Stable Flocking Motion of Mobile Agents Following a Leader in Fixed and Switching Networks

Transcription

1 International Journal of Automation and Computing 1 (2006) 8-16 Stable Flocking Motion of Mobile Agents Following a Leader in Fixed and Switching Networks Hui Yu, Yong-Ji Wang Department of control science & Engineering, Huazhong University of Science & Technology, Wuhan , PRC Abstract: Multiple mobile agents with double integrator dynamics, following a leader to achieve a flocking motion formation, are studied in this paper A class of local control laws for a group of mobile agents is proposed From a theoretical proof, the following conclusions are reached: (i) agents globally align their velocity vectors with a leader, (ii) they converge their velocities to the leaders velocity, (iii) collisions among interconnected agents are avoided, and (iv) agent s artificial potential functions are minimized We model the interaction and/or communication relationship between agents by algebraic graph theory Stability analysis is achieved by using classical Lyapunov theory in a fixed network topology, and differential inclusions and nonsmooth analysis in a switching network topology respectively Simulation examples are provided Keywords: Cooperative control, flocking, multi-agent systems, nonsmooth analysis 1 Introduction Decentralized control for the coordination of networks of multiple autonomous agents has attracted a considerable amount of attention in recent decades Scientists from rather diverse disciplines, including animal behavior, physics & biophysics, social sciences, and computer science have been fascinated by the emergence of flocking, swarming, and schooling in groups of agents via local interaction Effort has been directed to try to understand how a group of autonomous moving creatures, such as flocks of birds, schools of fish, crowds of people [1,2], or man-made mobile autonomous agents, can cluster in formations without centralized coordination Such problems have also been studied in ecology and theoretical biology in the context of animal aggregation, and social cohesion in animal groups (see for example [3] and [4]) A computer model mimicking animal aggregation was proposed by Reynolds [5] Following Reynolds work several other computer models appeared in literature, leading to the creation of a new area in computer graphics known as artificial life [5,6] At the same time, several researchers in the area of statistical physics and complexity theory, have addressed flocking and schooling behavior, in the context of nonequilibrium phenomena in many-degree-of-freedom dynamic systems and self organization in systems of self- Manuscript received March 1, 2005; revised September 15, 2005 This work was supported in part by the NSFC (No ), and the NSFC International Collaborative Project (No ) Corresponding author address: yuhui@ctgueducn propelled particles [7 9] Similar problems have seen major effort in systems and control theory, in the context of cooperative control, distributed control of multiple vehicles, and formation control; see for example [10 19] The main goal of the above papers is to develop a decentralized control strategy, so that a global objective, such as a tight formation with desired inter-vehicle distances, is achieved Reynolds [5] aimed at generating a computer animation model of the motion of bird flocks and fish schools The author called the generic simulated flocking creatures boids A basic flocking model consisted of three simple steering behaviors, which described how an individual agent maneuvered, based on the position and velocities of its nearby flockmates: Separation: steer to avoid crowding local flockmates Alignment: steer towards the average heading of local flockmates Cohesion: steer to move toward the average position of local flockmates The superposition of these three rules resulted in all agents moving as a flock while avoiding collisions Generalizations of this model include a leader follower strategy, in which one agent acts as a group leader and other agents follow, and the aforementioned cohesion, separation, and alignment rules, resulting in leader following Vicsek et al [7], proposed such a model which although developed independently, turns out to be a special case of Reynolds [5], where all agents move with the same speed (no dynamics), and only follow an alignment rule In [7], each agent heading is updated as the average of the headings of the agent and its nearest neighbors, plus some additional noise Numerical simu-

