Minimizing The Maximum Distance Traveled To Form Patterns With Systems of Mobile Robots

TL;DR

This paper addresses the problem of forming patterns with mobile robots while minimizing the maximum distance traveled, providing necessary conditions for optimality and solutions for three-robot systems, and introducing a triangle similarity metric.

Contribution

It introduces the first known optimal solution for pattern formation with minimal maximum travel distance for three robots and proposes a new triangle similarity metric.

Findings

Optimal solutions satisfy specific necessary conditions.

A new triangle similarity metric ranging from 0 to 1.

Application of the metric beyond pattern formation.

Abstract

In the pattern formation problem, robots in a system must self-coordinate to form a given pattern, regardless of translation, rotation, uniform-scaling, and/or reflection. In other words, a valid final configuration of the system is a formation that is \textit{similar} to the desired pattern. While there has been no shortage of research in the pattern formation problem under a variety of assumptions, models, and contexts, we consider the additional constraint that the maximum distance traveled among all robots in the system is minimum. Existing work in pattern formation and closely related problems are typically application-specific or not concerned with optimality (but rather feasibility). We show the necessary conditions any optimal solution must satisfy and present a solution for systems of three robots. Our work also led to an interesting result that has applications beyond pattern formation. Namely, a metric for comparing two triangles where a distance of $0$ indicates the triangles are similar, and $1$ indicates they are \emph{fully dissimilar}.