On global feedback stabilization of decentralized formation control

We address the problem of global stabilization in decentralized formation control. Formation control is concerned with problems in which autonomous agents are required to stabilize at a given distance of other agents. In this context, a graph associated to a formation encodes both the information flow in the system and the distance constraints, by fixing the lengths of the edges. While globally stabilizing control laws for the case of n = 3 agents in a cyclic formation have been proposed, the case of n = 4 agents has so far resisted attempts to obtain globally stabilizing control laws. We show that a large class of control laws, including all control laws shown to work in the three agents case, cannot satisfactorily stabilize a four agents formation. The proof relies on applying ideas from singularity theory and dynamical systems theory which can be used to address global stabilization of a broad class of decentralized control systems.