Impossibility of Self-Organized Aggregation without Computation

TL;DR

This paper proves that no universal controller can guarantee aggregation for all robot configurations under severe hardware constraints, and provides a new controller with a rigorous proof of its effectiveness.

Contribution

It demonstrates the impossibility of universal aggregation controllers for minimal robots and introduces a new controller with a formal proof of success.

Findings

No universal controller exists for all robot initial states.

The original aggregation proof for two robots was incorrect.

A new controller with a rigorous proof was proposed.

Abstract

In their seminal work, Gauci et al. (2014) studied the fundamental task of aggregation, wherein multiple robots need to gather without an a priori agreed-upon meeting location, using minimal hardware. That paper considered differential-drive robots that are memoryless and unable to compute. Moreover, the robots cannot communicate with one another and are only equipped with a simple sensor that determines whether another robot is directly in front of them. Despite those severe limitations, Gauci et al. introduced a controller and proved mathematically that it aggregates a system of two robots for any initial state. Unfortunately, for larger systems, the same controller aggregates empirically in many cases but not all. Thus, the question of whether a controller exists that aggregates for any number of robots remains open. In this paper, we show that no such controller exists by investigating the geometric structure of controllers. In addition, we disprove the aggregation proof of the paper above for two robots and present an alternative controller alongside a simple and rigorous aggregation proof.