Simultaneous velocity consensus and shape control for a finite number of point agents on the unit circle

In this paper we study a second order, distributed control system for a finite number of point agents on the unit circle that achieves simultaneously velocity consensus and distance based formation shape control. Based on tools from Riemannian geometry, we propose a system of second order differential equations on the N-dimensional torus that achieves these two goals. We prove convergence of the trajectories to single closed geodesics on a torus and investigate the stability properties of the distributed algorithm.