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.