  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,
their trajectories remain bounded within a compact set, and their turn-rates
adhere to the saturation limits. We also characterize the explicit nature of
the boundary of this 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 that 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.
