Document Open Access Logo

Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility

Authors Raphael Gerlach , Sören von der Gracht , Christopher Hahn , Jonas Harbig , Peter Kling

PDF

File

Thumbnail PDF
PDF
LIPIcs.OPODIS.2024.13.pdf
  • Filesize: 1.37 MB
  • 28 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Self-organization
  • Theory of computation → Self-organization
Keywords
  • Swarm Algorithm
  • Swarm Robots
  • Distributed Algorithm
  • Pattern Formation
  • Limited Visibility
  • Oblivious

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

In the general pattern formation (GPF) problem, a swarm of simple autonomous, disoriented robots must form a given pattern. The robots' simplicity imply a strong limitation: When the initial configuration is rotationally symmetric, only patterns with a similar symmetry can be formed [Masafumi Yamashita and Ichiro Suzuki, 2010]. The only known algorithm to form large patterns with limited visibility and without memory requires the robots to start in a near-gathering (a swarm of constant diameter) [Christopher Hahn et al., 2024]. However, not only do we not know any near-gathering algorithm guaranteed to preserve symmetry but most natural gathering strategies trivially increase symmetries [Jannik Castenow et al., 2022]. 
Thus, we study near-gathering without changing the swarm’s rotational symmetry for disoriented, oblivious robots with limited visibility (the OBLOT-model, see [Paola Flocchini et al., 2019]). We introduce a technique based on the theory of dynamical systems to analyze how a given algorithm affects symmetry and provide sufficient conditions for symmetry preservation. Until now, it was unknown whether the considered OBLOT-model allows for any non-trivial algorithm that always preserves symmetry. Our first result shows that a variant of Go-To-The-Average always preserves symmetry but may sometimes lead to multiple, unconnected near-gathering clusters. Our second result is a symmetry-preserving near-gathering algorithm that works on swarms with a convex boundary (the outer boundary of the unit disc graph) and without "holes" (circles of diameter 1 inside the boundary without any robots).

Cite As Get BibTex

Raphael Gerlach, Sören von der Gracht, Christopher Hahn, Jonas Harbig, and Peter Kling. Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility. In 28th International Conference on Principles of Distributed Systems (OPODIS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 324, pp. 13:1-13:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.OPODIS.2024.13

Author Details

Raphael Gerlach
  • Institute of Mathematics, Paderborn University, Germany
Sören von der Gracht
  • Institute of Mathematics, Paderborn University, Germany
Christopher Hahn
  • Department of Informatics, University of Hamburg, Germany
Jonas Harbig
  • Heinz Nixdorf Institute, Paderborn University, Germany
Peter Kling
  • Department of Informatics, University of Hamburg, Germany

Funding

Sören von der Gracht and Jonas Harbig: This work was partially supported by the German Research Foundation (DFG) under the project numbers ME 872/14-1 & 453112019.

