THE collective control of multi-agent systems is a rapidly

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1 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 1 Stabilization of Planar Collective Motion: All-to-all Communication Rodolphe Sepulchre, Member, IEEE, Derek A. Pale, Student Member, IEEE, and aomi Ehrich Leonard, Senior Member, IEEE Abstract This paper proposes a design methodolog to stabilize isolated relative equilibria in a model of all-to-all coupled, identical, steered particles moving in the plane at unit speed. Isolated relative equilibria either correspond to parallel motion of all particles with fied relative spacing or to circular motion of all particles with fied relative phases. The stabilizing feedbacks derive from Lapunov functions that prove eponential stabilit and suggest almost global convergence properties. The results of the paper provide a low-order parametric famil of stabilizable collectives that offer a set of primitives for the design of higherlevel tasks at the group level. I. ITRODUCTIO THE collective control of multi-agent sstems is a rapidl developing field, motivated b a number of engineering applications that require the coordination of a group of individuall controlled sstems. Applications include formation control of unmanned aerial vehicles (UAVs) [1], [2] and spacecraft [3], cooperative robotics [4] [6], and sensor networks [7], [8]. A specific application motivating the results of the present paper is the use of autonomous underwater vehicles (AUVs) to collect oceanographic measurements in formations or patterns that maimize the information intake, see e.g. [9], [1]. This can be achieved b matching the measurement densit 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. The design focus of collective stabilization problems is on achieving a certain level of snchron among possibl man but individuall controlled dnamical sstems. The primar design issue is not how to control the individual dnamics, but rather how to interconnect them to achieve the desired level of snchron. This motivates the use of simplified models for the individuals. For instance, snchron in populations of coupled oscillators has been studied primaril b means of phase models; these models onl retain the phase information of individual oscillators as the fundamental information pertaining to snchron measures of the ensemble [11]. Even these simplified models raise challenging issues for the analsis or design of their interconnection. The have recentl motivated a number of new developments in stabilit analsis [12], [13]. This paper presents research results of the Belgian Programme on Interuniversit Attraction Poles, initiated b the Belgian Federal Science Polic Office. The scientific responsibilit rests with its authors. D. A. Pale was supported b the ational Science Foundation Graduate Research Fellowship, the Princeton Universit Gordon Wu Graduate Fellowship, and the Pew Charitable Trust grant E. Leonard was partiall supported b OR grants and The interconnected control sstem is high dimensional, both in the number of state variables and control variables. It is nevertheless characterized b a high level of smmetr, which is maimal when all the individual models are assumed identical. Smmetr properties make these models well suited to the reduction techniques of geometric control [14]. Smmetr and geometr pla a central role in the analsis of cclic pursuit strategies for kinematic uniccle models studied in [15]. Motivated b the issues above, we consider in the present paper the model of identical, all-to-all coupled, planar particles introduced in [16]. 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 velocit vector, the state of each particle includes its position in the plane. The snchron of the collective motion is thus measured both b the relative phasing and the relative spacing of particles. In previous work [17], [18], we observed that the norm of the average linear momentum of the group is a ke control parameter: it is maimal in the case of parallel motions of the group and minimal in the case of circular motions around a fied point. We eploited the analog with phase models of coupled oscillators to design steering control laws that stabilize either parallel or circular motions. Epanding on this idea, the design methodolog in the present paper is to construct potentials that reach their minimum at desired collective formations and to derive the corresponding gradientlike steering control laws as stabilizing feedbacks. We treat separatel phase potentials that control the relative orientation of particles and spacing potentials that control their relative spacing. We show that the design can be made somewhat sstematic and versatile, resulting in a simple but general controller structure. The stabilizing feedbacks depend on a restricted number of parameters that control the shape and the level of smmetr of parallel or circular formations. This low-order parametric famil of stabilizable collectives offers a set of primitives that can be used to solve path planning or optimization tasks at the group level. The results of the paper rest on two idealistic assumptions: all-to-all communication and identical individuals. either of these assumptions is realistic in a practical environment, not in engineering models nor in models of natural groups. The allto-all assumption is instrumental to the results of the paper but this strong assumption is completel relaed in a companion paper where we etend the present results to communication topologies where communication is limited. The assumption of identical individuals is fundamental to the smmetr properties

2 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 2 of the closed-loop vector field, which is instrumental to the proposed stabilit analsis. It is known that the individual dnamics ma ehibit much more complicated behaviors awa from this ideal situation. Man earlier studies nevertheless suggest that snchron is robust and that an ensemble phenomenon resulting from a specific interconnection structure will persist in spite of individual discrepancies. The analsis of the celebrated Kuramoto model (see [19] for a recent review) eemplifies both the robustness of snchronization at the ensemble level and its mathematical msteries at the individual level in a population of non-identical, all-to-all coupled oscillators. In this sense, the ideall engineered models considered in this paper ma help in capturing gross dnamical properties of more realistic, multi-agent, simulation models or of biologists observations of animal groups. The rest of the paper is organized as follows: Section II reviews the geometric properties of the considered model. Section III introduces a basic phasing potential that controls the group linear momentum. In Section IV we introduce a spacing potential that is