1 Collective Circular Motion of Multi-Agent Systems in Synchronize an Balance Formations With Secon-Orer Rotational Dynamics Anoop Jain an Debasish Ghose arxiv:161.3479v [cs.sy] 17 Mar 16 Abstract This paper consiers the collective circular motion of multi-agent systems in which all the agents are require to traverse ifferent circles or a common circle at a prescribe angular velocity. It is require to achieve these collective motions with the heaing angles of the agents synchronize or balance. In synchronization, the agents an their centroi have a common velocity irection, while in balancing, the movement of agents causes the location of the centroi to become stationary. The agents consiere are initially moving at unit spee aroun iniviual circles at ifferent angular velocities. It is assume that the agents are subjecte to limite communication constraints, an exchange relative information accoring to a time-invariant unirecte graph. We present suitable feeback control laws for each of these motion coorination tasks by consiering a seconorer rotational ynamics of the agent. Simulations are given to illustrate the theoretical finings. Inex Terms Synchronization, balancing, multi-agent systems, secon-orer rotational ynamics, esire angular velocity, limite communication. I. ITRODUCTIO There are various engineering applications such as tracking, surveillance, environmental monitoring, searching, sensing an ata collection, where it is require for the multi-agent systems to perform a particular collective motion [1] [4]. A multiagent system might comprise groun vehicles, air vehicles, unerwater vehicles or a combination of these. In this article, we focus on achieving collective circular motion that can be applie in the scenario where vehicles are require to enclose, capture, secure or monitor a target or a search region. Motivate by these applications, the collective motion where all the agents traverse i) ifferent circles, or ii) a common circle at the prescribe angular velocity along with their heaing angles in synchronize or in balance states, are consiere in this paper. Synchronization refers to the situation when all the agents, at all times, move in a common irection. A complementary notion of synchronization is balancing, in which all the agents move in such a way that their centroi, which is the average position of all the agents, remains fixe. It is evient in synchronize formation that agents an their A. Jain is a grauate stuent at the Guiance, Control an Decision System Laboratory GCDSL) in the Department of Aerospace Engineering, Inian Institute of Science, Bangalore, Inia email: anoopj@aero.iisc.ernet.in). D. Ghose is a Professor at the Guiance, Control an Decision System Laboratory GCDSL) in the Department of Aerospace Engineering, Inian Institute of Science, Bangalore, Inia email: ghose@aero.iisc.ernet.in). This work is partially supporte by Asian Office of Aerospace Research an Development AOARD). centroi move in the same irection. ote that, in this paper, collective motion an formation are use interchangeably. Earlier work in [] an [6] has focuse on achieving synchronize an balance formations in a group of agents uner all-to-all an limite communication scenarios, respectively. In these papers, it is consiere that the angular velocities of initial rotations of all the agents are the same an remains constant at all times. Recently, the effect of heterogeneity in various aspects have been stuie in the literature. For example, [8] consiers nonientical linear velocities of the agents, an [9] consiers heterogeneous control gains. In a similar spirit, in this paper, we consier that the angular velocities of the initial rotational motion of the agents are nonientical an are allowe to vary with time. This more general scenario is aresse in this paper. In a similar context, the authors in [7], by assuming an all-to-all coupling among agents, propose feeback controls to stabilize synchronize an balance circular formations at a esire angular velocity. However, unlike [7], in this paper, we further assume that the communication among agents is restricte an can be moele as a time-invariant an unirecte graph. Some relate work, but with all-to-all communication, has been presente in [1]. There exists an ample literature relate to the stuy of collective circular formation control. In [11], control laws are propose to stabilize collective circular motion of nonholonomic vehicles aroun a virtual reference beacon, which is either stationary or moving. In [1], authors propose a istribute circular formation control law for ring-networke nonholonomic vehicles with local coorinate frames. In [13], Chen an Zhang propose a ecentralize control algorithm to form a class of collective circular motion, in which the vehicles are evenly istribute over the motion circle, an have the same rotational raius. The latter assumption is relaxe in [14], where the agents move in circles aroun a common center, but with ifferent raii. In [1] an, [16], the control algorithms to stabilize the collective motion aroun a circular orbit, which has either a fixe raius an time-varying center [1], or a fixe center an time-varying raius [16], are propose. An extension of these results is given in [17], where a new framework base on affine transformations is iscusse to achieve more complex time-varying formations. In [18], the splay circular formation, characterize by equally space arrangement of multiple robots, is stabilize by using a moifie Kuramoto moel [19]. The stabilization of circular motion uner cyclic pursuit is given in the seminal paper [], an also iscusse in [1] uner ynamically ajustable
