[Paper Review] Optimal Asynchronous Rendezvous for Mobile Robots with Lights

Takashi Okumura, Koichi Wada+1 more|arXiv (Cornell University)|Jul 14, 2017

Optimization and Search Problems14 references3 citations

TL;DR

This paper presents optimal rendezvous algorithms for two asynchronous mobile robots with 2-color lights, achieving rendezvous in rigid movement and in non-rigid movement when the minimum move distance δ is known. The algorithm uses a two-color light system (A and B) and ensures convergence through coordinated color changes based on distance thresholds, proving that 2 colors are sufficient in these settings—resolving an open question on minimal light requirements in asynchronous systems.

ABSTRACT

We study a Rendezvous problem for 2 autonomous mobile robots in asynchronous settings with persistent memory called light. It is well known that Rendezvous is impossible when robots have no lights in basic common models, even if the system is semi-synchronous. On the other hand, Rendezvous is possible if robots have lights with a constant number of colors in several types lights. In asynchronous settings, Rendezvous can be solved by robots with 4 colors of lights in non-rigid movement, if robots can use not only own light but also other robot's light (full-light), where non-rigid movement means robots may be stopped before reaching the computed destination but can move a minimum distance δ> 0 and rigid movement means robots can reach the computed destination. In semi-synchronous settings, Rendezvous can be solved with 2 colors of full-lights in non-rigid movement. In this paper, we show that in asynchronous settings, Rendezvous can be solved with 2 colors of full-lights in rigid movement and in non-rigid movement if robots know the value of the minimum distance δ. We also show that Rendezvous can be solved with 2 colors of full-lights in general non-rigid movement if we consider some reasonable restricted class of asynchronous settings.

Motivation & Objective

• To determine the minimal number of light colors required for rendezvous in fully asynchronous mobile robot systems.
• To close the gap in known solvability results for rendezvous under asynchronous schedulers with limited light capabilities.
• To show that 2 colors of full-light are sufficient for rendezvous in both rigid and non-rigid movement when the minimum move distance δ is known.
• To investigate whether rendezvous can be solved with 2 colors in general non-rigid asynchronous systems, proposing a restricted model (LC-atomic) where it holds.

Proposed method

• Proposes a two-color light system (A and B) where robots coordinate based on observed distance and light states.
• Uses a distance-based decision rule: robots switch from A to B when distance exceeds 2δ, and back to A when distance drops below 2δ.
• Introduces a cycle-based protocol where robots perform Look-Compute-Move cycles, with color changes triggered only after observing stable configurations.
• Employs a time-synchronization abstraction via cycle start times to ensure consistent state transitions despite asynchrony.
• Analyzes convergence under rigid movement and non-rigid movement with known δ, proving that robots eventually meet when both are in state A and distance is ≤2δ.
• Introduces the LC-atomic model—a restricted asynchronous setting where rendezvous with 2 colors is achievable, even in general non-rigid movement.

Experimental results

Research questions

• RQ1 Can rendezvous be achieved with only 2 colors of lights in asynchronous systems with rigid movement?
• RQ2 Is 2-color full-light sufficient for rendezvous in non-rigid movement when the minimum move distance δ is known?
• RQ3 Can rendezvous be solved with 2 colors in general non-rigid asynchronous systems, or is a restricted model required?
• RQ4 What is the minimal number of light colors needed for rendezvous in asynchronous settings with persistent memory (lights)?
• RQ5 Does the presence of δ knowledge enable optimal (2-color) rendezvous in non-rigid movement?

Key findings

• Rendezvous is solvable with 2 colors of full-light in asynchronous systems under rigid movement.
• Rendezvous is solvable with 2 colors of full-light in non-rigid movement when the minimum move distance δ is known.
• The proposed algorithm ensures convergence when both robots are in state A and distance is at most 2δ, with convergence guaranteed in finite time.
• In the LC-atomic model—a restricted asynchronous setting—2-color rendezvous is achievable even in general non-rigid movement.
• The paper establishes that 2 colors are optimal for rendezvous in asynchronous systems under rigid and δ-known non-rigid movement, closing a gap in prior work.
• It remains open whether 2-color rendezvous is possible in general non-rigid asynchronous systems without additional restrictions.