minimum in circular formations. These results provide the basic control laws to achieve either parallel or circular formations, the onl possible relative equilibria of the model. Due to smmetr, the dimension of the equilibrium set is high and can be reduced with the help of smmetr breaking controls laws that derive from further potentials. We show in Sections V, VI and VII how to stabilize isolated relative equilibria of the model, both in circular formations (Sections V and VI) and in parallel formations (Section VII). Eponential stabilization of isolated circular relative equilibria is presented in Section VI. Section VIII illustrates how to combine the results of the previous sections in a low-parameter catalog of stabilizable collectives. Conclusions are presented in Section IX. II. A MODEL OF STEERED PARTICLES I THE PLAE We consider a continuous-time model of identical particles (of unit mass) moving in the plane at unit speed and subject to steering control [1]: ṙ k = e iθ k θ k = u k, k = 1,...,. In comple notation, the vector r k = k + i k C R 2 denotes the position of particle k and the angle θ k S 1 denotes the orientation of its (unit) velocit vector e iθ k = cos θ k + i sin θ k. The scalar u k is the steering control for particle k. The model (1) reflects second-order dnamics of particles with forcing onl in the direction normal to velocit (steering control), i.e., r k = θ k (iṙ k ) with ṙ k a unit vector. We use a bold variable without inde to denote the corresponding -vector, e.g. θ = (θ 1,..., θ ) T and u = (u 1,..., u ) T. In the absence of steering control ( θ k = ), each particle moves at unit speed in a fied direction and its motion is decoupled from the other particles. We stud the design of various feedback control laws that result in coupled dnamics and closed-loop convergence to different tpes of organized or collective motion. We assume identical control for each particle. In that sense, the collective motions that (1) we analze in the present paper do not require differentiated control action for the different particles (e.g. the presence of a leader for the group). The model (1) has been recentl studied b Justh and Krishnaprasad [1]. These authors have emphasized the Lie group structure that underlies the state space. The configuration space consists of copies of the group SE(2). When the control law onl depends on relative orientations and relative spacing, i.e., on the variables θ kj = θ k θ j and r kj = r k r j, j, k = 1,...,, the closed-loop vector field is invariant under an action of the smmetr group SE(2) and the closedloop dnamics evolve on a reduced quotient manifold. This (3 3)-dimensional manifold is called the shape space and it corresponds to the space of all relative orientations and relative positions. Equilibria of the reduced dnamics are called relative equilibria and can be onl of two tpes [1]: parallel motions, characterized b a common orientation for all the particles (with arbitrar relative spacing), and circular motions, characterized b circular orbits of the particles around a fied point. Both tpes of motion have been observed in simulations in a number of models that are kinematic or dnamic variants of the model (1), see for instance [2]. A simplification of the model (1) occurs when the feedback laws depend on relative orientations onl. The control has then a much larger smmetr group ( copies of the translation group), and the reduced model becomes a pure phase model θ = u where the phase variable θ belongs to the - dimensional torus T. This phase model still has an S 1 - smmetr if the feedback onl depends on phase differences. Phase-oscillator models of this tpe have been widel studied in the neuroscience and phsics literature. The represent a simplification of more comple oscillator models in which the uncoupled oscillator dnamics each have an attracting limit ccle in a higher-dimensional state space. Under the assumption of weak coupling, the reduction of higher-dimensional models to phase models b asmptotic methods (singular perturbations and averaging) has been studied in e.g. Ermentrout and Kopell [21] and Hoppensteadt and Izhikevich [22]. The results in this paper build upon an etensive literature on phase models of coupled oscillators [11], [19], [23], [24]. We will stress the close connection between collective motions in groups of oscillators and collective motions in groups of moving particles. Euler discretization of the continuous-time model (1) ields the discrete-time equations r k (t + 1) r k (t) = e iθ k(t) θ k (t + 1) θ k (t) = u k (t), k = 1,...,. The direction of motion of particle k is updated at each time step according to some feedback control u k. We consider continuous-time models in this paper, but we mention a few relevant references that stud their discrete-time counterpart. Couzin et al. [25] have studied such a model where the feedback control is determined from a set of simple rules: repulsion from close neighbors, attraction to distant neighbors, and preference for a common orientation. Their model includes stochastic effects but also ehibits collective motions reminiscent of either parallel motion or circular motion around (2)

3 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 3 a fied center of mass. Interestingl, these authors have shown coeistence of these two tpes of motion in certain parameter ranges and hsteretic transition from one to the other. A discrete stochastic phase model has been studied b Vicsek et al. [26]. III. PHASE STABILIZATIO We start our analsis with a basic but ke observation for the results of this paper: the (average) linear momentum Ṙ of a group of particles satisfing (1) is the centroid of the phase particles p θ, that is Ṙ = 1 ṙ k = 1 e iθ k p θ = p θ e iψ. (3) The parameter p θ is a measure of snchron of the phase variables θ [11], [19]. It is maimal when all phases are snchronized (identical). It is minimal when the phases balance to result in a vanishing centroid. The set of snchronized states is an isolated point modulo the action of the smmetr group S 1. It defines a manifold of dimension one. The set of balanced states, which we call the balanced set, is defined as all θ T for which p θ =. For odd, the equation p θ = has full rank everwhere and the set of balanced states defines a manifold of codimension two. For even, the balanced set is not a manifold of codimension two. The equation loses rank at points where there are two anti-snchronized, equall-sized clusters, i.e., each cluster consists of 2 snchronized phases and the phase of one cluster equals the phase of the other cluster plus π. In the particle model (1), snchronization of the phases corresponds to a parallel formation: all particles move in the same direction. In contrast, balancing of the phases corresponds to collective motion around a fied center of mass. Control of the group