Virtual leader approach to coordinated control of multiple mobile agents with asymmetric interactions

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1 Physica D 213 (2006) Virtual leader approach to coordinated control of multiple mobile agents with asymmetric interactions Hong Shi, Long Wang, Tianguang Chu Intelligent Control Laboratory, Center for Systems and Control, Department of Mechanics and Engineering Science, Peking University, Beijing , PR China Received 30 March 2005; received in revised form 27 October 2005; accepted 28 October 2005 Available online 28 November 2005 Communicated by W. van Saarloos Abstract This paper considers the collective dynamics of a group of mobile autonomous agents moving in Euclidean space with a virtual leader. We introduce a set of coordination control laws that enable the group to generate the desired stable flocking motion. The control laws are a combination of attractive/repulsive and alignment forces, and the control law acting on each agent relies on the state information of its flockmates and the external reference signal (or virtual leader ). Using the control laws, all agent velocities asymptotically approach the desired velocity, collisions can be avoided between the agents, and the final tight formation minimizes all agent global potentials. Moreover, we show that the velocity of the center of mass either is equal to the desired velocity or exponentially converges to it. Furthermore, when the velocity damping is taken into account, we can appropriately modify the control laws to generate the same stable flocking motion. Subsequently, for the case where not all agents know the desired common velocity, we show that the desired flocking motion can still be guaranteed. Numerical simulations are worked out to illustrate our theoretical results. Additionally, we consider the effect of white noise on the collective dynamics of the group, and demonstrate numerically that the desired flocking motion can be kept for weak noise and, as the noise intensity increases, the flocking motion can be destroyed. c 2005 Elsevier B.V. All rights reserved. PACS: a; b; j; r Keywords: Coordination; Collective dynamic behavior; Cooperative control; Flocking motion; Multi-agent systems; Networked systems; Swarm intelligence; Virtual leader 1. Introduction Flocking motion is ubiquitous in Nature, e.g., flocking of birds, schooling of fish, and swarming of bacteria. Such collective behavior has certain advantages such as avoiding predators and increasing the chance of finding food. The study of collective emergent behavior of multiple mobile autonomous agents has attracted much attention in many fields such as biology, physics, robotics and control engineering. Understanding the mechanisms and operational principles in them can provide useful ideas for developing formation control, distributed cooperative control and coordination of multiple mobile autonomous agents/robots. Recently, distributed control/coordination of the motion of Corresponding author. address: longwang@pku.edu.cn (L. Wang). multiple dynamic agents/robots has emerged as a topic of major interest in intelligent control [1 4]. This is partly due to recent technological advances in communication and computation, and wide applications of multi-agent systems in many engineering areas including cooperative control of unmanned aerial vehicles (UAVs), scheduling of automated highway systems, coordination/formation of underwater vehicles, attitude alignment of satellite clusters and congestion control in communication networks [5 8]. There has been considerable effort in modelling and exploring the collective dynamics, and trying to understand how a group of autonomous creatures or man-made mobile autonomous agents/robots can cluster in formations without centralized coordination and control [9 29]. In order to generate computer animation of the motion of flocks, Reynolds [9] modelled a flying bird as an object moving in a three-dimensional environment based on the /$ - see front matter c 2005 Elsevier B.V. All rights reserved. doi: /j.physd

2 52 H. Shi et al. / Physica D 213 (2006) positions and velocities of its nearby flockmates, and introduced the following three rules (named steering forces) [9]: (1) collision avoidance: avoid collisions with nearby flockmates, (2) velocity matching: attempt to match velocity with nearby flockmates, and (3) flock centering: attempt to stay close to nearby flockmates. Subsequently, Vicsek et al. [10] proposed a simple model of autonomous agents (i.e., points or particles). In the model, all agents move at a constant identical speed and each agent updates its heading as the average of the heading of itself with its nearest neighbors plus some additive noise. Moreover, the authors used numerical simulations to demonstrate that all agents eventually moved in the same direction, despite the absence of centralized coordination and control. In fact, Vicsek s model can be viewed as a special case of Reynolds model, as it only considers the regulation of velocity matching. Jadbabaie et al. [13] and Savkin [14] used two different methods to provide theoretical explanations for the observed behaviors in Vicsek s model. Stimulated by the simulation results in [9], Tanner et al. [15] considered a group of mobile agents moving in the plane with double-integrator dynamics. They introduced a set of control laws that enable the group to generate stable flocking motion, and provided theoretical justification. From [16], it is easy to see that, using these control laws, the group s final velocity relies solely on the initial velocities of all agents in the group. This means that these control laws cannot regulate the final speed and heading of the group. On the other hand, in reality, the motion of the group is sometimes inevitably influenced by some external factors. Hence, it is not enough to consider only the interactions among agents. In some cases, the regulation of agents has certain purposes such as achieving the desired common speed and heading, or arriving at a desired destination. Therefore, the cooperation/coordination of multiple mobile agents with some virtual leaders is an interesting and important topic. There have been some papers dealing with this issue in the literature. For example, Leonard and Fiorelli [1] viewed reference points as virtual leaders for manipulating the geometry of an autonomous vehicle group and directing the motion of the group. Refs. [13, 22,27] considered the cohesion/coordination of a group of mobile autonomous agents following an actual leader. In this paper, we investigate the collective behavior of multi-agent systems in high-dimensional space with point mass dynamics. By viewing the external control signals (or mission ) as virtual leaders, we show that all agents eventually move ahead at a desired common velocity and maintain constant distances between them. During the course of motion, each agent is influenced by the external control signal and the motion of other agents in the group. In [15,16], the authors used undirected graphs to describe the neighboring relations between agents, which means that they assume the neighboring relations are mutual. In other words, they only considered the case with bidirectional information exchange between agents. Moreover, they only considered all-identical or reciprocal coupling patterns in the swarm models. However, in some cases, the information exchange is not mutual, e.g., not all agents have the same sensing range, and due to the agent differences and the complexity of the interindividual interactions, the interaction strength between agents may vary from one pair to another, depending on relative distances of the individuals or other factors. Hence, the intensities of influence between two agents might be different and possibly even their information cannot be mutually exchanged at all. For example, for a group of agents with spherical sensing neighborhoods but with different radii of the neighborhoods or for a group of agents with conic sensing neighborhoods, the information exchange among agents might be unidirectional. In this paper, the results in [15,16] are extended to directed graphs. We consider the stability properties of the group in the