{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:54:51Z","timestamp":1760144091492,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,20]],"date-time":"2024-03-20T00:00:00Z","timestamp":1710892800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Slovak Grant Agency","award":["VEGA 1\/0201\/21","VEGA 1\/0436\/22","KEGA 027TUKE-4\/2022"],"award-info":[{"award-number":["VEGA 1\/0201\/21","VEGA 1\/0436\/22","KEGA 027TUKE-4\/2022"]}]},{"name":"Educational Grant Agency of the Ministry of Education, Science, Research and Sports of the Slovak Republic","award":["VEGA 1\/0201\/21","VEGA 1\/0436\/22","KEGA 027TUKE-4\/2022"],"award-info":[{"award-number":["VEGA 1\/0201\/21","VEGA 1\/0436\/22","KEGA 027TUKE-4\/2022"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article deals with the presentation of an alternative approach that uses methods of geometric mechanics, which allow one to see into the geometrical structure of the equations and can be useful not only for modeling but also during the design of symmetrical locomotion systems and their control and motion planning. These methods are based on extracting the symmetries of Lie groups from the locomotion system in order to simplify the resulting equations. In the second section, the special two-dimensional Euclidean group SE2 and its splitting into right and left actions are derived. The physical interpretation of the local group and spatial velocities is investigated, and by virtue of the fact that both of these velocities represent the same velocity from a physical point of view, the dependence between them can be found by means of the adjoint action. The third section is devoted to the modeling and analysis of the planar locomotion of the symmetrical serpentine robot; the positions and local group velocities of its links are derived, the vector fields for the local connections are given, and the trajectories of the individual variables in the lateral movement of the kinematic snake are shown. At the end of the article, the overall benefits of the scientific study are summarized, as is the comparison of the results from the simulation phase, while the most significant novelty compared to alternative publications in the field can be considered the realization of this study with a description of the relevant methodology at a detailed level; that is, the locomotion results confirm the suitability of the use of geometric mechanics for these symmetrical locomotion systems with nonholonomic constraints.<\/jats:p>","DOI":"10.3390\/sym16030376","type":"journal-article","created":{"date-parts":[[2024,3,20]],"date-time":"2024-03-20T13:09:36Z","timestamp":1710940176000},"page":"376","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Mathematical Modeling of Robotic Locomotion Systems"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7226-1240","authenticated-orcid":false,"given":"Erik","family":"Prada","sequence":"first","affiliation":[{"name":"Department of Industrial Automation and Mechatronics, Faculty of Mechanical Engineering, Technical University of Ko\u0161ice, Park Komensk\u00e9ho 8, 04200 Ko\u0161ice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2864-0651","authenticated-orcid":false,"given":"\u013dubica","family":"Mikov\u00e1","sequence":"additional","affiliation":[{"name":"Department of Industrial Automation and Mechatronics, Faculty of Mechanical Engineering, Technical University of Ko\u0161ice, Park Komensk\u00e9ho 8, 04200 Ko\u0161ice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8273-2384","authenticated-orcid":false,"given":"Ivan","family":"Virgala","sequence":"additional","affiliation":[{"name":"Department of Industrial Automation and Mechatronics, Faculty of Mechanical Engineering, Technical University of Ko\u0161ice, Park Komensk\u00e9ho 8, 04200 Ko\u0161ice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2274-3828","authenticated-orcid":false,"given":"Michal","family":"Kelemen","sequence":"additional","affiliation":[{"name":"Department of Industrial Automation and Mechatronics, Faculty of Mechanical Engineering, Technical University of Ko\u0161ice, Park Komensk\u00e9ho 8, 04200 Ko\u0161ice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6815-0262","authenticated-orcid":false,"given":"Peter J\u00e1n","family":"Sin\u010d\u00e1k","sequence":"additional","affiliation":[{"name":"Department of Industrial Automation and Mechatronics, Faculty of Mechanical Engineering, Technical University of Ko\u0161ice, Park Komensk\u00e9ho 8, 04200 Ko\u0161ice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1203-3446","authenticated-orcid":false,"given":"Roman","family":"Mykhailyshyn","sequence":"additional","affiliation":[{"name":"Walker Department of Mechanical Engineering, Cockrell School of Engineering, The University of Texas at Austin, Austin, TX 78712, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"319","DOI":"10.5802\/aif.233","article-title":"Sur la g\u00e9om\u00e9trie diff\u00e9rentielle des groupes de Lie de dimension infinie et ses applications \u00e0 l\u2019hydrodynamique des fluides parfaits","volume":"16","author":"Arnold","year":"1966","journal-title":"Ann. 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