linear momentum is thus achieved b minimizing or maimizing the potential U 1 (θ) = 2 p θ 2 (4) which suggests the gradient control u = KgradU 1, i.e. u k = K U 1 = K < p θ, ie iθ k >= K sin θ jk. (5) The inner product <, > is defined b < z 1, z 2 >= Re{ z 1 z 2 } for z 1, z 2 C. For vectors, we use the analogous boldface notation < z 1, z 2 > = Re{ z T 1 z 2 } for z 1, z 2 C. This all-to-all sinusoidal coupling (5) is the most frequentl studied coupling in the literature of coupled oscillators [11], [19], [24]. Its gradient nature enables the following global convergence analsis. Theorem 1: The potential U 1 = 2 p θ 2 reaches its unique minimum when p θ = (balancing) and its unique maimum when all phases are identical (snchronization). All other critical points of U 1 are isolated in the shape manifold T /S 1 and are saddle points of U 1. The phase model θ = u with the gradient control (5) forces convergence of all solutions to the critical set of U 1. If K <, then onl the set of snchronized states is asmptoticall stable and ever other equilibrium is unstable. If K >, then onl the balanced set where p θ = is asmptoticall stable and ever other equilibrium is unstable. Proof: The gradient dnamics θ = KgradU 1 forces convergence of all solutions to the set of critical points of U 1, characterized b the algebraic equations < p θ, ie iθ k >=, 1 k. (6) Critical points where p θ = are global minima of U 1. As a consequence, the balanced set is asmptoticall stable if K > and unstable if K <. From (6), critical points where p θ = p θ e iψ are characterized b sin(θ k Ψ) =, that is, M phases snchronized at Ψ mod 2π and M phases snchronized at (Ψ + π) mod 2π, with M < 2. At those points, p θ = 1 2M > 1. The value M = defines a snchronized state and corresponds to a global maimum of U 1. As a consequence, the set of snchronized states is asmptoticall stable if K < and unstable if K >. Ever other value 1 M < 2 corresponds to a saddle and is therefore unstable both for K > and K <. This is because the second derivative 2 U 1 θk 2 = 1 < p θ, e iθ k >= 1 cos(ψ θ k) p θ (7) takes negative values if θ k = Ψ and positive values if θ k = Ψ+π. As a consequence, a small variation δθ k at those critical points decreases the value of U 1 if θ k = Ψ and increases the value of U 1 if θ k = Ψ + π. The consequence of Theorem 1 is that parallel formations are stabilized b all-to-all sinusoidal coupling of the phases differences, i.e. the control law (5) with K <. With K >, the same control law stabilizes the center of mass of the particles to a fied point. The fact that the remaining equilibria are saddles suggests that the conclusions of Theorem 1 are almost global, that is, almost all solutions either converge to the snchronized state (K < ) or to the balanced set (K > ). We note that the conclusions of Theorem 1 can be equivalentl stated in a rotating frame, that is, for the phase model θ = ω 1 KgradU 1, ω R. (8) The convergence analsis is unchanged because of the propert < grad U 1, 1 >=, which is a consequence of the invariance of U 1 under the action of the smmetr group S 1. For ω =, the stead state of the phase model (8) gives rise to straight orbits in the particle model (1): snchronization then means parallel motion in a fied direction, with arbitrar but constant relative spacing, which is a relative equilibrium of the model. In contrast, balancing means straight orbits towards or awa from a fied center of mass, which does not correspond to a relative equilibrium of the model. For ω, the stead state of the phase model (8) gives rise to circular orbits of radius ρ = ω 1 in the particle model (1). In general, particles orbit different circles. Snchronization imposes parallel orientation of all velocit vectors whereas balancing imposes a fied center of mass. A collective motion with fied center of mass corresponds to a relative equilibrium of the model onl if all particles orbit the same circle. The four different tpes of collective motion associated with the phase model (8) are illustrated in Figure 1.

4 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL oting that (a) (c)!2!4! !1!2 5 1!4!2 2 4 (b) (d) 5!5 4 2!2!1!5 5 1!4!2 2 4 Fig. 1. The four different tpes of collective motion of = 12 particles obtained with the phase control (8). The position of each particle, r k, and its time-derivative, ṙ k, are illustrated b a circle with an arrow. The center of mass of the group, R, and its time-derivative, Ṙ, are illustrated b a crossed circle with an arrow. (a) ω = and K < ; (b) ω = and K > ; (c) ω and K < ; (d) ω and K >. Onl (a) is a relative equilibrium of the particle model (1). IV. STABILIZATIO OF CIRCULAR FORMATIOS In contrast to the phase control designed in the previous section, we now propose a spacing control that achieves global convergence to a circular relative equilibrium of the particle model (1). We start our analsis with the observation that under the constant control u k = ω, each particle travels at constant, unit speed on a circle of radius ρ = ω 1. The center of the circle traversed b particle k is c k = r k +iω 1 eiθ k. Multiplied b the constant factor iω, c k becomes s k = iω c k = e iθ k iω r k. (9) A circular relative equilibrium is obtained when all the centers coincide; this corresponds to the algebraic condition P s =, P = I 1 11T. (1) This suggests to choose a stabilizing control that minimizes the Lapunov function oting that S(r, θ) = 1 2 P s 2. (11) ṡ k = ie iθ k (u k ω ), (12) the time-derivative of S along the solutions of (1) is Ṡ =< P s, P ṡ >= < P k s, ie iθ k > (u k ω ) (13) where P k denotes the k-th row of the matri P and where we have used the fact that P is a projector, i.e. P 2 = P. Choosing the control law results in u k = ω κ < P k s, ie iθ k >, κ > (14) Ṡ = κ < P k s, ie iθ k > 2. (15) we obtain P k s = s k 1 1T s = e iθ k iω r k (Ṙ iω R), < P k s, ie iθ k > = < ω (r k R), e iθ k > < Ṙ, ieiθ k > = < ω r k, e iθ k > U 1 (16) where we denote b r k = r k R the relative position of particle k from the group center of mass R = 1 r k. Using (16), we rewrite the control law (14) as u k = κ U 1 +ω (1+κ < r k, ṙ k >), κ >, ω. (17) Lapunov analsis provides the following global convergence result. Theorem 2: Consider the particle model (1) with the spacing control (17). All solutions converge to a relative equilibrium defined b a circular formation of radius ρ = ω 1 with direction determined b the sign of ω. Proof: The Lapunov function S(r, θ) defined in (11) is positive definite and proper in the reduced shape space, that is, when all points (r, θ) that differ onl b a rigid translation r + 1r and a rigid rotation θ + 1θ are identified. From (15), S is nonincreasing along the solutions and, b the LaSalle Invariance principle, solutions for the reduced sstem on shape space converge to the largest invariant set Λ where κ < P k s, ie iθ k > (18) for k = 1,...,. In this set, θk = ω and s k is constant for all k = 1,.... This means that (18) can hold onl if P s. As a result, s = 1s for some fied s C, i.e., all particles orbit the same circle of radius ρ. V. PHASE SYMMETRY BREAKIG I CIRCULAR FORMATIOS The spacing control law of Section IV stabilizes particle motions to a unique set in the phsical plane modulo the smmetr group of rigid displacements. In contrast, the phase arrangement of the particles is arbitrar. To reduce the dimension of the equilibrium set, we combine the spacing potential S(r, θ) defined in (11) with a phase potential U(θ) which is minimum at the desired phase configuration. We require that U(θ) preserves the S 1 smmetr of rigid rotation, that is, < grad U, 1 >=. Theorem 3: Consider the particle model (1) and a smooth phase potential U(θ) that satisfies < grad U, 1 >=. The control law u k = ω (1 + κ < r k, ṙ k >) (U κu 1), ω (19) enforces convergence of all solutions to the set of relative equilibria defined b circular formations where all particles move around the same circle of radius ρ and direction given b the sign of ω with a phase arrangement in the critical set of U. Ever (local) minimum of U defines an asmptoticall

5 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 5 stable set of relative equilibria. Ever relative equilibrium where U does not reach a minimum is unstable. Proof: We use the composite Lapunov function V (r, θ) = κs(r, θ) + U(θ) (2) which is lower bounded since S and U takes values in a compact set. The time-derivative of V along the solutions of (1) is V = (κ < P k s, ie iθ k > (u k ω ) + U u k ) (21) which, using the propert < gradu, 1ω >=, becomes V = (κ < P k s, ie iθ k > + U )(u k ω ). (22) Because the control (19) is u k = ω κ < P k s, ie iθ k > U, (23) the Lapunov function V satisfies V = (u k ω ) 2 along the closed-loop solutions. B the LaSalle Invariance principle, solutions for the reduced sstem on shape space converge to the largest invariant set Λ where κ < P k s, ie iθ k >= U (24) for k = 1,...,. In the set Λ, the dnamics reduce to θ k = ω, which implies that U is constant. Therefore the right-hand side of (24) vanishes in the set Λ, which implies P s = since ṡ =. We conclude that solutions converge to a circular relative equilibrium and that the asmptotic phase arrangement is in the critical set of U. Consider the set E of circular relative equilibria of radius ρ with a phase configuration in the critical set of U. Because V (r, θ) = U(θ) in E, local minima of U correspond to local minima of the Lapunov function. An connected subset of E on which U reaches a strict minimum is therefore asmptoticall stable. ote that because E is a set of equilibria, U is constant on an connected subset. In contrast, consider = ( r, θ) E such that U( θ) is not a minimum and denote b E the connected component of E containing. U is constant in E. To show instabilit of, consider a compact neighborhood B( ) in the shape space such that B( )\E contains no other relative equilibrium. A solution with initial condition in B( ) either asmptoticall converges to E or leaves B( ) after a finite time. Let = ( r, θ) B( ) such that U(θ) < U( θ). Then the solution with initial condition cannot converge to E (and therefore leaves B after a finite time) since V decreases along solutions and V ( r, θ) < V ( r, θ). Because V ( ) is not minimum, can be chosen arbitrar close to, which proves instabilit of. Theorem 3 thus provides a global convergence analsis of closed-loop dnamics achieved with the control law (19). It shows that solutions either converge to circular relative equilibria with a phase configuration that (locall) minimizes the phase potential U or belong to the stable manifold of an unstable equilibrium. As an illustration, simulation results in (a) (b) (c) !5!1!15!2! !5!1!15!2! !5!1!15!2!25!2!1 1 2!2!1 1 2!2!1 1 2 Fig. 2. Circular formations achieved with the control (19) with = 12, ω = κ =.1, and U = KU 1. (a) Convergence to a balanced circular formation (K > ); (b) Convergence to a circular formation with no constraint on the asmptotic phase arrangement (K = ); (c) Convergence to a snchronized circular formation (K < ). Figure 2 illustrate the convergence result for the choice U = KU 1. For K =, the control law (19) achieves convergence to a circular formation but the asmptotic phase arrangement is not constrained. For K <, the control law (19) forces convergence to the snchronized circular formation. For K >, the control law (19) forces convergence to a balanced circular formation. ote that for K = κ, the epression of the control law simplifies to u k = ω (1 + κ < r k, ṙ k >). The stabilit analsis in Theorem 3 is entirel determined b the critical points of the phase potential U: (local) minima correspond to stable equilibria and other critical points correspond to unstable relative equilibria. When a critical point is nondegenerate, this analsis etends to the Jacobian linearization of the closed-loop sstem, providing (local) eponential stabilit conclusions. We sa that a critical point θ is nondegenerate (in T /S 1 ) if all eigenvalues of the Hessian 2 U θ ( θ) are different 2 from zero, ecept for the zero eigenvalue (with eigenvector 1) associated to the rotational smmetr.

6 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 6 Theorem 4: Consider as in Theorem 3 the particle model (1), a smooth phase potential U(θ) that satisfies < grad U, 1 >=, and the control law (19). A relative equilibrium determined b a nondegenerate critical point θ of U is eponentiall stable if θ is a (local) minimum and eponentiall unstable otherwise. Proof: The proof is a local version of the proof of Theorem 3 around the relative equilibrium = ( r, θ). In the coordinates (s, θ), the quadratic approimation of V at is δv = 1 2 (κ < P s, P s > +δθt Hδθ) where δθ = θ ( θ + ω t1) and H = 2 U θ 2 ( θ). Likewise, the control law (23) linearizes to δu k = κ < P k s, ie i( θ k +ω t) > H k δθ, where H k is the kth row of H. Then, δv = κ < P k s, ie i( θ k +ω t) > δu k + (Hδθ) T δu = (κ < P k s, ie i( θ k +ω t) > +H k δθ)δu k = (δu k ) 2 along the closed-loop solutions of the linearized sstem. The stabilit analsis then proceeds as in the proof of Theorem 3: if θ is a minimum, then H and the LaSalle invariance principle proves asmptotic stabilit of the linearized (periodic) closed-loop sstem, which implies eponential stabilit of the equilibrium. In contrast, if θ is not a minimum, then there eist initial conditions for which δv <. The corresponding solutions of the linearized sstem cannot converge to the equilibrium and must diverge eponentiall. The above result ields a sstematic and general design methodolog b reducing the design of eponentiall stabilizing control laws to the construction of phase potentials. Specificall, control laws that eponentiall stabilize isolated relative circular equilibria