Critical transition for colliding swarms
Jason Hindes, Victoria Edwards, M. Ani Hsieh, Ira B. Schwartz
CCritical transition for colliding swarms
Jason Hindes , Victoria Edwards , , M. Ani Hsieh and Ira B.Schwartz U.S. Naval Research Laboratory, Washington, DC 20375, USA Mechanical Engineering and Applied Mechanics, University of Pennsylvania,Philadelphia, Pennsylvania 19104, USAE-mail: [email protected]
Abstract.
Swarming patterns that emerge from the interaction of many mobileagents are a subject of great interest in fields ranging from biology to physics androbotics. In some application areas, multiple swarms effectively interact and collide,producing complex spatiotemporal patterns. Recent studies have begun to addressswarm-on-swarm dynamics, and in particular the scattering of two large, collidingswarms with nonlinear interactions. To build on early numerical insights, we develop amean-field approach that can be used to predict the parameters under which collidingswarms are expected to form a milling state. Our analytical method relies on theassumption that, upon collision, two swarms oscillate near a limit-cycle, where eachswarm rotates around the other while maintaining an approximately constant anduniform density. Using this approach we are able to predict the critical swarm-on-swarm interaction coupling, below which two colliding swarms merely scatter, for nearhead-on collisions as a function of control parameters. We show that the criticalcoupling corresponds to a saddle-node bifurcation of a stable limit cycle in the uniform,constant density approximation. Our results are tested and found to agree with largemulti-agent simulations.
Swarming occurs when spatiotemporal patterns and behaviors emerge from theinteraction of large numbers of coupled mobile systems, typically with fairly limitedcapabilities and local dynamics. Examples have been discovered in nature over manyspatiotemporal scales from colonies of bacteria, to swarms of insects[1, 2, 3, 4], flocks ofbirds [5, 6, 7], schools of fish[8, 9], crowds of people[10], and active-matter systems moregenerally[11]. Understanding the principles behind swarming patterns and describinghow they emerge from simple models has been the subject of significant work inphysics, applied mathematics, and engineering sciences [12, 13, 14, 15, 16, 17, 18,19, 20, 21, 22, 23, 24]. Parallel with this work, and because of the robustness,scalability, and collective-problem solving capabilities of natural swarms, much researchhas focused on designing and building swarms of mobile robots with a large andever expanding number of platforms, as well as virtual and physical interaction a r X i v : . [ n li n . PS ] J a n mechanisms[11, 25, 26, 27, 28]. Applications for such systems range from exploration[26],mapping[29], resource allocation [30, 31, 32], and swarms for defense [33, 34, 35]Since the overall cost of robotic systems has decreased significantly in recent years,it has become possible to use artificial swarms in the real world [36, 37, 27, 26]. Thisintroduces the possibility of having multiple swarms occupying the same physical space,resulting in mutual interactions and perturbations of one another’s dynamics[38]. Asthe potential for such swarm-on-swarm interactions increases, an understanding of howmultiple swarms collide and merge becomes necessary.Though much is known about the behaviors and stability of single isolated swarmswith physically-inspired, nonlinear interactions[39, 40, 41, 42, 43], much less is knownabout the intersecting dynamics of multiple such swarms, even in the case where oneswarm is a single particle, as in predator–prey modeling[44]. Recent numerical studieshave shown that when two flocking swarms collide, the resulting dynamics typicallyappear as a merging of the swarms into a single flock, milling as one uniform swarm,or scattering into separate composite flocks moving in different directions[45, 46, 38].Though interesting, a more detailed analytical understanding of how and when thesebehaviors occur is needed, especially when designing robotic swarm experiments, andcontrolling their outcomes.To make progress, we consider a generic system of mobile agents moving underthe influence of self-propulsion, friction, and pairwise interaction forces. In the absenceof interactions, each swarmer tends to a fixed speed, which balances its self-propulsionand friction but has no preferred direction[47]. A simple model that captures the basicphysics is ¨ r i = (cid:2) α i − β | ˙ r i | (cid:3) ˙ r i − λ i (cid:88) j (cid:54) = i ∂ r i U ( | r j − r i | ) (1)where r i is the position-vector for the i th agent in two spatial dimensions, α i is a self-propulsion constant, β is a friction constant, and λ i is a coupling constant[39, 40, 41, 42,43]. The total number of swarming agents is N . Beyond providing a basis for theoreticalinsights, Eq.