Stabilization of Planar Collective Motion with Limited Communication

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1 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 1 Stabilization of Planar Collective Motion with imited Communication Rodolphe Sepulchre, Member, IEEE, Dere A. Paley, Student Member, IEEE, and Naomi Ehrich eonard, Senior Member, IEEE Abstract This paper proposes a design methodology to stabilize relative equilibria in a model of identical, steered particles moving in the plane at unit speed. Relative equilibria either correspond to parallel motion of all particles with fixed relative spacing or to circular motion of all particles around the same circle. Particles exchange relative information according to a communication graph that can be undirected or directed and time-invariant or time-varying. The emphasis of the paper is to show how previous results assuming all-to-all communication can be extended to a general communication framewor. I. INTRODUCTION The development of versatile and scalable methodology to control collective motion for multi-agent systems is driven by a growing number of engineering applications that depend on the coordination of a group of individually controlled systems. The various applications, which impose diverse operational constraints and performance requirements, include formation control of unmanned aerial vehicles (UAVs) [1], [2] and spacecraft [3], cooperative robotics [4] [6], and sensor networs [7], [8]. In this paper, we are strongly motivated by the application of autonomous underwater vehicles (AUVs) as mobile sensors to collect oceanographic measurements in formations or patterns that yield maximally information-rich data sets, see e.g. [9], [10]. This can be achieved by matching the measurement density in space and time to the characteristic scales of the oceanographic process of interest. Coordinated, periodic trajectories such as the ones studied in this paper, provide a means to collect measurements with the desired spatial and temporal separation. Motivated by these issues, we consider in the present paper the design of stable collective motion patterns in a model of identical planar particles introduced in [11]. The particles move at constant speed and are subject to steering controls that change their direction of motion. In addition to a phase variable that models the orientation of the velocity vector, the state of each particle includes its position in the plane. The synchrony of the collective motion is thus measured both by the relative phasing and the relative spacing of particles. In recent wor [12], we proposed a versatile design methodology that specifies collectives as minima of artificial potentials. These potentials are used as yapunov functions for the design of gradient-lie control laws that mae them decreasing along the solutions of the closed-loop system. The resulting control laws exhibit a number of desirable properties but they require all-to-all communication, that is, the control law of a given particle requires

2 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 2 relative information from all other particles. All-to-all communication is an assumption that is unrealistic in a number of applications. In large sensor networs, each agent is expected to exchange information with a limited number of neighbors only. The interplay between the information flow among particles and the individual agent dynamics then becomes a central issue of the collective design. Even if all-to-all communication is available, limited use of communication may be advantageous for performance. The objective of the present paper is to address this issue in the framewor of the planar particle model [11] and to show that the results previously obtained in the framewor of all-to-all communication [12] can be recovered in a systematic way under quite general assumptions on the networ communication. Two different approaches are considered to address the limited communication topology. A first proposed approach is to generalize the design of artificial potentials in such a way that the resulting control laws respect the communication constraints. Encoding the communication constraints in a graph, we propose the design of potentials based on the graph aplacian and show that these potentials reduce to the previously proposed all-to-all potentials in the particular case of a complete graph. This approach requires the communication topology to be time-invariant, connected, and bidirectional. This generalization is relevant, for instance, in design framewors where the topology of the communication is a design parameter but where the number of communication lins is limited by technological constraints. A second approach is to generalize the design of collectives in a framewor where the communication is possibly time-varying, unidirectional, and not fully connected at any given instant of time. The possibility to achieve synchronization under such general assumptions on the communication has been extensively addressed in the recent literature on consensus algorithms [13] [17]. In particular, the new analysis tools proposed in [14], [18] are appropriate to study convergence issues in the present framewor under wea assumptions on the communication graph. The application of these tools currently imposes restrictions on the system dynamics and, more fundamentally, on the Euclidean nature of the underlying state-space. As a consequence, only local results have been available for non-euclidean state spaces such the those encountered in phase-oscillators models [14], [19] or in the present particle model [20]. However, a recent result [21], [22] shows how consensus estimators designed for Euclidean space [23] can be used in the closed-loop system dynamics to obtain globally convergent consensus algorithms in phase models. We use this idea to generalize our previously proposed all-to-all design [12] to general communication topologies. In this approach, the weaened assumptions on the communication topology are compensated for by an increased exchange of information between communicating particles. Communicating particles indeed not only exchange their relative configuration variables but also their relative estimates of averaged quantities used in centralized designs. The rest of the paper is organized as follows: Section II reviews the geometric properties of the considered particle model. Section III focuses on a simpler phase model that controls the relative directions of particles only. In Section IV we consider the stabilization of parallel and circular formations, which are the two types of relative equilibria of the full particle model. The next two sections elaborate on these basic results by proposing symmetry-breaing control laws that either reduce the dimension of the equilibrium set in shape space (Section V) or achieve stabilization in the absolute space by connecting the networ to a reference particle (Section VI). Section

