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Impossibility of Self-Organized Aggregation without Computation

Roy Steinberg, Kiril Solovey

TL;DR

The paper tackles whether a single bimodal controller can guarantee self-organized aggregation for any number of memoryless, communication-free differential-drive robots with binary LoS sensing. It first strengthens the two-robot case by exposing a flaw in Gauci et al.'s proof and introducing a provably aggregating controller $u^*$, with a three-case aggregation argument. It then proves an impossibility result: for all bimodal controllers, there exist initial states and $n>2$ for which aggregation cannot be guaranteed, and further shows that nonaggregation states can have positive probability under realistic disturbances. Empirical results corroborate the theory, showing robust two-robot aggregation with $u^*$ and illustrating persistent nonaggregation in larger rings under perturbations and noise. The findings imply that achieving large-scale aggregation requires additional robot capabilities beyond the basic, memoryless, noncommunicating setup.

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.

Impossibility of Self-Organized Aggregation without Computation

TL;DR

The paper tackles whether a single bimodal controller can guarantee self-organized aggregation for any number of memoryless, communication-free differential-drive robots with binary LoS sensing. It first strengthens the two-robot case by exposing a flaw in Gauci et al.'s proof and introducing a provably aggregating controller , with a three-case aggregation argument. It then proves an impossibility result: for all bimodal controllers, there exist initial states and for which aggregation cannot be guaranteed, and further shows that nonaggregation states can have positive probability under realistic disturbances. Empirical results corroborate the theory, showing robust two-robot aggregation with and illustrating persistent nonaggregation in larger rings under perturbations and noise. The findings imply that achieving large-scale aggregation requires additional robot capabilities beyond the basic, memoryless, noncommunicating setup.

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.
Paper Structure (11 sections, 3 theorems, 1 equation, 5 figures)

This paper contains 11 sections, 3 theorems, 1 equation, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Definition
  3. Aggregation for two robots
  4. Issue in a previous proof and a possible remedy
  5. An alternative provably-aggregating controller
  6. Impossibility of Aggregation for $n>2$ robots
  7. Nonsingularity of nonaggregating states
  8. Experimental Results
  9. Aggregation for $n=2$
  10. Nonaggregation for $n>2$ robots
  11. Future Work

Key Result

Lemma 1

The bimodal controller $u^*=(-a,a,b,b)$, where $a,b \in (0,1]$ is aggregating for $n=2$.

Figures (5)

  • Figure 1: An example of a no-sensing scenario with two moving robots which violates an assumption made in the aggregation proof for two robots gauci2014self. Both robots move along the dashed circle of radius $R$ centered around the ICR with the same speed and never see one another or aggregate.
  • Figure 2: Illustration for Case 1 in the Lemma \ref{['lem:2robots']} proof. Robot $0$ and $1$ are in blue and red, respectively, with corresponding rays $v_0,v_1$ as dashed lines.
  • Figure 3: Counterexamples for bimodal controllers. (a) Based on daymude2021deadlock, each robot equipped with an $\{\text{SB,CB}\}$-XX has no other robot in sight and is blocked by another robot from behind. (b) A different counterexample for a CB-RS controller, where the center robot is stuck continuously rotating. (c) A XX-CB controller causes all peripheral robots to be stuck, as they attempt to move backwards but are blocked by their neighbor.
  • Figure 4: Motion of the first pair of robots in Theorem \ref{['thrm:N deadlock']}. [left] The robots meet at a deadlock at the origin if placed without perturbations. [right] With added perturbations, the two robots see no more than the highlighted area.
  • Figure 5: A plot demonstrating the effect of various noise sources and slippage on the aggregation percentage in the bot-in-the-middle scenario.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof