Trajectory-Constrained Collective Circular Motion with Different Phase Arrangements

A methodology to stabilize synchronized, balanced, or symmetric phase patterns of unicycle-type agents on a desired common circle, while restricting their trajectories to a certain region of interest, is proposed in this note. These phase patterns are characterized by the motion of the collective centroid of the group of agents and derived by optimizing the average linear momentum of the group. Under a mild assumption on initial states of the agents, we design control laws by exploiting the concept of barrier Lyapunov function in conjunction with bounded phase potential functions. We show that the agents asymptotically stabilize to a desired phase arrangement and their trajectories remain bounded during stabilization. We obtain bounds on the different quantities of interest in the postdesign analysis and show that these bounds depend on the initial conditions and can be altered by adjusting the controller gains. We also prove convergence when the control input is saturated to a prespecified value. Finally, we provide a discussion on the application and limitation of the proposed approach and characterize the feasible initial conditions.