3 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 3 VII illustrates an application of the results in the framewor of the ocean sampling project that motivated most of the developments proposed in the present paper. Section VIII illustrates an application of the results in the framewor of a proximity communication graph, which involves unidirectional and time-varying communication lins. Conclusions are presented in Section IX. II. MODE AND PROBEM STATEMENT A. A model of steered particles in the plane We consider a continuous-time model of N identical particles (of unit mass) moving in the plane at unit speed and subject to steering control [1]: ṙ = e iθ θ = u, = 1,..., N. (1) In complex notation, the vector r = x + iy C R 2 denotes the position of particle and the angle θ S 1 denotes the orientation of its (unit) velocity vector e iθ = cos θ + i sin θ. The scalar u is the steering control for particle. The model (1) reflects second-order dynamics of particles with forcing only in the direction normal to velocity (steering control), i.e., r = θ (iṙ ) with ṙ of unit length. We use a bold variable without index to denote the corresponding N-vector, e.g. θ = (θ 1,..., θ N ) T and u = (u 1,..., u N ) T. In the absence of steering control ( θ = 0), each particle moves at unit speed in a fixed direction and its motion is decoupled from the other particles. We study the design of various feedbac control laws that result in coupled dynamics and closed-loop convergence to different types of organized or collective motion. The model (1) has been recently studied by Justh and Krishnaprasad [1]. These authors have emphasized the ie group structure that underlies the state space. The configuration space consists of N copies of the group SE(2). When the control law only depends on relative heading directions and relative positions, the closed-loop vector field is invariant under an action of the symmetry group SE(2) and the closed-loop dynamics evolve on a reduced quotient manifold. This (3N 3)-dimensional manifold is called the shape space and it corresponds to the space of all relative headings and relative positions. We use shape variables θ j = θ θ j and r j e iθ = (r r j )e iθ, j, = 1,..., N, which are relative heading and relative position with respect to particle direction, respectively. Equilibria of the reduced dynamics are called relative equilibria and can be only of two types [1]: parallel motions, characterized by a common orientation for all the particles (with arbitrary relative spacing), and circular motions, characterized by circular orbits of the particles around a fixed point. B. Graph representation of the communication limitations It is common in a multi-agent framewor to describe the communication between particles by means of a graph. The N nodes of the graph represent the N particles of the model. A (weighted) edge from to j indicates a communication lin from particle to particle j. The resulting weighted digraph (directed graph) is denoted by G = (V, E, A), where V = {v 1,..., v N } is the set of nodes, E V V is the set of edges, and A is a weighted

4 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 4 adjacency matrix with nonnegative elements a j. The element a j is zero when there is no communication lin from to j. We assume that there are no self-cycles i.e. a = 0, I = 1,..., N. The digraph G is called strongly connected if and only if any two distinct nodes of the graph can be connected via a path that follows the direction of the edges of the digraph. The in-degree and out-degree of node v are defined as d in = N j=1 a j and d out = N j=1 a j, respectively. The digraph G is said to be balanced if the in-degree and the out-degree of each node are equal, that is, a j = j j a j, I. The graph aplacian associated to the graph G is a square matrix defined as i l j = a i, j = a j, j. The -th row of is denoted by. By construction, the aplacian matrix has zero row sums, that is 1 = 0. Several properties of the graph translate into matrix properties of its aplacian. In particular, the following equivalences (see e.g. [17], [18]) are relevant in the framewor of the present paper: The graph G is undirected iff is symmetric; The graph G is strongly connected iff 1 spans the ernel of, i.e. x = 0 x = 1x 0 ; The graph G is balanced iff + T 0. For further bacground on spectral graph theory, see, for example, [24]. A particular generalization of the all-to-all communication topology that we consider in the present paper is the fixed and connected topology corresponding to a d 0 -circulant graph in which each node is connected to d 0 other nodes, for a fixed d 0 where 2 d 0 N 1. Both the adjacency and aplacian matrices of a circulant graph are circulant, i.e. they are completely defined by their first row. Each subsequent row of a circulant matrix is the previous row shifted one position to the right with the first element equal to the last element of the previous row. For example, the all-to-all topology is (N 1)-circulant and the ring topology is 2-circulant. The following lemma (see for instance [28] for a proof) will be useful in various proofs. emma 1: et be the aplacian of a d 0 -circulant graph with N vertices. Set φ = ( 1) 2π N Then the vectors for = 1,..., N. f (l) = e i(l 1)φ, l = 1,..., N (2) define a basis of N orthogonal eigenvectors of. The unitary matrix F whose columns are the N (normalized) eigenvectors 1 N f (l) diagonalizes, that is, = F ΛF, where Λ = diag{0, λ 2,..., λ N } 0 is the (real) diagonal matrix of the eigenvalues of. It is both of theoretical and practical interest to consider time-varying communication topologies: in a networ of moving particles, existing lins can fail and new lins can appear as other particles leave or enter an effective range of detection. Time-varying communication topologies will be described by a time-varying δ-digraph G(t) = (V, E(t), A(t)), where the elements of A(t) are bounded and satisfy some threshold δ > 0, that is, a j (t) = 0 in

5 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 5 the absence of a communication lin and a j (t) δ in the presence of a communication lin. The set of neighbors of node v at time t is denoted by N (t) := {v j V : a j (t) δ}. Consider a graph G(t) = (V, E(t), A(t)). A node v is said to be connected to node v j (j ) in the interval I = [t a, t b ] if there is a path from v to v j which respects the orientation of the edges for the directed graph (V, t I E(t), A(τ)dτ). G(t) is said to be uniformly connected if there exists an index and a time horizon I T > 0 such that for all t node is connected to all the other nodes across [t, t + T ]. A. All-to-all communication III. PHASE SYNCHRONIZATION AND PHASE BAANCING All control laws in this paper are based on gradient-lie stabilization of collectives specified as critical points of well-chosen potentials. The inner product <, > is defined by < z 1, z 2 >= Re{z 1z 2 } for z 1, z 2 C. For vectors, we use the analogous boldface notation < w, z > = Re{w z} for z 1, z 2 C N. We first recall a basic result from our previous paper [12], which characterizes the critical points of the potential U m (θ) defined as U m = N 2 p mθ 2 (3) where p mθ is the mth moment of the phase distribution on the circle, defined as p mθ = 1 N e imθ, m = 1, 2,.... (4) mn =1 Theorem 1: [ Synchronization and balancing] et 1 m N. The potential U m = N 2 p mθ 2 reaches its unique minimum when p mθ = 0 (balancing modulo 2π m ) and its unique maximum when the phase difference between any two phases is an integer multiple of 2π m in the shape manifold T N /S 1 and are saddle points of U m. Proof: See Theorem 5 in the companion paper [12]. (synchronization modulo 2π m ). All other critical points of U m are isolated The significance of Theorem 1 for the collective model (1) is as follows: for m = 1, the potential U 1 (θ) reaches its maximum when all phases synchronize. This state is characteristic of parallel formations in the collective model (1), i.e., all particles move in the same direction. In contrast, the potential U 1 (θ) reaches its minimum when all phases balance. This state is characteristic of a fixed center of mass in the collective model because p θ := p 1θ = 1 N N =1 ṙ. The parameter p θ is a classical measure of synchrony of the phase variables θ in the literature on coupled oscillators. In the context of the Kuramoto model, p θ has been called the complex order parameter [25]. For m > 1, the interpretation of the critical points of U m is similar modulo 2π m, that is, when two phases differing by 2π m are identified. The use of potentials U m for m > 1 is postponed until Section V where we address the stabilization of particular symmetric phase arrangements. To achieve stabilization of minima or maxima of the potential U m, it is natural to consider the gradient control u = KgradU m, i.e. u = K U m θ = K < p mθ, ie imθ >= K N N sin(m(θ θ j )), K 0. (5) j=1

