Consensus of Nonholonomic Systems Using a Geometric PD Controller

Formation control, incorporating the nonholonomic constraints upfront, is a challenging task. At the individual agent level, Brockett's theorem states that nonholonomic systems cannot be stabilized to a point using smooth, and time-invariant feedback control law. We present a solution to the formation problem in a geometric framework by incorporating the constraint distribution at an individual agent level that obeys each agent's nonholonomic constraints. The objective of the consensus is then tackled by introducing a suitable Morse function. Critical points of the chosen Morse function form a consensus manifold. We take inspiration from the consensus law on Euclidean space and draw a comparison. The negative gradient of the Morse function forms the proportional control leading the trajectory to the set of critical points which is same as the consensus manifold. In addition, we propose a corollary that enables a priori specified orientation regulation in a set of wheeled mobile robots. These findings are verified using several numerical experiments with random initial conditions.