Geometric decompositions of collective motion

Abstract

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.

A Multi-Layer Swarm Control Model for Information Propagation and Multi-Tasking

“…In [48], they developed a geometrical framework based on the principal axis of the swarm to analyze the motion of a swarm. The framework results in decomposition of the motion into kinematic modes such as translations, rotations, expansions, and compressions.…”

“…This resembles a communication through behavior which might be useful in modeling information propagation in biological and robotic swarms [47,46]. Additionally, the use of PCA leverages the value of relative positions in that it extracts more geometrical information about the swarm that might be beneficial for different applications [48,69].…”

A Distributed Active Perception Strategy for Source Seeking and Level Curve Tracking

“…An important insight we analytically show in this paper is that the field measurements are communicated via the distributed perception algorithm. This might be useful in modeling information propagation in biological and robotic swarms [14], [20], [31], [33].…”

Flocks, Games, and Cognition: A Geometric Approach

“…While there is a long history of smoothing techniques in biological data analysis, we used an efficient method based on the theory of optimal control of linear systems with quadratic cost functionals, to obtain smoothed trajectories for each bird [12], [13]. The passage from individual-scale to flock-scale analysis is based on a recent development of the idea of kinematic modes in many-particle systems [14]. Using the geometric language of fiber bundles, the velocity of the flock as a whole is split into several mutually orthogonal components (kinematic modes).…”

“…Taking fractions of the different energy modes with respect to the kinetic energy relative to center of mass yields a signature of a flocking event on a standard (probability) simplex. In [14] this process was applied to pigeon flock data from [5].…”