6 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 6 The resulting closed-loop dynamics correspond to either a descent or ascent algorithm depending on the sign of the gain K, leading to the following result as a direct consequence of Theorem 1. As a convention throughout the paper, we associate negative gains (K < 0) to synchronization controls and positive gains (K > 0) to balancing controls. Corollary 1: The phase model θ = u with the gradient control (5) forces convergence of all solutions to the critical set of U m. If K < 0, then only the set of synchronized states is asymptotically stable and every other equilibrium is unstable. If K > 0, then only the balanced set where p mθ = 0 is asymptotically stable and every other equilibrium is unstable. The all-to-all sinusoidal coupling (5) with m = 1 is the most frequently studied coupling in the literature of coupled oscillators [25] [27]. However, it requires all-to-all communication, an assumption that we relax in the following sections. B. Time-invariant and undirected communication graphs et P = I N 1 N 11T. Then P e imθ = e imθ mp mθ 1. One deduces the equality P e imθ 2 =< e imθ, P e imθ > = N(1 m 2 p mθ 2 ) (6) which implies that the quadratic form < e imθ, P e imθ > has the same critical points as the phase potential U m (θ). Because P is ( 1 N times) the aplacian matrix of the complete graph, the identity (6) suggests to replace the optimization of U m (θ) in the previous section by the optimization of W m (θ) = 1 m Q (e imθ ) := 1 2md max < e imθ, e imθ > (7) where we denote by Q (s) the quadratic form in s C N and d max = max I {d out }. Notice that W m(θ) = ( N N N 1 2m mu m(θ) ) in the case of a complete graph, in which case both potentials have the same critical points. Suppose that is the aplacian of an undirected graph with unit edge weights, which implies = T a j {0, 1}. Then the significance of optimizing the potential W m instead of U m is that the gradient control u = K W m = K < ( + T ) e imθ, ie imθ >= K sin(m(θ θ j )) (8) θ 2d max d max is adapted to the topology of the communication graph. That is, the control u only uses the relative information from the communicating neighbors j N, denoted by j. The synchronized state modulo 2π m is a global minimum of W m provided that G is balanced. However, W m may possess other (local) minima and the (local) maxima which do not necessarily correspond to balanced states. A meaningful generalization of Theorem 1 holds for d 0 -circulant graphs. Theorem 2: If is the aplacian of a connected and balanced graph, then the global minimum of W m (θ) is synchronized modulo 2π m, i.e. eimθ = e iθ0 1 where θ 0 S 1. If the graph is d 0 -circulant, then the global maximum of W m (θ) is balanced modulo 2π m, i.e. 1T e imθ = 0. Proof: If is the aplacian of a balanced graph, then < z, z> 0 for all z C N. If the graph is also connected, then < z, z> = 0 if and only if z = 1z 0, which proves that the global minimum of W m is j and

7 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 7 reached (only) for e imθ = e iθ0 1. If the graph is also d 0 -circulant, then emma 1 provides the following equivalent expression for the phase potential (7): W m (θ) = 1 2md 0 < v, Λv >= 1 2md 0 where v = F e imθ. Because v = e imθ = N, an upper bound is provided by N v 2 λ 1. (9) =2 W m (θ) 1 < v, Λv >= N λ max. (10) 2md 0 2md 0 This bound is attained by selecting e imθ as the maximal eigenvector of F. This vector is orthogonal to the mimimal eigenvector 1, that is, satisfies 1 T e imθ = 0. C. General communication graphs The goal of this section is to extend the stabilization of synchronized or balanced states under weaer assumptions on the communication graph. We first recall a general result that holds in the Euclidean space C N. Proposition 1: et G(t) be a uniformly connected δ-digraph and (t) the corresponding aplacian matrix. In the linear time-varying system ẋ = (t)x, x C N (11) the set of synchronized states is uniformly exponentially stable. In particular, each solution of (11) exponentially converges to a synchronized state x1. If G(t) is balanced, then the asymptotic consensus x is the average of the initial conditions 1 N N =1 x (0). Proposition 1 shows that synchronization is achieved under wea assumptions on the communication graph when the state evolves in an Euclidean space. There is a vast recent literature on the convergence properties of consensus algorithms of the type (11). We refer the reader to the recent paper [14] for a proof of Proposition 1 (the setup in [18] is even more general since it considers communication delays). When the graph G is fixed and undirected, (11) is a gradient system for the quadratic form 1 2 < x, x >. It is tempting to consider the analog of (11) on the torus. The gradient expression (8) suggests the generalization θ = K d max < e imθ, ie imθ >, K < 0. (12) The dynamics (12) linearize to (11) in the neighborhood of a synchronized state, showing that local synchronization is indeed achieved by the dynamical system (12) under the sole assumption that G(t) is uniformly connected. A stronger convergence result is proven in [14] by mapping the dynamics (12) into the Euclidean space. This change of coordinates requires to restrict the set of initial phases in an open interval (θ 0 π 2, θ 0 + π 2 ). An alternative proof is provided in [19]. For initial conditions in a broader interval, simulation results indicate that, in contrast with the consensus dynamics (11), synchronization is not achieved in general by the dynamical system (12). This fact is not surprising when considering that the proof of Proposition 1 rests on a contraction property for the convex hull of the N states x i C, a proof concept that does not extend to the torus.