case with directed information exchange. In [16], we have shown that, when the topology of the neighboring relations is fixed and moreover the coupling matrix is symmetric, stable flocking motion can be achieved and the velocity of the center of mass (CoM) is invariant and is equal to the group final velocity. This means that the final velocity of the group is determined by the masses and the initial velocities of all agents in the group and does not rely on the control laws. In this paper, we investigate the motion of the group with fixed information topology and asymmetric coupling matrix. In order to generate the desired stable flocking, we introduce a set of control laws such that each agent regulates its velocity on the basis of the desired velocity and the velocities of a fixed set of neighbors and regulates its position such that its global potential becomes a minimum. One feature of this paper is that the self-organized global behavior is achieved via partially local feedback, i.e., the desired emergent dynamics is produced through local interactions and information exchange between the dynamic agents. This paper is organized as follows. In Section 2, we formulate the problem to be investigated. By using some specific control laws, we analyze the system stability, the motion of the CoM, and the convergence rate of the system in Section 3. We present some different control laws that can also generate the desired stable flocking motion in Section 4. For the case where not all agents know the desired velocity, we introduce a set of control laws and study the system stability in Section 5. Some numerical simulations are presented in Section 6. We investigate the effect of noise on the motion of the agent group in Section 7. Finally, we briefly summarize our results in Section 8. Some basic concepts and results in algebraic graph theory are provided in the Appendix A. 2. Problem formulation Consider N agents moving in an n-dimensional Euclidean space; each has point mass dynamics described by ẋ i = v i, m i v i = u i, i = 1,..., N, where x i R n is the position vector of agent i, v i R n is its velocity vector, m i > 0 is its mass, and u i R n is the (force) control input acting on agent i. Our objective is to make the entire group move at a desired velocity and maintain constant distances between the agents. In (1)

3 H. Shi et al. / Physica D 213 (2006) what follows, we will investigate the motion of the agent group in two different cases, that is, we consider the group motion in an ideal case (i.e., velocity damping is ignored) and a nonideal case (i.e., velocity damping is taken into account). For the two different cases, we propose two different control laws such that the entire group moves at a desired common velocity, and at the same time, freedom from collisions between the agents can be ensured, and the group final configuration minimizes all agent global potentials. We first consider the ideal case, that is, we ignore the velocity damping. In this case, in order to achieve our control objective, we try to regulate each agent velocity to the desired velocity, reduce the velocity differences between agents, and at the same time, regulate their distances such that their global potentials become minima. Hence, we choose the control law u i for agent i to be u i = α i + β i + γ i, (2) where α i is used to regulate the potentials among agents, β i is used to regulate the velocity of agent i to the weighted average of its flockmates, and γ i is used to regulate the momentum of agent i to the desired final momentum (all to be designed later). α i is derived from the social potential field which is described by an artificial social potential function V i, a function of the relative distances between agent i and its flockmates. Freedom from collisions and cohesion in the group can be guaranteed by this term. β i reflects the alignment or velocity matching with neighbors among agents. γ i is designed to regulate the momentum among agents based on the external signal (the desired velocity). By using such a momentum regulation, we can obtain the explicit convergence rate of the CoM of the system. Remark 1. The design of α i and β i indicates that, during the course of motion, agent i is influenced only by its neighbors, whereas γ i reflects the influence of the external signal on the agent motion. Certainly, in some cases, the velocity damping cannot be ignored. For example, objects moving in a viscous environment and mobile objects with high speeds such as supersonic aerial vehicles are subjected to the influence of velocity damping. Then, in this case, the model in Eq. (1) should be in the following form: ẋ i = v i, m i v i = u i k i v i, i = 1,..., N, where k i > 0 is the velocity damping gain, k i v i is the velocity damping term, and u i is the control input for agent i. Here we assume that the damping force is in proportion to the magnitude of the velocity and the damping gains k i, i = 1,..., N, are not equal to each other. In order to achieve our control objective, we need to compensate for the velocity damping. Hence, we modify the control law u i to u i = α i + β i + γ i + k i v i. (4) (3) 3. Main results In this section, we investigate the stability properties of multiple mobile agents with point mass dynamics described in Eq. (1). We will present explicit control input in Eq. (2) for the terms α i, β i, and γ i. The control law acting on each agent is based on two kinds of information topologies that are the position information topology and the velocity information topology. We will employ matrix analysis and algebraic graph theory as basic tools for our discussion. Some basic concepts and results are given in the Appendix A, and more information can be found in [30,31]. Due to the complexity of the agent interactions, we will define two kinds of structure topologies to describe the information flows between the agents. Throughout this paper, we assume that each agent is equipped with two onboard sensors: the position sensor which is used to sense the position information of the flockmates and the velocity sensor which is used to sense the velocity information of its neighbors, and assume that all sensors can sense instantaneously. Correspondingly, we will define two kinds of graphs to describe the neighboring relations between the agents. In what follows, we will use an undirected graph G to describe the position sensor information flow and use a weighted directed graph D to describe the velocity sensor information flow. Following [15], we make the following definitions. Definition 1 (Position Neighboring Graph). The position neighboring graph, G = (V, E), is an undirected graph consisting of a set of vertices, V = {n 1,..., n N }, indexed by the agents in the group, and a set of edges, E = {(n i, n j ) V V : n j n i }, containing unordered pairs of vertices that represent the position neighboring relations. Definition 2 (Velocity Neighboring Graph). The velocity neighboring graph, D = (V, E, A), is a weighted directed graph consisting of a set of vertices, V = {n 1,..., n N }, indexed by the agents in the group, and a set of arcs, E = {(n i, n j ) V V : n j n i }, containing ordered pairs of vertices that represent the velocity neighboring relations. A = [a i j ] R N N is the weight matrix which consists of the coefficients of interaction between the agents. Note that, in the velocity neighboring graph D, an arc (n i, n j ) represents a unidirectional velocity information exchange link from n i to n j, which means that agent i can obtain the velocity information of agent j. Let I = {1, 2,..., N}. For the velocity neighboring graph, we write N i { j : a i j > 0} I \ {i} for the set which contains all velocity neighbors of agent i, i.e., agent i can only sense the velocities of the agents that are contained in N i, where a i j is the weight of arc (n i, n j ). Definition 3 ([15]: Potential Function). Potential V i j is a differentiable, nonnegative, radially unbounded function of the distance x i j between agents i and j, where x i j = x i x j denotes the relative position vector between agents i and j, such that V i j ( x i j ) as x i j 0 and V i j attains its unique minimum when agents i and j are located at a desired distance.

