Collective motions of heterogeneous swarms
Klementyna Szwaykowska, Luis Mier-y-Teran Romero, Ira B. Schwartz
11 Collective motions of heterogeneous swarms
Klementyna Szwaykowska, Luis Mier-y-Teran Romero, and Ira B. SchwartzU.S. Naval Research LaboratoryCode 6792Plasma Physics DivisionNonlinear Dynamical Systems SectionWashington, DC [email protected], [email protected], [email protected]
Abstract — The emerging collective motions of swarms ofinteracting agents are a subject of great interest in applicationareas ranging from biology to physics and robotics. In thispaper, we conduct a careful analysis of the collective dynamicsof a swarm of self-propelled heterogeneous, delay-coupledagents. We show the emergence of collective motion patternsand segregation of populations of agents with different dynamicproperties; both of these behaviors (pattern formation andsegregation) emerge naturally in our model, which is basedon self-propulsion and attractive pairwise interactions betweenagents. We derive the bifurcation structure for emergence ofdifferent swarming behaviors in the mean field as a function ofphysical parameters and verify these results through simulation. N OTE TO P RACTITIONERS
Our research deals with understanding the emerging be-haviors of groups of simple, interacting agents. The mo-tivation for studying this subject is twofold: first, under-standing the mechanisms that govern collective motions ofbiological organisms in processes like wound healing, cancergrowth, flocking and herding, etc. Second, the applicationof our insights to synthesis of controllers for swarms ofautonomous robotic agents to perform surveillance or mon-itoring in uncertain environments. Swarming behavior istypically modeled for groups of identical agents, under theassumption that sensing and processing times are negligiblysmall. We incorporate the real-world complications of (1)finite sensing/processing time, which appears as a delayin our model of agent motion, and (2) differences in thedynamical capabilities of swarming agents. We conduct atheoretical analysis of the collective motions of the swarm.We show the emergence of large-scale patterns in the swarmmotion as a function of the physical parameters or the swarm,as well as segregation of the agents into separate groupswhere all agents in a given group have identical dynamics.I. I
NTRODUCTION
The dynamics of aggregates, or swarms, of interactingmobile agents form an active area of study for biological,physical, and synthetic systems. Simple rules of interactionbetween agents can lead to a wide range of complex ag-gregate behaviors, even in the absence of leader agents andglobal motion strategy [1]. The emergence of rich collective behaviors from simple interactions is, in fact, a wide-spreadphenomenon in many application domains. In biology, theformation of aggregates is common on a wide range ofspatio-temporal scales, for organisms ranging from bacteriato fish to birds and humans [2]–[6]. In robotics, aggregates oflocally interacting agents have been proposed as a means tocreate scalable sensor arrays for surveillance and exploration[7], [8]; and for formation of reconfigurable modal systems,in which a group of simple agents can be used to accomplisha task that would be impossible for any agent individually,as in [9]–[11].Understanding the dynamical characteristics of swarm be-havior is essential for algorithm design and implementation.There is a wide range of existing works which model thedynamics of swarms on the level of individual agents [4]–[6],[12], as well as using continuum models [3], [13], [14]. It hasbeen shown that, under the right conditions, swarms convergeto