Non-collision conditions in multi-agent robots formation using local potential functions

An analysis of the convergence and non-collision conditions of a formation control strategy for multi-agent robots based on local potential functions is presented. The goal is to coordinate a group of agents, considered as points in plane, to achieve a particular formation. The control law is designed using local attractive forces only where every agent knows the position of another two agents reducing the requirements of the control law implementation. The control law guarantees the convergence to the desired formation but does not avoid inter-agent collisions. A set of necessary and sufficient non-collision conditions based on the explicit solution of the closed-loop system is derived. The conditions allow to conclude from the initial conditions whether or not the agents will collide. The formal proof is presented for the case of three agents. The result is extended to the case of formations of three unicycles.