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 less 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：连续多机器人运动规划(MRMP)的若干基于取样的方法中,一个基础结构是高光谱图(TR),它来自通过高光谱产品为个体机器人建造的多光谱图。我们研究TR编码一个接近最佳的MRMP解决方案的条件 -- -- 满足这些条件意味着各种受欢迎的规划者,包括DRRT* 和适用于连续域的离散方法M* 和 CBS 的近最佳性。我们开发了这种类型的第一次有限抽样分析,其中说明了样本的数量、其确定性分布和每个个体机器人应使用的连接光度的大小,以保障使用TR的接近最佳性。这大大改进了以往一次的零光谱分析,其中样本数量往往不尽相同。我们新的有限抽样规模分析有助于保证在有限的时间内实际采用高质量的解决方案。为了实现我们的新结果,我们首先开发了一个取样方案,我们称之为交错格的网格,每个个体光谱图应该使用它来保证使用近光谱的连接度图,从而保证使用TRM的接近最佳性。这极大地改进了先前的机器人的样本,而我们要求更深入地进行模拟的机器人的实验。