organized steady-state behaviors; and furthermore, thatenvironmental noise and/or processing delay acting on agentdynamics can lead to formation of new steady-state motions,or phase transitions between between co-existing steadystates [1], [15], [16]. Noise is used to model effects of envi-ronmental disturbance and unknown interaction dynamics inrobotic systems. Delays are important in biological modelingof population dynamics, blood cell production, and geneticnetworks [17]–[19], etc.; and in mathematical models ofrobot networks where communication and processing delaysmust be taken into account [20].Most existing works assume that the swarm is madeup of agents with identical dynamics. However, real-worldswarms often include agents with varying dynamical prop-erties, which leads to new collective behaviors. In biologicalsystems, heterogeneity arises quite naturally when, for ex-ample, motion or sensing capabilities in an age-structuredswarm vary significantly with age. A more striking exampleis that of of predator-prey interactions between a herd ofprey animals and an individual or small group of predators,where there are distinct time-scale differences in the motionof predator and prey animals [6]. Another systems whereheterogeneity plays a significant role is the segregation ofintermingled cell types, as during growth and developmentof an organism. It has been shown that segregation can be a r X i v : . [ n li n . AO ] S e p achieved simply by introducing heterogeneity in intercelladhesion properties [21], [22], or by increasing the intercellattraction between self-propelled cells of a single type [23].An approach based on the cell segregation model in [23]is used in [24] to design a potential-based controller thatachieves segregation in swarms of self-propelled autonomousrobots. Heterogeneity also appears in robotic systems whenindividual robots with disparate capabilities are used togetherto achieve a common goal, as in [11]. Certain robots in theswarm may lack capabilities that are costly to implement.Stranieri et al [25], for example, show that flocking behaviorcan be achieved when a fraction of the agents lack the abilityto align their velocities with those of their neighbors. Addi-tionally, heterogeneity may arise over time as some agentsin the swarm malfunction. For example, [26] introducesan observer to judge the overall “health” of a swarm, asindividual agents lose speed from energy dissipation.In this paper we carry out a systematic analysis of the mo-tion of a swarm composed of heterogeneous agents, using themethodology outlined in [1]. We extend the model in [1] andanalyze the dynamical behaviors of a heterogeneous swarmof delay-coupled agents, where the swarm is divided intotwo distinct populations with different motion capabilities.The inspiration for our model comes from the idea of usingswarms of autonomous mobile agents as sensor platformsto survey/monitor an area of interest. Such agents may havedifferent dynamical properties if, for example, some agentsbut not others are equipped with a particular sensor packagethat interferes with their motion. The package may be tooexpensive or otherwise impractical to mount on all swarmagents. Overall, allowing for heterogeneity in dynamicalbehaviors of swarm agents gives greater flexibility in systemdesign, and is therefore desirable not only from a theoreticalbut also from a practical point of view.The research presented here gives a general approach ofmodeling and analysis that can be used to understand theeffects of individual agent dynamics on the collective motionof swarms. We know that swarms of self-propelled delay-coupled agents exhibit self-ordering and pattern formation,and that the collective patterns formed depend on the modelparameters [1], [15]; furthermore, we observe in simulationthat heterogeneity in the swarm composition leads to segrega-tion of the individual swarm populations. We will show howcollective motion patterns (translation, ring formation, androtation about a common center of mass) and segregationof individual populations emerge in a basic but generalswarming model.II. P ROBLEM S TATEMENT
