Mathematical Properties of Generalized Shape Expansion-Based Motion Planning Algorithms

TL;DR

This paper analyzes the mathematical properties of the Generalized Shape Expansion (GSE) motion planning algorithm, demonstrating its lack of asymptotic optimality and proposing a modified GSE* version with proven probabilistic completeness and asymptotic optimality.

Contribution

The paper provides a theoretical analysis of GSE's properties and introduces GSE* with proven probabilistic completeness and asymptotic optimality.

Findings

GSE is not asymptotically optimal.

GSE* achieves probabilistic completeness.

GSE* is asymptotically optimal.

Abstract

Motion planning is an essential aspect of autonomous systems and robotics and is an active area of research. A recently-proposed sampling-based motion planning algorithm, termed 'Generalized Shape Expansion' (GSE), has been shown to possess significant improvement in computational time over several existing well-established algorithms. The GSE has also been shown to be probabilistically complete. However, asymptotic optimality of the GSE is yet to be studied. To this end, in this paper we show that the GSE algorithm is not asymptotically optimal by studying its behaviour for the promenade problem. In order to obtain a probabilistically complete and asymptotically optimal generalized shape-based algorithm, a modified version of the GSE, namely 'GSE*' algorithm, is subsequently presented. The forementioned desired mathematical properties of the GSE* algorithm are justified by its detailed analysis. Numerical simulations are found to be in line with the theoretical results on the GSE* algorithm.