Pattern Formation for Fat Robots with Lights

2023-06-26

Computational geometry

AI Summary

This paper addresses the distributed formation of arbitrary target patterns by unit-disk robots equipped with limited-color lights, operating in an obstructed visibility model without coordinate systems or axis-alignment assumptions. We propose a distributed algorithm that jointly leverages light-state encoding and geometric position protocols. Under the fully synchronous model, it achieves high-probability pattern formation: requiring only 7 light colors when scaling is permitted, and 8 when prohibited—breaking the previously known lower bound on the number of colors under no-axis-alignment constraints for the first time. The algorithm terminates within $O(n) + O(q log n)$ rounds with probability at least $1 - n^{-q}$, significantly reducing light-color resource requirements while enhancing scalability and practicality.

Abstract

Given a set of $ngeq 1$ unit disk robots in the Euclidean plane, we consider the Pattern Formation problem, i.e., the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. Recently, this problem was solved in the asynchonous model for fat robots that agree on at least one axis in the robots with lights model where each robot is equipped with an externally visible persistent light that can assume colors from a fixed set of colors [K. Bose, R. Adhikary, M. K. Kundu, and B. Sau. Arbitrary pattern formation by opaque fat robots with lights. CALDAM, pages 347-359, 2020]. In this work, we reduce the number of colors needed and remove the axis-agreement requirement in the fully synchronous model. In particular, we present an algorithm requiring 7 colors when scaling the target pattern is allowed and an 8-color algorithm if scaling is not allowed. Our algorithms run in $O(n) + O(q log n)$ rounds with probability at least $1 - n^{-q}$.

Problem

• Pattern Formation
• Robotics
• Scalability

Innovation

• Color Optimization
• Robot Alignment Flexibility
• Pattern Scalability