2 Hui Yu et al/ Stable Flocking Motion of Mobile Agents Following a Leader in Fixed and Switching Networks 9 lations in Vicsek et al [7] indicate a coherent collective motion, in which the headings of all agents converge to a common value; a surprising result in the physics community that was followed by a series of papers (see for example [20,21]) HGTanner [18,19] developed a class of local control laws for a group of mobile agents in a fixed and dynamic network topology respectively, that resulted in multi-agent flocking behavior In this paper, we introduce a leader agent, and construct local control laws that allow a group of mobile agents to align their velocities with the leader and achieve a constant relative inter-agent distance, while avoiding collisions with each other We theoretically establish the stability properties of the interconnected closed loop system, by combining results from classical control theory, mechanics, and nonsmooth analysis theory We first consider the case where the topology of control interactions between agents is fixed Each agent regulates its position and orientation based on a fixed set of neighbors In this case, control inputs for an agent are smooth We show that under a fixed control interconnection topology, a system of mobile agents is capable of coordinating itself so that all agents following a leader achieve flocking behavior Then, we turn our attention to the case where agent interaction is local, and limited within a certain neighborhood around each agent The time varying nature of an interconnection topology introduces discontinuities in control inputs; system stability is then analyzed using nonsmooth Lyapunov stability As in the smooth case, the connectivity properties of a switching graph are instrumental in establishing global asymptotic stability This paper is organized as follows: in section 2, we define the problem addressed in this paper, and sketch a solution approach In section 3 we propose the coordination control strategies adopted in this paper In section 4 and section 5 we introduce a control scheme that triggers flocking, and analyze the stability of a closed loop system in fixed and switching topology cases respectively Results are verified in section 6 via numerical simulations Section 7 summarizes results and highlights our key points 2 Problem formulation Consider N + 1 agents moving on a plane, with dynamics described by: ḃ i = v i, v i = u i, i = 0, 1, 2,, N Where b i = (x i, y i ) T is the position vector of agent i, v i = (ẋ i, ẏ i ) is its velocity vector, and u i = (u xi, u yi ) T is control (acceleration) input Since it will often be of interest to consider the case in which a group of mobile agents moves with a leader, which is driven at a velocity v 0 (t); the position vector of the ith agent relative to the leader is denoted r i, and the velocity of the ith agent relative to the leader ṙ i = v i v 0 Equations of motion therefore become: d dt ( ri ṙ i ) = ( ṙi u i v 0 ) (1) for i = 1, 2,,N The relative position vector between agent i and j is denoted r ij = r i r j The objective is for the whole group to achieve flocking behavior, by following a leader agent using local control laws Control input for agent i is a combination of two components: u i = a i + α i, i = 1, 2,,N (2) The first component, a i, is derived from the field produced by an artificial potential function, U i, which depends on the relative distance between agent i and its flockmates This term is responsible for collision avoidance, and cohesion in the group The second component, α i, aligns the velocity vector of agent i to follow the leader Definition 1 (Flocking) A group of mobile agents is said to (asymptotically) flock, when all agents attain the same velocity vector, distances between agents are stabilized, and no collisions occur between them The problem is to design control input (2) so that a group of mobile agents flocks, following a leader in the sense of Definition 1 3 Coordination strategies In this section, we will refine the acceleration input of (2) into specific expressions for components a i and α i To represent the control interconnections between agents, we use a graph with a vertex corresponding to each agent Edges capture the dependence of agent controllers on the state of other agents Adjacency in the graph will therefore induce a (logical) neighboring relationship between agents Definition 2 (Neighboring graph) The neighboring graph, G = {V, E}, is an undirected graph consisting of: a set of vertices (nodes), V = {n 0, n 1, n 2,, n N }, indexed by a leader, and agents in the group, and a set of edges, E = {(n i, n j ) V V n i n j }, containing unordered pairs of nodes that represent neighboring relations The set of all neighbors of agent i, is called the neighboring set, denoted: N i {j i j} {0, 1, 2,, N}\{i} (3)