(1) has been implemented in experiments with several robotics platformsincluding autonomous ground, surface, and aerial vehicles[48, 49, 50].An example interaction potential that we consider in detail is the Morse potential, U ( r ) = Ce − r/l − e − r (2)– a common model for soft-core interactions with local repulsion and attraction ranges,scaled as l and 1, respectively[42, 51]. In the following, we assume that two interactingswarms are subject to the same underlying physics, Eqs.(1-2), but with different initialconditions and potentially different control parameters. In particular, we assumethat within each swarm the parameters are homogeneous, e.g., α i ∈ { α (1) , α (2) } and λ i ∈ { λ (1) , λ (2) } , where the superscripts (1) and (2) denote the first and second swarms,respectively. The assumption that the two swarms satisfy the same basic physics makessense if the swarms are composed of similar agents, and should be a reasonable, baselineassumption for biological and active-matter swarm collisions. As in [45, 46], we are interested in the collision of two flocking swarms composed ofapproximately equal numbers of agents (in the absence of any communication delay).The swarms are each prepared in a flocking state with initial velocities and positionsthat are a large distance D from the collision region, such that r i = d (1) i − D ˆx and ˙ r i = (cid:112) α (1) /β ˆx if i ∈ (1), and r i = d (2) i + D (cid:112) α (2) /α (1) (cos( θ ) ˆx + sin( θ ) ˆy ) and˙ r i = (cid:112) α (2) /β ( − cos( θ ) ˆx + sin( θ ) ˆy ) if i ∈ (2). The internal flocking coordinates, d (1) i and d (2) i , represent local minimum energy configurations (MECs), − ∂ d i U ( | d j − d i | ) = i ∀ i [52]. Before colliding, the swarms are assumed to be far apart, D (cid:29)
1, with a collisionangle θ . Note that given this setup, the net force on every agent is initially zero (aconsequence of the MEC and the finite-range of interactions), and the swarms collidenear the origin.Qualitatively, for relatively small θ the two flocks typically scatter or mill dependingon the coupling strength and relative velocities. In the former the swarms leave thecollision region in separate flocking states with perturbed velocities. In the latter theyform a milling state, and circulate around a stationary center of mass. Figure 1(a)gives an example scattering diagram for the collision of symmetric flocks with equalparameters. The two final swarm states are specified with blue and red for scattering andmilling, respectively; the green portions indicate the formation of a combined flockingstate, which is comparatively infrequent for the parameters shown. For small θ we cansee that the combined milling state (MS) appears for couplings above a certain value λ min – the smallest coupling needed to form a MS. This critical coupling will be a primaryfocus in what follows.In order to visualize collisions that result in milling states, we show four time-snapshots in Figure 1(b) when λ = λ min . Agents in the two swarms are drawn withdifferent colors, and their velocities shown with arrows. In the first snapshot (upperleft), the swarms approach collision with configurations and velocities identical to thosespecified in the first paragraph of this section– namely, the MEC with constant velocity.In the second snapshot (upper right) the swarms rotate around each other with aconstantly changing heading, roughly uniform velocity distribution, and a configurationapproximately equal to the MEC. Over time each swarm’s density elongates in thedirection of rotation (third snapshot, lower left), as the velocity distribution becomesless homogeneous. Finally, on long times scales the two swarms blend into one and forma MS with agents from each uniformly distributed across the whole. This qualitativesequence holds more generally within the red region of Fig. 1(a).In order to predict the critical coupling, λ min , our approach is to find an analyticaldescription of the collision dynamics that is applicable for the first two snapshots inFigure 1(b), where two approximately MEC flocks approach, and then rotate around acommon center. Our conjecture is that if such rotations are approximately stable, then aMS occurs upon collision (and visa versa). Though we will analyze two-flock collisions inthis way assuming Morse-potential interactions, Eq.(2), our method should be applicable (a) (b) -3 -1 1 31234 -3 -1 1 31234-3 -1 1 30246-1135-6 -2 2 670.2 Figure 1.
Collision of two symmetric flocks. (a) Scattering digram indicating thefinal, aggregate swarm state as a function of the collision angle and coupling: scattering(blue), milling (red), and single-flock (green). The critical coupling is specified with adashed vertical line, and separates the scattering and milling regions. (b) Four time-snapshots for λ = λ min showing each swarm with different colors: red squares and bluecircles. Velocities are drawn with arrows. Swarm parameters are α = 1, β = 5, C = 10 / l = 0 .
