 (Obstacle set). For each agent i at time t, the detected obstacle set is defined as the subset Ω t i {B 1(O 1, ρ 1 ),, B k (O k, ρ k ), } in the detected range of the sensor. 3.1 Distributed potential function control algorithm Based on potential function method, a presumed solution can be obtained without taking the input constraints into consideration. To reflect the dynamic environment, we define a new velocity potential function, which includes the velocities of the target and obstacles. There are two objectives for this proposed algorithm. The first one is to move the agents to their final destination while avoiding collision with other agents. The second objective is to avoid collision with the obstacles. Each obstacle has the same physical influence area, represented by a 2D circle of center O k and radius ρ b (ρ b > ρ k ). Let ρ ik > 0 denote the relative distance

3 48 International Journal of Automation and Computing 8(1), February 2011 between the i-th agent and the k-th obstacle. First, only one obstacle is considered. The proposed approach can then be extended to the case of multiple obstacles. According to that whether the agents are within the area of the obstacle s influence or not, we divide the environment of the obstacle into two parts: safety area (ρ ik > ρ b ) and obstacle avoidance area (ρ ik ρ b ) Target tracking in safety area (ρ ik > ρ ) b In safety area, agents are outside the area of the obstacle s influence. Only the target and agents act on the potential field. The objective is to move the agents to track a moving target while avoiding collision with other agents. To make the agents track a target, we define the potential function [10,14] for agent i: Û a(q i) = 1 2 ξ 1 et i e i (5) Ǔ a(q i) = 1 2 ξ 2ėT i ė i (6) where Ûa(qi) is the position attractive potential function, and Ǔa(qi) is the velocity attractive potential function. ei = q i q tar denotes the position tracking error vector, and ė i = q i q tar denotes the velocity tracking error vector. q i and q i are respectively the position and velocity of agent i. q tar and q tar are the position and velocity for the target, respectively. ξ 1 > 0 and ξ 2 > 0 are scaling factors for position and velocity attractive potentials, respectively. Meanwhile, we should also consider the interactions between agents. If agents are closely spaced, a repulsive force should act on the agents; if agents are far-spaced, an attractive force should act on the agents. In a word, each agent should keep a desired distance with its neighbors. Let q ij = q i q j denote the actual distance between agent i and agent j, and d (d > 0) is the desired distance between them. q j is the position of agent j. Any agent j has the same physical influence area, represented by a 2D circle of center q j and radius r (r > d). The definition of the interior potential function is given as follows: Definition 2. A differential potential function should satisfy the following: 1) The value of function is negligibly small as q ij d, and is strictly decreasing on (0, d]. 2) The function is strictly increasing on (d, r], and cuts off at r. Then, the potential function can be defined as with U a (q i) = j N i d q ij E j(τ)dτ (7) ξ 3 (q ij d)(q ij r), q ij (0, r] E j(q ij) = q ij 0, q ij (r, ] where N i is the set of neighbors of agent i. ξ 3 > 0 is the scaling factor for interior potential. Combining (7) with (5) and (6), we have the total potential function for agent i U a(q i) = 1 2 ξ 1e T i e i ξ 2ė T i ė i + j N i d q ij E j(τ)dτ. (8) The distributed control input of agent i can be divided into two parts: the negative gradient of the position potential with respect to q i and the negative gradient of the velocity potential with respect to q i. The corresponding result is given as follows: u i = qi Û a(q i) qi Ǔ a(q i) qi Ua (q i) = (9) ξ 1 (q i q tar) ξ 2 ( q i q tar)+ ξ 3 (q ij d)(q ij r)(q j q i), q j N i qij 2 ij (0, r] ξ 1 (q i q tar) ξ 2 ( q i q tar), q ij (r, ). Then, the control input for system (2) can be described as follows: u = (u 1,, u N ) T. (10) Theorem 1. By using the control input in (10), all the agents can track the moving target. Meanwhile, the agents cannot collide with each other. Proof. Rearranging (2) and noting that q i = p i, we have q i = p i (11) ṗ i = u i where p i denotes the velocity of agent i. Defining x i = q i q tar and re-arranging (12), we have ẋ = y y i = p i p tar (12) ẏ = u = xp (x) Ay (13) where x = [x 1, x 2,, x N ] T, y = [y 1, y 2,, y N ] T, u = [u 1, u 2,, u N ] T. P (x) = U a (x) + ( N ξ 1 x i )/2, A = ξ 2 I N + L I N. We build a Lyapunov function as H(x, y) = P (x) y i. (14) Differentiating H(x, y) with respect to time t, we have Ḣ(x, y) = y T (L I 2)y. (15) For all y 0, y T (L I 2)y < 0. Thus, Ḣ(x, y) < 0. The system is asymptotically stable. Therefore H(x(t), y(t)) H(x(0), y(0)) <. (16)