Consider a swarm of delay-coupled self-propelled agents,or robots, comprised of two distinct populations (1 and 2),following a single motion strategy, but with heterogeneousdynamics. The agents in Population 2 are less “maneuver-able” in the sense that they are not able to accelerate asrapidly as those in Population 1. This setup models co-deployment of small, fast agents, and larger, slower agentsin a given area. Let κ and κ be the inverse mass of agents in Populations 1 and 2, respectively. We scale units so that κ = 1 and κ = κ ∈ (0 , .Let r ki ∈ R denote the position of the i th agent inPopulation k ( k = 1 , ); let N and N denote the numberof agents in Populations 1 and 2, respectively; and let N = N + N be the total number of agents in the swarm. Theagents have self-propulsion and are globally attracted to eachother in a symmetric fashion, with coupling coefficient a ,however, there is a delay τ in sensing of agent positions. Fornotational convenience, we introduce the following notation:let κ = 1 and κ = κ . The motion of the agents is governedby the following set of delay differential equations (dotsdenote differentiation with respect to time): ¨ r i = κ (cid:0) − (cid:13)(cid:13) ˙ r i (cid:13)(cid:13)(cid:1) ˙ r i (1a) − aκ N (cid:32) N (cid:88) j (cid:54) = i,j =1 ( r i ( t ) − r j ( t − τ ))+ N (cid:88) j =1 ( r i ( t ) − r j ( t − τ )) (cid:33) ¨ r i = κ (cid:0) − (cid:13)(cid:13) ˙ r i (cid:13)(cid:13)(cid:1) ˙ r i (1b) − aκ N (cid:32) N (cid:88) j =1 ( r i ( t ) − r j ( t − τ ))+ N (cid:88) j (cid:54) = i,j =1 ( r i ( t ) − r j ( t − τ )) (cid:33) . The first term in the above equations represents the self-propulsion of swarm agents, while the second models pair-wise attraction between all agents in the swarm. This sim-plified model does not include short-range repulsion or othercollision-avoidance strategies; however, earlier studies withhomogeneous swarms indicate that the collective dynamicsof the swarm are not significantly altered by the introductionof short-range repulsion terms.The goal is now to characterize the steady-state motionsof this system. Following the approach in [1], we begin byconsidering the dynamics in the limit where the number ofagents goes to infinity.III. M
EAN -F IELD APPROXIMATION
Since basic collective swarm motions. as observed in sim-ulation, consist of translation and rotation, the steady-statemotions of the centers of mass of the individual populationsare a means to characterize the motion of the overall group.Let R and R ∈ R denote the position of the centers ofmass of Populations 1 and 2, respectively: R k ( t ) = 1 N k N k (cid:88) i =1 r ki ( t ) , k = 1 , . (2)As in [1], we assume that the deviations of the robots fromthe centers of mass of their respective populations are small.We analyze the steady-state motions of the swarm in thelimit as N k → ∞ for k = 1 , . The positions of the agents in each population can bewritten relative to the respective center of mass as r ki ( t ) = R k ( t ) + δr ki ( t ) . (3)Note that N (cid:88) i =1 δr i ( t ) = N (cid:88) i =1 δr i ( t ) = 0 . (4)For convenience, we introduce the notation ¯ k = (cid:40) for k = 11 for k = 2 . (5)Substituting (3) into (1a) and simplifying the resulting ex-pression using (4) allows us to write the equations of motionin terms of R k and δr k : ¨ R k + δ ¨ r ki = κ k (cid:18) − (cid:13)(cid:13)(cid:13) ˙ R k + δ ˙ r ki (cid:13)(cid:13)(cid:13) (cid:19) ( ˙ R k + δ ˙ r ki ) − aκ k N (cid:0) ( N k − R k ( t ) − R k ( t − τ )+ δr ki ( t )) + δr ki ( t − τ )+ N ¯ k ( R k ( t ) − R ¯ k ( t − τ ) + δr ki ( t )) (cid:1) . (6)Summing the equations for δr ki over i = 1 , . . . , N k , anddividing through by N k , we get the equation of motion forthe centers of mass of Population k : ¨ R k = κ k (cid:18) − (cid:13)(cid:13)(cid:13) ˙ R k ( t ) (cid:13)(cid:13)(cid:13) (cid:19) ˙ R k ( t ) − κ k N k N k (cid:88) i =1 (cid:32) (cid:13)(cid:13) δ ˙ r ki (cid:13)(cid:13) ˙ R k ( t )+ (cid:104)(cid:13)(cid:13) δ ˙ r ki ( t ) (cid:13)(cid:13) + 2 (cid:104) ˙ R k ( t ) , δr ki ( t ) (cid:105) (cid:105) δr ki ( t ) (cid:33) − aκ k N (cid:16) ( N − R k ( t ) − ( N k − R k ( t − τ ) − N ¯ k R ¯ k ( t − τ ) (cid:17) , (7)where (cid:104)· , ·(cid:105) denotes the dot product in R .We now take the limit N → ∞ , while keeping the fractionof agents in Population 1, c = N /N, (8)constant. Under the assumption of small deviations of theagents from the centers of mass of the respective popula-tions, terms in δr ki can be neglected. We get the followingequations for the motion of the center of mass of Population k ( k = 1 , ): ¨ R k = κ k (cid:18) − (cid:13)(cid:13)(cid:13) ˙ R k ( t ) (cid:13)(cid:13)(cid:13) (cid:19) ˙ R k ( t ) − aκ k (cid:16) R k ( t ) − cR ( t − τ ) − (1 − c ) R ( t − τ ) (cid:17) . (9)Let [ X k , Y k ] = R k and [ U k , V k ] = ˙ R k denote, respectively,the position and velocity of the center of mass of population k = 1 , . Let superscript τ denote a delay τ , so that X τ ( t ) = X ( t − τ ) . The equations of motion can be written in termsof X k , Y k , U k , and V k as ˙ X k = U k (10a) ˙ Y k = V k (10b) ˙ U k = κ k (1 − U k − V k ) U k − aκ k ( X k − cX τ − (1 − c ) X τ ) (10c) ˙ V k = κ k (1 − U k − V k ) V k − aκ k ( Y k − cY τ − (1 − c ) Y τ ) . (10d)The system in (10) has an invariant stationary solution givenby X = X = X , Y = Y = Y U = U = 0 , V = V = 0 , (11)as well as a translating solution where the center of masstravels in a straight line at constant velocity. A. Bifurcation of the stationary solution
About the stationary solution, the system exhibits a num-ber of Hopf bifurcations for different values of the parameters a , c , κ , and τ . To find the locations of these bifurcationpoints, consider the linearization of the dynamics (10) aboutthe stationary solution (without loss of generality, we choose X = Y = 0 ). The linearized dynamics are ˙ X k = U k (12a) ˙ Y k = V k (12b) ˙ U k = κ k U k − aκ k (cid:16) X k − cX τ − (1 − c ) X τ (cid:17) (12c) ˙ V k = κ k V k − aκ k (cid:16) Y k − cY τ − (1 − c ) Y τ (cid:17) . (12d)Let ξ = [ X , Y , U , V , X , Y , U , V ] T . The abovesystem takes the form ˙ ξ = L ξ , where L is a linear operator.Let ν denote an eigenvector of L ; then a solution starting at ν can be expressed as e λt ν . This equation can only be satisfiedif the matrix M ( λ ; a, c, κ, τ ) is singular, where M = λI −L .That is, λ must satisfy M = D , where D ( λ ; a, c, κ, τ ) = ( λ − λ + a )( λ − κλ + aκ ) − (( κ + c − κc ) λ − κλ + aκ ) ae − λτ . (13)Hopf bifurcations of the mean-field equations occur when Re( λ ) = 0 . Setting λ = iω gives D ( iω ; a, c, κ, τ ) = 0 which allows us to solve for parameter values of where Hopfbifurcations occur. Solutions in terms of a and τ , for differentvalues of c and κ , are shown by the solid blue lines in Fig.1. Below the first Hopf bifurcation curve, the