control gains. Moreover, the cyclic pursuit problem of vehicles with heterogenous constant linear velocities is consiere in []. In [3], a Lyapunov guiance vector fiel approach is use to guie a team of unmanne aircraft to fly a circular orbit aroun a moving target with prescribe inter-vehicle angular spacing. The circumnavigation problem for a team of unicycle-type agents, with the goal of achieving specific circular formations an circling on ifferent orbits centere at a target of interest, is stuie in [4]. It is to be note that in the literature escribe above, most of the attention is towars achieving a particular type of collective circular formation. However, in the present work, the emphasis is given towar achieving the same along with a particular arrangement of the heaing angles of the agents which coul be a synchronize, balance or a combination of both usually calle as symmetric phase pattern). These formations serves as the motion primitives, an can be utilize to get more general motion patterns []. The main contribution of this paper is to propose a limite communication base control strategy to stabilize aforementione collective circular motion of a group of agents with their phase arrangements either in synchronize or in balance formation, while allowing the angular velocities of iniviual, initial circular motions, performe by the agents, to be ifferent. With this purpose, in this paper, we consier ientical agents moving in a planar space at constant unit linear spee with secon-orer rotational ynamics. Thus, the ynamics of each agent is represente by a state vector, which inclues the position, heaing angle an angular velocity of each agent as its elements. The secon-orer rotational moel is particularly relevant in the context of planar rigi-boy motion, where a ynamic vehicle moel must account not only for motion of the agent s center of mass, but also for rotational motion about the center of mass []. We use secon-orer rotational moel to erive feeback controls that are aequate to regulate the orientations as well as the angular velocities of the agents. The outline of the paper is as follows. In Section II, we escribe the system moel an formulate the problem. In Section III, control laws are propose to stabilize collective motion of agents on ifferent circles at esire angular velocity with their phase arrangement either in synchronize or in balance states. The control laws to stabilize collective motion aroun a common circle of esire raius as well as center with their phase arrangement, again either in synchronize or in balance states, is propose in Section IV. The control strategy to stabilize symmetric balance patterns is propose in Section V. In Section VI, we combine the results of the previous sections an propose control algorithms to stabilize symmetric circular formations suitable for mobile sensor network applications. Finally, Section VII conclues the paper. II. SYSTEM DESCRIPTIO AD PROBLEM STATEMET A. Agent Moel Similar to [7] an [], the collective secon-orer rotational ynamics of ientical agents, moving in a planar space, each assume to have unit mass an unit linear spee, 6 1 4 a) 3 Fig. 1. Unirecte circulant graphs for = 6. Both a) an b) are circulant graphs but only b) is connecte. is represente as 6 1 4 b) ṙ k = e iθ k 1a) θ k = ω k 1b) ω k = u k, k = 1,...,. 1c) Here, complex notations are use to escribe the position an velocity of each agent. For k = 1,...,, the position of the k th agent is r k C, while the velocity of the k th agent is ṙ k = e iθ k = cosθ k + isinθ k C, where, θ k is the orientation of the unit) velocity vector of the k th agent from the real axis, an i = 1 represents the stanar complex number. The orientation θ k, of the velocity vector, which is also referre to as the phase of the k th agent [19], represents a point on the unit circle S 1. In 1), ω k R is the angular velocity of the k th agent, which is etermine by the feeback control u k R. If the control law u k is constant an equal to ω k, then the k th agent travels at constant unit linear spee on a circle of raius ρ k = ω k 1. The irection of rotation aroun the circle is etermine by the sign of ω k. If ω k >, then the k th agent rotates in the anticlockwise irection, whereas, if ω k <, then the k th agent rotates in the clockwise irection. Let the initial motion of all the agents with ynamics 1) be governe by the open-loop control u k =, k. In this situation, the k th agent moves in a circular orbit of raius ω k 1 with angular velocity ω k. Our aim is to seek a feeback control u k, k such that the collective motion of agents, subjecte to limite communication constraints represente by a timeinvariant unirecte graph, converge to a circular motion at esire angular velocity an hence esire raius of the circling orbit since raius of rotation = angular velocity 1 for an agent circling at unit linear spee) with their phase angles either in synchronize or in balance states. We assume that agents can exchange information only about their orientations accoring to the unerlying interaction network, an they are globally provie the information about the esire angular velocity Ω. In aition, when it is require for the agents to move aroun a common circle, the information about the esire center c of the common circle) is also globally provie to them. ote that issue of collision avoiance among agents is not consiere in this work. B. otations We introuce a few aitional notations that are use in this paper. We use bol face letters r = r 1,...,r ) T C, θ = θ 1,...,θ ) T T, where, T is the -torus, which is equal to S 1... S 1 -times), an ω = ω 1,...,ω ) T R, 3
