Pattern Formation Problem for Synchronous Mobile Robots in the Three Dimensional Euclidean Space

TL;DR

This paper extends the understanding of pattern formation in synchronized mobile robots from 2D to 3D space, providing a necessary and sufficient condition based on symmetry groups for the robots to form a target pattern.

Contribution

It generalizes the symmetricity concept to 3D space and characterizes pattern formation conditions for FSYNC robots using rotation groups.

Findings

Necessary and sufficient condition for pattern formation in 3D space.

Extension of symmetricity concept to 3D with rotation groups.

Algorithm for pattern formation in 3D space.

Abstract

We consider a swarm of autonomous mobile robots each of which is an anonymous point in the three-dimensional Euclidean space (3D-space) and synchronously executes a common distributed algorithm. We investigate the pattern formation problem that requires the robots to form a given target pattern from an initial configuration and characterize the problem by showing a necessary and sufficient condition for the robots to form a given target pattern. The pattern formation problem in the two dimensional Euclidean space (2D-space) has been investigated by Suzuki and Yamashita (SICOMP 1999, TCS 2010), and Fujinaga et al. (SICOMP 2015). The symmetricity $\rho(P)$ of a configuration (i.e., the positions of robots) $P$ is intuitively the order of the cyclic group that acts on $P$. It has been shown that fully-synchronous (FSYNC) robots can form a target pattern $F$ from an initial configuration $P$ if and only if $\rho(P)$ divides $\rho(F)$. We extend the notion of symmetricity to 3D-space by using the rotation groups each of which is defined by a set of rotation axes and their arrangement. We define the symmetricity $\varrho(P)$ of configuration $P$ in 3D-space as the set of rotation groups that acts on $P$ and whose rotation axes do not contain any robot. We show the following necessary and sufficient condition for the pattern formation problem which is a natural extension of the existing results of the pattern formation problem in 2D-space: FSYNC robots in 3D-space can form a target pattern $F$ from an initial configuration $P$ if and only if $\varrho(P) \subseteq \varrho(F)$. For solvable instances, we present a pattern formation algorithm for oblivious FSYNC robots. The insight of this paper is that symmetry of mobile robots in 3D-space is sometimes lower than the symmetry of their positions and the robots can show their symmetry by their movement.