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On a Class of Hierarchical Formations of Unicycles and Their Internal Dynamics

This paper studies a class of hierarchical formations for an ordered set of n + 1 unicycle robots: the first robot plays the role of the leader and the formation is induced through a constraint function F, so that the position and orientation of the ith robot depends only on the pose of the preceding ones. We study the dynamics of the formation with respect to the leader's reference frame by introducing the concept of reduced internal dynamics, we characterize its equilibria and provide sufficient conditions for their existence. The discovered theoretical results are applied to the case in which the constraint F induces a formation where the ith robot follows a convex combination of the positions of the previous i - 1 vehicles. In this case, we prove that if the curvature of the leader's trajectory is sufficiently small, the positions and orientations of the robots, relative to the leader's reference frame, are confined in a precise polyhedral region.