3 to represent the vectors of length for the agents positions, heaings an angular velocities, respectively. ext, we efine the inner prouct z 1,z of two complex numbers z 1,z C as z 1,z = Re z 1 z ), where z 1 represents the complex conjugate of z 1. For vectors, we use the analogous bolface notation w,z = Rew w z) for w,z C, where w enotes the conjugate transpose of w. The norm of z C is efine as z = z,z 1/. The vectors an 1 are use to represent by =,,...,) T R, an 1 = 1,1,...,1) T R, respectively. C. Representation of Limite Communication Topology In the framework of multiagent systems, communication among agents is escribe by means of a graph. A graph is a pair G = V,E ), where V = {v 1,...,v } is a set of noes or vertices an E V V is a set of eges or links. Elements of E are enote as v j,v k ) which is terme an ege or a link from v j to v k. An unirecte link between noes v j an v k inicates that the information can be share from noe v j to noe v k an vice versa. A graph G is calle an unirecte graph if it consists of only unirecte links. The noe v j is calle a neighbor of noe v k if the link v j,v k ) exists in the graph G. In this article, the set of neighbors of noe v j is represente by j. A complete graph is an unirecte graph in which every pair of noes is connecte, that is, v j,v k ) E, j,k. The Laplacian of a graph G, enote by L = [l jk ] R, is efine as [6], j, if j = k l jk = 1, if k j otherwise where, j is the carinality of the set j. This efinition allows the representation of the several properties of a graph in the form of matrix properties of its Laplacian L. It is well known that the Laplacian L of an unirecte an connecte graph G is P1) symmetric an positive semi-efinite, an P) has an eigenvalue of zero associate with the eigenvector 1, that is, L x = iff x = 1x. In this article, we will also use the notion of a circulant graph. A graph G is circulant if an only if its Laplacian L is a circulant matrix, that is, L is completely efine by its first row [7]. Each subsequent row of a circulant matrix is the previous row shifte one position to the right with the first entry of the row equal to the last entry of the previous row. An example of an unirecte circulant graph, consisting of 6 noes, is shown in Fig. 1. ote that the Laplacian for the graphs in Figs. 1a) an 1b) are, respectively, given by L a = circ1,,, 1,,) an L b = circ, 1,,,, 1), where, circz) represents the circulant matrix with z being its first row. As both L a an L b are circulant matrices, both the graphs shown in Fig. 1 are circulant, but only the graph shown in Fig. 1b) is connecte. ow, we state the following lemma from [8], which escribes various properties of an unirecte circulant graph that will be useful in proving the results in this paper. Lemma 1: Let L be the Laplacian of an unirecte circulant graph G = V,E ) with vertices. Set φ k = k 1)π/ for k = 1,...,. Then, the vectors f l) = e il 1)φ, l = 1,...,, ) efine a basis of orthogonal eigenvectors of L. The unitary matrix F whose columns are the normalize) eigenvectors 1/ ) f l) iagonalize L, that is, L = FΛF, where Λ = iag{,λ,...,λ } is the real) iagonal matrix of the eigenvalues of L. III. COTROL DESIG The esign of control laws is escribe in this section. At first, a phase potential W 1 θ) is escribe, the minimization of which correspons to synchronize formation, an its maximization correspons to balance formation. Then, a potential function Gω) whose minimization results in the collective motion of all the agents at a esire angular velocity, is propose. Finally, the control law u k is obtaine by minimizing a composite potential function consisting of W 1 θ) an Gω) as escribe below. A. Achieving Synchronize an Balance Formations The average linear momentum of a group of agents plays an important role in stabilizing their synchronize an balance formations. It is maximize in synchronize formation an minimize in balance formation. From 1), the average linear momentum, p θ, of a group of -agents, is given by, p θ = 1 e iθ k = p θ e iψ, 3) which is also referre to as the phase orer parameter [19]. The phase arrangement θ is synchronize if the moulus of the phase orer parameter 3) equals one, that is, p θ = 1. The phase arrangement θ is balance if the phase orer parameter 3) equals zero, that is, p θ = [19]. Thus, the stabilization of synchronize an balance formations is accomplishe by consiering the potential Uθ) = /) p θ, 4) which reaches its unique minimum when p θ = balance) an its unique maximum when all phases are ientical synchronize). Base on this potential function, the esign of control law for the all-to-all communication among agents may be accomplishe [1]. However, in orer to account for limite communication among agents, we moify the potential function 4) in the following manner [6]. Let P = I 1/)11 T, where, I is an -ientity matrix, be a projection matrix which satisfies P = P. Let the vector e iθ be represente by e iθ = e iθ 1,...,e iθ ) T C. Then, Pe iθ = I 1 ) 11T e iθ = e iθ p θ 1. ) Using ), one can obtain Pe iθ = e iθ,pe iθ = 1 p θ ), 6) which is zero minimum) when p θ = 1 synchronize formation), an equates to maximum) when p θ = balance
4 formation). Since P is 1/) times the Laplacian of the complete graph, the ientity 6) suggests that the optimization of Uθ) in 4) may be replace by the optimization of W 1 θ) = Q L e iθ ) = 1 e iθ,l e iθ, 7) which is a Laplacian quaratic form associate with L, an is positive semi-efinite. ote that, for a connecte graph, the quaratic form 7) vanishes only when e iθ = e iθ c 1, where θ c S 1 is a constant see property P), that is, the potential W 1 θ) is minimize in the synchronize formation. The time erivative of W 1 θ), along the ynamics 1), is ) ) W1 W1 Ẇ 1 θ) = θ k = ω k. 8) ote that W 1 = 1 = 1 j=1 j=1 e iθ j,l j e iθ e iθ j,l j e iθ ) + e iθ j, L je iθ ) ) e iθ j,ie iθ k = 1 ie iθ k,l k e iθ j k = ie iθ k,l k e iθ = sinθ j θ k ), 9) j k where, L k is the k th row of the Laplacian L. Substituting 9) in 8), we get Ẇ 1 θ) = ie iθ k,l k e iθ ω k. 1) For the reasons which will be aresse in Section V, let us efine the m th phase orer parameter p mθ an the phase potential W m θ), respectively, as p mθ = 1 m W m θ) = 1 e imθ k, 11) e imθ,l e imθ, 1) where, m {1,,3,...