Exact robot navigation in geometrically complicated but topologically simple spaces

Abstract

Navigation functions on forests of stars, geometrically complicated C-spaces (configuration spaces) that are topologically indistinguishable from a simple disc punctured by disjoint smaller discs representing model obstacles, are constructed. For reasons of mathematical tractability, each C-space obstacle is approximated by a Boolean combination of linear and quadratic polynomial inequalities (with sharp corners allowed), and a calculus of implicit representations is used to effectively represent such obstacles. Evidence is provided of the effectiveness of this technology of implicit representations in the form of several simulation studies.