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The core innovation is a reproducing kernel Hilbert space (RKHS) parametrization method for rhythmic dynamics modeling, ensuring the existence of stable closed-loop orbits within the generated trajectories. By leveraging transverse contraction theory, we provide theoretical guarantees for the orbital stability of the learned dynamics. To address computational inefficiencies associated with linear matrix inequalities (LMI) constraints, we relax the semi-infinite constraints and simplify the parametrization, transforming the problem into iterative solutions of convex quadratic optimization problems, which can be efficiently solved. We validate the proposed algorithm through simulations and a series of real-world rhythmic tasks. The simulation results indicate that our method significantly outperforms existing approaches in accurately replicating demonstrated behaviors. Additionally, real-world experiments consistently show high performance in completing rhythmic tasks, demonstrating the method\u2019s potential to address challenges in reproducing periodic movements and advancing rhythmic motion replication.<\/jats:p>","DOI":"10.1177\/02783649251369026","type":"journal-article","created":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T16:06:37Z","timestamp":1758125197000},"page":"775-800","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Learning orbitally stable dynamics via transverse contraction criteria for modeling periodic tasks"],"prefix":"10.1177","volume":"45","author":[{"given":"Haoyu","family":"Zhang","sequence":"first","affiliation":[{"name":"University of Chinese Academy of Sciences"},{"name":"Chinese Academy of Sciences"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7565-8788","authenticated-orcid":false,"given":"Long","family":"Cheng","sequence":"additional","affiliation":[{"name":"University of Chinese Academy of Sciences"},{"name":"Chinese Academy of Sciences"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zeyu","family":"Liu","sequence":"additional","affiliation":[{"name":"University of Chinese Academy of Sciences"},{"name":"Chinese Academy of Sciences"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu","family":"Zhang","sequence":"additional","affiliation":[{"name":"University of Chinese Academy of Sciences"},{"name":"Chinese Academy of Sciences"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2025,9,17]]},"reference":[{"key":"e_1_3_5_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2018.8619106"},{"key":"e_1_3_5_3_1","doi-asserted-by":"publisher","DOI":"10.1561\/2200000036"},{"key":"e_1_3_5_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.robot.2008.10.024"},{"key":"e_1_3_5_5_1","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2021.0830"},{"key":"e_1_3_5_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/120887679"},{"key":"e_1_3_5_7_1","volume-title":"International Conference on Learning Representations","author":"Beik-Mohammadi H","year":"2024","unstructured":"Beik-Mohammadi H, Hauberg S, Arvanitidis G, et al. 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