3 10 International Journal of Automation and Computing 1 (2006) 8-16 The neighboring set of agent i, N i, can represent the set of agents with which i is allowed to communicate (giving rise to a fixed logical interconnection network), or the set of agents which i can sense, transmit or receive information, to or from In the latter case, the neighboring set may express physical proximity, since sensing and communication capabilities can also be spatially-related; giving rise to a dynamic, distancedependent interconnected network These two cases motivate stability analysis in Section 4 and 5, respectively The control law of agent i, u i, can be defined as: u i = ri U ij γṙ i + v 0, i = 1, 2,,N, (4) }{{} }{{} α i a i where γ is a positive constant, and U ij depends on the relative distance between neighbors, defined as follows: Definition 3 (Potential function) Potential U ij is a differentiable, nonnegative, radially unbounded function of the distance r ij between agents i and j, such that: 1) U ij ( r ij ) as r ij 0, and 2) U ij attains its unique minimum when agents i and j are located at a desired distance Cohesion and separation is achieved using artificial potential fields (see, Fig1) One possible choice could be: U ij ( r ij ) = 1 r ij 2 + ln r ij 2 Fig1 Example of an inter-agent potential function Having defined U ij we can express the total potential of agent i as U i = U ij ( r ij ) = χ N0 (i)u i0 ( r i0 )+ U ij ( r ij ) (5) \{0} where χ N0 (i) = { 1, i N0 0, i N 0 4 Fixed interconnection topology If the interconnection topology of a group is represented by a time invariant but connected graph, then control law (4) creates an asymptotically stable equilibrium manifold, on which the group satisfies the conditions for flocking given by Definition 1 Each agent maintains a fixed set of neighbors, implying that a neighboring graph is constant The main consequence of time invariance is that the mechanical energy of a group is differentiable, agent control laws are smooth, and classic Lyapunov theory can be applied If we consider the following positive semi-definite function: W = 1 2 (2χ N0 (i)u i0 + \{0} U ij + ṙ T i ṙi) (6) Using LaSalle s invariant principle, we can show that the closed loop system of agents (1) (4) flocks, provided that a neighboring graph is connected: Theorem 1 (Flocking in a fixed network) Consider a system of N mobile agents, following a leader with dynamics (1), and each steered by control law (4) Then, all agent velocity vectors become asymptotically the same as the leader, the relative distance that agents maintain between them becomes constant, collisions between interconnected agents are avoided, and the system approaches a configuration that minimizes all agent potentials Proof The level sets of W, define compact sets in the space of agent relative velocities and relative distances The set {r ij, ṙ i } such that W c, for c > 0 is closed by continuity Boundedness follows from connectivity: from W c we have U ij c Connectivity ensures that a path connecting nodes i and j has a length at most N Therefore, r ij U 1 ij (cn), and similarly, ri Tr i c yields r i c Therefore, the set: Ω = {(ṙ i, r ij ) W c} is compact If the derivative of W defined in (6) is: Ẇ = ṙ T i = (χ N0 (i)ṙi T r i U i0 + \{0} ri U ij + ṙ T i (u i v 0 )) ṙi T (χ N0 (i) ri U i0 + \{0} ri U ij +u i v 0 )

4 Hui Yu et al/ Stable Flocking Motion of Mobile Agents Following a Leader in Fixed and Switching Networks 11 = ṙi T ( = ri U ij + u i v 0 ) γṙi T ṙ i (7) and Ẇ is negative definite and equal to zero, if and only if, ṙ i = 0 for all i By LaSalle s invariant principle, if the initial conditions of the system lie in Ω, its trajectories will converge to the largest invariant set inside region S = {ṙ i Ẇ = 0} This implies that v i = v 0 and ṙ ij = 0 (which means that the relative distance between agent i and j is constant) for all i asymptotically Note that Ω can be made arbitrarily large, ensuring the semi-global asymptotic stability of the invariant set Interconnected-agents cannot collide, since this would result U i and the system departing Ω, which is a contradiction since Ω is positively invariant Furthermore, system trajectories converge to S = {ṙ i Ẇ = 0} in a steady state In S, agent velocity dynamics become: r1 U 1 v 0 v 0 r1 U 1 v = + v = rn U N v 0 v 0 rn U N This implies that the potential U i of each agent i is minimized In fact