Team assembling problem for asynchronous heterogeneous mobile robots

We investigate the team assembling problem for a swarm of heterogeneous mobile robots which requires the robots to autonomously partition themselves into teams satisfying a given specification A=(a1,a2,…,ak), where ai is the number of robots with color (i.e., robot type) i in one team. A robot, which is represented by a point in the two-dimensional Euclidean space, is asynchronous, oblivious, and anonymous in the sense that robots with the same color are indistinguishable and all robots execute the same algorithm to determine their moves. It has neither any access to the global coordinate system nor any explicit communication medium. We show that GCD(a1,a2,…,ak)=1 is a necessary and sufficient condition for the robots to have an algorithm to solve the team assembling problem in a self-stabilizing manner, i.e., starting from any arbitrary initial configuration, the robots form teams according to the specification.