mean-fieldpredicts a stationary state which corresponds to a ring statein the full swarm dynamics. This is similar to the ringstate described in [1], where swarm agents circle about astationary center of mass in either direction, with constantradius and speed. The first Hopf bifurcation in the mean-field approximation gives rise to a rotating state analogousto the one in [1], in which the centers of mass of the swarm τ (a) κ = 0 . and c = 0 . τ (b) κ = 0 . and c = 0 . τ (c) κ = 0 . and c = 0 . τ (d) κ = 0 . and c = 0 . τ (e) κ = 0 . and c = 0 . τ (f) κ = 0 . and c = 0 . τ (g) κ = 0 . and c = 0 . τ (h) κ = 0 . and c = 0 . τ (i) κ = 0 . and c = 0 . Fig. 1:
The solid blue lines show τ vs a Hopf bifurcation curves for the center-of-mass heterogeneous swarm dynamics, fordifferent values of the parameters c and κ . The location of the pitchfork bifurcation where the translating state disappears isshown by the dashed red curve. The point where the Hopf curve intersects the pitchfork bifurcation curve is the Bogdanov-Takens point. The “ ∆ ” and “+” in (b) show the points in parameter space corresponding to the simulations in Fig. 2 andFig. 4, respectively.populations rotate about a common stationary point. Higher-order Hopf bifurcations lead to formation of rotating stateswith higher angular frequency, but these states appear tobe unstable, based on our simulations with homogeneousswarms. The introduction of heterogeneity leads to a sepa-ration between the agents in the two populations in both ofthese steady state motions. B. Ring State
The ring state in the heterogeneous swarm is similar tothat described in [1] for homogeneous agent swarms; thatis, agents move in either direction about a stationary centerof mass, with constant speed and radius. The heterogeneityintroduces a split in the rings formed by the agents of thetwo populations, however, so that they become separated (seeFig. 2).It can be shown that the angular frequency ω i and radius ρ i of the particles in population i = 1 , satisfy ρ = 1 / √ a ω = √ a (14) ρ = 1 / √ aκ ω = √ aκ (15)(see Appendix for details). Note that the radius for eachpopulation depends only on the strength of the couplingconstant and the acceleration factor; that is, the radii of thetwo populations are not coupled and are independent of thetime delay τ .The above calculations were verified using a full-swarmsimulation with 300 agents, and different values of theparameters a , κ , c , and τ . The results of comparing the ringradii and angular velocities obtained from simulation andtheory are shown in Fig. 3. C. Rotating State
The rotating state, like the ring state, is also present inthe case of a homogeneous swarm [1]. In the rotating state, −0.5 0 0.5 1 1.5−0.500.511.5 x y (a) t = 10 . −0.5 0 0.5 1 1.5−0.500.511.5 x y (b) t = 31 . −0.5 0 0.5 1 1.5−0.500.511.5 x y (c) t = 315 Fig. 2:
Simulated swarm of N = 300 agents, with agents in Population 1 (blue) and in Population 2 (red),converging to the ring state. In this simulation, κ = 0 . , c = 0 . , a = 1 . , and τ = 2 . . This point in parameterspace is marked by a “ ∆ ” in Fig. 1b.the swarm populations collapse to their respective centers ofmass and rotate about a common center point with constantphase offset (see Fig. 4).Our numerical simulations of the full swarm dynamicssuggest that the radii of the rotating populations are equal.Let ρ denote the radius of the rotating state, ω the angularfrequency, and let ∆ θ = θ − θ denote the phase offset. Itcan be shown (see Appendix for details) that these quantitiesmust satisfy the following relations: sin ∆ θ = (2 c − P ( c, κ, ω ) sin ωτ (16) ω = aκ (cid:104) − cos ωτ (17) + c ((2 c −
