On the internal dynamics of formations of unicycle robots

This paper studies a general class of formations of unicycle robots. One of the robots plays the role of the leader and the formation is induced through a constraint function F that depends on the position and the orientation of the vehicles. We investigate the flexibility of such formations with respect to the leader's reference frame by introducing the notion of formation internal dynamics, we characterize its equilibria and provide sufficient conditions for their existence. The theory is specialized to a particular constraint function F that leads to a formation where robot i follows a convex combination of the positions of the preceding i-1 robots. Sufficient conditions are presented in this scenario that guarantee that the position and orientation of the vehicles with respect to the leader's reference frame are confined in a specific polyhedral region, regardless of the trajectory of the leader, provided that its curvature is sufficiently small.