Navigation functions for focally admissible surfaces

Abstract

This work presents a sharper condition for the applicability of Navigation Functions (NF). The condition depends on the placement of the destination with respect to the focal surfaces of obstacles. The focal surface is the locus of centers of principal curvatures. If each obstacle encompasses at least one of its focal surfaces, then the world is navigable using a Koditschek-Rimon NF (KRNF). Moreover, the Koditschek-Rimon (KR) potential is non-degenerate for all destinations which are not on a focal surface. So, for almost all destinations there exists a non-degenerate KR potential. This establishes a link between the differential geometry of obstacle surfaces and KRNFs. Channel surfaces (e.g. Dupin cyclides) and certain Boolean operations between shapes are examples of admissible obstacles. We also prove a weak converse result about the inexistence of a KRNF for obstacles with some concave point, for large tuning parameters. Finally, our results support non-trivial simulations in a forest, a pipeline and a cynlinder rig, with some notes about allowable types of non-smoothness.