[{"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."}],"publication":"arXiv:2503.07576","department":[{"_id":"101"}],"type":"preprint","keyword":["dynamical systems","coupled systems","distributed computing","robot swarms","autonomous mobile robots","symmetry","equivariant dynamics"],"date_created":"2025-03-11T08:21:05Z","file":[{"date_created":"2025-03-11T08:27:32Z","creator":"svdg","file_id":"58954","content_type":"application/pdf","file_name":"Analyzing_Symmetries_of_Swarms_of_Mobile_Robots_Using_Equivariant_Dynamical_Systems.pdf","access_level":"open_access","file_size":812198,"relation":"main_file","date_updated":"2025-03-11T08:27:32Z"}],"date_updated":"2025-03-11T08:53:02Z","author":[{"id":"32655","orcid":"0009-0002-4750-2051","first_name":"Raphael","last_name":"Gerlach","full_name":"Gerlach, Raphael"},{"id":"97359","full_name":"von der Gracht, Sören","first_name":"Sören","orcid":"0000-0002-8054-2058","last_name":"von der Gracht"}],"year":"2025","title":"Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems","language":[{"iso":"eng"}],"project":[{"name":"Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme","_id":"106","grant_number":"453112019"}],"citation":{"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} }","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.","short":"R. Gerlach, S. von der Gracht, ArXiv:2503.07576 (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.","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>.","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."},"file_date_updated":"2025-03-11T08:27:32Z","oa":"1","external_id":{"arxiv":["2503.07576"]},"has_accepted_license":"1","status":"public","ddc":["004"],"user_id":"97359","_id":"58953","page":"23"},{"citation":{"mla":"Li, Shouwei, et al. “A Continuous Strategy for Collisionless Gathering.” <i>Theoretical Computer Science</i>, vol. 852, 2021, pp. 41–60, doi:<a href=\"https://doi.org/10.1016/j.tcs.2020.10.037\">10.1016/j.tcs.2020.10.037</a>.","ama":"Li S, Markarian C, Meyer auf der Heide F, Podlipyan P. A continuous strategy for collisionless gathering. <i>Theoretical Computer Science</i>. 2021;852:41-60. doi:<a href=\"https://doi.org/10.1016/j.tcs.2020.10.037\">10.1016/j.tcs.2020.10.037</a>","bibtex":"@article{Li_Markarian_Meyer auf der Heide_Podlipyan_2021, title={A continuous strategy for collisionless gathering}, volume={852}, DOI={<a href=\"https://doi.org/10.1016/j.tcs.2020.10.037\">10.1016/j.tcs.2020.10.037</a>}, journal={Theoretical Computer Science}, author={Li, Shouwei and Markarian, Christine and Meyer auf der Heide, Friedhelm and Podlipyan, Pavel}, year={2021}, pages={41–60} }","apa":"Li, S., Markarian, C., Meyer auf der Heide, F., &#38; Podlipyan, P. (2021). A continuous strategy for collisionless gathering. <i>Theoretical Computer Science</i>, <i>852</i>, 41–60. <a href=\"https://doi.org/10.1016/j.tcs.2020.10.037\">https://doi.org/10.1016/j.tcs.2020.10.037</a>","ieee":"S. Li, C. Markarian, F. Meyer auf der Heide, and P. Podlipyan, “A continuous strategy for collisionless gathering,” <i>Theoretical Computer Science</i>, vol. 852, pp. 41–60, 2021.","short":"S. Li, C. Markarian, F. Meyer auf der Heide, P. Podlipyan, Theoretical Computer Science 852 (2021) 41–60.","chicago":"Li, Shouwei, Christine Markarian, Friedhelm Meyer auf der Heide, and Pavel Podlipyan. “A Continuous Strategy for Collisionless Gathering.” <i>Theoretical Computer Science</i> 852 (2021): 41–60. <a href=\"https://doi.org/10.1016/j.tcs.2020.10.037\">https://doi.org/10.1016/j.tcs.2020.10.037</a>."},"status":"public","volume":852,"user_id":"15415","_id":"22510","page":"41-60","abstract":[{"text":"Over the past decades, the Gathering problem, which asks to gather a group of robots in finite time given some restrictions, has been intensively studied. In this paper, we are given a group of n autonomous, dimensionless, deterministic, and anonymous robots, with bounded viewing range. Assuming a continuous time model, the goal is to gather these robots into one point in finite time. We introduce a simple convergence criterion that defines a new class of algorithms which perform gathering in O(nd) time, where d is the diameter of the initial robot configuration. We show that some gathering algorithms in the literature belong to this class and propose two new algorithms that belong to this class and have quadratic running time, namely, Go-To-The-Relative-Center algorithm (GTRC) and Safe-Go-To-The-Relative-Center algorithm (S-GTRC). We prove that the latter can perform gathering without collision by using a slightly more complex robot model: non oblivious, chiral, and luminous (i.e. robots have observable external memory, as in [8]). We also consider a variant of the Gathering problem, the Near-Gathering problem, in which robots must get close to each other without colliding. We show that S-GTRC solves the Near-Gathering problem in quadratic time and assumes a weaker robot model than the one assumed in the current state-of-the-art.","lang":"eng"}],"publication":"Theoretical Computer Science","department":[{"_id":"63"}],"keyword":["Local algorithms","Distributed algorithms","Collisionless gathering","Mobile robots","Multiagent system"],"type":"journal_article","date_created":"2021-06-28T09:24:15Z","intvolume":"       852","date_updated":"2022-01-06T06:55:35Z","publication_status":"published","author":[{"last_name":"Li","first_name":"Shouwei","full_name":"Li, Shouwei"},{"full_name":"Markarian, Christine","first_name":"Christine","last_name":"Markarian"},{"last_name":"Meyer auf der Heide","first_name":"Friedhelm","full_name":"Meyer auf der Heide, Friedhelm","id":"15523"},{"last_name":"Podlipyan","first_name":"Pavel","full_name":"Podlipyan, Pavel"}],"publication_identifier":{"issn":["0304-3975"]},"year":"2021","title":"A continuous strategy for collisionless gathering","doi":"10.1016/j.tcs.2020.10.037","language":[{"iso":"eng"}]},{"citation":{"mla":"Li, Shouwei, et al. “The Impact of the Gabriel Subgraph of the Visibility Graph on the Gathering of Mobile Autonomous Robots.” <i>Theoretical Computer Science</i>, vol. 852, 2021, pp. 29–40, doi:<a href=\"https://doi.org/10.1016/j.tcs.2020.11.009\">10.1016/j.tcs.2020.11.009</a>.","bibtex":"@article{Li_Meyer auf der Heide_Podlipyan_2021, title={The impact of the Gabriel subgraph of the visibility graph on the gathering of mobile autonomous robots}, volume={852}, DOI={<a href=\"https://doi.org/10.1016/j.tcs.2020.11.009\">10.1016/j.tcs.2020.11.009</a>}, journal={Theoretical Computer Science}, author={Li, Shouwei and Meyer auf der Heide, Friedhelm and Podlipyan, Pavel}, year={2021}, pages={29–40} }","ama":"Li S, Meyer auf der Heide F, Podlipyan P. The impact of the Gabriel subgraph of the visibility graph on the gathering of mobile autonomous robots. <i>Theoretical Computer Science</i>. 2021;852:29-40. doi:<a href=\"https://doi.org/10.1016/j.tcs.2020.11.009\">10.1016/j.tcs.2020.11.009</a>","ieee":"S. Li, F. Meyer auf der Heide, and P. Podlipyan, “The impact of the Gabriel subgraph of the visibility graph on the gathering of mobile autonomous robots,” <i>Theoretical Computer Science</i>, vol. 852, pp. 29–40, 2021.","apa":"Li, S., Meyer auf der Heide, F., &#38; Podlipyan, P. (2021). The impact of the Gabriel subgraph of the visibility graph on the gathering of mobile autonomous robots. <i>Theoretical Computer Science</i>, <i>852</i>, 29–40. <a href=\"https://doi.org/10.1016/j.tcs.2020.11.009\">https://doi.org/10.1016/j.tcs.2020.11.009</a>","chicago":"Li, Shouwei, Friedhelm Meyer auf der Heide, and Pavel Podlipyan. “The Impact of the Gabriel Subgraph of the Visibility Graph on the Gathering of Mobile Autonomous Robots.” <i>Theoretical Computer Science</i> 852 (2021): 29–40. <a href=\"https://doi.org/10.1016/j.tcs.2020.11.009\">https://doi.org/10.1016/j.tcs.2020.11.009</a>.","short":"S. Li, F. Meyer auf der Heide, P. Podlipyan, Theoretical Computer Science 852 (2021) 29–40."},"volume":852,"user_id":"15415","_id":"22511","page":"29-40","status":"public","department":[{"_id":"63"}],"type":"journal_article","keyword":["Local algorithms","Distributed algorithms","Collisionless gathering","Mobile robots","Multiagent system"],"date_created":"2021-06-28T09:34:45Z","abstract":[{"lang":"eng","text":"In this paper, we reconsider the well-known discrete, round-based Go-To-The-Center algorithm due to Ando, Suzuki, and Yamashita [2] for gathering n autonomous mobile robots with limited viewing range in the plane. Remarquably, 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 acceptable under the assumption that each robot is represented by a single point. Otherwise, collisions should be avoided. In this paper, we consider a continuous Go-To-The-Center algorithm 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(n2) for gathering in two dimensions Euclidean space, and Θ(n) for the one dimension. Our main contribution is the introduction and evaluation of a continuous algorithm which performs Go-To-The-Center considering only the neighbors of a robot with respect to the Gabriel subgraph of the visibility graph, i.e. Go-To-The-Gabriel-Center algorithm. 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 Go-To-The-Center and the Go-To-The-Gabriel-Center algorithms: Whereas lots of collisions occur during a run of the Go-To-The-Center algorithm, typically only one, namely the final collision occurs during a run of the Go-To-The-Gabriel-Center algorithm. We can prove this “collisionless property” of the Go-To-The-Gabriel-Center algorithm for one dimension. In two-dimensional Euclidean space, we conjecture that the “collisionless property” holds for almost every initial configuration. We support our conjecture with measurements obtained from the simulation where robots execute both continuous Go-To-The-Center and Go-To-The-Gabriel-Center algorithms.\r\n"}],"publication":"Theoretical Computer Science","doi":"10.1016/j.tcs.2020.11.009","language":[{"iso":"eng"}],"intvolume":"       852","date_updated":"2022-01-06T06:55:35Z","publication_status":"published","publication_identifier":{"issn":["0304-3975"]},"author":[{"last_name":"Li","first_name":"Shouwei","full_name":"Li, Shouwei"},{"id":"15523","full_name":"Meyer auf der Heide, Friedhelm","last_name":"Meyer auf der Heide","first_name":"Friedhelm"},{"full_name":"Podlipyan, Pavel","first_name":"Pavel","last_name":"Podlipyan"}],"title":"The impact of the Gabriel subgraph of the visibility graph on the gathering of mobile autonomous robots","year":"2021"}]