4 54 H. Shi et al. / Physica D 213 (2006) Functions V i j, i, j = 1,..., N, are the artificial social potential functions that govern the interindividual interactions. Cohesion and separation can be achieved by artificial potential fields [6]. One example of such potential functions is the following: V i j ( x i j ) = a ln x i j 2 + b x i j 2, where a and b are some positive constants. It is easy to see that V i j attains its unique minimum a(1 + ln(b/a)) when x i j = b/a. Hence, when the distance x i j between agents i and j is b/a, the potential function V i j attains its unique minimum. In this paper, we consider a group of mobile agents with fixed topology. In order to make the final potential of each agent a global minimum and, at the same time, ensure freedom from collisions in the group, we assume that the position neighboring graph G is complete. This means that each agent can obtain the position information of all the other agents in the group. By the definition of V i j, the total potential of agent i can be expressed as V i = V i j ( x i j ). (5) Agent dynamics in the ideal case is different from that in the nonideal case, i.e., agents have different motion equations in the two cases. Hence, in what follows, we will discuss the motion of the agent group in two different cases. Note that, in this section, we always assume that all agents can detect the external signal, that is, they all know the desired common velocity. In the case where not all agents know the mission, we will discuss the flocking control problem in a separate section Ideal case In this case, we take the control law u i for agent i to be w i j (v i v j ) x i V i j m i (v i v 0 ), (6) where v 0 R n is the desired common velocity and is a constant vector, w i j 0, and w ii = 0, i, j = 1,..., N, represent the interaction coefficients. w i j > 0 if agent i can obtain the velocity information of agent j, and is 0 otherwise. We write W = [w i j ] R N N for the interaction coefficient matrix (coupling matrix) associated with the velocity neighboring graph D. Hence, when D is connected, W + W T is symmetric and irreducible. The control law in Eq. (6) implies that we adopt the local velocity regulation and the global potential regulation to achieve our control objective. Such a regulation is due to the complexity of the interactions between agents (or particles) in Nature. In the discussion to follow, we will need the concept of weight balance condition, defined below: Weight Balance Condition [23]: consider the weight matrix W = [w i j ] R N N ; for all i = 1,..., N, we assume that Nj=1 w i j = N j=1 w ji. The weight balance condition has a graphical interpretation: consider the directed graph associated with a matrix; weight balance means that, for any node in this graph, the weight sum of all incoming edges equals the weight sum of all outgoing edges [31]. The weight balance condition can find physical interpretations in engineering systems such as water flow, electrical current, and traffic systems. Proposition 1. Let D be a weighted directed graph such that the weight balance condition is satisfied. Then D is strongly connected if and only if it is weakly connected. Proof. We only need to prove that if D is weakly connected, then it is strongly connected. We will prove it by contradiction. The proof follows a similar line to the proof of Lemma in [30]. Assume that D is weakly connected, but not strongly connected; then we denote all strongly connected components of D as D 1,..., D m, where m is an integer and m > 1. It follows that all arcs joining two different strongly connected components must have the same direction, and therefore, we can define a directed graph D where the strongly connected components of D are defined as its vertices and there exists an arc from D i to D j in D if and only if there is an arc in D starting in D i and ending in D j. Obviously the directed graph D cannot contain any cycle since otherwise the number of strongly connected components of D will be equal to or less than m 1. Hence, there is a strongly connected component D i0 such that any arc ending on a vertex in it must start at a vertex in it. Since D is weakly connected, there must be at least one arc starting in D i0 and ending on a vertex not in D i0. Thus, in D i0, the sum of in-degrees of all vertices is less than the sum of out-degrees. This means that there must be a vertex in D such that the weight balance condition cannot be satisfied. Thus we obtain the contradiction. This completes the proof. Throughout this paper, we assume that the coupling matrix satisfies the weight balance condition. Hence, if D is weakly connected, then it must be strongly connected Stability analysis Theorem 1. Taking the control law in Eq. (6), all agent velocities in the group described in Eqs. (1) asymptotically approach the desired common velocity, avoidance of collisions between the agents is ensured, and the group final configuration minimizes all agent global potentials. This theorem becomes clearly true after Theorem 2 is proved, so we proceed to present Theorem 2. We define the error vectors: e i p = xi v 0 t, and ev i = v i v 0, where t is the time variable and v 0 is the desired common velocity. Then ev i represents the velocity difference vector between the actual velocity and the desired velocity of

5 H. Shi et al. / Physica D 213 (2006) agent i. It is easy to see that ė i p = ei v and ėi v = vi. Hence, the error dynamics is given by ė i p = ei v, ė i v = 1 m i u i, i = 1,..., N. By the definition of V i j, it follows that V i j ( x i j ) = V i j ( e i j p ) Ṽ i j, where e i j p e i p e j p, and hence Ṽ i = V i and e i p Ṽ i j = x i V i j. Thus, the control input for agent i in the error system has the following form: w i j (e i v e j v ) (7) e i p Ṽ i j m i e i v. (8) Consider the following positive semi-definite function: J = 1 2 (Ṽ i + m i ev it ei v ). (9) It is easy to see that J is the sum of the total artificial potential energy and the total kinetic energy of all agents in the error system. Define the level set of J in the space of agent velocities and relative distances in the error system: Ω = {(e i v, ei j p ) : J c, c > 0}. (10) In what follows, we will prove that the set Ω is compact. In fact, the set {ev i, ei p j } with J c is closed by continuity. Moreover, boundedness can be proved by the connectivity of the position neighboring graph. More specifically, from J c, we have Ṽ i j c. On the other hand, since the potential function V i j is radially unbounded, Ṽ i j is also radially unbounded. Thus there is a positive constant d such that e i p j d for all i, j I. By similar analysis, we have ev itei v 2c/m i; thus ev i 2c/m i. By the symmetry of Ṽ i j with respect to e i p j and by e i p j = ep ji, it follows that Ṽ i j e i j p = Ṽ i j e i p and therefore d dt 1 2 Ṽ i = = Ṽ i j, (11) e j p ( e i p Ṽ i ) T ev i. Theorem 2. Taking the control law in Eq. (8), all agent velocities in the system described in Eqs. (7) asymptotically approach zero, avoidance of collisions between the agents is ensured, and the group final configuration minimizes all agent global potentials. Proof. Choosing the Lyapunov function J defined as in Eq. (9) and calculating the time derivative of J along the solution of the error system (7), we have J = w i j e it v (ev i e v j ) m i ev it ei v j N i = e T v (L I n)e v e T v (M I n)e v = 1 2 et v ((L + LT ) I n )e v e T v (M I n)e v, (12) where e v = (ev 1T,..., ent v ) T is the stack vector of all agent velocity vectors in the error system; L = [l i j ] R N N with w i j, i j, l i j = (13) w ik, i = j k=1,k i is the Laplacian matrix of the weighted velocity neighboring graph if we set the corresponding edge weight of the graph to be w i j ; M = diag(m 1,..., m N ); I n is the identity matrix of order n and stands for the Kronecker product. By the definition of matrix L and the weight balance condition, it is easy to see that L + L T is symmetric, each row sum is equal to 0, the diagonal entries are positive, and all the other entries are nonpositive. By matrix theory [31], all eigenvalues of L + L T are nonnegative. Hence, matrix L + L T is positive semi-definite. Furthermore, it is easy to see that matrix M is positive definite. Thus J 0, and J = 0 implies that e 1 v = = e N v = 0. This occurs only when v1 = = v N = v 0. It follows immediately that ė i v = vi = 0 for all i I. On the basis of LaSalle s invariance principle [32], we have that the system trajectories converge to the largest positively invariant subset of the set defined by E = {e v : J = 0}. In E, the agent velocity dynamics in the error system is ė i v = 1 m i e i p Ṽ i j = 1 m i e i p Ṽ i. (14) Thus, in the steady state, all agent velocities in the error system no longer change and they equal zero, and moreover, from Eq. (14), the potential Ṽ i of each agent i is globally minimized. Freedom from collisions between the agents can be ensured, since otherwise it will result in Ṽ i. Furthermore, in the steady state, since ev i = 0 for all i I, we get d dt ei p e p j 2 = 2(e i p e p) j T (ev i e v j ) = 0, and hence the distances between agents are invariant. From the proof of Theorem 2, it follows that, in the steady state, all agent actual velocities no longer change and they are equal to the desired velocity. Remark 2. In the velocity neighboring graph D, if all the nonzero interaction coefficients equal 1, then the weight balance condition implies that, for each vertex, the number of arcs starting at it is equal to the number of arcs ending on it. The graphs satisfying such properties have been defined as the balanced graphs [8]. Hence, the balanced graph can be viewed as a special example of a directed graph satisfying the weight balance condition.

