Controlling and Stabilizing a Rigid Formation Using a Few Agents

We show in this paper that a small subset of agents of a formation of $n$ agents in Euclidean space can control the position and orientation of the entire formation. We consider here formations tasked with maintaining interagent distances at prescribed values. It is known that when the interagent distances specified can be realized as the edges of a rigid graph, there is a finite number of possible configurations of the agents that satisfy the distance constraints, up to rotations and translations of the entire formation. We show here that under mild conditions on the type of control used by the agents, a small subset of them forming a clique can work together to control both position and orientation of the formation as a whole. Mathematically, we investigate the effect of certain permissible perturbations of a nominal dynamics of the formation system. In particular, we show that any such perturbation leads to a rigid motion of the entire formation. Furthermore, we show that the map which assigns to a perturbation the infinitesimal generator of the corresponding rigid motion is locally surjective, which then leads to the controllability result.