1) sin ωτ + cos ωτ ) P ( c, κ, ω ) (cid:105) (18) ρ = (cid:113) − a (1 − c (1 − c ) P ( c, κ, ω ) cos ωτ ) sin ωτω | ω | . (19) R ad i u s o f R i ng κ =0.3 κ =0.6 κ =0.9 Population 1 Population 2 A ngu l a r V e l o c i t y A bou t R i ng κ =0.3 κ =0.6 κ =0.9Population 1 Population 2 Fig. 3:
Comparison of theoretical and simulated radiusand angular velocity in the ring state. Theoretical valuesare shown by the solid lines, while values obtained fromsimulations are shown by the red crosses. The simulationswere run for a swarm of N = 300 agents, with fraction inPopulation 1 c = 0 . and time delay τ = 1 . . −1 0 1 2−1.5−1−0.500.511.5 x y Fig. 4:
Swarm in the rotating state at t = 323 . . Agents inPopulation 1 are shown in blue, while those in Population 2are in red. The dotted circle shows the trajectory of the twoswarm populations about a common stationary point. Thesimulation was run with N = 300 agents, with N = 60 and N = 240 . The parameter values are: κ = 0 . , c = 0 . , a = 1 . , and τ = 5 . . This point in parameter space ismarked by a “+” in Fig. 1b.where P ( c, κ ) = (1 − κ )(1 − cos ωτ )(1 + k ) c − − κ ) c (1 − c ) sin ωτ . (20)The above relations may be used to derive theoretical valuesfor the radius, angular velocity, and phase offset betweenPopulations 1 and 2. A comparison of the theoretical valuesand those observed in full-swarm simulations is shown inFig. 5, for different values of the parameters κ , c , a , and τ . Note that the above relations, derived from the mean-fieldapproximation, give a good approximation to values obtainedfrom the full swarm simulation; however, in some cases, the use of the mean-field approximation leads to significant errorin computed values. −2 −1.5 −1 −0.5 0 0.5−2−1.5−1−0.500.5 ∆θ (theoretical) ∆ θ ( s i m u l a t i on ) (a) Real vs. theoretical phase dif-ference between the two swarmpopulations. ω (theoretical) ω ( s i m u l a t i on ) (b) Real vs. theoretical angularfrequency for the rotating state. R ad i u s ( s i m u l a t i on ) (c) Real vs. theoretical radius ofthe rotating state.
Fig. 5:
Comparison of theoretical and simulated phasedifference, angular velocity, and radius in the rotating state.Theoretical values are along the x-axis, while values obtainedfrom simulations are along the y-axis. The simulations wererun for a swarm of 300 agents, for κ = 0 . , . , . , c = 0 . , a = 0 . , . , . , . , and τ = 4 . , . . D. Translating state
The system in (10) has a steady-state translating solution,where ˙ U = ˙ U = ˙ V = ˙ V = 0 , U = U = U , V = V = V , and X ( t ) = X ( t ) = X + U t (21a) Y ( t ) = Y ( t ) = Y + V t. (21b) U and V must satisfy: U + V = 1 − aτ, (22)which is possible only if aτ ≤ . In fact, the system (10)has a pitchfork bifurcation along the parameter-space curve c κ Fig. 6:
Contour plot of the coupling coefficient at theBogdanov-Takens point a BT as a function of κ and c .Different values of a BT are shown by the colors (see colorbaron the right). τ = 1 /a (see Fig. 1), where the stationary solution givesrise to the translating state (the other branch of the pitchforkcorresponds to an unphysical solution with imaginary speed).The same bifurcation curve exists in the homogeneous sys-tem ( κ = 1 ) [1].The point where the pitchfork bifurcation coincides withthe first Hopf curve is a Bogdanov-Takens (BT) point (seeFig. 1). In the homogeneous swarm case, this point is locatedat a = 1 / , τ = 2 ; for the heterogeneous swarm the locationof the point depends on the acceleration factor κ and on thefraction c of agents in Population 1. The BT point is at a BT = κ − c (1 − κ )) (23) τ BT = 1 a BT . (24)The value of coupling coefficient at the Bogdanov-Takenspoint a BT as a function of κ and c is shown in Fig. 6.IV. C ONCLUSION