}, an e imθ = e imθ 1,...,e imθ ) T. ow, we state the following lemmas from [8], which escribes various properties of the phase potential W m θ) that will be useful in proving the results in this paper. Lemma : Critical points of the Laplacian phase potential W m θ)) Let L be the Laplacian of an unirecte an connecte graph G = V,E ) with vertices. Consier the Laplacian phase potential W m θ) efine in 1). If e imθ for all m is an eigenvector of W m θ), then mθ is a critical point of W m θ), an mθ is either synchronize or balance. The potential W m θ) reaches its global minimum if an only if mθ is synchronize. If G is circulant, then W m θ) reaches its global maximum in a balance phase arrangement. Proof: The critical points of W m θ) are given by the algebraic equations W m = ie imθ k,l k e imθ =, 1 k. 13) Let e imθ be an eigenvector of L with eigenvalue λ R. Then L e imθ = λe imθ, an W m = ie imθ k,l k e imθ = λ ie imθ k,e imθ k =, k, 14) which implies that mθ is a critical point of W m θ). Since graph G is unirecte, the Laplacian L is symmetric, an hence its eigenvectors associate with istinct eigenvalues are mutually orthogonal [9]. Since G is also connecte, 1 spans the kernel of L. Therefore, the eigenvector associate with λ = is e imθ = e iθ c 1 for any θ c S 1, which implies mθ is synchronize. Also e imθ,l e imθ = if an only if e imθ = e iθ c 1 for any θ c S 1. This proves the first part of the theorem. ext, we assume that G is a circulant graph, then Lemma 1 provies the following equivalent expression of the phase potential 1). W m θ) = 1 F e imθ,λf e imθ. 1) Let w = F e imθ, above equation can be written as W m θ) = 1 w,λw = 1 k= w k λ k. 16) Since matrix F is unitary, w = e imθ =. Thus, W m θ) = 1 w,λw λ max, 17) where, λ max is the maximum eigenvalue of L. The maximum of W m θ) is attaine by selecting e imθ as the eigenvector of L associate with the maximum eigenvalue. Since e imθ is orthogonal to 1, that is, it satisfies 1 T e imθ =, an thus correspons to the phase balancing of mθ see 11)). This completes the proof. Remark 1: For m = 1, the phase potential W 1 θ) minimizes when all the phases synchronize. This state correspons to the situation when all the agents, at all times, move in a common irection. On the other han, the potential W 1 θ) maximizes when all the phases balance. This state correspons to the motion of all the agents about a fixe point such that p θ =. This fixe point is actually the position R of the centroi of the agents since Ṙ = 1/) k ṙk = p θ. The use of the phase potential W m θ) for m > 1 will be elaborate in Section V, where we aress the stabilization of symmetric phase arrangements. B. Achieving Desire Angular Velocity The agents, initially rotating at ifferent angular velocities, are require to stabilize their collective motion at esire angular frequency Ω an hence achieve the esire raius ρ = Ω 1 ). For this, we choose a potential function Gω) = 1 which is minimize when ω k = Ω, k. ω k Ω ), 18)
The time erivative of Gω), along the ynamics 1), yiels Ġω) = ω k Ω ) ω k = ω k Ω )u k 19) C. Composite Potential Function an Control Law In this subsection, the control law u k, for the k th agent, is propose by constructing a composite potential function, which ensures that all the agents travel aroun iniviual circles at a esire angular velocity Ω with their phases either in balance or in synchronize states. Theorem 1: Let L be the Laplacian of an unirecte an connecte circulant graph G = V,E ) with vertices. Consier system ynamics 1) with control law u k = K ω k Ω ) W 1 ), k. ) For K >, all the agents converge to a collective motion in which they travel aroun iniviual circles of the same raius ρ = Ω 1 with their phase angles in balance state. Proof: Consier a composite potential function ) V 1 θ,ω) = K λ max W 1 θ) + Gω); K >. 1) ote that W 1 θ) /)λ max Lemma for m = 1) which ensures that V 1 θ,ω). Using 8) an 19), the time erivative of the potential function V 1 θ,ω) along the ynamics 1), is V 1 θ,ω) = K ) W1 ω k + ω k Ω )u k. ) With the control law ), the time erivative of V 1 θ,ω) results in V 1 θ,ω) = K k Ω ) ω KΩ From 1), we note that W 1 = Using 4), 3) becomes V 1 θ,ω) = K W 1 3) j k sinθ j θ k ) =. 4) ω k Ω ). ) Since θ,ω) T R is compact, it follows from the LaSalle s invariance theorem [3] that, for K >, all the solutions of 1) with the control law ) converge to the largest invariant set containe in { V 1 θ,ω) = }, that is, the set b = {θ,ω) ω k = Ω, k}. 6) In b, ω k = Ω, k, which implies that each agent rotates with angular spee Ω. Moreover, since u k = ω k =, k in the set b, it implies from ) that W 1 / ) =, k, which efines the critical points of W 1 θ). The set b is itself the largest invariant set since ) W1 = ie iθ k,l k e iθ t t = Ω e iθ k,l k e iθ + ie iθ k, iω L k e iθ =, k, 7) on this set. ow, we analyze the critical points of W 1 θ). Analysis of the critical points of W 1 θ): The critical points of W 1 θ) are given by the algebraic equations W 1 = ie iθ k,l k e iθ =, 1 k. 8) For m = 1, since 8) is the same as 13), it follows from Lemma that W 1 θ) minimizes in the synchronize state an maximizes in balance state. Moreover, since maximization of the potential W 1 θ) correspons to the global minimum of V 1 θ,ω), balance formation is asymptotically stable. This completes the proof. Theorem : Let L be the Laplacian of an unirecte an connecte circulant graph G = V,E ) with vertices. Consier system ynamics 1) with control law u k = K ω k Ω ) + W 1 ), k. 9) For K <, all the agents converge to a collective motion in which they travel aroun iniviual circles of the same raius ρ = Ω 1 with their phase angles in synchronize state. Proof: Consier a composite potential function V θ,ω) = KW 1 θ) + Gω); K < 3) Using 1) an 19), the time erivative of the potential function V along the ynamics 1), is ) W1 V θ,ω) = K ω k + ω k Ω )u k 31) With the control law 9), the time erivative of V results in V θ,ω) = K k Ω ) ω KΩ Using 4), 3) becomes V θ,ω) = K W 1. 3) ω k Ω ). 33) Since θ,ω) T R is compact, it follows from the LaSalle s invariance theorem [3] that, for K <, all the solutions of 1) with the control law 9) converge to the largest invariant set containe in { V θ,ω) = }, that is, the set s = {θ,ω) ω k = Ω, k}. 34) In s, ω k = Ω, k, which implies that each agent rotates with angular spee Ω. Moreover, since u k = ω k =, k in the set s, it implies from ) that W 1 / ) =, k, which efines the critical points of W 1 θ). The set s is itself the largest invariant set since 7) hols. Following Lemma, since minimization of the potential W 1 θ) correspons to the global minimum of V θ,ω), balance formation is asymptotically