although cohesion is ensured for a connected graph, the fixed topology of the graph cannot guarantee collision avoidance unless the neighboring graph is complete when two agents are not linked, they cannot be aware of being close to each other Remark 1 Collision avoidance between all agents can only be guaranteed with this control scheme, when all agents are interconnected to each other This requires a neighboring graph to be complete 5 Dynamic interconnection topology In this section, we relax the assumption that an interconnection topology is fixed, and allow neighboring relations to depend on the physical proximity of agents Therefore, two agents are considered interconnected, when their distance is below a certain threshold R: i j r ij < R A control law for agent i is still expressed by (4), with the only difference being that N i now changes dynamically as a function of distance between agents Such changes introduce discontinuities in (4), which in turn give rise to a set of discontinuous differential equations for the dynamics of agent i The stability of discontinuous dynamics can be analyzed using differential inclusions [22], and nonsmooth analysis [23,24] Function U ij can be nonsmooth at distance r ij = R, and constant U ij = V R for R ij > R, to capture the fact that beyond this distance there is no agent interaction One example of such a nonsmooth potential function is depicted in Fig2, and as follows: U ij = 1 r ij 2 + log r ij 2, r ij < R V R, r ij R Fig2 A nonsmooth inter-agent potential function Having defined U ij, we can express the total potential of agent i as: U i = ((N + 1) N i )V R + U ij ( r ij ) (8) Stability analysis is based on a nonsmooth version of LaSalle s invariant principle (Shevitz and Paden, 1994), using a nonnegative Lyapunov-like function: Q = 1 2 (2U i0 + U ij + (v i v 0 ) T (v i v 0 )) (9) j=1 Now we can generalize Theorem 1 to the case where an interconnection topology switches arbitrarily between connected neighbor graphs: Theorem 2 (Flocking in switching networks) Consider a system of N mobile agents, and a leader with dynamics (1), each steered by control law (4), and assume that a neighboring graph is connected Then, all pairwise velocity differences converge asymptotically to zero, relative distances between agents are maintained constant asymptotically, collisions between agents are avoided, and the system approaches a configuration that minimizes all agent potentials Proof We differentiate function Q as it is expressed in (9) For all points where r ij R for any

5 12 International Journal of Automation and Computing 1 (2006) 8-16 (i, j) N N, the time derivative of Q is calculated as in (7) yielding: Q = γṙ T ṙ, r ij R, (i, j) N N If for some (i, j) N N we have r ij = R, then we need to consider the generalized time derivative of Q In order to apply the nonsmooth version of LaSalle s invariant principle introduced by [25], we need to first establish the regularity of Q (for a formal definition of regularity, see [23]) To this end, we use the following Lemmas: Lemma 1 [19] The functionis U ij is regular [23] everywhere in its domain Corollary 1 [19] The generalized gradient of U ij at R is empty: U ij (R) = (10) Therefore, Q is regular as a sum of regular functions Another interesting fact that results from U ij increasing at R is the following, which is useful in computing the generalized time derivative of Q Lemma 2 [19] The (partial) generalized gradient of U ij with respect to r i at R is empty: ri U ij (R) = (11) The regularity of Q as a sum of regular functions, and the property of finite sums of generalized gradients, ensures that: Q [ r1 U1j T,, rn UNj T, ṙt 1,, ṙt N ]T j N 1 j N N Then, for the generalized time derivative of Q we will have: ṙ 1 Q ξ T ṙ N ξ Q K[u 1 v 0 ] K[u N v 0 ] ṙ 1 = ṙ N ξ T K[ r1 U 1j ] γṙ 1 j N 1 ξ Q K[ rn U Nj ] γṙ N j N N ( ξi T ṙi) + ṙi T K[ ri U ij ] ξ i γṙ T i ṙi Where ξ i ri U ij and ri U i = ri U ij switch over time, depending on the neighboring set N i of each agent i By recalling that U ij (R) = (Lemma 2), and using differential inclusion algebra [26], we can obtain: Q ṙi T ri U ij ṙi T ri U ij γṙi T ṙ i = co{ ṙi T ṙ i } (12) For any graph, the right hand of (12) will be an interval of the form [e, 0], with e < 0 Therefore, q 0 for all Q Q, and the interval contains 0 only when ṙ i = 0 Applying the nonsmooth version of LaSalle s principle proposed by [25], it follows that for initial conditions in Ω, the Filippov trajectories of the system converge to a subset of S = {ṙ i ṙ i = 0}, in which ṙ i = v i v 0 = 0, i N and ṙ ij = 0, (i, j) N N This