Adaptive perturbation method for global stabilization of minimally rigid formations in the plane

This paper considers the problem of control of a minimally rigid formation shape in a plane for n agents. Because the formation graph contains cycles, undesired equilibria arise under the existing control laws and the global stability of the desired formation is not ensured. In this paper, a constructive adaptive perturbation method is proposed for globally stabilizing minimal rigid formations. Perturbations are added to the movement directions of appropriate n-2 agents in the sub-graph which contains the remaining edges of the minimally rigid formation graph by removing a spanning tree. It is shown that the proposed novel bidirectional control law can not only guarantee the global asymptotical stability of the desired formation, but also ensure that no collision happens between any two adjacent agents during the motion. Simulation results are provided to illustrate the effectiveness of the control algorithm.