8 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 8 Comparing the dynamics (12) to the (all-to-all) gradient dynamics (5), one is led to interpret (local) static estimate of the average quantity P l eimθ l N 1 P l eimθ l d max as a. In contrast to the consensus dynamics (11), the use of this static estimate in (12) fails to recover the global convergence results of the gradient dynamics (5). The approach recently proposed in [22] shows that the global convergence results can be recovered by using a local dynamic estimate instead. We summarize the main results of [22] in the following theorems. Theorem 3: [Dynamic Synchronization] et G(t) be a uniformly connected δ-digraph and (t) the corresponding aplacian matrix. Consider the phase model θ = u. Then the (dynamic) control u = K m < w, i >, K < 0 ẇ = ikw < w, i > N j=1 l j(t)w j e im(θj θ ), w (0) = 1 enforces exponential convergence of solutions θ(t) to the critical set of U m. Furthermore, only the set of synchronized states modulo 2π m is asymptotically stable and every other limit set is unstable. If G(t) is a balanced digraph, then the asymptotic consensus value for eimθ m, 1 N, is the average p mθ(0) = 1 mn N =1 eimθ (0). e Proof: Set x = w imθ m. Then x obeys the consensus dynamics ẋ = (t)x, which implies that the solution x(t) exponentially converges to the consensus variable x1. This implies that the dynamics θ = K m < w, i > exponentially converges to the (time-invariant) dynamics θ = K < x, ie imθ >. Consequently, θ(t) asymptotically converges to an equilibrium in the critical set of U m and only the set of synchronized states is asymptotically stable. If G(t) is balanced, then < 1, x > is a conserved quantity and x = 1 mn N =1 w (0)e imθ (0) = p mθ (0). Theorem 4: [Dynamic Balancing] et G(t) be a uniformly connected balanced δ-digraph and (t) the corresponding aplacian matrix. Consider the phase model θ = u. Then the (dynamic) control u = K m < w, i >, K > 0 ẇ = ik(w 1) < w, i > N j=1 l j(t)w j e im(θj θ ), w (0) = 1 enforces exponential convergence of solutions θ(t) to the critical set of U m. Furthermore, only the set of balanced states modulo 2π m is asymptotically stable and every other limit set is unstable. e Proof: Set x = w imθ m. Then x satisfies the dynamics ẋ = (t)x + idiag{u }e imθ = (t)x + d e imθ dt m. (15) As a consequence, the quadratic form V (x) = 1 2 < x, x > is nonincreasing along the solutions and satisfies N V = < x, (t)x> + u < x, ie imθ >= < x, (t)x > 1 N u 2 0. (16) K =1 We deduce from (16) that u is a time function in 2 (0, + ) along each solution of the closed-loop dynamics. But then (15) is the consensus dynamics (11) with an additive perturbation in 2 (0, + ), which does not destroy the convergence of x(t) to a consensus equilibrium 1 x. Since G(t) is balanced, one obtains from (15), =1 (13) (14) 1 N < 1, ẋ> = 1 N < 1, d e imθ dt m >. (17) Integrating both sides of (17) and using the fact that x (0) = eimθ (0) m, one concludes that 1 N N =1 x (t) = p mθ (t) for all t 0. This implies that the dynamics θ = K m < w, i > asymptotically converges to the (time-invariant)

9 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 9 dynamics θ = K < p mθ, ie imθ >. This implies that θ(t) asymptotically converges to an equilibrium in the critical set of U m and that only the set of balanced states modulo 2π m is asymptotically stable. Theorem 3 and Theorem 4 generalize the global convergence results of the all-to-all gradient control (5) under wea assumptions on the communication graph. This generalization is obtained at the price of increased communication between the communicating particles because particles must not only exchange their relative configuration variables but also their local estimate of the average p mθ. In both theorems, the variable w can be interpreted as a local estimate of p mθ in the local frame attached to particle while x = w e imθ m is the local estimate of p mθ in an absolute (reference) frame. Consensus filters that reconstruct possibly time-varying average quantities have been recently proposed in [23]. It should be noted that all the convergence results in this section still hold when adding a constant ω 0 R to u. The constant ω 0 is eliminated from the equations by using the change of coordinates θ = θ ω 0 t1 which amounts to expressing the dynamics of θ in a rotating frame. The feedbac control laws are invariant under this change of coordinates because they are expressed in the relative coordinates (θ θ j ). In geometric terms, the stabilization taes place in the shape manifold T N /S 1. We exploit this feature in the subsequent sections. IV. STABIIZATION OF PARAE AND CIRCUAR FORMATIONS A basic observation in the companion paper [12] is to reformulate the stabilization of parallel and circular formations in the collective model (1) as the tas of synchronizing the state variables s = iω 0 c = e iθ iω 0 r,, = 1,..., N. (18) For ω 0 = 0, the synchronization of s is equivalent to the synchronization of the phase variables θ. This corresponds to a parallel formation in the collective model (1). For ω 0 0, the state c = is ω 0 is the center of the circle of radius ρ 0 = ω 0 1 traversed by particle under the action of the constant control u = ω 0. A circular relative equilibrium is obtained when all the centers coincide, which corresponds to synchronization of the variables s. A. Time-invariant and undirected communication graphs In our previous paper [12], synchronization of the particle centers (18) is achieved by minimizing the quadratic form P s 2 =< s, P s >, which, for ω 0 = 0, reduces to the phase synchrony measure (6). Viewing P as 1 N times the aplacian of a complete graph, one is led to the following extension. Theorem 5: [aplacian control] et G be a connected and balanced graph and be the aplacian of G. Then the quadratic potential Q (s) = 1 2d max < s, sbs> reaches its global minimum in the set of synchronized states s = 1s 0. If G is undirected, this set is (locally) exponentially stabilized by the control law u = ω 0 + If ω 0 0, then the convergence is global. K d max < s, ie iθ >, K < 0. (19) Proof: The proof is entirely analogous to the proof of Theorem 2 in the companion paper [12] except that the all-to-all aplacian (N)P is replaced by the aplacian of an arbitrary connected and undirected graph. If