75, and N = 100. to a broad range of second-order dynamical swarms given position-dependent, nonlinearinteractions with finite attractive and repulsive length scales. First, we would like to find a low-dimensional approximation for the flocking statedynamics. A clue comes from Figure 2(a), which plots the fraction of nodes at a givendistance r from the center of mass (CM) of a single moving flock for different valuesof the repulsion strength, C . We can see that the radial distribution is approximately linear in r . Moreover, since the potential is radial, we expect the steady-state angulardistribution to be uniform; the inlet panel shows an example flocking state with such aspatial distribution of agents. Together, these imply a roughly uniform density in theflocking state, ρ = N/πR , where R is the maximum radius. Given the uniform-densityassumption, the predicted fraction of agents at a given r is f ( r ) = 2 r ∆ r/R , where ∆ r is the bin-size used to plot the distribution. This prediction is drawn with lines forcomparison in Figure 2(a).Assuming a uniform density, we can describe a flock in general by its CM-dynamicsand the boundary radius, R . In particular, every agent, including those on the boundary,move with constant speed, (cid:112) α/β , where α is the self-propulsion constant for the flock.A self-consistent formula can be derived for R , and used to compute it, by satisfyingforce-balance on the boundary. For example, consider an agent with d i = R ˆ x . The (a) (b) Figure 2.
Uniform constant density approximation for flocking states (UCDA).(a) fraction of agents a distance r from the center of mass for C =1 . , . , .
25 (green-diamonds) when l = 0 .
75, where C is the repulsive-force strength with length-scale l . The dashed, solid, and dotted linesindicate UCDA predictions from solving Eq.(3). The inlet panel shows an exampleflocking state with the UCDA boundary drawn in black for C = 1 .
1. (b) Flockingstate boundary, R = max { r } , from simulations: ( l = 0 .
85, magenta-xs), ( l = 0 .
75, blue-circles), ( l = 0 .
60, green-diamonds) and ( l = 0 .
50, red-squares) compared to UCDApredictions shown with lines near each series. Other swarm parameters are α = 1, β = 5, λ = 2, and N = 100. x -component of the interaction force must be zero,0 = (cid:90) π (cid:90) (cid:32) Cl e − Rl √ u − u cos φ − e − √ u − u cos φ (cid:33) · u cos φ − (cid:112) u − u cos φ · ududφ , (3)where u ≡ r/R . Note that the y -component of the force is trivially zero due to theuniform-angular distribution of agents. Comparisons between simulations and numericalsolutions to Eq.(3) are shown in Fig.2(b) for a range of control parameters, and indicategood agreement.Next, we can approximate the initial collision dynamics of two flocks by assumingthat the uniform density configuration is maintained within each flock, with a boundarygiven by Eq.(3). Namely, the mean-field collision-model that we will analyze below isof two interacting, constant-density disks composed of self-propelled particles. Consideran agent positioned at the CM of each swarm, r (1) ( t ) and r (2) ( t ). Such agents feelself-propulsion, friction, and interaction forces, just as in Eq.(1). However, the non-zero contribution for the latter only comes from the other flock , since the interactionforce from its own cancels, e.g., due to the MEC assumption. Moreover, the interactionforce from the opposing flock is felt gradually as the two swarms approach, because ofthe finite-range interactions and the initially large separation between the flocks. Tofind the non-zero contribution, we simply need to integrate the interaction force over aconstant-density disk of radius R , centered on the opposing swarm’s CM. If we assumethat the two swarms are roughly equally sized, each with N/ r (1) = (cid:2) α (1) − β | ˙ r (1) | (cid:3) ˙ r (1) − λ (1) N E ( r (2) , r (1) ; R ) (4a)¨ r (2) = (cid:2) α (2) − β | ˙ r (2) | (cid:3) ˙ r (2) − λ (2) N E ( r (1) , r (2) ; R ) (4b) E ( r (2) , r (1) ; R ) = (cid:90) π (cid:90) R r (2) + d − r (1) | r (2) + d − r (1) | · rdrdφπR · (cid:32) Cl e −| r (2) + d − r (1) | /l − e −| r (2) + d − r (1) | (cid:33) (4c) d = r cos φ ˆ x + r sin φ ˆ y , (4d)where d is an internal-coordinate inside the constant-density disk centered on theopposing swarm’s CM. Equations (3-4d) constitute the mean-field dynamical systemthat we call the uniform constant density approximation (UCDA). The integrals inEq.(4c) can be evaluated using e.g., trapezoid rule with 100 discretization points. Ournext step is to study stable oscillations of r (1) ( t ) and r (2) ( t ) in the UCDA and compareto swarm collision dynamics. (a) (b) Figure 3.
Collision dynamics resulting in milling. (a) Center-of-mass trajectoriesfor two colliding swarms when λ = λ min , shown with solid-blue and dashed-red lines.Arrows give the direction of motion. The dashed-black line indicates the bifurcatinglimit cycle in the uniform constant density approximation. Other swarm parametersare α = 1, β = 5, l = 0 . N = 100, and C = 1 .
