A navigation function is an artificial potential energy function on a robot configuration space (C-space) which encodes the task of moving to an arbitrary destination without hitting any obstacle. In particular, such a function possesses no spurious local minima. In this paper we construct navigation functions on forests of stars: geometrically complicated C-spaces that are topologically indistinguishable from a simple disc punctured by disjoint smaller discs, representing model obstacles. For reasons of mathematical tractability we approximate each C-space obstacle by a Boolean combination of linear and quadratic polynomial inequalities (with sharp corners allowed), and use a calculus of implicit representations to effectively represent such obstacles. We provide evidence of the effectiveness of this technology of implicit representations in the form of several simulation studies illustrated at the end of the paper
