Synchronization and balancing around simple closed polar curves with bounded trajectories

Abstract

The problem of synchronization and balancing around simple closed polar curves is addressed for unicycle-type multi-agent systems. Leveraging the concept of barrier Lyapunov function in conjunction with bounded Lyapunov-like curve-phase potential functions, we propose distributed feedback control laws and show that the agents asymptotically stabilize to the desired closed curve in synchronized and balanced curve-phase patterns, and their trajectories remain bounded within a compact set. Our control design methodology is based on the proposition of two models, namely the parametric-phase control model and the curve-phase control model. We also characterize the trajectory-constraining set based on the magnitude of the safe distance of the exterior boundary from the desired curve. We further establish a connection between the perimeters and areas of the trajectory-constraining set with the perimeter and area of the desired curve. We obtain bounds on different quantities of interest in the post-design analysis and provide simulation results to illustrate the theoretical findings.