Constrained Stabilization on the n-Sphere with Conic and Star-shaped Constraints

The problem of constrained stabilization on the n-sphere under star-shaped constraints is considered. We propose a control strategy that allows one to almost globally steer the state to a desired location while avoiding star-shaped constraints on the n-sphere. Depending on the state's proximity to the unsafe regions, the state is either guided towards the target location along the geodesic connecting the target to the state or steered towards the antipode of a predefined point lying in the interior of the nearest unsafe region. We prove that the target location is almost globally asymptotically stable under the proposed continuous, time-invariant feedback control law. Non-trivial simulation results on the 2-sphere and the 3-sphere demonstrate the effectiveness of the theoretical results.