Geometrical methods for mismatched formation control

Formation shape control for a collection of point agents is concerned with devising decentralized control laws which will move the formation so that certain inter-agent distances reach prescribed desired values. Standard algorithms such as that proposed by [1] perform steepest descent of a smooth error function, ensuring that the formations will always converge to equilibrium points for the gradient flow. The convergence to equilibrium points of these algorithms depends critically on the fact that there is no mismatch in two neighboring agents' understandings of what the desired distance between them is supposed to be. If mismatches occur then the limiting dynamics will typically become periodic, as has been explored in several recent papers such as, e.g., [2]-[5]. The goal then becomes to develop methods to count such relative equilibria and characterize their local stability properties. In this paper we apply basic Lie group methods to analyze the relative equilibria in the presence of mismatches, thus simplifying earlier proofs in the literature.