6 3 1 1 p θ 1.8.6.4. Limite communication All to all communication Angular frequencies ra/sec).8.6.4...4.6 Limite communication Angular frequencies ra/sec).8.6.4...4.6 All to all communication 1 1 3 1 1.8 1 3.8 1 3 a) b) c) ) Fig.. Balancing of = 6 agents, connecte by a graph as shown in Fig.1b), on iniviual circles, each having the esire raius ρ = Ω 1 = m. a) Trajectories of the agents uner the control laws ) with K = 1. b) Average linear momentum approaches zero with time. c) Consensus of angular spees frequencies) at esire value Ω =. ra/sec limite communication). ) Consensus of angular spees at esire value Ω =. ra/sec all-to-all communication) 1 1 1 Heaing Angles eg) 3 3 1 1 Limite communication Heaing Angles eg) 3 3 1 1 All to all communication Angular frequencies ra/sec).8.6.4...4.6 Limite communication Angular frequencies ra/sec).8.6.4...4.6 All to all communication 1 1 1 1 1 1 1.8 1 3.8 1 3 a) b) c) ) e) Fig. 3. Synchronization of = 6 agents, connecte by a graph as shown in Fig.1b), on iniviual circles, each having the esire raius ρ = Ω 1 = m. a) Trajectories of the agents uner the control laws 9) with K = 1. b) Consensus in heaing angles limite communication). c) Consensus in heaing angles all-to-all communication). ) Consensus of angular spees at esire value Ω =. ra/sec limite communication). e) Consensus of angular spees at esire value Ω =. ra/sec all-to-all communication). Since the agents continue to rotate aroun their respective circles in synchronize fashion, their heaing angles keep increasing with time. stable. This completes the proof. Remark : In the previous analysis, the invariant sets b an s, efine by 6) an 34), respectively, look similar, however, they are ifferent since b contains phases θ corresponing to the balance formation, while s contains phases θ corresponing to the synchronize formation. Example 1: In this example, Theorems 1 an are emonstrate through simulation of = 6 agents connecte via a graph as shown in Fig. 1b). Let the initial positions, initial heaing angles an nonientical initial angular velocities of the agents be r) = 1, 1),1,3), 1, ),,1),1,), 4,1)) T, θ) = 3,4,1,7,9,6 ) T, an ω) =.,.3,.4,.,.6,.8) T, respectively. Although the initial locations of the agents are given for representing the trajectories of the agents in the simulation, the locations themselves are not important so far as the objective of synchronization is concerne. Even with ifferent locations, the convergence properties will be the same, although the trajectories will be ifferent. Fig. shows balancing of the agents aroun iniviual circles at esire angular velocity ω =. ra/sec, an hence esire raius ρ = Ω 1 = m of circular orbits as expecte. On the other han, Fig. 3 shows synchronization of agents on iniviual circles at the same esire angular frequency velocity. In all figures in this paper, the trajectory of the centroi is shown in black. ote that, in these figures, the convergence of the angular spees frequencies) to the esire value Ω is faster uner all-to-all communication scenario as expecte. IV. MOTIO O A COMMO CIRCLE In this section, we achieve synchronization an balancing of a group of agents aroun a common circle of esire raius ρ = Ω 1 as well as center c, which is fixe. This situation is shown in Fig. 4, where, the k th agent rotates in the anticlockwise irection on a circle of raius ρ = Ω 1 an center c. Without loss of generality, we assume Ω >. Therefore, at equilibrium, all the agents, which may initially be rotating in clockwise an anticlockwise irections, move in the anticlockwise irection on a common circle of esire raius as well as center. The position r k of the k-th agent in Fig. 4, is given by r k = c iρ e iθ k. 3) In orer to stabilize collective motion of all the agents aroun a common circle of raius ρ = Ω 1 > an center c, which is fixe, we introuce an error variable e k = r k c iρ e iθ k), k, an choose a potential function as, Sr,θ) = 1 k e = 1 = 1 r k c + iρ e iθ k r k c + iρ e iθ k,r k c + iρ e iθ k,36)
7 which is non-negative an becomes zero whenever r k = c iρ e iθ k, k. It means that the minimization of Sr,θ) correspons to the situation when all the agents move on a common circle of raius ρ = Ω 1 > centere at the fixe point c. The time erivative of the potential function 36) along the ynamics 1), yiels Ṡr,θ) = r k c iρ e iθ k,e iθ k 1 ρ ω k ) 37) Using linearity of inner prouct [3] in 37), we get Ṡr,θ) = r k c,e iθ k 1 ρ ω k ) ρ ie iθ k,e iθ k 1 ρ ω k ) 38) Since ie iθ k,e iθ k =, 38) is simplifie to Ṡr,θ) = r k c,e iθ k 1 ρ ω k ). 39) Theorem 3: Let L be the Laplacian of an unirecte an connecte circulant graph G = V,E ) with vertices. Consier system ynamics 1) with control law u k = κρ ω k Ω ) + Ω κ r k c,e iθ k + K W ) 1. 4) For K > an κ >, all the agents converge to a circular formation in which they travel aroun a common circle of raius ρ = Ω 1 > an center c in the anticlockwise irection with their phase angles in balance state. Proof: Consier a composite potential function U 1 r,θ,ω) = κsr,θ)+ρ K λ max W 1 θ) ) +ρ 3 41) Using 1), 19) an 39), the time erivative of the potential function U 1 r,θ,ω) along the trajectories of 1), is U 1 r,θ,ω) = κ + K r k c,e iθ k 1 ρ ω k ) ) W1 ρ ω k ) + ρ 3 Using 4), 4) can be rewritten as U 1 r,θ,ω) = κ r k c,e iθ k 1 ρ ω k ) + ρ 3 ω k Ω )u k 4) + K W 1 Gω); K > It,κ is > easy. to check that 49) is satisfie only if ) ω k Ω )u k 43) Uner control law 4), the time erivative of U 1 r,θ,ω) results in U 1 r,θ,ω) = κρ 4 ω k Ω ). 44) Since r,θ,ω) R T R is compact, it follows from the LaSalle s invariance theorem [3] that, for K > an κ >, Y axis e iθk r k ρ c Ω ie iθk k-th agent X axis Fig. 4. Orientation of the k th agent on a circle of esire raius ρ = Ω 1 > an center c. all the solutions of 1) with the control law 4) converge to the largest invariant set containe in { U 1 r,θ,ω) = }, that is, the set Γ = {r,θ,ω) ω k = Ω, k}. 4) In Γ, ω k = Ω, k, which implies that each agent rotates with angular spee Ω. Moreover, since u k = ω k =, k, in the set Γ, it implies from 4) that κ r k c,e iθ k + K W 1 =, 46) for all k. Let Γ b be the largest invariant set in Γ. Taking the time-erivative of 46) in the set Γ yiels κ r k c,e iθ k + K ) W1 =. 