Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds

TL;DR

This paper establishes tight separation bounds for unlabeled multi-robot motion planning in polygonal workspaces, ensuring solvability and providing an efficient algorithm under these conditions.

Contribution

It introduces precise separation bounds that guarantee solutions for dense robot configurations and presents an algorithm with proven runtime complexity under these bounds.

Findings

Separation bounds of 4 units between start-start and target-target positions.

Separation of 3 units between start and target positions.

Algorithm guarantees a solution if separation bounds are met.

Abstract

We consider the unlabeled motion-planning problem of $m$ unit-disc robots moving in a simple polygonal workspace of $n$ edges. The goal is to find a motion plan that moves the robots to a given set of $m$ target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end. If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time $O(n \log n + mn + m^2)$ if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance $4$ apart, any two target positions are at least distance $4$ apart, and any pair of a start and a target positions is at least distance $3$ apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary.