Formation shape control: Global asymptotic stability of a four-agent formation

This paper considers formation shape control of a team of four agents in the plane, motivated by an example from [1]. We utilize bidirectional, gradient-based interagent distance control laws which are designed so that the agents cooperatively achieve a specified desired formation shape. When every interagent distance is actively controlled (i.e. the information architecture is a complete graph), there may exist equilibrium formation shapes with incorrect interagent distances. We prove that any incorrect equilibrium formation shape is locally unstable for formations with internal angles that satisfy an acuteness condition and thereby establish that the desired formation shape, for all practical purposes, is globally asymptotically stable.