A New Framework for Optimal Path Planning of Rectangular Robots Using a Weighted $L_p$ Norm

Abstract

This paper introduces a new framework for modeling the optimal path planning problem of rectangular robots. Typically constraints for the safe, obstacle-avoiding path involve a set of inequalities expressed using logical OR operations, which makes the problem difficult to solve using existing optimization algorithms. Inspired by the geometry of the unit sphere of the weighted $L_p$ norm, the authors find exact and approximate constraints for safe configurations using only logical AND operations. The proposed method does not require integer programming nor computation of a Minkowski sum in the configuration space. In particular, the authors analyze two different cases of obstacle geometry: circular obstacles and rectangular obstacles. Using the weighted $L_p$ norm requires six inequalities to represent the exact constraints for collision avoidance of circular obstacles using AND operations, and eight inequalities for rectangular obstacles. Four shortest path planning examples are analyzed to validate the effectiveness of the proposed method.