Symmetry and reduction in collectives: cyclic pursuit strategies

Galloway, Kevin S.; Justh, Eric W.; Krishnaprasad, P. S.

doi:10.1098/rspa.2013.0264

Cited by 30 publications

(52 citation statements)

References 18 publications

(32 reference statements)

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“…relative bearings or distances) which can plausibly be sensed by an individual [6]. The associated closed-loop dynamical system is the object of investigation: certain solutions of interest (e.g.…”

Section: (A) Allelomimesis As a Collective Strategymentioning

confidence: 99%

Optimality, reduction and collective motion

Justh

Krishnaprasad

2015

Proc. R. Soc. A.

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The planar self-steering particle model of agents in a collective gives rise to dynamics on the N-fold direct product of SE(2), the rigid motion group in the plane. Assuming a connected, undirected graph of interaction between agents, we pose a family of symmetric optimal control problems with a coupling parameter capturing the strength of interactions. The Hamiltonian system associated with the necessary conditions for optimality is reducible to a Lie-Poisson dynamical system possessing interesting structure. In particular, the strong coupling limit reveals additional (hidden) symmetry, beyond the manifest one used in reduction: this enables explicit integration of the dynamics, and demonstrates the presence of a 'master clock' that governs all agents to steer identically. For finite coupling strength, we show that special solutions exist with steering controls proportional across the collective. These results suggest that optimality principles may provide a framework for understanding imitative behaviours observed in certain animal aggregations.

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Section: (A) Allelomimesis As a Collective Strategymentioning

confidence: 99%

Optimality, reduction and collective motion

Justh

Krishnaprasad

2015

Proc. R. Soc. A.

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“…Though [1] primarily addressed the general n-agent cyclic-pursuit system, that paper also included a brief sketch of some interesting low-dimensional special cases. Of particular interest was the two-agent 'mutual CB pursuit' case, with integrable shape dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…In our companion paper [1], we developed a framework for analysing collectives of autonomous agents engaged in dyadic pursuit interactions. Modelling the agents as self-steering particles on the Lie group SE(2) (the group of rigid motions in the plane) with interactions governed by a static directed cycle graph, we demonstrated a symmetry reduction to a labelled shape space and provided a convenient parametrization of space.…”

Section: Introductionmentioning

confidence: 99%

Symmetry and reduction in collectives: low-dimensional cyclic pursuit

2016

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We investigate low-dimensional examples of cyclic pursuit in a collective, wherein each agent employs a constant bearing (CB) steering law relative to exactly one other agent. For the case of three agents in the plane, we characterize relative equilibria and pure shape equilibria of associated closed-loop dynamics. Re-scaling time yields a reduction of phase space to two dimensions and effective tools for stability analysis. Study of bifurcation of a family of collinear equilibria dependent on a single CB control parameter reveals the presence of a rich collection of trajectories that are periodic in shape and undergo precession in physical space. For collectives in three dimensions, with an appropriate notion of CB pursuit strategy and corresponding steering law, the two-agent case proves to be explicitly integrable. These results suggest control schemes for small teams of mobile robotic agents engaged in area coverage tasks such as search and rescue, and raise interesting possibilities for behaviour in biological contexts.

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“…Bottom-up models of such interactions have been devised using feedback control laws [4][5][6] and subject to mathematical analysis. Methods of statistical mechanics have been employed to examine data and create models emphasizing averaged effects as opposed to individual movement [7,8].…”

Section: Introductionmentioning

confidence: 99%

Geometric decompositions of collective motion

Mischiati

Krishnaprasad

2017

Proc. R. Soc. A.

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Collective motion in nature is a captivating phenomenon. Revealing the underlying mechanisms, which are of biological and theoretical interest, will require empirical data, modelling and analysis techniques. Here, we contribute a geometric viewpoint, yielding a novel method of analysing movement. Snapshots of collective motion are portrayed as tangent vectors on configuration space, with length determined by the total kinetic energy. Using the geometry of fibre bundles and connections, this portrait is split into orthogonal components each tangential to a lower dimensional manifold derived from configuration space. The resulting decomposition, when interleaved with classical shape space construction, is categorized into a family of kinematic modes-including rigid translations, rigid rotations, inertia tensor transformations, expansions and compressions. Snapshots of empirical data from natural collectives can be allocated to these modes and weighted by fractions of total kinetic energy. Such quantitative measures can provide insight into the variation of the driving goals of a collective, as illustrated by applying these methods to a publicly available dataset of pigeon flocking. The geometric framework may also be profitably employed in the control of artificial systems of interacting agents such as robots.

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Symmetry and reduction in collectives: cyclic pursuit strategies

Cited by 30 publications

References 18 publications

Optimality, reduction and collective motion

Optimality, reduction and collective motion

Symmetry and reduction in collectives: low-dimensional cyclic pursuit

Geometric decompositions of collective motion

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