implies that all agents will achieve a common velocity vector the same as the leader, and maintain a constant relative distance with other agents asymptotically Furthermore, system trajectories converge to S = {ṙ i Ẇ = 0} in a steady state In S, agent velocity dynamics become: r1 U 1 v = + rn U N v 0 v 0 v v 0 v 0 = r1 U 1 rn U N This implies that the potential U i of each agent i is minimized 6 Simulation In this section, we verify numerically the stability results obtained in Section 4 and 5 In a simulation example, a group consists of 5 mobile agents and a leader, with identical second order dynamics The leader is driven by a sine acceleration signal in the x- and y-axis direction respectively Initial positions are generated randomly within a square area [0, 5; 0, 5] m Initial velocities were selected randomly with arbitrary directions, and a magnitude in the range (0, 1) [m/s] An interconnection graph was also generated randomly, with the only requirement being that it was connected Fig3 and 4 depict snapshots of the system s evolution over a 15 second simulation time, for a fixed topology and dynamic topology respectively The position of each agent is represented by a small dot, the leader marked as a circle, and neighboring relations by line segments

6 Hui Yu et al/ Stable Flocking Motion of Mobile Agents Following a Leader in Fixed and Switching Networks 13 connecting agents Velocity vectors are depicted as arrows, with their base point being the position of the corresponding agent Dotted lines show the trajectory trails for each agent The simulation verifies that the system converges to an invariant set that corresponds to a tight formation, a common heading direction the same as the leader, and a constant relative distance between agents The shape of the formation which the group converges to, is determined by artificial potential functions (a) The initial configuration (b) An initial maneuver brings agents closer (c) Potential force ensures cohesion (d) Control force ensures alignment (e) Velocity vectors converge (f) A steady state is achieved Fig3 Successive simulation time snapshots of flocking, using a fixed interconnection topology

7 14 International Journal of Automation and Computing 1 (2006) 8-16 (a) The initial configuration (b) An initial maneuver brings agents closer (c) Potential force ensures cohesion (d) Control force ensures alignment (e) Velocity vectors converge (f) A steady state is achieved Fig4 Successive simulation time snapshots of flocking, using a dynamic interconnection topology

8 Hui Yu et al/ Stable Flocking Motion of Mobile Agents Following a Leader in Fixed and Switching Networks 15 7 Conclusion This paper provides a theoretical framework for the controller design and stability analysis of distributed flocking algorithms for multi-agent dynamic systems We showed how a group of autonomous mobile agents following a leader agent, can cooperate to exhibit flocking behavior Flocking requires all agents to have a common heading the same as a leader, to stay close to each other while avoiding collisions, and to achieve a stabilized distance with other agents We model flocking, and introduce local controllers that establish a stable and coordinated flocking motion These local controllers are based on a fixed or dynamic communication (or sensing) network, which allows the exchange of state information between interconnected agents Stability of group motion follows from the connectivity properties of the underlying network topology References [1] D J Low Following the crowd Nature, vol 407, no 6803, pp , 2000 [2] T Vicsek A question of scale Nature, vol 411, no 6836, pp 421, 2001 [3] D Grunbaum, A Okubo Modeling social animal aggregations Frontiers in Theoretical Biology, Lecture Notes in Biomathematics, Springer-Verlag, vol 100, pp , 1994 [4] G Flierl, D Grunbaum, S Levin, D Olson From individuals to aggregations: the interplay between behavior and physics Journal of Theoretical Biology, vol 196, no 2, pp , 1999 [5] C Reynolds Flocks, birds, and schools: a distributed behavioral model Computer Graphics, vol 21, no 1, pp 25 34, 1987 [6] D Terzopoulos Artificial life for computer graphics Communications of the ACM, vol 42, no 8, pp 32 42, 1999 [7] T Vicsek, A Czirok, E B Jacob, I Cohen, O Schochet Novel type of phase transitions in a system of self-driven particles Physical Review Letters, vol 75, no 2, pp , 1995 [8] J Toner, Y Tu Flocks, herds, and schools: A quantitative theory of flocking Physical Review E, vol 58, no 6, pp , 1998 [9] H Levine, W J Rappel Self organization in systems of self-propelled