---
_id: '58953'
abstract:
- lang: eng
  text: In this article, we investigate symmetry properties of distributed systems
    of mobile robots. We consider a swarm of n robots in the OBLOT model and analyze
    their collective Fsync dynamics using of equivariant dynamical systems theory.
    To this end, we show that the corresponding evolution function commutes with rotational
    and reflective transformations of R^2. These form a group that is isomorphic to
    O(2) x S_n, the product group of the orthogonal group and the permutation on n
    elements. The theory of equivariant dynamical systems is used to deduce a hierarchy
    along which symmetries of a robot swarm can potentially increase following an
    arbitrary protocol. By decoupling the Look phase from the Compute and Move phases
    in the mathematical description of an LCM cycle, this hierarchy can be characterized
    in terms of automorphisms of connectivity graphs. In particular, we find all possible
    types of symmetry increase, if the decoupled Compute and Move phase is invertible.
    Finally, we apply our results to protocols which induce state-dependent linear
    dynamics, where the reduced system consisting of only the Compute and Move phase
    is linear.
author:
- first_name: Raphael
  full_name: Gerlach, Raphael
  id: '32655'
  last_name: Gerlach
  orcid: 0009-0002-4750-2051
- first_name: Sören
  full_name: von der Gracht, Sören
  id: '97359'
  last_name: von der Gracht
  orcid: 0000-0002-8054-2058
citation:
  ama: Gerlach R, von der Gracht S. Analyzing Symmetries of Swarms of Mobile Robots
    Using Equivariant  Dynamical Systems. <i>arXiv:250307576</i>. Published online
    2025.
  apa: Gerlach, R., &#38; von der Gracht, S. (2025). Analyzing Symmetries of Swarms
    of Mobile Robots Using Equivariant  Dynamical Systems. In <i>arXiv:2503.07576</i>.
  bibtex: '@article{Gerlach_von der Gracht_2025, title={Analyzing Symmetries of Swarms
    of Mobile Robots Using Equivariant  Dynamical Systems}, journal={arXiv:2503.07576},
    author={Gerlach, Raphael and von der Gracht, Sören}, year={2025} }'
  chicago: Gerlach, Raphael, and Sören von der Gracht. “Analyzing Symmetries of Swarms
    of Mobile Robots Using Equivariant  Dynamical Systems.” <i>ArXiv:2503.07576</i>,
    2025.
  ieee: R. Gerlach and S. von der Gracht, “Analyzing Symmetries of Swarms of Mobile
    Robots Using Equivariant  Dynamical Systems,” <i>arXiv:2503.07576</i>. 2025.
  mla: Gerlach, Raphael, and Sören von der Gracht. “Analyzing Symmetries of Swarms
    of Mobile Robots Using Equivariant  Dynamical Systems.” <i>ArXiv:2503.07576</i>,
    2025.
  short: R. Gerlach, S. von der Gracht, ArXiv:2503.07576 (2025).
date_created: 2025-03-11T08:21:05Z
date_updated: 2025-03-11T08:53:02Z
ddc:
- '004'
department:
- _id: '101'
external_id:
  arxiv:
  - '2503.07576'
file:
- access_level: open_access
  content_type: application/pdf
  creator: svdg
  date_created: 2025-03-11T08:27:32Z
  date_updated: 2025-03-11T08:27:32Z
  file_id: '58954'
  file_name: Analyzing_Symmetries_of_Swarms_of_Mobile_Robots_Using_Equivariant_Dynamical_Systems.pdf
  file_size: 812198
  relation: main_file
file_date_updated: 2025-03-11T08:27:32Z
has_accepted_license: '1'
keyword:
- dynamical systems
- coupled systems
- distributed computing
- robot swarms
- autonomous mobile robots
- symmetry
- equivariant dynamics
language:
- iso: eng
oa: '1'
page: '23'
project:
- _id: '106'
  grant_number: '453112019'
  name: 'Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme'
publication: arXiv:2503.07576
status: public
title: Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical
  Systems
type: preprint
user_id: '97359'
year: '2025'
...