6 56 H. Shi et al. / Physica D 213 (2006) Remark 3. From the proof of Theorem 2, it is easy to see that, when the coupling matrix is symmetric, i.e., the interindividual interaction is reciprocal, the desired stable flocking motion can also be obtained, by using the control law in Eq. (6). Remark 4. Note that, in fact, our control objective can be achieved as long as the position neighboring graph is connected, but, in this case, freedom from collisions between all agents cannot be avoided, and all agent final potentials cannot be guaranteed to be the global minimum The motion of the CoM In what follows, we will analyze the motion of the CoM of system (1). The position vector of the CoM in system (1) is defined as x = m i x i. m i Thus, the velocity vector of the CoM is v = m i v i. m i On using control law (6), we obtain v 1 = ( ) w i j (v i v j ) m j N i i ] + x i V i j + m i (v i v 0 ). By the symmetry of function V i j with respect to x i j and the weight balance condition, we get v = v + v 0. (15) Suppose the initial time t 0 = 0, and v (0) = v0. By solving Eq. (15), we get v = v 0 + (v 0 v0 )e t. Thus, it follows that, if v 0 = v0, then the velocity of the CoM is invariant and equals v 0 for all time; if v 0 v0, then the velocity of the CoM exponentially converges to the desired velocity v 0 with a time constant of 1 s. Therefore, from the analysis above, we have the following theorem. Theorem 3. Taking the control law in Eq. (6), if the initial velocity of the CoM is equal to the desired velocity, then it is invariant for all time; otherwise it will exponentially converge to the desired velocity with a time constant of 1 s. Remark 5. Note that, by the calculation above, we can see that, when the coupling matrix satisfies the weight balance condition, the velocity variation of the CoM does not rely on the neighboring relations or the magnitudes of the interaction coefficients. Even if the velocity neighboring graph is not connected, the velocity of the CoM still equals the desired velocity or exponentially converges to it, and the final velocities of all connected agent groups equal the desired velocity as well. It is obvious that when there is no external signal acting on the group and the motion of each agent is only based on the state information of its flockmates, the velocity of the CoM is invariant Convergence rate analysis From Eq. (12), it is easy to see that the interaction coefficients can influence the decay rate of the total energy J; hence it can also influence the convergence rate of the system. In what follows, we will present some qualitative analysis. Let us again consider the dynamics of the error system. From the analysis in Theorem 2, we know that J 0, and J = 0 occurs only when ev 1 = = en v = 0, that is, only when all agents have reached the desired velocity. In other words, if there exists one agent whose velocity is different from the desired velocity, then the energy function J is strictly monotone decreasing with time. Of course, before the group forms the final tight configuration, there might be a case where all agent velocities have reached the desired value, but due to the regulation of the potentials among agents, it instantly changes into a case where not all agents have the desired velocity. Hence, the decay rate of the energy is equivalent to the convergence rate of the system. From Eq. (12), it follows that J = 1 2 et v ((L + LT + 2M) I n )e v. Since L+L T is positive semi-definite and M is positive definite, L+L T +2M is positive definite. By matrix theory [31], we have 1 2 (λ max + 2m max )e T v e v J 1 2 λ min et v e v m min e T v e v, where λ max > 0 denotes the largest real eigenvalue of matrix L + L T, λ min > 0 denotes the smallest real eigenvalue of matrix L + L T + 2M, m min min i I {m i }, and m max max i I {m i }. In fact, in the case where the velocity neighboring graph D is connected, matrix L + L T is irreducible and the eigenvector associated with the single zero eigenvalue is 1 N = [1,..., 1] T R N. On the other hand, it is known that the identity matrix I n has an eigenvalue µ = 1 of n multiplicity and n linearly independent eigenvectors p 1 = [1, 0,..., 0] T, p 2 = [0, 1, 0,..., 0] T,..., p n = [0,..., 0, 1] T. By matrix theory [31], the eigenvalues of (L + L T ) I n are nonnegative, λ = 0 is an eigenvalue of multiplicity n and the associated eigenvectors are q 1 = [p 1T,..., p 1T ] T,..., q n = [p nt,..., p nt ] T.