4 J. Yan et al. / Target Tracking and Obstacle Avoidance for Multi-agent Networks with 49 From (14) and (16), Ua (x) > 0, and P (x) = Ua (x) + ( N xi )/2, we can find ξ ξ 1 x i < H(x(0), y(0)) (17) which guarantees the target tracking control. We prove the process of collision avoidance between agents by contradiction. Assume that there exists a time t that at least two agents collide with each other. Then H(x(t), v(t)) = U a (x) ξ 1 x i + ξ 2 y i Ua (x) E j(0) = which is in contradiction with the condition H(x(t), v(t)) H(x(0), v(0)) <. Therefore, there is no agent colliding with its neighbors Obstacle avoidance control (ρ ik ρ b ) Once agents are within the area of the obstacle s influence, they need to avoid the obstacle. In conventional potential function method, the repulsive potential function was defined as [10] 1 Urd(q k i) = 2 ξ 4 (ρ 1 ik ) 2, if ρ ik ρ b 0, else (18) where ρ b > 0 is the influence threshold, and ρ ik = q i O k is the relative distance between the i-th agent and the k-th obstacle, O k is the position of the obstacle, and ξ 4 > 0 is the scaling factor for position repulsive potential. The definition of (18) is based on the relative position between agent and obstacle. In a static environment, the repulsive potential can reflect the whole information of the obstacle. However, in a dynamic environment where the obstacle keeps moving, (18) cannot reflect the whole information of the obstacle. One result of this demerit is that the agents are possible to collide with the obstacles when the relative velocities between agents and obstacles are large to some extent. To solve this problem, a new velocity potential field is constructed. If the relative velocity between the obstacle and agent is large, the velocity potential should be large enough to generate appropriate repulsive force, so that the agent will not collide with the obstacle even the agent and obstacle are nearly-spaced and to keep a large relative velocity. Meanwhile, there is no need to construct a velocity repulsive potential field when the agents move away from the obstacle. Therefore, the velocity potential function can be constructed as follows: ξ 5 (Ȯk q i) 2 sin θ ik, Urv(q k i) = if ρ ik ρ b and θ ik [ π 2, π 2 ) (19) 0, else where ξ 5 > 0 is scaling factor for velocity repulsive potential, Ȯ k is the velocity of obstacle k, q oa(ik) = (q oax(ik), q oay(ik) ) T is the relative position vector from obstacle k to agent i, q oa(ik) = Ȯk q i is the relative velocity from obstacle k to agent i, θ ik / [ π/2, π/2) denotes that the agent moves away from the obstacle, and θ ik is the angle of q oa(ik) with respect to q oa(ik), which can be depicted in Fig. 1. Combining (18) with (19), we can get the total repulsive potential function for agent i as follows: U k r(i) = U k rd(i) + U k rv(i). (20) Then, we can get the distributed control input for agent i: u k r(i) = qi U k r(i). (21) To get qi Urv(i), k we define the heading angle of the obstacle as 0. φ k denotes the heading angle of the obstacle, then φ k = 0 can be obtained. Rearranging Fig. 1 and nothing that φ k = 0, we can get Fig. 2. u k rd(i) is the position repulsive potential control input, u k rv(i) is the velocity repulsive potential control input, and u k r(i) is the total repulsive potential control input. In Fig. 2, another new control input u k r(i) (φ k = 0) can be designed as follows: ũ k r(i) = qi U k rv(i) = (22) qi [ξ 5 (Ȯk q i) 2 sin θ ik ] = qi [ξ 5 (Ȯk q i) 2 q oay(ik) q oa(ik) ] = ξ 5 ( q obs q i) 2 q oax(ik) ( q q 1.5 oay(ik), q oax(ik) ) = oa(ik) ξ 5 q 2 q oax(ik) oa(ik) [ sin θ ik, cos θ ik ]. q oa(ik) Fig. 1 Velocity state of the agent i and obstacle (The white ball denotes the agent and the gray ball denotes the obstacle)