47) t t Using 7), 47) becomes r k c,e iθ k = t r k c,iω e iθ k Since e iθ k,e iθ k = 1, 48) can be written as r k c,iω e iθ k + e iθ k,e iθ k =. 48) = 1. 49) r k = c iρ e iθ k, k, ) which is the position of the k th agent rotating aroun a circle of raius ρ an center at c see Eq. 3)). This implies that, in the set Γ b, all the agents converge to a common circle of raius ρ = Ω 1 > an center c. Moreover, by substituting ) in 46), we get W 1 / ) =, k, which efines the critical points of W 1 θ) belonging to the set Γ b. Since maximization of the potential W 1 θ) correspons to the global minimum of U 1 r,θ,ω), balance formation is asymptotically stable Lemma ). This completes the proof. Theorem 4: Let L be the Laplacian of an unirecte an connecte circulant graph G = V,E ) with vertices. Consier system ynamics 1) with control law 4). For K < an κ >, all the agents converge to a circular formation in which they travel aroun a common circle of raius ρ = Ω 1 > an center c in the anticlockwise irection with their phase angles in synchronize state. Proof: Consier a composite potential function U r,θ,ω) = κsr,θ) ρ KW L θ) + ρ 3 Gω); K <,κ >. 1) Uner the control 4), the time erivative of U r,θ,ω) along
8 1 1 1 1 1 1 3 a) 1 1 1 1 1 1 3 b) Angular frequencies ra/sec).8.6.4...4.6.8 1 3 c) Fig.. Synchronization an balancing of = 6 agents, connecte by a graph as shown in Fig.1b), aroun the common circle of esire raius ρ = Ω 1 = m an esire center c =,). a) Balance formation uner the control law 4) with K =. an κ =.1. b) Synchronize formation uner the control law 4) with K = 1 an κ =.1. c) Consensus of angular velocities at esire value Ω =. ra/sec for balance formation. the trajectories of 1), yiels U r,θ,ω) = U 1 r,θ,ω) = κρ 4 ω k Ω ). ) Since U r,θ,ω) = U 1 r,θ,ω), the proof follows the same steps as use to prove Theorem 3. However, in this case, let Γ s be the largest invariant set in Γ efine in 4). Thus, it can be conclue that all the agents converge to a common circle of raius ρ = Ω 1 > an center c in the set Γ s. Moreover, since minimization of the potential W 1 θ) correspons to the global minimum of U r,θ,ω), synchronize formation is asymptotically stable Lemma ) in the set Γ s. This completes the proof. Example : In this example, the simulation results are presente for the same 6 agents as consiere in Example 1. Fig. epicts the synchronization an balancing of the agents aroun a common circle at esire angular spee Ω =. ra/sec an hence esire raius ρ = m) an esire center c =,). Balance formation is shown in Fig. a), an synchronize formation is shown in Fig. b). In Fig. c), the convergence of the angular spees of the agents to a esire value Ω =. ra/sec, is shown in balance formation only since the plot for synchronize formation is similar. ote that these figures are obtaine by setting ifferent values of gains K an κ. The selection of these gains is arbitrary an epens upon a particular problem of interest. By selecting gains K an κ appropriately, we can make the system of agents to stabilize in a esire formation, at faster or slower convergence rates. It is worth noting that, in Fig. a), the final position of the centroi of the group coincies with the center c =,) of the common circle. This is ue to the fact that, in balance formation, the linear momentum p θ = p 1θ =, which causes 3) to reuce to c = 1/) r k when summe over all k on both the sies, which is the average position of all the agents, or the position of their centroi. V. SYMMETRIC BALACED PATTERS AROUD THE COMMO CIRCLE In this section, we aim to achieve symmetric balance patterns of the agents aroun a common circle of esire raius ρ = ω 1 > as well as center c. These patterns are efine by the formations in which the agents are in phase balancing along with a symmetrical arrangement of their phase angles aroun the esire common circle []. Let 1 M be a ivisor of. A symmetric arrangement of phases consisting of M clusters uniformly space aroun the common circle, each with /M synchronize phases, is calle an M, )-pattern. For instance, the 1, )-pattern correspons to the synchronize state an the, )-pattern correspons to the so-calle splay state, which is characterize by phases uniformly space aroun the common circle. We now state the following lemma [8] that is useful in proving the results of this section. Lemma 3: Let 1 M be a ivisor of. An arrangement θ of phases is an M,)-pattern if an only if, for all m {1,...,M 1}, the phase arrangement mθ is balance an the phase arrangement Mθ is synchronize. As mentione in the above lemmas, the m th harmonic of the potential W m θ) efine in 1) thus plays an important role in stabilizing symmetric phase patterns. Moreover, for various values of m, ifferent symmetric balance patterns arise as shown in Fig for = 6. See [] for a more etaile escription. Following Lemma 3, we choose an M,) phase potential as W M, M 1 ) K m θ) = m=1 m λ max W m θ) K M M W Mθ), 3) with K m > for 1 m M 1, an K M <. The global minimum of W M, θ) achieve only) when W m θ) is maximize for 1 m M 1, an W M θ) is minimize. This correspons to the situation when θ,θ,...,m 1)θ are balance an Mθ is synchronize Lemma ), an hence the minimization of 3) gives rise to an M,)-pattern Lemma 3). Substituting for W m θ) from 1) into 3), yiels W M, θ) = 1 M 1 K ) m m=1 m λ max e imθ imθ,l e K M M e imθ,l e imθ, 4) whose time erivative along the ynamics 1), is given by Ẇ M, θ) = M m=1 K m ie imθ k,l k e imθ ω k, )
