On Probabilistic Completeness of the Generalized Shape Expansion-Based Motion Planning Algorithm

Mathematical Properties of Generalized Shape Expansion-Based Motion Planning Algorithms

“…We indicate a summary of the proof of probabilistic completeness of the GSE algorithm. Further details are provided in [26]. We show this fact by establishing a few properties of the GSE visibility function f : 2 Xfree → 2 Xfree defined as…”

“…Simulation studies have been carried out in 2-D and 3-D environments having 4 and 16 obstacles representing low to high obstacle densities using MATLAB R2020a on Intel Core i7 2.2GHz processor. Simulations and comparison studies related to probabilistic completeness of the GSE have already been provided in [26] and hence, are omitted here for brevity. First, feasible shortest paths generated using the GSE and GSE ⋆ algorithms over 50 iterations in the 2-D and 3-D environments each having both 4 and 16 obstacles are illustrated in Figs.…”

“…. We note that the GSE algorithm is probabilistically complete, as described in Section III-B (details available in [26]). Since the graph generated by the GSE algorithm is a subgraph of the graph generated by the GSE ⋆ algorithm, the GSE ⋆ algorithm is probabilistically complete.…”

“…This paper is evolved from [26], where the probabilistic completeness property of the GSE algorithm was established. However asymptotic optimality of the GSE algorithm has not yet been studied so far.…”