Minimizing The Maximum Distance Traveled To Form Patterns With Systems of Mobile Robots
This work addresses the problem of efficient robot coordination for pattern formation, which is incremental as it builds on existing feasibility-focused research by adding optimality constraints.
The paper tackles the pattern formation problem for mobile robots by minimizing the maximum distance traveled by any robot to achieve a desired pattern, focusing on systems of three robots and establishing necessary conditions for optimal solutions.
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}.