10 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 10 ω 0 0, then s taes values in C N and the convergence is global because the quadratic form Q(s) has no other minima. If ω 0 = 0, then the control (19) reduces to (12) and the result is only local, except for complete graphs. B. General communication graphs For a complete graph, the control law (19) with K = dmax K d max+1 taes the form u = ω 0 + K < s (p θ iω 0 R), ie iθ >, K < 0 (20) where the position R = 1 N N =1 r and the velocity p θ = 1 N N =1 eiθ are quantities averaged over the entire group. In a manner analogous to the result of Section III-C, these average quantities can be estimated by a consensus filter, leading to a generalization of the control law of Theorem 5 to arbitrary communication graphs. Theorem 6: [ Dynamic aplacian control] et G(t) be a uniformly connected δ-digraph and (t) the corresponding aplacian matrix. Then the (dynamic) control u = ω 0 K < w iω 0 v, i >, K < 0 ẇ = iw u N j=1 l j(t)w j e i(θj θ ), w (0) = 1 v = 1 iv u N j=1 l j(t)v j e i(θj θ ) N j=1 l j(t)r j e iθ, v (0) = 0 (21) enforces exponential convergence of solutions to a relative equilibrium of the collective model (1). If ω 0 = 0, the relative equilibrium is a parallel formation. For ω 0 0, the relative equilibrium is a circular formation of radius ρ 0 = ω 0 1 with direction determined by the sign of ω 0. Proof: Set x = w e iθ and y = v e iθ + r. Then x and y obey the consensus dynamics ẋ = (t)x and ẏ = (t)y, respectively, which implies that the solutions x(t) and y(t) exponentially converge to the consensus variables x1 and ȳ1, respectively. This implies that the dynamics θ = u exponentially converge to the (timeinvariant) dynamics θ = ω 0 + K < s ( x iω 0 ȳ), ie iθ >. (22) For ω 0 = 0, (22) is identical to the dynamic synchronization result of Theorem 3. For ω 0 0, the quadratic form S = 1 2 s x + iω 0 ȳ 2 (23) satisfies Ṡ = K < s x+iω 0 ȳ, ie iθ > 2 0 along the solutions of (22). One concludes that all solutions converge to a circular formation of radius ρ 0 = ω 0 1 centered at c 0 = i x ω 0 + ȳ. If G(t) is balanced then c 0 = ip θ(0) + R(0), where R(t) = 1 N N j=1 r j(t). In Theorem 6, the variable w can be interpreted as a local estimate of p mθ in the local frame attached to particle while x = w e imθ m is the local estimate of p mθ in an absolute (reference) frame. The variable v can be interpreted as a local estimate of the center of mass R in the local frame attached to particle while y = v e iθ + r is the local estimate of R in an absolute (reference) frame. ω 0

11 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 11 V. STABIIZATION OF SYMMETRIC PATTERNS An interesting feature of the phase potentials derived in Section III and the spacing potentials derived in Section IV is that they can be combined additively. The phase potentials W m, 1 m N, defined in (7) satisfy Ẇ m = N =1 W m u θ and, because they are invariant to a rigid rotation θ θ + ω 0 1, they also satisfy Ẇ m = N =1 W m θ (u ω 0 ) for an arbitrary ω 0 R. iewise, the spacing potential Q (s) defined in Theorem 5 satisfies Q = 1 d max N < s, ie iθ > (u ω 0 ). =1 As a consequence, a linear combination of the potentials also satisfies V = V (s, θ) = KQ (s) M K m W m (θ), (K < 0) (24) ( N K M < s, ie iθ > + d max =1 The corresponding gradient-lie control u = ω 0 + K d max < s, ie iθ > + K m W m θ M ) K m W m θ (u ω 0 ). is thus the same linear combination of the individual gradient-lie controls. A critical point that jointly minimizes the individual potentials KQ and K m W m, 1 m M, also minimizes the yapunov function V. This suggests a versatile design method for the decentralized stabilization of relative equilibria that jointly minimize various combinations of the potentials proposed in the previous sections. A. Stabilization of symmetric balanced patterns et 1 M N be a divisor of N. An (M, N)-pattern is a symmetric arrangement of N phases consisting of M clusters uniformly spaced around the unit circle, each with N/M synchronized phases. For any N, there exist at least two symmetric patterns: the (1, N)-pattern, which is the synchronized state, and the (N, N)-pattern, which is the so-called splay state, characterized by N phases uniformly spaced around the circle. An (M, N)-pattern circular formation of radius ρ 0 is an isolated relative equilibrium of the collective model (1). emma 2: Each (M, N)-pattern is a critical point of the potentials W m, 1 m N. Proof: Modulo a uniform rotation and with the notations of emma 1, a (M, N)-pattern is characterized by the phase arrangement θ = N M φ

