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.

中文翻译：

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基于广义形状扩展的运动规划算法的数学性质

运动计划是自主系统和机器人技术的重要方面，也是研究的活跃领域。最近提出的一种基于采样的运动计划算法，称为“通用形状扩展”（Generalized Shape Expansion，GSE），与几种现有的公认算法相比，在计算时间上具有显着的改进。GSE也被证明是概率完整的。但是，GSE的渐近最优性尚待研究。为此，在本文中，我们通过研究针对长廊问题的行为，证明GSE算法不是渐近最优的。为了获得概率完全和渐近最优的基于广义形状的算法，随后提出了GSE的修改版本，即“ GSE *”算法。GSE *算法的上述所需数学特性通过其详细分析得到了证明。发现数值模拟与GSE *算法的理论结果一致。