An underlying structure in several sampling-based methods for continuous multi-robot motion planning (MRMP) is the tensor roadmap (TR), which emerges from combining multiple PRM graphs constructed for the individual robots via a tensor product. We study the conditions under which the TR encodes a near-optimal solution for MRMP -- satisfying these conditions implies near optimality for a variety of popular planners, including dRRT*, and the discrete methods M* and CBS when applied to the continuous domain. We develop the first finite-sample analysis of this kind, which specifies the number of samples, their deterministic distribution, and magnitude of the connection radii that should be used by each individual PRM graph, to guarantee near-optimality using the TR. This significantly improves upon a previous asymptotic analysis, wherein the number of samples tends to infinity. Our new finite sample-size analysis supports guaranteed high-quality solutions in practice within finite time. To achieve our new result, we first develop a sampling scheme, which we call the staggered grid, for finite-sample motion planning for individual robots, which requires significantly fewer samples than previous work. We then extend it to the much more involved MRMP setting which requires to account for interactions among multiple robots. Finally, we report on a few experiments that serve as a verification of our theoretical findings and raise interesting questions for further investigation.

翻译：摘要：连续多机器人运动规划（MRMP）中若干基于采样的方法隐含一种张量路图（TR）结构，该结构源于通过张量积组合为各机器人独立构建的PRM图。我们研究了TR编码MRMP近优解的条件——满足这些条件意味着多种主流规划器（包括dRRT*，以及应用于连续域的离散方法M*和CBS）具有近优性。我们首次建立了此类有限样本分析，明确规定了每个PRM图应使用的样本数量、确定性分布及连接半径的取值，以保证基于TR的近优性。这显著改进了先前样本数量趋于无穷时的渐近分析。新的有限样本量分析能够在有限时间内保证实际应用中获得高质量解。为取得这一新成果，我们首先为单机器人有限样本运动规划开发了一种称为交错网格的采样方案，该方案所需的样本量远少于先前工作。随后将其扩展至需考虑多机器人交互的更复杂的MRMP场景。最后，我们通过若干实验验证理论发现，并提出了值得进一步探究的有趣问题。