References

  1. Muhammad Zaki Almuzakki and Bayu Jayawardhana. Cooperative nearest-neighbor control of multi-agent systems: Consensus and formation control problems. IEEE Control Systems Letters, 7:1873-1878, 2023. URL: https://doi.org/10.1109/LCSYS.2023.3282423.
  2. Kaustav Bose, Ranendu Adhikary, Manash Kumar Kundu, and Buddhadeb Sau. Arbitrary pattern formation on infinite grid by asynchronous oblivious robots. Theor. Comput. Sci., 815:213-227, 2020. URL: https://doi.org/10.1016/J.TCS.2020.02.016.
  3. Kaustav Bose, Manash Kumar Kundu, Ranendu Adhikary, and Buddhadeb Sau. Arbitrary pattern formation by asynchronous opaque robots with lights. Theor. Comput. Sci., 849:138-158, 2021. URL: https://doi.org/10.1016/J.TCS.2020.10.015.
  4. Jannik Castenow, Matthias Fischer, Jonas Harbig, Daniel Jung, and Friedhelm Meyer auf der Heide. Gathering anonymous, oblivious robots on a grid. Theor. Comput. Sci., 815:289-309, 2020. URL: https://doi.org/10.1016/J.TCS.2020.02.018.
  5. Jannik Castenow, Jonas Harbig, Daniel Jung, Peter Kling, Till Knollmann, and Friedhelm Meyer auf der Heide. A unifying approach to efficient (near)-gathering of disoriented robots with limited visibility. In Eshcar Hillel, Roberto Palmieri, and Etienne Rivière, editors, 26th International Conference on Principles of Distributed Systems, OPODIS 2022, December 13-15, 2022, Brussels, Belgium, volume 253 of LIPIcs, pages 15:1-15:25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.OPODIS.2022.15.
  6. Yao Chen, Jinhu Lu, Xinghuo Yu, and David J. Hill. Multi-agent systems with dynamical topologies: Consensus and applications. IEEE Circuits and Systems Magazine, 13(3):21-34, 2013. URL: https://doi.org/10.1109/MCAS.2013.2271443.
  7. Pascal Chossat and Reiner Lauterbach. Methods in Equivariant Bifurcations and Dynamical Systems, volume 15 of Advanced series in nonlinear dynamics. World Scientific, 2000. Google Scholar
  8. Serafino Cicerone, Gabriele Di Stefano, and Alfredo Navarra. Asynchronous arbitrary pattern formation: the effects of a rigorous approach. Distributed Comput., 32(2):91-132, 2019. URL: https://doi.org/10.1007/S00446-018-0325-7.
  9. Reuven Cohen and David Peleg. Convergence properties of the gravitational algorithm in asynchronous robot systems. SIAM J. Comput., 34(6):1516-1528, 2005. URL: https://doi.org/10.1137/S0097539704446475.
  10. Shantanu Das, Paola Flocchini, Nicola Santoro, and Masafumi Yamashita. Forming sequences of geometric patterns with oblivious mobile robots. Distributed Comput., 28(2):131-145, 2015. URL: https://doi.org/10.1007/S00446-014-0220-9.
  11. Bastian Degener, Barbara Kempkes, Tobias Langner, Friedhelm Meyer auf der Heide, Peter Pietrzyk, and Roger Wattenhofer. A tight runtime bound for synchronous gathering of autonomous robots with limited visibility. In Rajmohan Rajaraman and Friedhelm Meyer auf der Heide, editors, SPAA 2011: Proceedings of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures, San Jose, CA, USA, June 4-6, 2011 (Co-located with FCRC 2011), pages 139-148. ACM, 2011. URL: https://doi.org/10.1145/1989493.1989515.
  12. Florian Dörfler and Francesco Bullo. Synchronization in complex networks of phase oscillators: A survey. Automatica, 50(6):1539-1564, 2014. URL: https://doi.org/10.1016/j.automatica.2014.04.012.
  13. Miroslaw Dynia, Jaroslaw Kutylowski, Pawel Lorek, and Friedhelm Meyer auf der Heide. Maintaining communication between an explorer and a base station. In Yi Pan, Franz J. Rammig, Hartmut Schmeck, and Mauricio Solar, editors, Biologically Inspired Cooperative Computing, IFIP 19th World Computer Congress, TC 10: 1st IFIP International Conference on Biologically Inspired Computing, August 21-24, 2006, Santiago, Chile, volume 216 of IFIP, pages 137-146. Springer, 2006. URL: https://doi.org/10.1007/978-0-387-34733-2_14.
  14. Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro, editors. Distributed Computing by Mobile Entities, Current Research in Moving and Computing, volume 11340 of Lecture Notes in Computer Science. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7.
  15. Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro. Moving and computing models: Robots. In Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro, editors, Distributed Computing by Mobile Entities, Current Research in Moving and Computing, volume 11340 of Lecture Notes in Computer Science, pages 3-14. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7_1.
  16. Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Giovanni Viglietta. Distributed computing by mobile robots: uniform circle formation. Distributed Comput., 30(6):413-457, 2017. URL: https://doi.org/10.1007/S00446-016-0291-X.
  17. Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Peter Widmayer. Arbitrary pattern formation by asynchronous, anonymous, oblivious robots. Theor. Comput. Sci., 407(1-3):412-447, 2008. URL: https://doi.org/10.1016/J.TCS.2008.07.026.
  18. Nao Fujinaga, Yukiko Yamauchi, Hirotaka Ono, Shuji Kijima, and Masafumi Yamashita. Pattern formation by oblivious asynchronous mobile robots. SIAM J. Comput., 44(3):740-785, 2015. URL: https://doi.org/10.1137/140958682.
  19. Raphael Gerlach, Sören von der Gracht, Christopher Hahn, Jonas Harbig, and Peter Kling. Symmetry preservation in swarms of oblivious robots with limited visibility, 2024. URL: https://doi.org/10.48550/arXiv.2409.19277.