are automaticall derived from phase potentials U that have nondegenerate minima at the desired location. The phase potential U 1 of (4) achieves this objective for the stabilization of the snchronized circular formation. The net section focuses on the construction of more general phase potentials that can be used to isolate specific balanced circular formations. VI. EXPOETIAL STABILIZATIO OF ISOLATED CIRCULAR RELATIVE EQUILIBRIA A. Stabilization of higher momenta When the phase potential U 1 reaches its minimum, the phase arrangement of the particles is onl stabilized to the balanced set which is high dimensional. More general phase potentials are introduced in this section in order to reduce the dimension of this equilibrium set. A natural generalization of the potential U 1 is a potential U m = 2 p mθ 2 (25) which depends on the mth moment p mθ of the phase distribution on the circle, defined as p mθ = 1 m e imθ k, m = 1, 2,.... (26) ote that p 1θ = p θ. The net proposition is a direct generalization of Theorem 1. Theorem 5: Let m. The potential U m = 2 p mθ 2 reaches its unique minimum when p mθ = (balancing modulo 2π ) and its unique maimum when the phase difference m between an two phases is an integer multiple of 2π m (snchronization modulo 2π m ). All other critical points of U m are isolated in the shape manifold T /S 1 and are saddle points of U m. Proof: Critical points of U m are the roots of U m =< p mθ, ie imθ k >=, k = 1,...,. (27) Critical points for which p mθ = are global minima of U m. Critical points for which p mθ = p mθ e iψm are characterized b M phases satisfing mθ k = Ψ m mod 2π and M phases satisfing mθ k = (Ψ m + π) mod 2π, with M < 2. At those critical points, m p mθ = 1 2M > 1. U m is maimized when M =. Critical points for which 1 M < 2 are saddles because the second derivative 2 U m θ 2 k = 1 m < p mθ, e imθ k > = 1 cos(ψ m mθ k )m p mθ (28) takes negative values if mθ k = Ψ m and positive values if mθ k = Ψ m + π. As a consequence, a small variation δθ k at those critical points decreases the value of U m if mθ k = Ψ m and increases the value of U m if mθ k = Ψ m + π. We show in the net section how linear combinations of the potentials U m enable the stabilization of specific sets of isolated relative equilibria characterized b various discrete smmetr groups. B. Smmetric balanced patterns Let 1 M be a divisor of. An (M, )- pattern is a smmetric arrangement of phases consisting of M clusters uniforml spaced around the unit circle, each with /M snchronized phases. For an, there eist at least two smmetric patterns: the (1, )-pattern, which is the snchronized state, and the (, )-pattern, which is the spla state, characterized b phases uniforml spaced around the circle. Figure 3 provides an illustration of all smmetric balanced patterns for = 12. Smmetric balanced patterns are etremals of the potentials U m. As a consequence, the are characterized as minimizers of well-chosen potentials; these can be written as linear combinations of the U m. This result for M =, i.e. the spla state, was also presented in [27]. We first prove a technical lemma that we will use in making this characterization eplicit.

7 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 7 Fig. 3. The si possible different smmetric patterns for = 12 corresponding to M = 1, 2, 3, 4, 6 and 12. The top left is the snchronized state and the bottom right is the spla state. The number of collocated phases is illustrated b the width of the black annulus denoting each phase cluster. Lemma 1: Consider the following sum where m, M, P (M) m e i 2πm M j. (29) If m (M) M, then P m = M, otherwise P m (M) =. Proof: If m 2πm M then ei M j = 1 for all j which proves the first part of the lemma. To prove the second part, we treat (29) as the sum of a geometric series and evaluate it for all m that satisf m M /. Multipling both sides of equation (29) b e i 2πm M gives, P m (M) e i 2πm M = M e i 2πm M (j+1) =P m (M) Rearranging terms and solving for P (M) m P (M) m = e i 2πm M e i 2πm M + e i 2πm M (M+1). e i2πm 1 e i 2πm ields, M 1, (3) which shows that P m (M) = since the numerator of (3) vanishes for all m that satisf m M /. Theorem 6: Let 1 M be a divisor of. Then θ T is an (M, )-pattern if and onl if it is a global minimum of the potential U M, = K m U m (31) m=1 with K m > for m = 1,..., M 1 and K M <. Proof: The global minimum of U M, is reached (onl) when K m U m is minimized for each m. If K m > for m = 1,..., M 1 and K M <, following (25) and (26) this means the global minimum corresponds to p mθ = for m = 1,..., M 1 and p Mθ = 1 M. We show that this implies an (M, )-pattern configuration. From the condition p Mθ = 1 M and Theorem 5, we can conclude that there are M clusters such that the kth cluster is of size k at phase Θ k = 2π M k, k = 1,..., M, where M k =. We would like to show that k = M have p mθ = 1 m and for all k = 1,..., M. We k e i 2πm M k =, m = 1,..., M 1 (32) Mp Mθ = 1 k e i2πk = 1. (33) Equations (32) and (33) are a sstem of linear equations in the unknown variables = ( 1,..., M ) T. amel, (32) and (33) can be written as A = b where A = A T C M M with [A] kj = e i 2π M kj, j, k = 1,..., M and b = (,...,, ) T R M. The inverse of A is given b A 1 = 1 M ĀT, where the bar denotes the comple conjugate. To see this, observe that [AĀT ] kj = l=1 e i 2π(k j) M l, j, k = 1,..., M. (34) For j = k, equation (34) evaluates to M b Lemma 1. For j k, we have k j < M and equation (34) evaluates to zero b Lemma 1. Therefore, the solution to the sstem of equations (32) and (33) is = 1 M ĀT b. Since both the Mth row and column of ĀT are all ones, we find that k = M for all k = 1,..., M. et, we show that an (M, )-pattern configuration minimizes each K m U m for m = 1,..., M. For an (M, )- pattern, the size of cluster k is /M and its phase is given b Θ k = 2π M k, where k = 1,..., M. Recall that the mth moment p mθ of the phase distribution is given b (26). Evaluated at an (M, )-pattern, the mth moment becomes p (M) mθ = = 1 m M 1 Mm e imθj e i 2πm M j, m = 1,..., M. (35) B Lemma 1, p (M) mθ = 1 m for m M and zero otherwise. Therefore, for phases in an (M, )-pattern, p mθ = for m = 1,..., M 1 and p Mθ = 1 M. Corollar 1: Let M =. Then θ T is an (, )- pattern, i.e. the spla state, if and onl if it is a global minimum of the potential U, = 2 m=1 K m U m (36) with K m > for m = 1,..., 2, where 2 is the largest integer less than or equal to 2. Proof: B Theorem 6, the phases are in the spla state if and onl if the correspond to the global minimum of the potential (31) with M =. Since p mθ = imposes two constraints on the sstem for each m, minimizing the potential U M, imposes 2 constraints on the phase arrangement. However, for potentials with S 1 smmetr, the dimension of the shape space is 1. Therefore, we need onl set p mθ = for m = 1,..., 2 since that imposes independent constraints.

8 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 8 Due to the characterization of Theorem 6, stabilizing control laws for (M, )-pattern circular formations are directl and sstematicall provided b Theorem 3. Theorem 7: Each (M, )-pattern circular formation of radius ρ is an isolated relative equilibrium of the particle model (1) and is eponentiall stabilized b the control law u k = ω (1 + κ < r k, ṙ k >) (U M, κu 1 ). (37) Proof: The theorem is a consequence of Theorems 3, 4, and 6. We onl need to prove that each (M, )-pattern defines a nondegenerate critical point of the potential U (M,). Let the negative gradient of the potential U M, be defined in terms of the coupling function, Γ(θ kj ), i.e. M, U = m=1 K m m sin mθ kj Γ(θ kj ), (38) where K m > for m = 1,..., M 1 and K M <. Also, let Γ (θ kj ) be the derivative of Γ(θ kj ) with respect to θ kj, given b Γ (θ kj ) = K m cos mθ kj. (39) m=1 As shown in [28], the linearization of coupling functions of this form about an (M, )-pattern has eigenvalues that can be described as the union of two sets. The first set consists (M) of the eigenvalue λ with multiplicit M. These eigenvalues are associated with intra-cluster fluctuation. The second set consists of M eigenvalues λ (M) p, p =,..., M 1. These eigenvalues are associated with inter-cluster fluctuation. Both sets of eigenvalues can be epressed as functions of the Fourier coefficients of Γ (θ kj ). For a general coupling function, the Fourier epansion of Γ (θ kj ) is Γ (θ kj ) = (a l cos lθ kj + b l sin lθ kj ). (4) l=1 The formulas for calculating the (real part of) the eigenvalues are as follows [28]: λ (M) = a Ml (41) Re{λ (M) p } = l=1 ( a Ml a M(l 1)+p + ) a Ml p. (42) 2 l=1 ote that onl the a l coefficients determine stabilit and that Re{λ (M) p } = Re{λ (M) M p }. The a l coefficients are given b integrating which gives and a l = 1 π π π Γ (θ kj ) cos lθ kj dθ kj (43) a l = K l, l = 1,..., M (44) a l =, l = or l > M. (45) As a result, a Ml = for l > 1, which, using (41), ields λ (M) = K M <. In addition, using (42), we find that !5!1!15!2!25!2!1 1 2 Fig. 4. The result of a numerical simulation starting from random initial conditions and stabilizing the spla state formation using the control j k (37) with M = = 12, ω = κ =.1, K m = ω for m = 1,...,, and j k 2 K m = for m >. 2 = K M Kp+K M p 2 < for p = 1,..., M 1 and =. The zero eigenvalue corresponds to rigid rotation of all phases [28]. Since the coupling function (38) is the gradient of the potential U M,, its Jacobian is equivalent to the Hessian of U M, and, consequentl, all the eigenvalues are real. Therefore, each (M, )-pattern defines a nondegenerate critical point of the potential U M, since the Hessian has rank 1. Theorem 7 does not eclude convergence to circular formations that correspond to other critical points of the phase potential U (M,). However, no other local minima were identified and simulations suggest large regions of attraction of the (M, )-pattern circular formations. Figure 4 illustrates a simulation of the spla state stabilization for = 12 particles using the control law (37). λ (M) p λ (M) VII. SPATIAL SYMMETRY BREAKIG I PARALLEL FORMATIOS As shown in Theorem 1, the phase control u = gradu 1 suffices to stabilize parallel equilibria. However, the asmptotic relative positions of particles are arbitrar. In this section, we show how to reduce the dimension of the equilibrium set in close analog to the design of the spacing control in Section IV. We choose an arbitrar isolated parallel relative equilibrium in the shape space b imposing on the relative position r k of particle k with respect to the center of mass a fied length ρ k and a fied orientation ψ k relative to the group direction. Assuming that all phases are snchronized at the relative equilibrium, this gives in comple notation r k = r k R = ρ k e i(θ k ψ k ) d 1 k eiθ k (46) where d k = ei ψ k ρ k is a constant comple number. The relative positions with respect to the center of mass must balance, that

9 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 9 is, r k =, which imposes the constraint n d 1 k =. (47) Motivated b the derivation of the spacing control in Section IV, we observe that the definition t k = d k 1 + d k ( r k + e iθ k ) allows one to specif the parallel relative equilibrium of interest b the conditions P t =, p θ = 1, P = I 1 11T. (48) Indeed, the condition p θ = 1 imposes phase snchronization, that is, θ = 1θ for some θ S 1, whereas the condition P t = implies t = 1t for some t C, which in turn gives r k + e iθ = (1 + d 1 k )t. Summing over k ields t = e iθ, which corresponds to the desired relative equilibrium. The desired relative equilibrium thus minimizes the Lapunov function ( ) 1 V (r, θ) = κ 2 P t 2 U 1 (θ), κ > (49) which, in analog with the results of Section IV, suggests the control law u k = (1 + κ) U 1 d k κ < P k t, d k + 1 ieiθ k >, κ >. (5) Lapunov analsis provides the following result. Theorem 8: Consider the particle model (1) with the control law (5). The parallel relative equilibrium defined b (48) is Lapunov stable and a global minimum of the Lapunov function (49). Moreover, for ever κ >, there eists an invariant set in which the Lapunov function is nonincreasing along the solutions. In this set, solutions converge to a parallel relative equilibrium that satisfies < P k t, d k d k + 1 ieiθ >= (51) for some θ S 1 and for k = 1,...,. Proof: In the coordinates (t, θ), the sstem (1) becomes d ṫ k = k d k +1 (P ke iθ + ie iθ k u k ) θ k = u k. (52) Writing the control law (5) as u k = U1 + v k = < P k e iθ, ie iθ k > +v k, and using the identit P k e iθ ie iθ k < P k e iθ, ie iθ k >=< P k e iθ, e iθ k > e iθ k one obtains d k ṫ k = d k +1 (< P ke iθ, e iθ k > e iθ k + ie iθ k v k ) θ k = U1 + v k. (53) Setting v k =, the time-derivative of the Lapunov function (49) along solutions of (53) is V v= = κ gradu (54) d k κ < P k t, d k + 1 eiθ k >< P k e iθ, e iθ k > Fi an arbitrar compact neighborhood of the parallel equilibrium (46) that contains no critical point of U 1 other than the snchronized state. Then there eists a constant c 1 > such that the inequalit d k < P k t, d k + 1 eiθ k > c 1 P t (55) holds in for k = 1,...,. We show that a similar constant c 2 > eists for the inequalit < P k e iθ, e iθ k > c 2 gradu 1 2. (56) Because the set is