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Abstract
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.
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