particles Physical Review E, vol 63, no 2, pp , 2001 [10] N Leonard, E Friorelli Virtual leaders, artificial potentials and coordinated control of groups IEEE Conference on Decision and Control, Orlando, Florida USA, pp , 2001 [11] R Olfati, R M Murray Distributed structural stabilization and tracking for formations of dynamic multi-agents IEEE Conference on Decision and Control, Las Vegas, NV, pp , 2002 [12] H R John, W Hongyang Social potential fields: A distributed behavioral control for autonomous robots Robotics and Autonomous Systems, vol 27, no 3, pp , 1999 [13] Y Liu, K M Passino, M M Polycarpou Stability analysis of m-dimensional asynchronous swarms with a fixed communication topology IEEE Transactions on Automatic Control, vol 48, no 1, pp 76 95, 2003 [14] V Gazi, K M Passino Stability analysis of swarms IEEE Transactions on Automatic Control, vol 48, no 4, pp , 2003 [15] A Jadbabaie, J Lin, A S Morse Coordination of groups of mobile autonomous agents using nearest neighbor rules IEEE Transactions on Automatic Control, vol 48, no 6, pp , 2002 [16] J P Desai, J P Ostrowski, V Kumar Modeling and control of formations of nonholonomic mobile robots IEEE Transactions on Robotics and Automation, vol 17, no 6, pp , 2001 [17] Z Lin, M Broucke, B Francis Local control strategies for groups of mobile autonomous agents IEEE Transactions on Automatic Control, vol 49, no 4, pp , 2004 [18] H G Tanner, A Jadbabaie, G J Pappas Stable Flocking of Mobile Agents, Part I: Fixed Topology 42nd IEEE Conference on Decision and Control, Maui, Hawaii USA, pp , 2003 [19] H G Tanner, A Jadbabaie, G J Pappas Stable Flocking of Mobile Agents, Part II: Dynamic Topology 42nd IEEE Conference on Decision and Control, Maui, Hawaii USA, pp , 2003 [20] J Toner, Y Tu Long range order in a two dimensional xy model: How birds fly together Physical Review Letters, vol 75, no 6, pp , 1995 [21] J Toner, Y Tu Flocks, herds, and schools: A quantitative theory of flocking Physical Review E, vol 58, no 6, pp , 1998 [22] A F Filippov Differential equations with discontinuous right-hand side Translations of the American Mathematical Society, vol 42, no 2, pp , 1964 [23] F H Clarke Optimization and Nonsmooth Analysis, Classics in applied mathematics, 5, SIAM, Philadelphia, 1990 [24] F H Clarke, Y S Ledyaev, R J Stern, P R Wolenski Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178, Springer, New York, 1998 [25] D Shevitz, B Paden Lyapunov stability theory of nonsmooth systems IEEE Transactions on Automatic Control, vol 39, no 9, pp , 1994 [26] B Paden, S Sastry A calculus for computing filipov s differential inclusion with application to the variable structure control of robot manipulators IEEE Transactions on Circuits and Systems, vol 34, no 1, pp 73 82, 1987 Hui Yu was born in Yichang, Hubei, P R China, in 1967 He received his BSc degree in Mathematics Education from Central China Normal University, Wuhan, P R China, a M S degree in System analysis and integration from East China Normal University, Shanghai, P R China, in 1991 and 1999, respectively From July 1999 to September 2004, he was a lecturer at the Department of Mathematics, China Three Gorges University, Yichang, Hubei, P R China Since September 2004, he has been a doctoral candidate at the Department of Control Science & Engineering, Huazhong University of Science & Technology, Wuhan, P R China His research interests are in the distributed coordination of mobile autonomous multiagents, as well as the application of mobile sensor networks and robotics

9 16 International Journal of Automation and Computing 1 (2006) 8-16 Yong-Ji Wang was born in Ji an, Jiangxi, PR China, in 1955 He received his BSc degree in Electrical Engineering from Shanghai Railway University, Shanghai, PR China, an MS degree and a PhD degree in automation from Huazhong University of Science and Technology, Wuhan, PR China, in 1982, 1984 and 1990, respectively Since 1984, he has been at Huazhong University of Science and Technology, Wuhan, PR China, where he is currently a Professor of Electrical Engineering His main interest is in intelligent control, neural network, predictive, adaptive control, and most recently, coordination and control of large groupings of mobile autonomous agents Dr Wang is a member of the IEEE, USA, a standing member of the council of the Electric Automation Committee of the Chinese Automation Society, and a member of the council of the Intelligent Robot Committee of the Chinese Artificial Intelligence Society He is an area editor (Asia and Pacific) of the Int J of Simulation Identification and Control