In this paper we have analyzed the collective motions ofa swarm of delay-coupled heterogeneous agents. The swarmmotions are characterized by the emergence of large-scalepatterns (translation, ring formation, and rotation), and theautomatic segregation of populations of agents with differentdynamical properties. Separation of the swarm into distinctpopulations is a direct consequence of swarm heterogeneity,and is not observed under homogeneous swarm dynamics.The patterns observed in simulation were shown to arisein the motions of the swarm center of mass, in the limitas the number of agents in each population goes to infinity.We derive expressions for the speed of the swarm in thetranslating state as a function of time delay and couplingcoefficient; for the radii and angular velocities of both agentpopulations in the ring and rotating states; and for the fixedphase offset between populations in the rotating state. Wehave verified these calculations with simulations of the full-swarm dynamics. In spite of discrepancies, it is remarkablethat our model reduction, which starts with N second-orderdelay-differential equations and yields one equation of thesame type, is able to quantitatively capture so many aspectsof the full swarm dynamics. A real-world model of swarming physical agents mustincorporate a collision-avoidance strategy. This could beimplemented, for example, by adding a short-range repulsionto the agent dynamics. Such interactions may affect thecollective motion of the swarm to some degree, but ourpreliminary simulations of homogeneous swarms indicatesthat the qualitative behavior of the swarm is not affected byshort-range repulsion forces. This will be addressed morecarefully in a future paper.In our model, we have assumed that the motion of eachagent in the swarm depends on the positions of all otheragents. In future work, we will relax this assumption tomodel the effects of non-global coupling on the collectiveswarm motion; we will also add noise to the swarm dynam-ics. We know that adding noise causes switching between co-existing stable states (ring and rotating state) in homogeneousswarms [1]. We will investigate how switching behaviorchanges when the swarm is made up of heterogeneousagents.Our work presents new insights into the collective motionsof aggregates of heterogenous, self-propelled agents, whetherbiological or artificial. Our results are important from apractical design standpoint for artificial systems, as whena swarm of robots is used to survey/monitor a given areaof interest. In addition to their relevance in the study ofswarming and herding motions in biological systems, ourresults on heterogeneity play a predictive role where thedynamics of individual agents are to large degree beyondour control. A
CKNOWLEDGMENTS
This research was performed while KS held a NationalResearch Council Research Associateship Award at the U.S.Naval Research Laboratory. This research is funded by theOffice of Naval Research contract no. N0001412WX2003and the Naval Research Laboratory 6.1 program contract no.N0001412WX30002. A
PPENDIX R ING S TATE