12 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 12 This means that the vector e im θ = e im N M φ is the -th eigenvector of, with = 1 + (m N M 1) mod N. But if e im θ is an eigenvector of, then θ is a critical point of < e im θ, e im θ >, which proves the claim. The following result was proven in [12] for complete graphs and extended in [29] to d 0 -circulant graphs. Theorem 7: [aplacian stabilization of (M, N)-pattern circular formations] Consider a connected and d 0 - circulant graph G = (V, E, A), the associated aplacian, and the associated phases potentials W m (θ) = 1 2md max < e imθ, e imθ >. et 1 M N be a divisor of N. If θ T N is an (M, N)-pattern then it is a (local) minimum of the potential with K m > 0 for m = 1,..., M 1 and W M,N = K M < 1 M M K m W m (25) M 1 mk m < 0. (26) An (M, N)-pattern circular formation of radius ρ 0 = ω 0 1 is (locally) exponentially stabilized by the control law control u = ω 0 + K < s, ie iθ > + d max Proof: By emma 2, θ is a critical point of W M,N u = M M K m W m θ. (27). We show that θ is exponentially stabilized by the gradient K m W m θ. (28) et B R N e be the incidence matrix of graph G where e is the cardinality of the set of edges E and B l = B jl = ±1 if j and 0 otherwise. Note = BB T. The Hessian H M,N (θ) of W M,N is determined by 2 W m θ 2 = m < e imθ, e imθj > (29) d 0 and, for j, 2 W m θ j θ = j Evaluating (29) and (30) at θ yields the weighted aplacian m d 0 < e imθ, e imθj >, j, 0, otherwise. (30) H M,N ( θ) = BΦ M,N ( θ)b T (31) where the weight matrix Φ M,N (θ) R e e is defined by Φ M,N (θ) = 1 M mk m diag(cos(mb T θ)). (32) d 0 To complete the first part of the proof of Theorem 7, we find conditions on the gains K m to ensure H M,N ( θ) has only one zero eigenvalue and all other eigenvalues are negative. Using (32), we obtain the condition M mk m cos(m( θ j θ )) < 0, j. (33)

13 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 13 Choosing the gain K M according to (26) ensures that (33) is satisfied. The phase arrangement θ is a nondegenerate critical point of the potential W M,N since the Hessian H M,N is equal to the Jacobian of the gradient control (28). The proof of the second part of Theorem 7 is a straightforward adaptation of Theorem 3 in [12]. Under the control (27) the yapunov function V (s, θ) in (24) satisfies V = N =1 (u ω 0 ) 2 0 along the closed-loop solutions. By the asalle Invariance principle, solutions for the reduced system on shape space converge to the largest invariant set Λ where K d 0 < s, ie iθ >= M K m W m θ (34) for = 1,..., N. In the set Λ, the dynamics reduce to θ = ω 0 which implies that W M,N is constant. Therefore the right-hand side of (34) vanishes in the set Λ, which implies s = 0 since ṡ = 0. We conclude that solutions converge to an (M, N)-pattern circular formation of radius ρ 0 = ω 0 1. Theorem 7 does not exclude convergence to circular formations that correspond to other critical points of the phase potential W M,N. Simulations indicate that the basin of attraction of certain symmetric patterns is indeed small in the case of incomplete graphs. In contrast, convergence to a different critical point is not observed in simulations for a complete graph, suggesting a large basin of attraction in the case of all-to-all communication. The all-to-all control law maes use of the averaged quantities p mθ, 1 m M and 1 N N =1 s. For general graphs, these averaged quantities can be reconstructed by consensus filters so that the all-to-all result is recovered with the following dynamic control scheme. Theorem 8: [Dynamic stabilization of (M, N)-pattern circular formations.] et G(t) be a balanced and uniformly connected δ-digraph and (t) the corresponding aplacian. Consider the collective model (1). Then the set of (M, N)-pattern circular formations is (locally) exponentially stabilized by the (dynamic) control u = ω 0 K < w iω 0 v, i > M K m m < w (m), i >, ẇ = iw u N j=1 l j(t)w j e i(θj θ), w (0) = 1 v = 1 iv u N j=1 l j(t)v j e i(θj θ ) N j=1 l jr j e iθ, v (0) = 0 ẇ (m) = im(w (m) 1)(u ω 0 ) N ẇ (M) = imw (M) (u ω 0 ) N j=1 l j(t)w (m) j j=1 l j(t)w (M) j with gains K < 0, K m > 0, m = 1,..., M 1, and K M < 0. e im(θj θ ), w (m) (0) = 1, m = 1,..., M 1 e im(θj θ ), w (M) (0) = 1 Proof: Set x = w e iθ and y = v e iθ + r. Then x and y obey the consensus dynamics ẋ = (t)x and ẏ = (t)y, respectively, which implies that the solutions x(t) and y(t) exponentially converge to the consensus variables x1 and ȳ1, respectively. Similarly, set x (M) = w (M) which implies that x (M) (t) converges to consensus variables x (M) 1. e im(θ ω 0 t) M, then (35) ẋ (M) = (t)x (M) (36) Next we prove exponential stability of (M, N)-pattern circular formations with the modified control ũ = ω 0 + K < s ( x iω 0 ȳ), ie iθ M 1 > K m m < w(m), i > K M M < x(m), ie im(θ ω 0t) >. (37)