compact, it is sufficient to establish (56) in the vicinit of critical points of U 1, that is when the right hand side of (56) vanishes. B assumption, the snchronized state is the onl critical point of U 1 in the set. Locall around that state, we write U1 θ j = θ j θ av + h.o.t. where θ av = 1 i=1 θ i and h.o.t. stands for higher-order terms. We then obtain gradu 1 2 = (θ j θ av ) 2 + h.o.t. (57) The left-hand side of (56) can be rewritten as 1 1 cos(θ j θ k ) = 1 (θ j θ k ) 2 + h.o.t. (58) 2 Using θ j θ k = θ j θ av +θ av θ k and the triangle inequalit ield 1 (θ j θ k ) 2 1 (θ j θ av ) 2 + (θ k θ av ) (θ j θ av ) 2 which, from (58), provides the inequalit 1 1 cos(θ j θ k ) + 1 (θ j θ av ) 2 + h.o.t. (59) Comparing (57) and (59), we see that (56) holds for (an) c 2 > +1 in the vicinit of the snchronized state and therefore also for some uniforml bounded constant c 2 in the compact set. Using the inequalities (55) and (56) in (54) ields V v= κ(1 c 1 c 2 P t ) gradu 1 2 (6) This implies that for ever κ >, there eists a neighborhood of the parallel equilibrium (46) where the Lapunov function V satisfies V v= ɛ gradu 1 2 (61) for some ɛ >. With v k = u k U1 defined from (5), one obtains V ɛ gradu 1 2 ( ) U1 d k κ < P k t, d k + 1 ieiθ k > v k = ɛ gradu 1 2 vk 2. (62)

10 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 1 For a fied κ >, let V κ be the largest value such that (62) holds in the set Ω κ = {(t, θ) V (t, θ) V κ }. Then the set Ω κ is invariant and solutions in Ω κ converge to the largest invariant set where θ = θ 1 for some θ S 1 and v k =, k = 1,...,. This is a set of parallel equilibria satisfing (51). The phase control u = gradu 1 stabilizes the set of parallel equilibria, which is of dimension 2( 1) in the shape space. Awa from singularities, the algebraic constraints (51) are independent. As a result, the control law (5) isolates a subset of parallel equilibria of dimension 2 in the shape space. The result of Theorem 8 is thus weaker than the results for circular equilibria for two reasons: the result is onl local and the control law does not isolate the desired parallel equilibrium for > 2. A simple calculation indeed shows that the Jacobian linearization of (52) at the parallel equilibrium (46) possesses 2 uncontrollable spatial modes with zero eigenvalue. This means that the Jacobian linearization of the closed-loop sstem will possess 2 zero eigenvalues for an smooth static state feedback. For > 2, no smooth static state feedback can achieve eponential stabilit of an isolated relative parallel equilibrium. VIII. STABILIZABLE COLLECTIVES In this section, we focus on the control structure (19) and discuss the role of ke parameters. The constant ω determines the tpe of relative equilibrium. For ω, the control (19) produces circular motion with radius ρ = ω 1 and sense of rotation determined b the sign of ω. The potential U determines the stead-state phase arrangement. For ω = κ = and U = KU 1, K <, the control (19) produces parallel motion. In the following subsections, we investigate removing the SE(2) smmetr of the control (19), i.e. its invariance to rigid translation and rotation in the plane [1]. We stabilize circular motion about a fied beacon and parallel motion along a fied reference direction. We define behavior primitives to enable the group to track piecewise-linear trajectories with fied wapoints. A. SE(2) Smmetr Breaking The control (19) depends onl on the relative spacing, r kj = r k r j, and relative phase, θ kj = θ k θ j, variables. As a result, the model (1) with control (19) is invariant to rigid translations and rotations in the plane, which corresponds to the action of the smmetr group SE(2). In this subsection, we investigate breaking this smmetr first b adding a fied beacon to break the R 2 smmetr and, second, b adding a heading reference to break the S 1 smmetr. We break the R 2 translation smmetr of the spacing control (19) b stabilizing circular motion with respect to a fied beacon. Let R C be the location of a fied beacon and (re)define the vector from the beacon to particle k b r k = r k R. We obtain the following etension of Theorem 3. Corollar 2: Consider the particle model (1) and a smooth phase potential U(θ) that satisfies < grad U, 1 >=. The control law (19) where r k = r k R, R is the location of a fied beacon, and U 1 is removed, i.e., u k = ω (1 + κ < r k R, ṙ k >) U, ω (63) enforces convergence of all solutions to a circular formation of radius ρ = ω 1 about R. Moreover, the asmptotic phase arrangement is a critical point of the potential U. Proof: We use the Lapunov function V (r, θ) which is the sum of the composite Lapunov function V (r, θ) defined b (2) in the proof of Theorem 3 and a new term that is minimized when the center of the circular relative equilibrium is at R : V (r, θ) = V (r, θ) T s + iω R 2. (64) Recall that V (r, θ) = κ 2 P s 2 + U(θ) with s given b (9) and P the projector defined in (1). We compute ( V = κ < P k s + 1 1T s+ Since iω R, ie iθ k > + U ) (u k ω ). P k s + 1 1T s + iω R o = e iθ k iω (r k R ), using the control law (63), we once again obtain V = (u k ω ) 2 along the solutions of the closedloop sstem. Solutions converge to the largest invariant set Λ where κ < ω r k, ie iθ k >= U (65) for k = 1,...,. Proceeding as in the proofs of Theorems 2 and 3, we use the result that on Λ the control is u k = ω for k = 1,...,. This implies that the right hand side of (65) is constant for each k. Since θ k = ω, r k and the left hand side of (65) is constant onl if it is zero. This implies a circular relative equilibrium centered at R. et, we break the S 1 rotational smmetr of the control (19) b introducing a heading reference θ, where θ = ω. We couple the dnamics of the particle group to the reference heading b adding a new coupling term to onl one of the particles in the group. This ields the following etension of Theorem 3. Corollar 3: Consider the particle model (1) and a smooth phase potential U(θ) that satisfies < grad U, 1 >=. Let u k, k = 1,..., 1 be given b (19) and u = ω (1 + κ < r, ṙ >) (U κu 1) θ + d sin(θ θ ) (66) where θ = ω and d >. This control enforces convergence of all solutions to relative equilibria as in Theorem 3. In addition, relative equilibria with phase arrangement minimizing U and satisfing θ = θ define an asmptoticall stable set.