7 H. Shi et al. / Physica D 213 (2006) Hence, e T v ((L + LT ) I n )e v = 0 if and only if e 1 v = = en v. Hence, when not all agents have reached the common velocity, for any solution of the error system, e v must be in the subspace spanned by the eigenvectors of (L + L T ) I n corresponding to the nonzero eigenvalues. Thus, J 1 2 (λ 2 + 2m min )e T v e v, where λ 2 > 0 denotes the second-smallest real eigenvalue of matrix L + L T. However, if the velocity neighboring graph is not connected, then there must be a permutation matrix P R N N such that P T (L + L T )P is a block diagonal matrix. Without loss of generality, we assume that L + L T = diag((l 1 + L T 1 ),..., (L r + Lr T)), and M = diag(m 1,..., M r ) is the corresponding agent mass matrix, where 1 < r N represents the number of connected velocity neighboring graphs, L i + Li T is the Laplacian matrix associated with the ith connected component D i, and hence J = 1 2 r r êv it ((L i + Li T + 2M i ) I n )êv i m i minê it v ê i v m mine T v e v where m i min min k D i {m k }, êv i is the stack vector of all agent velocity vectors in the connected subgroup D i, and m min is defined as before. And when not all agents in the same connected subgroups have reached the common velocity, we have J 1 2 min i I {λi 2ê it v ê i v } m mine T v e v, where λ i 2 denotes the second-smallest real eigenvalue of matrix L i +Li T, and I = {1,..., r}. Therefore, we have the following conclusion: The convergence rate of the system relies on the interaction coefficients as well as agent masses, and it is always not faster than the convergence rate of the CoM. Furthermore, if the initial velocity of the CoM is not equal to the desired velocity, then the fastest convergence rate of the system does not exceed the exponential convergence rate with convergence exponent 1. Remark 6. Note that when the group has achieved the final steady state, the control input above equals zero Nonideal case Sometimes, the velocity damping should not be ignored. Then, in this case, in order to make the group generate the desired stable flocking motion, the velocity damping must be cancelled by some terms in the control laws. Hence, we modify the control law as in Eq. (4), where α i, β i, and γ i are defined as in Eq. (6), that is, the control law acting on agent i is w i j (v i v j ) x i V i j m i (v i v 0 ) + k i v i, (16) where w i j, v 0, and k i are defined as before. Then, the total force acting on agent i is w i j (v i v j ) x i V i j m i (v i v 0 ). All the results in the ideal case can be analogously extended to the nonideal case. That is, following Theorems 1 and 2, we can easily obtain the desired stable flocking motion: when the velocity damping is taken into account, on using control law (16), all agent velocities in the group described in Eq. (3) asymptotically approach the desired value, freedom from collisions between all agents can be ensured, and the group final configuration minimizes all agent global potentials. Furthermore, following Theorem 3 and the convergence rate analysis above, we conclude that the convergence rate of the system relies on the interaction coefficients and agent masses, and when the initial velocity of the CoM is not equal to the desired velocity, the fastest convergence rate of the system does not exceed the exponential convergence rate with convergence exponent 1. Note also that, because the velocity damping is cancelled by some terms in the control law, it cannot influence the convergence rate of system (3). Remark 7. In the steady state, the group keeps on moving at a desired velocity. During this period, the control laws role is only canceling the velocity damping. 4. Discussion on various control laws In the sections above, we introduced a set of control laws that enable the group to generate the desired stable flocking motion. However, it should be clear that control law (6) is not a unique control law for producing the desired motion for the group. In this section, we provide some more useful control laws. For simplicity, we only present the control laws for the group moving in the ideal case, since in the nonideal case, we only need to add the terms k i v i (i = 1,..., N) to cancel the velocity damping. In the following, we will propose three different control laws that can achieve our control objective. The analysis and proofs are quite similar for these control laws, so we only present the control laws and their corresponding Lyapunov functions. (1) In the control laws above, γ i is used to regulate the momentum of agent i. However, we can also use γ i to directly regulate the velocity of agent i to the desired value. Hence, we take the control law acting on agent i to be w i j (v i v j ) x i V i j (v i v 0 ). (17) We still consider the error system (7) and choose Lyapunov function (9). By a similar calculation, we get J = 1 2 et v ((L + LT ) I n )e v e T v e v.

8 58 H. Shi et al. / Physica D 213 (2006) Using the same analysis method as in Theorem 2, we obtain that J 0, and J = 0 implies that ev 1 = = en v = 0. The rest of the analysis is similar to Theorem 2, and thus is omitted. Remark 8. Note that control law (17) can make the group generate the desired stable flocking motion. But we cannot explicitly estimate the convergence rate of the CoM, using this control law. (2) Suppose that α i and β i rely on agent i s mass. The control law acting on agent i has the following form: m i w i j (v i v j ) m i x i V i j m i (v i v 0 ). (18) In this case, for the error system (7), we choose the following Lyapunov function: J = 1 2 (Ṽ i + ev it ei v ). (19) Following the analysis method in Theorem 2, we can show that the desired stable flocking motion will be achieved. Definition 4. Define the center of the system of agents as x = ( N x i) /N. Definition ( 5. The average velocity of all agents is defined as N v = v i) /N. It is obvious that the velocity of the system center is just the average velocity of all agents. Using the control law in Eq. (18), we have v = v + v 0. Suppose the initial time t 0 = 0 and v(0) = v 0. We get v = v 0 + ( v 0 v 0) e t. It is obvious that, if v 0 = v 0, then the velocity of the system center is equal to the desired velocity v 0 for all time, and if v 0 v 0, then the velocity of the system center exponentially converges to the desired velocity with a time constant of 1 s. (3) Suppose that α i and β i rely on agent i s mass, and γ i is used to regulate the velocity of agent i to the desired velocity. The control law u i is then taken to be m i w i j (v i v j ) m i x i V i j (v i v 0 ). (20) We consider the error system (7) and choose the corresponding Lyapunov function (19). Then, J = 1 2 et v ((L + LT ) I n )e v e T v (M 1 I n )e v, where M 1 is the inverse of matrix M. The rest of the analysis is similar, and thus is omitted. Remark 9. Note that neither the convergence rate of the CoM nor that of the system center can be explicitly estimated using the control law in Eq. (20). From the analysis above, we conclude that the control law in Eq. (6) is the best one among the various control laws. On the one hand, control law (6) can be given certain physical explanations; on the other hand, the corresponding Lyapunov function has certain physical meaning. More importantly, the convergence rate of the CoM of the system can be accurately estimated, by using the control law in Eq. (6). 