14 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 14 Because the closed-loop systems obtained with the control laws u and ũ only differ by a perturbation in 2 [0, ), exponential stability of the modified system implies convergence of the solutions to the equilibrium in the original system as well. Set x (m) = w (m) e im(θ ω 0 t) m, m = 1,..., M 1, then x (m) (t) obeys the dynamics ẋ (m) = (t)x (m) + i diag{u ω 0 }e im(θ ω0t1) = (t)x (m) + d dt (eim(θ ω0t1) ), m = 1,..., M 1. (38) m Consider the yapunov function V = K N S =1 M 1 K m x (m) 2 K M 2 eim(θ ω0t1) M x (M) 1 2 with S defined by (23). With the control law (37), V is nonincreasing along the solutions and satisfies M 1 V = K m < x (m), (t)x (m) > N (ũ ω 0 ) 2 0. This implies that ũ is a time function in 2 (0, + ) along each solution of the closed-loop dynamics. For each m {1, M 1}, (38) is a linear exponentially stable system with an additive perturbation in 2 (0, + ). This implies that each estimator x (m) (t), 1 m M 1, converges to a consensus equilibrium 1 x (m). Since G(t) is balanced, one obtains from (38), 1 N < 1, ẋ(m) > = 1 N < 1, d dt e im(θ ω0t1) m =1 >, m = 1,..., M 1. (39) Integrating both sides of (39) and using the fact that x (m) (0) = eimθ(0) m, one concludes that 1 N N =1 x(m) (t) = e imω0t p mθ (t) for all t 0. Using that each x (m) (t) converges to e imω0t p mθ (t), 1 m M, one concludes that, along each solution of the closed-loop system, the dynamics θ asymptotically converge to the (all-to-all) control law M ū = ω 0 + K < s x + iω 0 ȳ, ie iθ > K m < p mθ, ie iθ >. (40) As a consequence, the only limit sets of the closed-loop dynamics are circular formations with a phase arrangement in the critical set of the (all-to-all) potential W (M,N) NP. In particular, the (M, N) pattern circular formation is an exponentially stable equilibrium of the closed-loop system. Exponential stabilization of (M, N)-pattern circular formation was considered in our previous paper [12] in the framewor of complete graphs (all-to-all communication). Theorem 8 extends that result to the framewor of general communication graphs. As mentioned in our earlier paper, the result is only local. However, we have identified no other minimum of the (all-to-all) phase potential W (M,N) NP than the (M, N)-pattern. VI. SYMMETRY BREAKING All the control laws discussed so far are control laws in the shape space, that is, the closed-loop dynamics are invariant to rigid translations and rotations in the plane, which corresponds to the action of the symmetry group SE(2). It is of interest in some applications to brea this SE(2) symmetry. For phase synchronization, this means forcing the synchronization to a specific phase θ 0 S 1. For the control of circular formation, this means forcing

15 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 15 the formation to a specific circle in the plane. We show in this section how these can be achieved in a decentralized way. et the reference frame be described by the (virtual) reference particle with dynamics ṙ 0 = e iθ0 θ 0 = ω 0. (41) If ω 0 = 0, θ 0 serves as a fixed reference orientation. If ω 0 0, then the reference particle traces a circle of radius ρ 0 = ω 0 1 centered at c 0 = r 0 + i ω 0 e iθ0 serves as a reference center for the circular formation. =: is0 ω 0. In that case, θ 0 (t) serves as reference orientation, while c 0 In order to describe the information flow from the (virtual) reference particle to the particles, we augment the communication δ-digraph G(t) by an additional reference node 0, with directed edges from 0 to the informed particles. The augmented δ-digraph is denoted by G(t) and its aplacian matrix is denoted by (t). The aplacian (t) has the bloc structure (t) = l 10 (t). l N0 (t) (t) diag{l 0 (t)} Note that the uniform connectedness assumption can be satisfied for the extended δ-digraph G(t) even if the reference node is connected to only one other node.. A. Time-invariant and undirected communication graphs For a fixed graph, the potential S 0 (s, s 0 ) = N =1 l 0 s s 0 2 satisfies N Ṡ 0 = l 0 < s s 0, ie iθ > (u ω 0 ) =1 Augmenting the potential Q (s) by the symmetry-breaing potential S 0 in Theorem 5 results in the augmented control law u = ω 0 + K d max < s, ie iθ > K d max l 0 < s s 0, ie iθ >, K < 0. (42) As a direct corollary of Theorem 5, the symmetry-breaing control law (42) stabilizes a parallel formation with orientation θ 0 if ω 0 = 0 and a circular formation centered at c 0 if ω 0 0. The same modification in Theorem 7 stabilizes an (M, N)-pattern circular formation centered at s 0, breaing the translation symmetry of the control law. The rotation symmetry can also be broen by synchronizing the orientation of the (M, N)-pattern with the reference θ 0 (t). To this end, the phase potential W M,N (θ) must be augmented by the symmetry-breaing potential S 0 (e imθ, e imθ0 ). The resulting control law is u = ω 0 + K < s, ie iθ > K l 0 < s s 0, ie iθ > + d max d max M K m W m θ K M d max l 0 < e imθ0, ie imθ >. (43)

16 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 16 As a direct corollary of Theorem 7, the symmetry-breaing control law (43) stabilizes an (M, N)-pattern circular formation centered at c 0 and with all phases synchronized modulo 2π M with the reference θ 0(t). B. General communication graphs Symmetry-breaing can also be achieved in the dynamic control algorithms of the previous sections. In order to brea the translation symmetry in Theorem 6, one must simply consider the augmented graph G(t) in the estimation dynamics of w and v, that is ẇ = iw u N j=0 l j (t)w j e i(θj θ ), v = 1 iv u N j=0 l j (t)v j e i(θj θ ) N j=0 l j r j e iθ, = 1,..., N (44) with the definition of the control law unchanged for 1 N and u 0 = ω 0. The consensus dynamics for x = w e iθ and y = v e iθ + r are ẋ = (t)x and ẏ = (t)y, respectively. Because ẋ 0 0 and ẏ 0 0, the only possible consensus values are x 0 (0) and y 0 (0), respectively. Setting w 0 (0) = 1 and v 0 (0) = 0 results in x(0) = e iθ0 and y(0) = r 0 (0). If G(t) is uniformly connected, the (modified) control law of Theorem 6 then achieves exponential convergence to a circular formation centered at c 0 = i x ω 0 + ȳ = r 0 + ieiθ 0, which breas the translation symmetry of the formation. A similar modification in the control law of Theorem 8 additionally enforces phase synchronization modulo 2π m to the reference θ 0 (t), which also breas the rotational symmetry of the phase arrangement. ω 0 VII. DESIGN OF COMMUNICATION GRAPHS AND APPICATIONS In this section we consider design of communication graphs and illustrate how design can be used to meet performance requirements in applications. Design of a communication graph for our many particle system refers to the choice of which of the available communication lins should be used in the steering control law at any given time. In the case that the particle system models a multi-agent system, availability of communication lins depends upon the technology used for sensing and communication among particles. It may be advantageous for system performance, efficiency and/or simplicity to mae limited use of available communication. Further, different choices of communication graph for the different terms in the steering control law yield multi-scale networ patterns as we describe below. The development in this paper is strongly motivated by the application to mobile sensor networs used for autonomous ocean sampling [9], [10]. In this particular application, each agent in the networ is an autonomous underwater glider, moving at near constant speed relative to the water, carrying sensors to measure the ocean (temperature, salinity, currents) and communicating only with a central on-shore computer (via Iridium satellite). Each agent can communicate at fairly regular intervals with the central computer; however, the particles do not synchronize their communication. In [30] a methodology is presented that enables control strategies for collective motion using the idealized particle model with decentralized steering control for planning together with feedbac provided via the asynchronous communication. Experimental results illustrate the methodology.