  20. Semyon Aronovich Geršgorin. Über die Abgrenzung der Eigenwerte einer Matrix. Izvestija Akademii Nauk SSSR, Serija Matematika, 7(6):749-754, 1931. Google Scholar
  21. Martin Golubitsky, Ian Stewart, and David G. Schaeffer. Singularities and Groups in Bifurcation Theory, volume 69 of Applied mathematical sciences. Springer, 1988. URL: https://doi.org/10.1007/978-1-4612-4574-2.
  22. Christopher Hahn, Jonas Harbig, and Peter Kling. Forming large patterns with local robots in the OBLOT model. In Arnaud Casteigts and Fabian Kuhn, editors, 3rd Symposium on Algorithmic Foundations of Dynamic Networks, SAND 2024, June 5-7, 2024, Patras, Greece, volume 292 of LIPIcs, pages 14:1-14:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.SAND.2024.14.
  23. Morris W. Hirsch and Stephen Smale. Differential equations, dynamical systems, and linear algebra, volume 60 of Pure and applied mathematics. Acad. Press, [nachdr.] edition, 1988. Smale, Stephen (VerfasserIn). Google Scholar
  24. Chang-kwon Kang, Farbod Fahimi, Rob Griffin, D Brian Landrum, Bryan Mesmer, Guangsheng Zhang, Taeyoung Lee, Hikaru Aono, Jeremy Pohly, Jesse McCain, et al. Marsbee-swarm of flapping wing flyers for enhanced mars exploration. Technical report, NASA, 2019. Google Scholar
  25. Anaïs Khuong, Guy Theraulaz, Christian Jost, Andrea Perna, and Jacques Gautrais. A computational model of ant nest morphogenesis. In Tom Lenaerts, Mario Giacobini, Hugues Bersini, Paul Bourgine, Marco Dorigo, and René Doursat, editors, Advances in Artificial Life: 20th Anniversary Edition - Back to the Origins of Alife, ECAL 2011, Paris, France, August 8-12, 2011, pages 404-411. MIT Press, 2011. URL: http://mitpress.mit.edu/sites/default/files/titles/alife/0262297140chap61.pdf.
  26. David G. Kirkpatrick, Irina Kostitsyna, Alfredo Navarra, Giuseppe Prencipe, and Nicola Santoro. Separating bounded and unbounded asynchrony for autonomous robots: Point convergence with limited visibility. In Avery Miller, Keren Censor-Hillel, and Janne H. Korhonen, editors, PODC '21: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, July 26-30, 2021, pages 9-19. ACM, 2021. URL: https://doi.org/10.1145/3465084.3467910.
  27. Moumita Mondal and Sruti Gan Chaudhuri. Uniform circle formation by swarm robots under limited visibility. In Dang Van Hung and Meenakshi D'Souza, editors, Distributed Computing and Internet Technology - 16th International Conference, ICDCIT 2020, Bhubaneswar, India, January 9-12, 2020, Proceedings, volume 11969 of Lecture Notes in Computer Science, pages 420-428. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-36987-3_28.
  28. Linda Pagli, Giuseppe Prencipe, and Giovanni Viglietta. Getting close without touching: near-gathering for autonomous mobile robots. Distributed Comput., 28(5):333-349, 2015. URL: https://doi.org/10.1007/S00446-015-0248-5.
  29. Arkadij Pikovskij, Michael Rosenblum, and Jürgen Kurths. Synchronization, volume 12 of Cambridge nonlinear science series. Cambridge Univ. Press, 1st paperback ed., repr edition, 2003. URL: http://www.loc.gov/catdir/description/cam041/2003283137.html.
  30. Thomas D. Seeley, P. Kirk Visscher, Thomas Schlegel, Patrick M. Hogan, Nigel R. Franks, and James A. R. Marshall. Stop Signals Provide Cross Inhibition in Collective Decision-Making by Honeybee Swarms. Science, 335(6064):108-111, January 2012. URL: https://doi.org/10.1126/science.1210361.
  31. Fernando Soto, Jie Wang, Rajib Ahmed, and Utkan Demirci. Medical Micro/Nanorobots in Precision Medicine. Advanced Science, 7(21), November 2020. URL: https://doi.org/10.1002/advs.202002203.
  32. Ichiro Suzuki and Masafumi Yamashita. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. Comput., 28(4):1347-1363, 1999. URL: https://doi.org/10.1137/S009753979628292X.
  33. Giovanni Viglietta. Uniform circle formation. In Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro, editors, Distributed Computing by Mobile Entities, Current Research in Moving and Computing, volume 11340 of Lecture Notes in Computer Science, pages 83-108. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7_5.
  34. Masafumi Yamashita and Ichiro Suzuki. Characterizing geometric patterns formable by oblivious anonymous mobile robots. Theor. Comput. Sci., 411(26-28):2433-2453, 2010. URL: https://doi.org/10.1016/J.TCS.2010.01.037.
  35. Yukiko Yamauchi. A survey on pattern formation of autonomous mobile robots: asynchrony, obliviousness and visibility. Journal of Physics: Conference Series, 473:012016, December 2013. URL: https://doi.org/10.1088/1742-6596/473/1/012016.
  36. Yukiko Yamauchi. Symmetry of anonymous robots. In Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro, editors, Distributed Computing by Mobile Entities, Current Research in Moving and Computing, volume 11340 of Lecture Notes in Computer Science, pages 109-133. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7_6.
  37. Yukiko Yamauchi and Masafumi Yamashita. Pattern formation by mobile robots with limited visibility. In Thomas Moscibroda and Adele A. Rescigno, editors, Structural Information and Communication Complexity - 20th International Colloquium, SIROCCO 2013, Ischia, Italy, July 1-3, 2013, Revised Selected Papers, volume 8179 of Lecture Notes in Computer Science, pages 201-212. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-03578-9_17.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact