The construction of analytic diffeomorphisms for exact robot navigation on star worlds

Abstract

A Euclidean Sphere World is a compact connected submanifold of Euclidean n n -space whose boundary is the disjoint union of a finite number of ( n − 1 ) (n - 1) dimensional Euclidean spheres. A Star World is a homeomorph of a Euclidean Sphere World, each of whose boundary components forms the boundary of a star shaped set. We construct a family of analytic diffeomorphisms from any analytic Star World to an appropriate Euclidean Sphere World "model." Since our construction is expressed in closed form using elementary algebraic operations, the family is effectively computable. The need for such a family of diffeomorphisms arises in the setting of robot navigation and control. We conclude by mentioning a topological classification problem whose resolution is critical to the eventual practicability of these results.