17 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 17 Because communication in this application is made through a central computer, there is, in principle, the opportunity to approximate all-to-all communication and therefore to design any ind of limited communication graphs as appropriate. Justh and Krishnaprasad have applied their coordinated control strategies to an experimental networ of unmanned aircraft. In this experiment, all-to-all communication is available, maing it therefore also possible to design communication graphs as desired [31]. Even in applications where technology limits communication, there is always the opportunity to use less than is available, and this will be desirable if it can be done so to advantage. In the ocean sampling networ application, and mobile sensor networ applications in general, the value of the collected measurement data can depend strongly on the collective motion pattern of the moving sensors. For instance, in the optimal mobile sensor coverage problem discussed in [10], patterns consisting of sensors moving around closed curves with coordinated phasing are practical solutions for maximizing information in data collected in a spatially and temporally varying field. Multi-scale patterns become significant when there are multiple significant scales in the sampled region. For example, in the ocean sampling problem, it is of interest to sample well both the boundary of a region for fluxes and the interior of a region for dynamics. Smaller scales become important with the emerging presence of features such as an eddy or a jet. There are many competing factors to consider in designing a communication graph for a given system. For example, the communication graph can influence the rate of convergence and region of attraction for stable patterns [16]. Fewer communication lins may minimize the complexity of the networ but could increase the sensitivity to failures. As illustrated in [32] for the example of the ring graph topology, the choice of communication graph can also be used as a way to isolate desired patterns. We propose a communication graph design concept, of particular relevance to sensor networ applications, that uses multiple graphs at a time to yield multi-scale (sensing) patterns. In this context the different graphs are used in the different terms in the control law so that subgroups can form and both intra and inter subgroup coordination can be regulated. In the following we present three examples to illustrate the design concept and versatility. Separate spacing and phase graphs We divide the N particles into B blocs (subgroups) of particles with n = N B particles in each subgroup. Although we assume that each subgroup is the same size in this example, this is not required. To each bloc we assign an all-to-all graph with aplacian np n = ni n 11 T. The bloc-all-to-all graph aplacian is = I B (np n ) where I B = diag{1 B } and is the Kronecer product. We assign to be the communication graph for the spacing control term so that in the case of circular control, particles within any subgroup move around the same circle but circles associated with the different subgroups are independently located. For the phase control, we assign the all-to-all graph NP ; this ignores subgroups and fixes the relative phases of all particles. Choosing the control u = ω 0 + K M,N < s, ie iθ WNP > (45) d max θ stabilizes an (M, N)-pattern over B separate circular formations of radius ρ 0 = ω0 1. Simulations of (1) subject to (45) are shown in Figure 1(a) for M = N = 6, B = 3, and K = ω 0 = 0.1. The phase gains are K 1 = 0.04 and K m = 0.01 for m = 2,..., 6. Note that the phases of the N = 6 particles are in the splay state and the relative

18 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTRO 18 (a) (b) (c) Fig. 1. Simulation results of (1) subject to aplacian control with separate spacing and phase graphs with reference particles and nested phase potentials: (a) the control (45) with M = N = 6, B = 3, K = ω 0 = 0.1 stabilizes three separate circular formations of radius ρ 0 = ω 1 0 and a splay arrangement of all 6 particle phases; (b) the control (46) fixes the centers of the circular formations to the reference locations c (b) 0 = i25(b 2), b = 1, 2, 3 denoted by crossed circles; (c) the control (47) stabilizes a nested (M, N)-pattern with Q = 3 and M = N = (2, 4, 4) T. Trajectories start from random initial conditions and are plotted after elapsed time equivalent to 50 revolutions; the last revolution of each particle is plotted in dar gray. positions of the B = 3 circles are arbitrary. Separate spacing and phase graphs with reference particles To each of the B blocs we associate one reference particle which traces out a circle of radius ρ 0 = ω 0 1 centered at c (b) 0, b = 1,..., B. At least one particle in each bloc is connected (by a directed path) to the corresponding reference particle. To each bloc we assign the augmented aplacian denoted by ñp n using the methodology from Section VI. The augmented bloc-all-to-all aplacian is = I B (ñp n). et s denote the corresponding augmented vector s which is defined by s = ( iω 0 c (1) 0, s 1,..., s n, iω 0 c (2) 0, s n+1,...) T. Choosing the control u = ω 0 + K < d M,N s, ie iθ WNP > (46) max θ stabilizes an (M, N)-pattern over B separate circular formations of radius ρ 0 = ω 1 0 centered at c(b) 0, b = 1,..., B. Simulations of (1) subject to (46) are shown in Figure 1(b) for c (b) 0 = i25(b 2), b = 1, 2, 3. All other parameters are the same as in the previous example. It can be observed in this case that the phases are still in the splay state but now the circles are positioned as desired. Separate spacing and phase graphs with reference particles and nested phase potentials et M = (M 1,..., M Q ) T and N = (N 1,..., N Q ) T where M q is a divisor of N q for q = 1,..., Q. A nested (M, N)-pattern