5 50 International Journal of Automation and Computing 8(1), February 2011 Fig. 2 Repulsive forces to the obstacle Rearranging (22) and nothing that φ ik 0, we have with qi U k r(i) = ξ 1 (ρ 1 ik u k r(i) = qi U k r(i) (23) )ρ 3 ik (qi O k), if θ ik / [ π 2, π 2 ) and ρ ik ρ b ξ 1 (ρ 1 ik )ρ 3 ik (qi O k)+ ξ 2 q 2 q oax(ik) oa(ik) [ sin(θ ik +φ q ik ), cos(θ ik +φ ik )], oa(ik) if θ ik [ π 2, π 2 ) and ρ ik ρ b 0, else Rearranging (9) and (23), we have the distributed control input for agent i: u k i = ξ 1 (q i q tar) ξ 2 ( q i q tar)+ ξ 3 (q ij d)(q ij r)(q j q i), j N i qij 2 if q ij (0, r] and ρ ik > ρ 0 ξ 1 (q i q tar) ξ 2 ( q i q tar), ξ 4 (ρ 1 ik ξ 4 (ρ 1 ik if q ij (r, ) and ρ ik > ρ 0 )ρ 3 ik (qi O k), if θ ik / [ π 2, π 2 ) and ρ ik ρ b )ρ 3 ik (qi O k) + ξ 5 q oa(ij) 2 q oax(ij) q oa(ij) [ sin(θ ik + φ ik ), cos(θ ik + φ ik )], if θ ik [ π 2, π 2 ) and ρ ik ρ b 0, else Then, the total control input for system (2) can be described as u k = (u k 1,, u k N ) T. (24) Remark 1. Although d denotes the desired distance between agent i and its neighbors, the distance d ij does not strictly converge toward d in fact. The reason is that the total potential function is a combination of some different potential functions, which are defined in Sections and Even now, 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 d ij > 0 for all agents, it means that there is no collision between agents. Remark 2. By using the control input in (24), all the agents will not collide with its neighbors. The proof is similar to Theorem 1 and is, hence, omitted. Remark 3. From the definition of the potential function in (24), we can have the following results: 1) The bigger the factors ξ 1 and ξ 2 are, the greater the tracking potential will be. 2) The bigger the factors ξ 3 and ξ 4 are, the greater the avoiding potential will be. Corollary 1. For multiple obstacles, we can replace U k r(i) with n U r(i). k Then, we can get the new solution k=1 u where there are n obstacles. u = (u 1,, u N ) T. (25) The representation of u i is similar to the solution in (24) and is, hence, omitted. 3.2 Distributed optimization control algorithm Taking the input constraints C into consideration, we can implement the distributed optimization control strategy. Using the dynamic model in system (1) and the presumed control vector in (25), the presumed state vector z p i of agent i can be obtained. Optimizing the presumed state vector at each sampling instant, a distributed control algorithm is designed. Remark 4. Note that the detected obstacle set is time dependent and evolves as long as the agent moves and discovers new obstacles. To ensure the collision avoidance with obstacles, for all t > 0, the distance between agent i and the detected obstacles must satisfy q i O k > ρ k, B k (O k, ρ k ) Ω t i. (26) Remark 5. The agents should also keep a safe distance from each other to avoid collision. This coupling constraint can be expressed as follows: For agent i and j N i, q i q j > 0. (27) Definition 3. To make the agents track a moving target while avoiding collision with obstacles and other agents, a distributed cost function for agent i {0,, N) can be defined as F i(z i, z p i, u i ) = ω q i q p i 2 + v q i q p i 2 (28)

6 J. Yan et al. / Target Tracking and Obstacle Avoidance for Multi-agent Networks with 51 where ω and v are positive constants. q p i and qp i, acquired from the solution in (25) and system (1), are the presumed position and velocity vectors, respectively. u i is the optimal control input. Obviously, F i = 0 is equivalent to q i = q i and q i = q p i for agent i, which means that agent i can catch up with the presumed state trajectories. Then, the optimization problem can be constructed as follows. Problem 1. For each agent i at time t, given the current presumed state vector z p i, we solve the following optimal problem: with subject to Ji (z i, z p i, u i ) = min J u i(z i, z p i, zp j, ui) (29) i J i(z i, z p i, zp j, ui) = ω qi qp i 2 + (30) v q i q p i 2 + ψ u i 2 ż i = az i + bu i (31) ż p i = azp i + bup i (32) u i C (33) q i O k > ρ k, B k (O k, ρ k ) Ω t i (34) qi q p j > 0, j Ni (35) where q p j is the presumed position vector for agent j. ψ is positive constant. The solutions of Problem 1 yield the optimal control law applied to system (2): u (t) = [u 1(t) T,, u N (t) T ] T. (36) Theorem 2 (Safety). By applying the result in (36), each agent will not collide with the obstacles and other agents. Proof. At each updating time t, the control input u i (t) are free decision variables, constrained on restrictions q i O k > ρ k and q i q p j > 0. From the definition of constraints function in (26) and (27), we can conclude that each agent will not collide with the obstacles and other agents. The route of the obstacle is specified by q obs = (0.65, t) T, 0 t < 260 s. The route of the target is specified by q tar = ( cos(0.1538t), 1.5 sin(0.1538t)) T, 0 t < 260 s q tar = ( sin(0.1538t), sin(0.1538t)) T, 0 t < 260 s. Some parameters in the simulation are given as follows: ξ 1 = 40, ξ 2 = 20, ξ 3 = 10, ξ 4 = 30, ξ 5 = 10, ρ k = 0.4, ρ b = 1.2, ω = 8, v = 6, ψ = 4. The dimension of the input constraint vector for all agents is two, and the input constraint set is defined as C = {(u 1, u 2) R 2 : 10 u j < 10, j = 1, 2}. The simulation studies are done for two cases: for one case, there is no obstacle, and in another case, one obstacle is considered. 