Gathering of seven autonomous mobile robots on triangular grids

TL;DR

This paper investigates the gathering problem of seven autonomous robots on triangular grids, demonstrating that increasing visibility range from 1 to 2 enables collision-free gathering under certain assumptions, with the algorithm being optimal in visibility range.

Contribution

The paper establishes the minimal visibility range needed for collision-free gathering of seven robots on triangular grids and provides an optimal algorithm for visibility range 2.

Findings

No collision-free gathering algorithm exists with visibility range 1.

An effective gathering algorithm is proposed for visibility range 2.

The algorithm is proven optimal regarding visibility range.

Abstract

In this paper, we consider the gathering problem of seven autonomous mobile robots on triangular grids. The gathering problem requires that, starting from any connected initial configuration where a subgraph induced by all robot nodes (nodes where a robot exists) constitutes one connected graph, robots reach a configuration such that the maximum distance between two robots is minimized. For the case of seven robots, gathering is achieved when one robot has six adjacent robot nodes (they form a shape like a hexagon). In this paper, we aim to clarify the relationship between the capability of robots and the solvability of gathering on a triangular grid. In particular, we focus on visibility range of robots. To discuss the solvability of the problem in terms of the visibility range, we consider strong assumptions except for visibility range. Concretely, we assume that robots are fully synchronous and they agree on the direction and orientation of the x-axis, and chirality in the triangular grid. In this setting, we first consider the weakest assumption about visibility range, i.e., robots with visibility range 1. In this case, we show that there exists no collision-free algorithm to solve the gathering problem. Next, we extend the visibility range to 2. In this case, we show that our algorithm can solve the problem from any connected initial configuration. Thus, the proposed algorithm is optimal in terms of visibility range.