9 1 1 1 1 1 1 1 1 1 1 1 1 3 a) 1 1 1 3 b) 1 1 1 3 c) Fig. 6. Different symmetric balance patterns of 6 agents, connecte by a graph as shown in Fig.1b), aroun the common circle of esire raius ρ = Ω 1 = m an esire center c =,), uner the control law 6) with K = 1,κ = K m =.1 for 1 m M 1, an K M =.. a),6) pattern. b) 3, 6) pattern. c) 6, 6) pattern: splay formation. Block 1 Block G1 G G = G1 G G3 Block 3 Fig. 7. Block interaction of 1 agents with 3 blocks, each containing 4 agents. where, L k is the k th row of the Laplacian L. Theorem : Let L be the Laplacian of an unirecte an connecte circulant graph G = V,E ) with vertices. Consier system ynamics 1) with control law u k = κρ ω k Ω ) + Ω κ r k c,e iθ k + K M m=1 G3 K m W m ) 6) with K m > for 1 m M 1, an K M <. For K > an κ >, all the agents converge to a circular formation in which they travel aroun a common circle of raius ρ = Ω 1 > an center c in the anticlockwise irection with their phase angles in the M, )-pattern. Proof: Consier a composite potential function V r,θ,ω) = κsr,θ) + ρ KW M, θ) + ρ 3 Gω) 7) with K m > for 1 m M 1, an K M <. Using 19), 39) an ), the time erivative of the potential function V r,θ,ω) along the ynamics 1), is V r,θ,ω) = κ + K M m=1 r k c,e iθ k 1 ρ ω k ) K m ie imθ k,l k e imθ ρ ω k ) + ρ 3 ote that ie imθ k,l k e imθ = ω k Ω )u k. 8) j k sinmθ j θ k )) = 9) Using 9), 8) can be rewritten as V r,θ,ω) = κ r k c,e iθ k 1 ρ ω k ) + ρ 3 + K M m=1 ) K m ie imθ k imθ,l k e ω k Ω )u k. 6) Uner the control law 6), the time erivative of V r,θ,ω) results in V r,θ,ω) = U 1 r,θ,ω) = κρ 4 ω k Ω ). 61) Since V r,θ,ω) = U 1 r,θ,ω), the proof follows the same steps as use to prove Theorem 3. However, in this case, let Ξ be the largest invariant set in Γ efine in 4. Thus, it can be conclue that all the agents converge to a common circle of raius ρ = Ω > an center c in the set Ξ. Moreover, it follows from Lemma that the maximization of W m θ) for 1 m M 1, an minimization of W M θ) correspons to the global minimum of the potential V r,θ,ω). Thus, the phase arrangements θ,θ,...,m 1)θ are balance an Mθ is synchronize in the set Ξ, an hence give rise to an M,) phase pattern Lemma 3). This completes the proof. Example 3: In this example, the simulation results are presente for the 6 agents consiere in Example 1. Fig. 6 shows the ifferent symmetric balance patterns of the agents aroun the common circle of esire raius ρ = Ω 1 = m an center c =,). The arrangement in Fig. 6c) is the splay state, in which the 6 agents are at equal angular separation of 6, as expecte. Remark 3: We may assume that agents move on ifferent altitues as in [8] when they are in synchronization aroun a common circular orbit. Approaches to collision avoiance among agents will be explore in future. VI. ACHIEVIG COORDIATED SUBGROUPS Motivate by mobile sensor network applications, as iscusse in [6], in this section, we propose control law that uses multiple graphs at a time to yiel multi-level sensing patterns in a group of agents. By using such a multi-level interaction among agents, a formation in which the agents are arrange in subgroups aroun ifferent circular orbits with a symmetric pattern of their phase angles, can be obtaine. In this context, the ifferent graphs, by consiering both intra-
1 an inter subgroup coorination, are use in the steering control law, so that subgroups can be forme an regulate. A particular application of such formations can be foun in the optimal mobile sensor coverage problem where it is require for the sensors to move aroun close curves with coorinate phasing for maximizing information in ata collecte in a spatially an temporally varying fiel for instance, from an ocean) [31]. In orer to use multi-level sensing networks, we ivie the group of agents into B blocks subgroups) an refer to each block by its block inex 1 b B. For the sake of convenience, we assume that there is no interaction between subgroups unless specifically mentione. Let the graph G b escribe the interaction between all the agents in block b. Collectively, the set of all interaction is efine by the graph G B b=1 G b. In this situation, the Laplacian Lˆ corresponing to graph G is a block-iagonal matrix, each block of which represents the Laplacian of the graph G b. For instance, consier a group of 1 agents ivie into 3 blocks containing 4 agents in each block. Let their interaction network be represente by a graph G = G 1 G G3 as shown in Fig. 7. The Laplacian matrix corresponing to this block interaction is given by L ˆ = L G1 4 4 4 4 4 4 L G 4 4 4 4 4 4 L G3 6) where, 4 4 represents a 4 4 zero matrix, an L G1 = L G3 = circ, 1,, 1) an L G = circ3, 1, 1, 1) are the Laplacian of the subgraphs G 1 = G 3 ) an G, respectively. Since Lˆ is a block iagonal matrix, its eigenvalues are the union of the eigenvalues of all block iagonal matrices [3]. In this illustration, although we assume that each subgroup is of the same size, however, this is not require. We further assume that each agent is assigne to one an only one block, so that B b=1 b =, where b { is the number } of agents in the b th block. Also, let F b = f1 b,..., f b b be the set of agent inices, an ρ b = { Ω b 1 } > an c b, respectively, be the esire raius, an center of the common circular orbit associate with the b th block. Base on these notations, we now consier the following two cases of coorinate subgroups with respect to the esire raius ρ b for a reason which will be apparent little later. A. Case 1: ρ b is ifferent for all 1 b B In this situation, we have the following corollaries to Theorems 3, 4 an. Corollary 1: For k F b, consier system ynamics 1) with control law } u k = κρ b ω k Ω b {Ω )+ b κ r k c b,eiθ k + K ie iθ k, Lˆ ) k e iθ, 63) where, Lˆ k is the k th row of the Laplacian Lˆ of a graph G = Bb=1 G b with G b being an unirecte an connecte circulant subgraph. For K > an κ >, all the agents belonging to the b th block converge to a circular formation in which they move on a circle of raius ρ b = { Ω b 1 } > an center c b in the anticlockwise irection with their phase angles in the balance state. Proof: Consier a composite potential function ) U1 b r,θ,ω) =κsb r,θ) + ρ b K λ max Ŵ 1 θ) { } 3 + ρ b G b ω); κ >,K >, 64) where, S b r,θ) = 1 r k c b + iρb eiθ k,r k c b + iρb eiθ k, Ŵ 1 θ) = 1/) e iθ, Lˆ e iθ, G b ω) = 1 ω k Ω) b, an λ max is the maximum eigenvalue of L ˆ. Since 64) has a structure similar to 41), the proof is similar to the proof of Theorem 3 an hence omitte. Similarly the corollaries to Theorems 4 an can be state, an are