Predicting, Avoiding, and Achieving Conjunctions on Spherical Manifolds

This paper presents a class of problems arising from applications in which vehicles move on spherical surfaces. It addresses the problem of prediction, avoidance, and achievement of positional conjunctions of such vehicles on spherical manifolds. To this end, the notion of collision triangles on spheres is developed, and analytical conditions governing the speed ratios and directions of motion of point objects that lead to conjunction are derived. These fundamental results are then extended to circular patch-shaped objects, which are realistic representations of vehicles moving on the surface of the sphere, thereby generalizing the notion of Euclidean collision cones to spherical manifolds. It is shown that these collision cones can be used to generate geodesic paths via guidance commands for speed and/or heading angle maneuvers. They can also be used to facilitate coverage path planning/information transfer between vehicles when the patches approximate sensing/communication footprints. The results are extended to multiple revolutions of the vehicles on the sphere, establishing a close link with Diophantine equations. These results address guidance problems in applications involving multiple satellites moving on spherical manifolds, as well as multiple aerial vehicles performing cooperative maintenance and inspection tasks on spherical domes and tanks.