The Construction of Analytic Diffeomorphisms for Exact Robot Navigation on

A navigation function is a scalar valued function on a robot configuration space which encodes the task of moving to a desired destination without hitting any obstacles. Our program of research concerns the construction of navigation functions on a family of configuration spaces whose “geometric expressiveness” is rich enough for navigation amidst real world obstacles. A sphere world is a compact connected subset of En whose boundary is the finite union of disjoint (n-1)-spheres. In previous work we have constructed navigation functions for every sphere world. In this paper we embark upon the task of extending the construction of navigation function to “star worlds.” A star world is a compact connected subset of En obtained by removing from a compact star shaped set a finite number of smaller disjoint open star shaped sets.This paper introduces a family of transformations from any star world into a suitable sphere world model, and demonstrates that these transformations are actually analytic diffeomorphisms. Since the defining properties of navigation functions are invariant under diffeomorphism, this construction, in composition with the previously developed navigation function on the corresponding model sphere world, immediately induces a navigation function on the star world.

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