not escribe here to avoi repetition. B. Case : ρ b = ρ for all 1 b B In such a situation, by assuming that the agents within a block can also interact with the agents in other blocks in aition to their inter block interaction, it is possible to use multilevel sensing network in a single control algorithm since the phase potentials corresponing to the ifferent communication topologies can be combine to get a single phase potential function. For instance, consier a group of 1 agents as shown in Fig. 7, now with two types of sensing networks comprising an intra- an an inter subgroup coorination. Assume that the intra subgroup interaction is all-to-all an the inter subgroup interaction is as shown Fig. 7. Let L = P= I 11 T ), an L ˆ, given by 6), be the Laplacian corresponing to the intraan inter subgroup coorination, respectively. Then, with such a multi-level interaction, the potential function in 64) becomes ) U b 1r,θ,ω) = κs b r,θ)+ρ K λ max W 1 θ) +ρ 3 Gω), where, 6) W 1 θ) = 1 e iθ, L ˆ + L )e iθ. 66) The potential U b 1r,θ,ω) is minimize by choosing the control u k = κρ ω k Ω ) + {Ω } [ κ r k c b,eiθ k + K ie iθ k, ) Lˆ k + L )] k e iθ, 67) which accounts for both intra- an inter subgroups coorination. ote that, unlike previous case of the ifferent raius of the circular orbits, here we can combine all the phase stabilizing potentials inepenently to get a single potential function since ρ b = ρ for all b = 1,...,B. Thus, uner the control 67), the phase arrangement in which the entire group, as well as each block, are in balance formation, is obtaine. The same behavior of agent s phase arrangement in synchronize an splay formations can be epicte uner suitable control laws. Remark 4: It is evient from the above iscussion that the secon orer rotational moel is aequate to control the phase
11 1 1 1 1 1 1 1 1 1 3 1 a) 1 1 3 3 1 b) 1 3 c) Fig. 8. Synchronization of 1 agents in 3 ifferent groups, with 4 agents in each group, uner the control laws 63) an 67) with K = 1 an κ =.. a) Different esire raius an the same esire center when the agents interact only within subgroups. b) Different esire raius an centers when the agents interact only within subgroups. c) Same esire raius an ifferent esire centers when the agents interact not only within subgroups but also among subgroups. 1 1 1 1 1 1 1 1 1 1 a) 1 1 3 1 1 b) 1 3 1 3 1 1 3 c) Fig. 9. Splay formation of 1 agents in 3 ifferent groups, with 4 agents in each group. a) Different esire raius an the same esire center when the agents interact only within subgroups. b) Different esire raius an centers when the agents interact only within subgroups. c) Same esire raius an centers when the agents interact not only within subgroups but also among subgroups. arrangement as well as the raius of the esire circular orbit. Thus, unlike [6], the class of collective circular motion stuie in this paper shows an interesting possibilities for the unmanne vehicles to expan an contract their formations about a esire location so as to better explore the search area, an thus more esirable for the applications to mobile sensor networks. Example : In this example, simulation results are presente for the above consiere = 1 agents whose initial positions, initial heaing angles an initial angular velocities are ranomly generate. At first, we assume that the agents interact only in blocks accoring to Fig. 7. In such a situation, synchronize an splay formations of the agents in three groups are shown in Figs. 8, an 9, respectively. The synchronize an splay formations of the agents, on the circles of ifferent esire raius an the same esire center, are shown in Figs. 8a) an 9a), respectively, while the same, on the circles of ifferent esire raius as well as centers is shown in Figs. 8b) an 9b), respectively. Since this case correspons to ifferent ρ, the agents are in synchronize an in splay formations within subgroups only. ext, we assume that the agents interact within subgroups accoring to Fig. 7 as well as among subgroups accoring to an all-to-all communication topology as iscusse above. In this situation, the synchronize an splay formations of the agents, on the circles of the same esire raius an ifferent esire centers, are shown in Figs. 8c) an 9c), respectively. Since the raius of all the circles corresponing to each group is same, the agents are collectively in synchronize an splay formations. In other wors, if the heaing phasors of all the agents were plotte on the same circle, then the resulting pattern woul be synchronize or in splay state. VII. C OCLUSIOS A Laplacian-base control esign methoology to stabilize synchronize an balance collective circular motions of a group of agents with secon-orer rotational ynamics uner limite communication topology, which is represente by a time-invariant unirecte graph, has been propose in this paper. In particular, the collective motion of agents aroun ifferent circles or aroun a common circle at a esire angular velocity have been investigate. The secon-orer feeback control laws have been erive from composite potential functions, which reach their minimum in the esire configuration of the agents. It has been shown that the secon-orer feeback controls are aequate to regulate the orientations as well as the angular velocities of the agents. The LaSalle s invariance principle has been use extensively to prove the asymptotic stability of the esire circular formation uner the propose control scheme. Moreover, the use of multi-level interaction networks to obtain various symmetric circular formations suitable for applications to mobile sensors has been iscusse in the context of better exploration of a search area. A piece of work in future to pursue time-varying communication topology an the collision avoiance. R EFERECES [1] Ren W, Bear RW. Distributive Consensus in Multi-Vehicle Cooperative Control: Theory an Applications, Springer-Verlag, 8. [] Mesbahi M, Egerstet M. Graph Theoritical Methos in Mutiagent etworks, Princeton Univ. Press, 1.
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