The paper deals with a general class of leader-follower formations of unicycle robots induced by a constraint function that depends on the position and the orientation of the vehicles. We study the flexibility of such formations by introducing the notion of formation internal dynamics, characterize its equilibria and give sufficient geometric conditions for their existence. In particular, we show that the displacement and the relative orientation of each follower with respect to the leader's reference frame are fixed if and only if the robots either move along circular paths or parallel straight lines. These equilibrium configurations always exist if the trajectory of the leader is a circle of sufficiently small curvature or a straight line.

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.

The paper focuses on leader-follower formations of nonholonomic mobile robots. A formation control alternative to those existing in the literature is introduced. We show that the geometry of the formation imposes a bound on the maximum admissible curvature of leader trajectory. A peculiar feature of the proposed strategy is that the followers position is not rigidly fixed with respect to the leader reference frame but varies in suitable cones centered in the leader reference frame. Our approach also applies to hierarchical multirobot formations described by rooted tree graphs. Simulation experiments confirm the effectiveness of the proposed control schemes.