Exact robot navigation in geometrically complicated but topologically simple spaces

Abstract

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. Robotics and Automation, Volume 3, 1990, pages 1937-1942 This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. NOTE: At the time of publication, author Daniel Koditschek was affiliated with Yale University. Currently, he is a faculty member in the Department of Electrical and Systems Engineering at the University of Pennsylvania. Comments Copyright 1990 IEEE. Reprinted from Proceedings of the IEEE International Conference onThis conference paper is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/386 Exact Robot Navigation in Geometrically Complicated but Topologically Simple SpacesElon Rimon a n d Daniel E. Koditschek Center for Systems Science Yale University, Department of Electrical Engineering Abstract A navigation function is an artificial potential energy function on a robot configuration space (C-space) which encodes the task of moving t o 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 repre...

On completeness of RFM solution structures

“…A brief English summary of the theory of R-functions is available as a technical report (Shapiro, 1991). Numerous applications of the theory are beginning to appear in English literature (Waberski, 1978;Rimon and Koditschek, 1990;Sourin and Pasko, 1995;Pasko et al, 1995;Rvachev and Sheiko, 1995;Rvachev et al, 1997;Bloomenthal, 1997;Shapiro and Tsukanov, 1998). …”

Escaping route method for a trap situation in local path planning

“…This problem has been alleviated by: 1) the redefinition of potential functions with no or a few local minima; and 2) the utilization of efficient search techniques with the capability of escaping from a local minimum. The first class of treatment includes: repulsive potential functions with angle distributions [5], the navigation function [6,7] and harmonic potential field [8]. However, these solutions are limited to simple or conservatively bounded obstacles such that a considerable portion of the workspace might be ignored.…”

Terrain Exploration by Autonomous Robots

“…The method described here for long-range route planning is a navigation-function method (Rimon and Koditschek 1988, 1989, and 1990. In navigation-function methods, the navigation route is determined in two steps.…”