5. Extensions and discussion In this section, we investigate the case where not all agents know the desired velocity, but we always assume that there exists at least one agent who can detect the external signal. In the case where there is no external signal acting on the group, the collective dynamic behaviors of the agent group have been analyzed in [17]. We first consider the case where the velocity neighboring graph is weakly connected. We divide the group into two subgroups. Subgroup One consists of all agents that can detect the reference signal, i.e., all agents who know the desired velocity belong to Subgroup One. Subgroup Two contains all agents that cannot detect the reference signal. Hence, each agent in Subgroup One regulates its state based on the reference signal and the information of its flockmates, whereas each agent in Subgroup Two regulates its state only on the basis of its flockmates. Without loss of generality, suppose that agents i, i = 1,..., N 1 (1 N 1 < N), are contained in Subgroup One, and agents j, j = N 1 + 1,..., N, are contained in Subgroup Two. Then, the control law acting on each agent i is taken to be w i j (v i v j ) x i V i j h i m i (v i v 0 ) (21) for i = 1,..., N, where h i is defined as { 1, if agent i is contained in Subgroup One, h i = 0, if agent i is contained in Subgroup Two. We still consider the error system (7). Using control law (21) and taking Lyapunov function (9), we have J = w i j e it v (ev i e v j ) h i m i ev it ei v j N i = 1 2 et v ((L + LT ) I n )e v e T v ( M I n )e v, where L, e v, and I n are defined as before, and M = diag(h 1 m 1,..., h N m N ). From the proof of Theorem 2, we obtain that matrix L + L T is positive semi-definite. By the connectivity of graph D, it follows that L+L T is irreducible and the eigenvector associated

9 H. Shi et al. / Physica D 213 (2006) with the single zero eigenvalue is 1 N. From the analysis in Section 3.1.3, we obtain that e T v ((L + LT ) I n )e v = 0 if and only if e 1 v = = en v. On the other hand, by the definition of h i and m i, it follows that matrix M is positive semi-definite, and e T v ( M I n )e v = 0 if and only if e i v = 0 for all i = 1,..., N 1. Hence, J 0, and J = 0 implies that e 1 v = = en v = 0. Following analysis similar to that in the previous sections, we can conclude that the desired stable flocking motion can be achieved. Here we omit the detailed proof. Hence, we have the following conclusion: when the velocity neighboring graph is weakly connected and there exists at least one agent who can detect the external reference signal, the desired stable flocking motion can be achieved by using the control law in Eq. (21). Remark 10. If there exists only one agent in the group who can detect the external reference signal, the group can still generate the desired stable flocking motion. This is of practical interest in control of multi-agent systems. Remark 11. Even if only one agent in the group cannot detect the external reference signal, it is difficult to explicitly estimate the convergence rate of the CoM. It should be noted that there is no actual leader among agents; all agents play the same role. However, we can view the external reference signal as a virtual leader. The results above suggest that, if we want to control a group of mobile agents to move at a given velocity, we only need to send our mission signal to any one of them. Then the signal can be propagated through the neighboring interactions. This is of practical interest in control of multiple mobile robots or a large population of animals (think of how one passes through a crowd of people and how a sheepdog steers a large group of sheep). Remark 12. For the case where the agent has limited sensing range and the interindividual interaction is reciprocal, the collective dynamic behaviors of the agent group have been analyzed in [19]. We should note that the condition on the connectivity of the velocity neighboring graph is strict. In what follows, we will present some further analysis for the case where the velocity neighboring graph is not connected. Obviously, in this case, the control objective can also be achieved as long as there exists at least one agent in each connected subgroup who can detect the reference signal. Next, we consider the case where there are two connected subgroups: Subgroup One and Subgroup Two. The velocity neighboring graphs associated with the two subgroups are denoted as D 1 and D 2, respectively. Without loss of generality, suppose that agents i, i = 1,..., N 2 (1 N 2 < N), are contained in Subgroup One, and agents j, j = N 2 +1,..., N, are contained in Subgroup Two. Moreover, we assume that there exists at least one agent k in Subgroup One who can detect the external signal, i.e., h k = 1 for some k {1,..., N 2 }, and that all agents in Subgroup Two cannot detect the external signal, i.e., h j = 0 for all j = N 2 + 1,..., N. Taking control law (21) and choosing Lyapunov function (9), we get J = 1 2 et v ((L + LT ) I n )e v e T v ( M I n )e v. (22) Because there is no velocity information exchange between Subgroup One and Subgroup Two, L + L T and M have the following forms: [ L + L T L1 + L T = L 2 + L T 2 ], M = [ ] M 1 0, 0 0 where L 1 R N 2 N 2 and L 2 R (N N 2) (N N 2 ) are the Laplacian matrices associated with D 1 and D 2, respectively, and M 1 = diag(h 1 m 1,..., h N2 m N2 ) is a diagonal matrix. By similar analysis, we obtain that J 0, and J = 0 implies that ev 1 = = en 2 v = 0, and e N 2+1 v = = ev N. This occurs only when v 1 = = v N 2 = v 0, and v N2+1 = = v N. This means that all agents in Subgroup One move at the desired velocity, and all agents in Subgroup Two always have the common velocity. Thus, in the steady state, the external input for Subgroup One equals 0, so we get N 2 j=n 2 +1 e i p Ṽ i j = 0, and due to the fact that N 2 i=n 2 +1 j=1 N 2 e i p Ṽ i j = j=n 2 +1 e i p Ṽ i j = 0, we obtain that the external input acting on Subgroup Two equals 0, too. Hence, the velocity of Subgroup Two is a constant vector. In the steady state, since the external input acting on Subgroup Two equals 0 for all the time, and on the other hand, the interaction force between Subgroup One and Subgroup Two is generated by a potential function which is a function of the distance between agents, the final velocity of Subgroup Two must be equal to v 0, that is, in the steady state, the velocity of Subgroup Two is also equal to the desired velocity. Hence, the desired stable flocking motion can also be achieved. 6. Numerical simulations In this section, we will present some numerical simulations for the system described by Eq. (1) in order to illustrate the theoretical results obtained in the previous sections. These simulations are performed with ten agents, labelled with dots, moving in the plane, whose initial positions, velocities and velocity neighboring relations are set randomly, but which satisfy: (1) all initial positions are set within a circle of radius R = 16 m centered at the origin; (2) all initial velocities are set with arbitrary directions and magnitudes within the range of [0, 4] m/s; and (3) the velocity neighboring graph is connected. All agents have different masses and they are set randomly in the range of (0, 1] kg. Note that, because the position neighboring graph is complete, in the following figures we will only present the velocity neighboring relations.