0. (b) Maximum x-coordinate reachedby the center of mass of the rightward moving (blue) flock when λ = λ min . Simulationresults are shown with blue circles for l = 0 .
75, green diamonds for l = 0 .
6, and redsquares for l = 0 .
5. Limit-cycle predictions from Eqs.(5a-5d) and Eq.(6) are drawn withlines near each series. Other swarm parameters are α = 1, β = 5, and N = 200. Stable oscillations in the UCDA come in the form of circular-orbit limit cycles whereboth flocks oscillate around a common center with the same frequency, a fixed phasedifference, and different amplitudes in general. We can compute the parameters forsuch limit cycles by substituting the ansatz r (1) ( t ) = R cos( ωt )ˆ x + R sin( ωt )ˆ y and r (2) ( t ) = R cos( ωt + γ )ˆ x + R sin( ωt + γ )ˆ y into Eqs.(4a-4d). The result is the followingfour root equations satisfying F i = 0 for i ∈ { , , , } : F = − R ω + λ (1) N E x (5a) F = − R ω (cid:2) α (1) − βR ω (cid:3) + λ (1) N E y (5b) F = − R ω sin γ (cid:2) α (2) − βR ω (cid:3) + R ω cos γ + λ (2) N E x (5c) F = R ω cos γ (cid:2) α (2) − βR ω (cid:3) + R ω sin γ + λ (2) N E y (5d)with E x = (cid:90) π (cid:90) R R cos γ + r cos φ − R d · (cid:16) Cl e − d/l − e − d (cid:17) · rdrdφπR (5e) E y = (cid:90) π (cid:90) R R sin γ + r sin φd · (cid:16) Cl e − d/l − e − d (cid:17) · rdrdφπR (5f) d = (cid:112) ( R cos γ + r cos φ − R ) +( R sin γ + r sin φ ) . (5g)Solutions to Eqs.(5a-5d) for L ≡ [ R , R , γ, ω ] can be shown to exactly match limitcycles within the UCDA; more importantly, they agree with the transient oscillationsfor collisions in the full system, Eqs.(1-2). For example, Fig.3(a) shows CM-trajectoriesin red and blue for two colliding swarms when λ = λ min . We can see that the trajectoriesapproach the UCDA limit-cycle, shown with a black-dashed line, before slowly decayinginto the origin. Using this picture as a basis, the maximum rotation radius duringcollisions can be compared directly to limit-cycle radii predictions from Eqs.(5a-5g).In Fig.3(b) we plot such a comparison using the maximum horizontal distance reachedby the CM of the rightward moving flock (as a proxy for the collision radius). Mean-field predictions and simulations quantitatively agree fairly well over a broad rangeof parameter values. Qualitatively, as the repulsive-force constant C increases, the twoswarms oscillate at larger distances from each other upon collision, particularly for largervalues of the repulsion scale, l . This increase in rotation distance, R , is accompaniedby a decrease in rotation frequency, ω ∼ R − .Next, we can consider stability. When control parameters are changed (one at atime), stable limit cycles satisfying Eqs.(5a-5g) disappear generically through saddle-node bifurcations (SNs). As stated previously in Sec.2, a post-collision MS in the fullsystem Eqs.(1-2) is not expected to form unless stable limit-cycles exist, and hence, λ min can be approximated by the SN value in the UCDA. We can find a general condition todetermine λ min at the SN through the following. Using the defined vector components F specified in Eqs.(5a-5d), we compute the derivatives of F with respect to the limit-cycleparameters, L . Finally, at the SN the Jacobian matrix J , defined as J mn ≡ ∂F m /∂L n ,has det J ( L ; λ min ) = 0 . (6)Combining Eq.(6) with Eqs.(5a-5d) gives a total of 5 root equations for the approximatecritical coupling and associated limit-cycle.In practice, if we consider symmetric collisions or asymmetry in the α ’s only(as we do in the remainder), the above results simplify. For example, in the case ofsymmetric collisions the relevant branch of stable limit cycles have R = R , γ = π , and ω = (cid:112) α/β/R . Moreover, the symmetric critical coupling predicts a scaling collapse: λ min N β/ α = 1 (cid:44) R (cid:90) π (cid:90) R rdrdφπR · Cl e − d/l − e − d d · (cid:34) − (2 R − r cos φ ) d − (2 R − r cos φ ) d · Cl e − d/l − e − dCl e − d/l − e − d (cid:35) . (7)where the left hand side is a function of the pairwise-interaction parameters only. Inaddition, λ min ∼ v , where v is the speed of each flock, (cid:112) α/β .Comparisons between the measured λ min from scattering diagrams, e.g., Fig.1(a),and the above predictions from Eq.(6) and Eqs.(5a-5d) are shown in Fig.4. In theleft subplot (a), we show results for collisions with symmetric parameters with a largevariety of N ’s and α ’s. As demonstrated with Eq.(7) our predicted scaling collapseholds. Qualitatively, the critical coupling increases monotonically with C , implyingthat the stronger the strength of repulsion, the larger the coupling needs to be in orderfor colliding swarms to form a MS. Also, note that our mean-field predictions are fairlyrobust to heterogeneities in the numbers in each flock, particularly for smaller values of C/l −
