The Impact of the Gabriel Subgraph of the Visibility Graph on the Gathering of Mobile Autonomous Robots

Abstract

In this work, we reconsider the well-known Go-To-The-Center algorithm due to Ando, Suzuki, and Yamashita [2] for gathering in the plane n autonomous mobile robots with limited viewing range. The above authors have introduced it as a discrete, round-based algorithm and proved its correctness. In [8], by Degener et al. it is shown that the algorithm gathers in $\varTheta \left( n^2\right)$ rounds. Remarkably, this algorithm exploits the fact, that during its execution, many collisions of robots occur. Such collisions are interpreted as a success because it is assumed that such collided robots behave the same from now on. This is o.k. under the assumption, those robots have no extent. Otherwise, collisions should be avoided.

In this paper, we consider a continuous Go-To-The-Center (GTC) strategy in which the robots continuously observe the positions of their neighbors and adapt their speed (assuming a speed limit) and direction. Our first results are time bounds of $O\left( n^2\right)$ for gathering in two-dimensional Euclidean space, and $\varTheta \left( n\right)$ for the one-dimensional case.

Our main contribution is the introduction and evaluation of a continuous algorithm which performs Go-To-The-Center considering only the neighbors of a robots w.r.t. the Gabriel subgraph of the visibility graph (GTGC). We show that this modification still correctly executes gathering in one and two dimensions, with the same time bounds as above. Simulations exhibit a severe difference of the behavior of the GTC and the GTGC strategy: Whereas lots of collisions occur during a run of the GTC strategy, typically only one, namely the final collision occurs during a run of the GTGC strategy. We can prove this “collisionless property” of the GTGC algorithm for the one-dimensional case. In the case of the two-dimensional Euclidean space, we conjecture that the “collisionless property” holds for almost every initial configuration.

This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center “On-The-Fly Computing” (SFB 901) and the International Graduate School “Dynamic Intelligent Systems”.

This work is submitted to Distributed & Mobile track of ALGOSENSORS 2016.

A    Appendix

1.1 A.1    Examples and experimental results

The Fig. 3 highlights the difference between GTC and GTGC. Initial configuration (Fig. 3(a)) is connected. Figure 3(b) and (c) represent the evolution of the group of the robots that uses GTC algorithm. Figure 3(d) and (e) represent robots that use GTGC algorithm at the same points in time $t_1 < t_2$. Robots 2, 6 and 3, 5 that use unit disc graph neighborhood have an early collision at time $t_2$ and behave as one since that point in time. On the other hand, robots that used GTGC algorithm had no collisions.

Fig. 3.

Visibility graphs during the gathering of the robots that use original GTC and modified GTGC. Blue circles represent robots; red lines represent trajectories of robots.

The Figs. 4, 5, 6 illustrate instances of the graphs used in the experiments.

Fig. 4.

An instance of the random unit Gabriel graph

Fig. 5.

An instance of the random clustered path graph with Gabriel edges and a single cluster in the middle.

Fig. 6.

An instance of the cross shaped graph that leads to the early collisions with GTGC

Table 1. GTC algorithm with the random unit disc graph as input

Table 2. GTC algorithm with a clustered path graph as an input