Stabilization of Planar Collective Motion: All-to-All Communication

Abstract

Abstract-This paper proposes a design methodology to stabilize isolated relative equilibria in a model of all-to-all coupled identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with fixed relative spacing or to circular motion of all particles with fixed relative phases. The stabilizing feedbacks derive from Lyapunov functions that prove exponential stability and suggest almost global convergence properties. The results of the paper provide a low-order parametric family of stabilizable collectives that offer a set of primitives for the design of higher-level tasks at the group level.

Stabilization of collective motion in three dimensions: A consensus approach

“…The present paper generalizes, to three-dimensional space, earlier work in the plane [8], [9]. Preliminary results in SE(3) have been presented in [10].…”

“…To give a geometric interpretation to (8) we compute the relative screw coordinates (expressed in a global frame) and we obtain an (instantaneous) pitch…”

“…When (11) are equal and constant for all the particles the resulting motion is characterized by a parallel formation (u k = 0) and a circular formation about the same point (u k = 0 and constant). The control laws are derived in [8], [9].…”

“…As a consequence s av = x av + r av × ω av and the approach followed in [8] does not yield shape control laws. To overcome this obstacle we first stabilize the motion about an axis of rotation whose direction is fixed.…”

Dynamics of Decision Making in Animal Group Motion

“…For instance, a continuous model of particles moving at constant speed in the plane with steering control (heading rate) designed to couple the dynamics of the particles has been used for stabilization of circular and parallel collective motion [4,5]. The use of the same kinds of models in the engineered and natural settings is no accident.…”