In the ring state, the agents in either population rotateabout a common stationary center of mass. To study thedynamics of the ring state, we must therefore re-introducethe full swarm dynamics. For convenience, we express thesein polar coordinates, with the origin located at the positionof the stationary center of mass, so that R ( t ) = R ( t ) ≡ .Then let ρ i k = (cid:13)(cid:13) δr ki (cid:13)(cid:13) , θ ki = ∠ δr ki (25)for k = 1 , . Setting R k = ˙ R k = ¨ R k = 0 in (6) gives: δ ¨ r ki = κ k N k N k (cid:88) j =1 (cid:13)(cid:13) δ ˙ r kj (cid:13)(cid:13) δ ˙ r kj + κ k (cid:16) − (cid:13)(cid:13) δ ˙ r ki (cid:13)(cid:13) (cid:17) δ ˙ r ki − aκ k N (cid:16) ( N − δr ki + δr k,τi (cid:17) . (26)In the ring state, R k = ˙ R k = ¨ R k = 0 requires that (cid:80) N k j =1 (cid:13)(cid:13) δ ˙ r kj (cid:13)(cid:13) δ ˙ r kj = 0 for k = 1 , [1]; so that, in the limit as N → ∞ , δ ¨ r ki = κ k (cid:16) − (cid:13)(cid:13) δ ˙ r ki (cid:13)(cid:13) (cid:17) δ ˙ r ki − aκ k δr ki . (27)Converting to polar coordinates leads to the following set ofequations: ¨ ρ ki = κ k ˙ ρ ki (cid:16) − ( ρ ki ˙ θ ki ) − ( ˙ ρ ki ) (cid:17) + (cid:16) ( ˙ θ ki ) − a (cid:17) ρ ki (28a) ρ ki ¨ θ ki = κ k ρ ki θ ki (cid:16) − ( ρ ki ˙ θ ki ) − ( ˙ ρ ki ) (cid:17) − ρ ki ˙ θ ki . (28b)Note that the equations governing the two populations areentirely uncoupled. In the ring state, ˙ ρ ki = ¨ ρ ki = 0 and theagents move with constant angular velocity so that ¨ θ ki = 0 for k = 1 , . Let ω ki denote the constant angular velocity ˙ θ ki of agent i in population k . Then (28) can be written as: (cid:0) ( ω ki ) − aκ k (cid:1) ρ ki (29a) ρ ki ω ki (cid:0) − ( ρ ki ω ki ) (cid:1) , (29b)and it follows that ρ ki = 1 / | ω ki | , ω ki = ±√ aκ k (30)for all agents in the swarm.A PPENDIX R OTATING S TATE
To find the parameters describing the rotating state of theswarm, we convert the equations for the swarm dynamics topolar coordinates. Suppose that the ring state is formed aboutthe stationary point ( X s , Y s ) T ∈ R , and choose the originof the polar coordinates to lie on ( X s , Y s ) T . Let ( ρ k , θ k ) denote the position, in polar coordinates, of the center ofmass of Population k , that is ρ k = (cid:112) ( X k − X s ) + ( Y k − Y s ) (31a) θ k = tan − Y k − Y s X k − X s . (31b)The equations of motions for the motion of the centers ofmass of the two swarm populations in polar coordinates, are ¨ ρ k = κ k (cid:16) − ρ k ˙ θ k − ˙ ρ k (cid:17) ˙ ρ k + ρ k ˙ θ k − aκ k (cid:16) ρ k − cρ τ cos( θ k − θ τ ) − (1 − c ) ρ τ cos( θ k − θ τ ) (cid:17) (32a) ρ k ¨ θ k = κ k (cid:16) − ρ k ˙ θ k − ˙ ρ k (cid:17) ρ k ˙ θ k − ρ k ˙ θ k − aκ k (cid:16) cρ τ sin( θ k − θ τ )+ (1 − c ) ρ τ sin( θ k − θ τ ) (cid:17) . (32b)In the rotating state, the radii of the populations and theangular frequencies are constant. Let ω k = ˙ θ k . Then ρ k ( t ) = ρ k (33a) θ k ( t ) = θ k + ω k t, (33b)and ¨ ρ k = ˙ ρ k = ¨ θ k = 0 . Furthermore, simulations of the fullswarm dynamics suggest that the radii of the two populations in the rotating state are equal; we therefore set ρ = ρ = ρ .Let ∆ θ = θ − θ denote the phase difference between thetwo populations. Substituting these equations into (32) andsimplifying the resulting expressions gives: ω k = aκ k (cid:0) − c cos( ω τ + ( ω k − ω ) t + θ k − θ ) − (1 − c ) cos( ω τ + ( ω k − ω ) t + θ k − θ ) (cid:1) (34a) (1 − ρ ω k ) ω k = aκ k (cid:0) c sin( ω τ + ( ω k − ω ) t + θ k − θ )+ (1 − c ) sin( ω τ + ( ω k − ω ) t + θ k − θ ) (cid:1) . (34b)Note that the time dependence on the right hand sides of allequations in (34) can be eliminated if and only if ω = ω .Let ω denote the common frequency of both populationsabout the center. Thus, we finally have the four equationsdescribing the behavior of the swarm in the ring state: ω = aκ k (cid:0) − c cos( ωτ + θ k − θ ) − (1 − c ) cos( ωτ + θ k − θ ) (cid:1) (35a) (1 − ρ ω ) ω = aκ k (cid:0) c sin( ωτ + θ k − θ )+ (1 − c ) sin( ωτ + θ k − θ ) (cid:1) . (35b)Relations (16)-(19) can be derived from (35) through somerather involved algebraic manipulations.R(35b)Relations (16)-(19) can be derived from (35) through somerather involved algebraic manipulations.R