4.1 Three agents track the moving target for no obstacle The position trajectories of the three agents and target are shown in Fig. 3. To show the feasibility, the target moves in a circle. Under the proposed algorithm, the agents can track the target. Meanwhile, we also find that agents 1 and 2 keep a pimping distance from the target, the reason is that each agent should avoid collision with the other agents. In order to ensure that there is no collision between agents, the distance between agents should be bigger than 0, as shown in Fig. 4 (a). Meanwhile, our method is supposed to solve collision avoidance problem not consensus problem; then, the desired distances between agents do not need to converge toward a constant. Fig. 4 (b) shows the control inputs. It is obvious to see that the control inputs are within the input constraint set. 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 while avoiding collision with a moving obstacle. The initial state of the agents are given as the follows (when t = 0): q 1 = (0, 0) T, q 2 = ( 0.4, 0.2) T, q 3 = ( 0.4, 0.2) T, q 1 = (0, 0) T, q 2 = (0, 0) T, q 3 = (0, 0) T. Fig. 3 Trajectories of the three agents and the target

7 52 International Journal of Automation and Computing 8(1), February 2011 Fig. 5 Trajectories of the three agents, target, and the obstacle Fig. 4 The distances between agents and the control inputs (d 12 denotes the distance between agents 1 and 2, d 13 is for agents 1 and 3, and d 23 is for agents 2 and 3) 4.2 Three agents track the moving target while there is an obstacle When there is a moving obstacle, the agents should avoid collision with the obstacle. As shown in Fig. 5, agents 3 and 2 are within the area of the obstacle s influence at points B and C, respectively. Then, they should avoid collision with the obstacle. Fig. 5 shows the position trajectories. To show this more clearly, Fig. 6 is used to show the distance between the agents and obstacle. It is obvious to see that the agents do not collide with the obstacle, because the distances are all bigger than 0. 5 Conclusion This paper studies the problem of target tracking and obstacle avoidance for multi-agent networks with input constraints. According to that whether the agents are within the area of the obstacle s influence or not, we divide the environment of the obstacle into two parts: safety area and obstacle avoidance area. Based on potential function method, a presumed solution can be obtained without taking the input constraints into consideration. Optimizing the presumed solution with considering the input constraints, a distributed control algorithm is proposed. Meanwhile, a new velocity potential is designed to solve the dynamic obstacles avoidance problem. Simulation results are provided to verify the effectiveness of the proposed approach. In this paper, the proposed approach is based on cooperative strategy. However, in real-life, some cases are noncooperative, such as a competition game. In the future, we will use non-cooperative game theory to analyze the target tracking and obstacle avoidance problems for multi-agent networks. 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) References [1] M. Defoort, T. Floquet, A. Kokosy, W. Perruquetti. Sliding mode formation control for cooperative autonomous mobile robots. IEEE Transactions on Industrial Electronics, vol. 55, no. 11, pp , [2] J. K. Kuchar, L. C. Yang. A review of conflict detection and resolution modeling methods. IEEE Transactions on Intelligent Transportation Systems, vol. 1, no. 4, pp , [3] N. Miyata, J. Ota, T. Arai, H. Asama. Cooperative transport by multiple mobile robots in unknown static environments associated with real-time task assignment. IEEE Transactions on Robotics and Automation, vol. 18, no. 5, pp , 2002.

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