Symmetry and reduction in collectives: cyclic pursuit strategies

Abstract

We specify and analyse models that capture the geometry of purposeful motion of a collective of mobile agents, with a focus on planar motion, dyadic strategies and attention graphs which are static, directed and cyclic. Strategies are formulated as constraints on joint shape space and are implemented through feedback laws for the actions of individual agents, here modelled as self-steering particles. By reduction to a labelled shape space (using a redundant parametrization to account for cycle closure constraints) and a further reduction through time rescaling, we characterize various special solutions (relative equilibria and pure shape equilibria) for cyclic pursuit with a constant bearing (CB) strategy. This is accomplished by first proving convergence of the (nonlinear) dynamics to an invariant manifold (the CB pursuit manifold), and then analysing the closedloop dynamics restricted to the invariant manifold. For illustration, we sketch some low-dimensional examples. This formulation-involving strategies, attention graphs and sensor-driven steering lawsand the resulting templates of collective motion, are part of a broader programme to interpret the mechanisms underlying biological collective motion.

Optimality, reduction and collective motion

“…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.…”

Symmetry and reduction in collectives: low-dimensional cyclic pursuit

“…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.…”

“…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.…”

Geometric decompositions of collective motion

“…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].…”