10 60 H. Shi et al. / Physica D 213 (2006) Fig. 1. Initial state. Fig. 3. Final configuration and velocities. Fig. 2. Final configuration and trajectories. The following simulations are all performed with the same group, and the group has the same initial state including all agent initial positions, velocities, and the fixed neighboring relations between agents. However, different control laws are taken in the form of Eq. (21) with the explicit potential function V i j ( x i j ) = 1 2 ln xi j x i j, i, j = 1,..., The interaction coefficient matrix W is generated randomly such that 10 j=1 w i j = 10 j=1 w ji, w ii = 0, and the nonzero w i j satisfy 0 < w i j < 1 for all i, j = 1,..., 10. We run all simulations for 150 s and choose suitable coordinate axes to show our simulation results. Fig. 1 presents the group initial state including the initial positions and velocities of all agents, and the velocity neighboring relations between agents. Figs. 2 and 3, 4 and 5, and 6 and 7 show the motion trajectories of all agents, the final configurations of the group, and all agent velocities in three different simulations, respectively. Fig. 8 depicts the motion trajectories of the CoM in the three simulations, and Figs. 9 and 10 are the velocity error plots along the x-axis and y-axis, respectively. Note that in Figs. 1 7 the dotted lines represent the agent trajectories, the solid lines represent the bidirectional neighboring relations, and the solid arrows Fig. 4. Final configuration and trajectories. Fig. 5. Final configuration and velocities. represent the unidirectional neighboring relations; in Figs. 9 and 10, the solid lines represent the velocity error curves, and the dashed lines represent the velocity error curves of the agent who can detect the external signal. In addition, the meanings of the other lines and arrows are all presented in the figures. Figs. 2 and 3 describe the group state in the case where the motion of the group is not influenced by any external signal and

11 H. Shi et al. / Physica D 213 (2006) Fig. 6. Final configuration and trajectories. Fig. 9. The X-velocity error curves. Fig. 7. Final configuration and velocities. Fig. 10. The Y -velocity error curves. Fig. 8. The trajectories and velocities of the CoM. only relies on the interactions between agents. It can be seen from them that, during the course of motion, all agents regulate their positions to minimize their potentials and eventually move ahead with a constant configuration. When we send a signal to the group and try to make all agents move at a desired velocity v 0 = [0.1, 0.1] T, Figs. 4 and 5 show the results in our simulation with the control laws taken in the form of Eq. (6), whereas Figs. 6 and 7 show the simulation results with the assumption that there is only one agent, labelled by a circle, who knows the desired velocity. It can be seen from them that all agents regulate their positions to minimize their potentials and eventually move ahead with a steady state configuration. Fig. 8 shows the motion trajectories of the CoM in three simulations, and Figs. 9(d) and 10(d) depict the corresponding curves of the errors between the velocities of the CoM and the desired velocity, where the star represents the initial position of the CoM, and (i), (ii), and (iii) represent the corresponding states of the CoM in the three simulations. Figs. 9(a) and 10(a) depict the error curves for exhibiting the differences between agent actual velocities and the velocity of the CoM, whereas Figs. 9(b) (c) and 10(b) (c) are the velocity error plots for describing the errors between agent actual velocities and the desired velocity. Figs. 9 and 10 explicitly demonstrate that, when there is no external signal acting on the group, the velocity of the CoM is invariant in time and all agent velocities asymptotically approach it; otherwise, the velocities

12 62 H. Shi et al. / Physica D 213 (2006) of all agents and the CoM asymptotically approach the desired velocity. Clearly, the convergence rate of the CoM is faster than the convergence rate of the system. Moreover, from Figs. 9 and 10, we can see an interesting phenomenon: increasing the number of agents receiving the external signal does not necessarily imply a faster convergence rate of the group to the desired velocity. This is consistent with some real situations in nature. For example, when we attempt to evacuate a crowd of people that is trapped in a large building (or arena) in a case of emergency, it is not a good idea to tell every person the emergent situation simultaneously, as this may cause panic and lead to a slower speed of escape. Figs. 2, 4 and 6 show that, for a given initial condition, a group of agents may exhibit different transient behaviors, depending on different choices of individuals for feeding in the external signal. This suggests the possibility of adjusting the transient process by appropriately selecting the subgroup of agents to which the external signal is sent. From Figs. 3, 5 and 7, we can see that the final tight configuration of the group is not unique, but in each case, the final potential of each agent is globally minimum according to the theoretical results. Hence, numerical simulations also indicate that the desired stable flocking motion can be achieved by using the control law in Eq. (21). 7. Effect of noise on flocking motion In the previous sections, we have discussed the collective motion of the agent group in the case where there is no external disturbance. However, in reality, the motion of the agent group is inevitably subjected to noise and disturbance in the environment. Hence, in this section we examine the effect of white noise on the collective dynamics of the group. We will study this problem by using some numerical simulations. Consider the case where the system is influenced by noise that is modelled by independent random forces η i (t) acting on individual agents. In this circumstance, the motion equation of agent i is ẋ i = v i, m i v i = u i + η i (t), i = 1,..., N, (23) where x i, v i, m i, and u i are defined as before, and η i (t) = (η i1 (t),..., η in (t)) T R n is the noise vector acting on agent i where η i j (t) ( j = 1,..., n) are independent white Gaussian noises with mean zero and variance σ 2. All the simulations in the following are performed with the same group moving in the plane as in Section 6, and the group has the same initial state including the neighboring relations between agents and the interaction coefficient matrix, but different control laws are taken in the form of Eq. (21) with different assumptions on h i, i = 1,..., N. More specifically, when there is no external signal acting on the agent group, the control law in Eq. (21) has the following form: w i j (v i v j ) x i V i j, (24) i.e., h i = 0 for all i = 1,..., N; when we send a control signal to the group and assume that all agents can detect it, then the control law acting on agent i is just the one in Eq. (6); when the control signal is sent to the agent group and there is only one agent that can detect it, then the control law in Eq. (21) should be in the following form: w i j (v i v j ) x i V i j h i m i (v i v 0 ), (25) where we assume that h 1 = 1, and h i = 0 for all i = 2,..., N. Integration is performed by means of a standard fourthorder Runge Kutta method with a fixed time step t = Noise is introduced by generating at each time step a random number η with Gaussian distribution. We denote by E i = e i v the magnitude of the velocity error of agent i, and denote by E = v v 0 the magnitude of the velocity error of the CoM. In addition, from [17], we obtain that, when there is no external signal acting on the agent group, the velocity of the CoM of the group is invariant and all agent velocities eventually converge to it, on using control law (24), so we denote by E i = v i v the magnitude of the error between the velocity of agent i and the velocity of the CoM. The following figures show the velocity error curves for different control laws in Eqs. (24), (6) and (25) and with different noise intensities, respectively, where the horizontal axis represents time and the vertical axis represents errors E i, E i, or E. Note that in all the following figures the dashed line in each part (c) is the velocity error curve of the agent which can detect the external signal, and part (d) is the velocity error plot of the CoM in the simulations, where (i), (ii), and (iii) represent the corresponding error curves of the CoM in parts (a), (b), and (c), respectively. In order to compare the error curves in a noisy environment with those without noise, we conduct simulations for both of the two cases. In the case without noise, we obtain the results shown in Fig. 11. For the case with noise, by carrying out many simulations, we obtain that, when the noise intensity σ 2 is less than , all the flocking motions in the previous sections can still be achieved by using control laws (24), (6) and (25), respectively. In these cases, the simulation results are similar to Fig. 11. Hence, for small noise intensities σ , the stochastic perturbations to agent velocities do not affect the characteristic features of the collective dynamics observed in the absence of noise. Increasing the noise intensity to , the velocity error curves are depicted in Fig. 12. We can see that, on using control law (24), the velocity of the CoM of the group fluctuates around its initial velocity value, all agent velocities can still converge to the velocity of the CoM and the entire group eventually moves at the velocity of the CoM, that is, flocking motion can still be achieved. Moreover, on using the control laws in Eqs. (6) and (25), the desired flocking motion is still achieved. Further increasing the noise intensity, we observe that all stable flocking motions are destroyed when σ equals 0.1, as shown in Fig. 13. We can see that, on using

