Symmetries and reduction for multi-agent control

Abstract

This paper computes all the continuous point-transformation symmetries for a planar multi-agent dynamical control system. The relative configuration among the agents is often the most important aspect of this problem, and this paper presents a method to “factor out” the location and orientation on the plane of a multi-agent formation problem. This is useful in the case where only the relative positioning of the agents is of concern. The method to compute the symmetries and determine the reduced coordinates is well-established. The contributions of this paper are (1) computing the symmetries for a multi-agent system, (2) highlighting the utility of the reduction for the formation control application such as formation stability and (3) analyzing the symmetries to determine a relatively broad class of systems which have the same symmetries and hence identical coordinates for reduction. Although it has been addressed in various ways, essentially all methods to determine the stability properties of the formation control problem need the dynamics to be formulated on a reduced, relative space. Hence, stability (and stability-like notions such as that which follow from LaSalle's Principle) analyses will be the main beneficiary of this work.