11 SUBMITTED TO IEEE TRASACTIOS O AUTOMATIC COTROL 11 Proof: Consider the potential W (r, θ) = V (r, θ) + d(1 cos(θ θ )) (67) with V (r, θ) given b (2). The time-derivative of W along the solutions of the particle model is Ẇ = V + d sin(θ θ )( θ θ ). (68) The control law (66) with θ = ω and u k given b (19) for k = 1,..., 1 results in Ẇ = (u k ω ) 2. Solutions therefore converge to the largest invariant set Λ where (24) holds for k = 1,..., 1. The 1 equations P k s =, k = 1,..., 1, impl P s = because the matri P has rank 1. Likewise, the 1 equations U =, k = 1,..., 1, impl gradu = because the Hessian of U has rank 1. The relation u = ω then reduces to d sin(θ θ ) = (69) which implies that θ = θ or θ + π. Therefore, relative equilibria with θ = θ which minimize V are asmptoticall stable since the also minimize the potential W. We note that in the case ω =, Corollar 3 proves asmptotic stabilit of parallel collective motion to a fied heading reference, θ. We use this result in the following subsection. B. Trajector tracking with behavior primitives We use the control (19) to define four behavior primitives which can be combined to track piecewise-linear trajectories following [18]. The behavior primitives include impulsive controls to align the particles with the reference input trajector and feedback controls to stabilize this trajector. The behaviors are referred to as circular-to-parallel, parallel-toparallel, parallel-to-circular, and circular-to-circular. In parallel motion, the group center of mass follows a fied reference heading. In the circular state, particles circular a fied beacon with a prescribed radius and sense of rotation. Circular-to-parallel. Starting from circular motion, this behavior stabilizes parallel motion along a fied reference heading. The inputs to this behavior are the reference heading, θ, and the gain, d. The impulse control which aligns the particles in the reference direction is θ k = θ θ k. (7) The feedback control that stabilizes parallel motion is of the form (19) for k = 1,..., 1 and (66) for k = with ω = κ = and U = KU 1, K <. Parallel-to-parallel. Starting from parallel motion, this behavior stabilizes parallel motion along a different reference trajector. The inputs to this behavior are the new reference heading, θ, and the gain, d. The impulsive control used to align the particles in the input direction is given b (7). The feedback control that stabilizes parallel motion is of the form (19) for k = 1,..., 1 and (66) for k = with ω = κ = and U = KU 1, K <. Parallel-to-circular. Starting from parallel motion, this behavior stabilizes circular motion about the location of the center of mass at the time the behavior is initiated. The input !5!1!15!2!25!3 A C B Fig. 5. Trajector tracking with = 12 starting from random initial conditions. The reference input is a piecewise-linear curve. The behavior sequence starts in the vicinit of A b stabilizing circular motion with ω = 1/25 and then follows A circular-to-parallel, B parallel-to-parallel, and C parallel-to-circular. This sequence repeats for the points C, D, and E and then ends with the circular-to-circular behavior at E with ω = 1/5. See tet for control parameters. to this behavior are the parameters, ω and κ >, the initial center of mass of the group, R, and the phase potential U(θ). The impulsive control used to align the particles in the input rotation direction is given b θ k = arg(iω r k ) θ k. (71) The feedback control used to stabilize circular motion is of the form (63), which is (19) with r k = r k R, where R is a fied beacon and U 1 is removed. Circular-to-circular. Starting from circular motion, this behavior stabilizes circular motion with a different radius, i.e. dilation/contraction, about the same fied reference. The input to this behavior are the parameters, ω and κ >, the initial center of mass of the group, R, and the phase potential U(θ). There is no impulsive control used to realign the particles. The feedback control used to stabilize circular motion is of the form (63), which is (19) with r k = r k R, where R is a fied beacon and U 1 removed. et, we use the behavior primitives to construct a behavior sequence that tracks a sample reference trajector. The admissible references are piecewise-linear paths specified b a list of desired heading and duration pairs. An eample of trajector tracking is shown in Figure 5. In this eample, twelve particles start from random initial conditions in the vicinit of the origin (point A). Each step in the behavior sequence is simulated for 2 time steps. The behavior sequence starts b stabilizing circular motion about the origin with ω = κ = 1/25, R =, and U(θ) =. The net behavior in the sequence is circular-to-parallel with reference heading θ = π/8 and gain d = 1/, which takes the sensor network from point A to point B in Figure 5. At point B, the behavior parallelto-parallel is used to track the reference input θ = 3π/8 to point C. Then the parallel-to-circular behavior stabilizes circular motion about a fied center of mass with ω = 1/25, κ = ω, and U(θ) =. The sequence is repeated for the E D