13 H. Shi et al. / Physica D 213 (2006) Fig. 11. The error curves (σ = 0). control laws (6) and (25), the velocities of the CoM slightly fluctuate around the desired velocity v 0. Figs. 14 and 15 show the corresponding velocity error plots when σ equals 1 and 10. In the two cases, the agent group cannot form a stable flocking motion. Comparing Fig. 14 with Fig. 15, we can see that increasing the noise intensity makes the velocity errors and the velocity differences between different agents larger, and at the same time, makes the amplitude of vibration of the velocity error larger. Moreover, for the same noise intensity, the system will have larger amplitude of vibration on using control law (25) than on using control law (6). This shows that the system has better disturbance rejection capability when using control law (6) than on using control law (25). In addition, from Fig. 15, we see that, on using control law (24), all agent velocities will deviate from the initial velocity of the CoM as the noise intensity increases, which means that the agent group will fluctuate arbitrarily. However, on using control laws (6) and (25), all agent velocities always fluctuate around the desired value v 0. Simulations demonstrate that, on using the different control laws in Eqs. (24), (6) and (25), flocking motions can be maintained for weak noise. The desired flocking motion cannot be kept with increasing noise intensity. For the case where there is an external signal acting on the group, the velocities of all agents will oscillate around the desired velocity; for the case without a reference signal, the velocities of all agents will oscillate around the velocity of the CoM, and the velocity of the CoM may deviate from its initial value. In the latter case, the entire agent group will move at some velocities affected by the external noise. Moreover, from the simulation results, we see, in the case where there is no external noise, that on using control law (25), the desired stable flocking motion can be obtained, and the group has a faster convergence rate than when using control law (6), in this case, the agent group has relatively weak capability of attenuating external disturbances. This means that increasing the number of agents receiving control signals can make the group more robust against external disturbances. 8. Conclusions Fig. 12. The error curves (σ = 0.03). Fig. 13. The error curves (σ = 0.1). We have investigated the collective behavior of multiple dynamic agents moving in the space with point mass dynamics, and presented some control laws which ensure that the group generates the desired stable flocking motion. The group dynamic properties are characterized in two different cases. When the velocity damping is negligible, using a set of coordination control laws, we can make the group generate the desired stable flocking motion. The control laws are a combination of attractive/repulsive and alignment forces, and they ensure that all agent velocities asymptotically approach the desired velocity, freedom from collisions between the agents is ensured, and the final tight formation minimizes all agent potentials. Moreover, we showed that, when the initial velocity of the CoM is not equal to the desired velocity, it will exponentially converge to the desired velocity. When the velocity damping is taken into account, we can appropriately

14 64 H. Shi et al. / Physica D 213 (2006) Acknowledgements This work was supported by National 973 Program (No. 2002CB ), NSFC (No and No ), and the Engineering Research Institute of Peking University. Appendix A. Algebraic graph theory preliminaries Fig. 14. The error curves (σ = 1). Fig. 15. The error curves (σ = 10). modify the control laws in order to generate the desired stable flocking. Subsequently, we investigated the motion of the group in the case where not all agents know the desired final velocity, and showed that the desired stable flocking motion can still be achieved using our control laws. Numerical simulations were worked out to further illustrate our theoretical results. Finally, we investigated the collective dynamic behaviors of the agent group in a noisy environment, and demonstrated by means of numerical simulations that the desired flocking motion can still be achieved for weak noise and, as the white noise intensity increases, the velocities of all agents will fluctuate around the desired value. Our method is general, integrating both matrix theory and algebraic graph theory, and is applicable to dealing with more complex agent dynamics, information topology and interaction mechanisms. In this section, we briefly summarize some basic concepts and results in algebraic graph theory that have been used in this paper. More comprehensive discussions can be found in [30]. An undirected graph G consists of a vertex set V = {n 1, n 2,..., n m } and an edge set E = {(n i, n j ) : n i, n j V}, where an edge is an unordered pair of distinct vertices of V. If n i, n j V and (n i, n j ) E, then we say that n i and n j are adjacent or that n j is a neighbor of n i, and denote this by writing n j n i. A graph is called complete if every pair of vertices are adjacent. A path of length r from n i to n j in a graph is a sequence of r + 1 distinct vertices starting with n i and ending with n j such that consecutive vertices are adjacent. If there exists a path between any two vertices of G, then G is connected. A directed graph D consists of a vertex set V = {n 1, n 2,..., n m } and an arc set E = {(n i, n j ) : n i, n j V}, where an arc, or directed edge, is an ordered pair of distinct vertices of V. Let D = (V, E, A) be a weighted directed graph. A = [a i j ] R m m is the weighted adjacency matrix, where a i j is the weight of arc (n i, n j ), a i j 0 for all i, j Ĩ = {1,..., m}: i j and a ii = 0 for all i Ĩ. The set of neighbors of vertex n i is defined as N i = { j Ĩ : a i j > 0}. The in-degree and out-degree of vertex n i are defined as deg in (n i ) = m j=1 a ji and deg out (n i ) = m j=1 a i j, respectively. The weighted graph D Laplacian matrix is defined as L(D) = A, where is the degree matrix of D which is a diagonal matrix and its ith diagonal element is ii = deg out (n i ). By definition, λ = 0 is an eigenvalue of the Laplacian matrix L(D) and 1 m is its associated right eigenvector. A path of length r from n 0 to n r in a directed graph is a sequence of r + 1 distinct vertices starting with n 0 and ending with n r such that (n k 1, n k ) is an arc of D for all k = 1,..., r. A weak path is a sequence of n 0,..., n r of distinct vertices such that for all k = 1,..., r, either (n k 1, n k ) or (n k, n k 1 ) is an arc. A directed graph is strongly connected if any two vertices can be joined by a path and is weakly connected if any two vertices can be joined by a weak path. Note that in this paper we assume that n i n j, meaning that there are no self-loops, and assume that each element of E is unique. References [1] N.E. Leonard, E. Fiorelli, Virtual leaders, artificial potentials and coordinated control of groups, in: Proc. 40th IEEE Conf. Decision and Control, vol. 3, 2001, pp [2] K. Warburton, J. Lazarus, Tendency distance models of social cohesion in animal groups, J. Theor. Biol. 150 (1991) [3] I. Suzuki, M. Yamashita, Distributed anonymous mobile robots: formation of geometric patterns, SIAM J. Comput. 28 (1999) [4] J.H. Reif, H. Wang, Social potential fields: a distributed behavioral control for autonomous robots, Robot. Auton. Syst. 27 (1999)