A Smooth Distributed Feedback for Formation Control of Unicycles

This paper investigates a formation control problem in which a group of kinematic unicycles is made to converge to a desired formation with parallel heading angles and come to a stop. A control law is presented, which solves this problem for almost all initial conditions in any given compact set. The proposed control law is local and distributed, meaning that each unicycle is only required to sense its relative displacement measured in its own body frame, and the relative heading angle with respect to each of its neighbors. No communication between the unicycles is required. The sensing graph is assumed to be connected, undirected, and time invariant. The idea used to solve the above-mentioned formation control problem is to rigidly attach to the body frame of each unicycle an appropriate fixed offset vector. Stabilizing the desired formation amounts to achieving consensus of the endpoints of the offset vectors, and simultaneously synchronizing the unicycles' heading angles. A control law achieving this goal is constructed by combining a bounded translational consensus controller with an attitude synchronizer. As a special case, the proposed solution solves the full unicycle synchronization problem, in which the unicycle positions are made to converge to each other, while the unicycle headings are made to align.