Singularities and global stability of decentralized formations in the plane

TL;DR

This paper investigates the stability of decentralized formation control in the plane, revealing that cyclic information flow graphs generally prevent global stabilization due to inherent singularities in the system dynamics.

Contribution

It demonstrates that most cyclic formations cannot be globally stabilized, explaining the limitations of existing control laws through the analysis of singularities.

Findings

Cyclic formations are generally not globally stabilizable.

Singularities in the dynamics lead to stable configurations that violate prescribed distances.

Acyclic formations can be globally stabilized with simple control laws.

Abstract

Formation control is concerned with the design of control laws that stabilize agents at given distances from each other, with the constraint that an agent's dynamics can depend only on a subset of other agents. When the information flow graph of the system, which encodes this dependency, is acyclic, simple control laws are known to globally stabilize the system, save for a set of measure zero of initial conditions. The situation has proven to be more complex when the graph contains cycles; in fact, with the exception of the cyclic formation with three agents, which is stabilized with laws similar to the ones of the acyclic case, very little is known about formations with cycles. Moreover, all of the control laws used in the acyclic case fail at stabilizing more complex cyclic formations. In this paper, we explain why this is the case and show that a large class of planar formations with cycles cannot be globally stabilized, even up to sets of measure zero of initial conditions. The approach rests on relating the information flow to singularities in the dynamics of formations. These singularities are in turn shown to make the existence of stable configurations that do not satisfy the prescribed edge lengths generic.