Control of minimally persistent formations in the plane

This paper studies the problem of controlling the shape of a formation of point agents in the plane. A model is considered where the distance between certain agent pairs is maintained by one of the agents making up the pair; if enough appropriately chosen distances are maintained, with the number growing linearly with the number of agents, then the shape of the formation will be maintained. The detailed question examined in the paper is how one may construct decentralized nonlinear control laws to be operated at each agent that will restore the shape of the formation in the presence of small distortions from the nominal shape. Using the theory of rigid and persistent graphs, the question is answered. As it turns out, a certain submatrix of a matrix known as the rigidity matrix can be proved to have nonzero leading principal minors, which allows the determination of a stabilizing control law.