Smooth Feedback Construction Over Spherical Polytopes

In this work, we investigate the partitioning and control problems on the 2-sphere, the set of all unit vectors in R-3. Specifically, we present a spherical-polytope-based partitioning for the 2-sphere and then propose a novel approach to construct a feedback control law over a given set of spherical polytopes. Instead of designing the control law directly on the sphere, we propose a smooth atlas on it based on the gnomonic projection. We further show that the gnomonic map projects the spherical polytopes to Euclidean polytopes. Moreover, the kinematics evolving on a spherical polytope can be transformed via feedback into a single integrator in the Euclidean space. Thanks to these properties, control algorithms that were originally developed for polytopes in Euclidean spaces can now be applied to spherical polytopes on the 2-sphere. We conclude this paper by showing a control construction on the sphere with cluttered obstacles.