1; predictions remain accurate for number heterogeneity as large as 20%.On the other hand, in Fig.4(b) we compare the measured λ min and predictions as afunction of asymmetry in the self-propulsion force constant for different N ’s. The firstswarm has α (1) = 1, while α (2) is varied. Contrary to the symmetric case the scalingcollapse disappears, apart from N . Moreover, the branch of stable limit cycles withequal radii R = R disappears in a cusp bifurcation (the solid-black line in in Fig.4(b)vanishes for α (2) (cid:38) . R < R and γ = − π/
2, shown with a dashed-black line.Interestingly, we can see that for larger values of α (2) − α (1) the critical coupling is nearlylinear in the difference, meaning that if one flock doubles its speed, then the couplingneeded to form a MS is expected to quadruple – again, a consequence of the flockspeed equalling (cid:112) α/β . Finally, note that as in (a), predictions remain accurate for asignificant range of differences in the numbers in each flock. (a) (b) Figure 4.
Critical coupling for forming milling states upon collision. (a) Symmetricparameter collisions for α = 1 (blue) and α = 2 (red): N = 10 (squares), N = 20(diamonds), N = 40 (circles), and N = 100 (triangles). Green stars denote α = 1 andmagenta x’s denote α = 2, when 40 agents collide with 60. (b) Asymmetric collisions for C = 10 / α (1) = 1. Blue points indicate equal numbers in each flock: N = 20(diamonds), N = 40 (circles), and N = 100 (triangles). Green stars denote collisionsbetween 40 agents with α (1) = 1 and 60 agents with α (2) . Solid and dashed linesindicate theoretical predictions for (a) and (b), respectively from solving Eqs.(5a-5d)and Eq.(6). Other swarm parameters are β = 5 and l = 0 . To summarize, in this work we studied the collision of two swarms with nonlinearinteractions, and focused in particular on predicting when such swarms would combineto form a mill. Unlike the full scattering diagram, which depends on whether ornot a particular set of initial conditions falls within the high-dimensional basin-of-attraction for milling (a hard problem in general), we concentrated on predicting theminimum coupling needed to sustain a mill for near head-on collisions. By noticingthat colliding swarms, which eventually form a mill, initially rotate around a commoncenter with an approximately constant density, we were able to transform the questionof a critical coupling into determining the stability of limit-cycle states within a mean-field approximation. Our bifurcation analysis agreed well with many-agent simulations.Future robotics experiments, similar to [48, 49], will be used to further test and verifyour analysis.Though our analysis dealt directly with soft-core interacting swarms, the basicapproach could be easily extended to a broader range of models, as long as the nonlinearforces between agents have a finite range. A straightforward way to improve the accuracyof our analysis would be to move beyond the uniform-density assumption, and replace itwith an exact steady-state density for flocking states with general interactions. Anothernext-step for improvement would be to include the density dynamics directly, whichmay provide further quantitative insights for controlling swarm collisions, including inother setups such as flocks-vs-mills. On the other hand, at a broader level our approachassumed that all agents are subject to the same basic physics, irrespective of the swarm0to which they are apart. This is a very classical and standard assumption, and meansthat when agents from different swarms enter each other’s sensing range, they stillexecute the same low-level controllers. The assumption makes sense if the swarms arecomposed of similar agents, e.g., birds, insects, etc. From a robotics perspective, onecould even think of scenarios where two swarms execute the controlled behavior inorder to “scope out” the other swarm. However, one would expect this initial phaseto be limited. Therefore, an interesting avenue for future work would be to studyhow disparate inter swarm interactions affect swarm-on-swarm behavior. Nonetheless,this work takes an important step towards understanding and analyzing the basiccollision dynamics of self-propelled swarms with nonlinear interactions, and providesnew methods by which they can be quantitatively predicted.
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