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Abstract-The paper proposes a new geometric approach to the stabilization of a hierarchical formation of unicycle robots. Hierarchical formations consist of elementary leader-follower units disposed on a rooted tree: each follower sees its relative leader as a fixed point in its own reference frame. Robots' linear velocity and trajectory curvature are forced to satisfy some given bounds. The major contribution of the paper is to study the effect of these bounds on the admissible trajectories of the main leader. In particular, we provide recursive formulas for the maximum velocity and curvature allowed for the main leader, so that the robots can achieve the desired formation while respecting their input constraints. An original formation control law is proposed and the asymptotic stabilization is proved. Simulation experiments illustrate the theory and show the effectiveness of the proposed designs.

Abstract-This paper studies a class of hierarchical formations for an ordered set of + 1 unicycle robots: the first robot plays the role of the leader and the formation is induced through a constraint function , so that the position and orientation of the th 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 induces a formation where the th robot follows a convex combination of the positions of the previous 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.