We study the effectiveness of metrics for Multi-Robot Motion-Planning (MRMP) when using RRT-style sampling-based planners. These metrics play the crucial role of determining the nearest neighbors of configurations and by that they regulate the connectivity of the underlying roadmaps produced by the planners and other properties like the quality of solution paths. After screening over a dozen different metrics we focus on the five most promising ones—two more traditional metrics, and three novel ones which we propose here, adapted from the domain of shape-matching. In addition to the novel multi-robot metrics, a central contribution of this work are tools to analyze and predict the effectiveness of metrics in the MRMP context. We identify a suite of possible substructures in the configuration space, for which it is fairly easy (i) to define a so called natural distance which allows us to predict the performance of a metric. This is done by comparing the distribution of its values for sampled pairs of configurations to the distribution induced by the natural distance; (ii) to define equivalence classes of configurations and test how well a metric covers the different classes. We provide experiments that attest to the ability of our tools to predict the effectiveness of metrics: those metrics that qualify in the analysis yield higher success rate of the planner with fewer vertices in the roadmap. We also show how combining several metrics together leads to better results (success rate and size of roadmap) than using a single metric.
(a) Tunnel scenario. The environment consists of a T-shaped free space and requires the robots in one side to exchange places with the robots on the other side. There are 6 translating disc robots of radius 2 and the width of each arm is 5, so the robots cannot exchange places within an arm without leaving it.
(b) Chambers scenario. The environment consists of three chambers. The structure of each chamber allows the robots to exit from the chamber in any order, not necessarily in the order they entered to the chamber (as opposed to the arms in the Tunnel scenario).
(c) 8-Puzzle scenario. The environment can be naturally partitioned into nine cells that form 33 grid. A robot can translate only between adjacent cells.
