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Biomim."],"published-print":{"date-parts":[[2026,8,31]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The spring-mass template acts as a fundamental bridge between animal locomotion and legged robotic platforms. However, controlling spring-mass dynamics involves a persistent trade-off: numerical integration offers accuracy but high computational cost, while approximate analytical solutions provide efficiency but suffer from linearization errors. To resolve this, we propose a physics-guided hybrid neural network framework for the spring-mass running template. Unlike closed-box predictors trained end-to-end, our architecture enforces flight phase physics by embedding exact analytical solutions for the tractable ballistic dynamics directly into the network structure, and learning only the non-integrable stance dynamics. This \u2018analytically augmented\u2019 design enhances interpretability without sacrificing accuracy. For control, we introduce a mixed-objective training strategy that combines supervised imitation with goal-conditioned learning, enabling precise trajectory tracking without the computational burden of per-step optimization. We evaluated the framework\u2019s prediction accuracy, control precision, and runtime efficiency through extensive simulations. Furthermore, we validate the robustness of the learned policies through a comprehensive basin of attraction analysis, demonstrating that the proposed networks stabilize contiguous manifolds of highly perturbed initial conditions compared to existing methods. Finally, we assessed the framework\u2019s real-world adaptability by validating our predictor against an archival experimental dataset from a physical one-legged hopper. Our results demonstrate that this hybrid approach outperforms existing analytical and data-driven methods in accuracy and robustness while maintaining the low inference times required for real-time embedded control.<\/jats:p>","DOI":"10.1088\/1748-3190\/ae7e2d","type":"journal-article","created":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T22:52:42Z","timestamp":1781650362000},"page":"046030","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["A physics-guided neural network framework for prediction and control of spring-mass running"],"prefix":"10.1088","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7754-4044","authenticated-orcid":true,"given":"Ahmet","family":"Safa Ozturk","sequence":"first","affiliation":[{"name":"Bilkent University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3535-5616","authenticated-orcid":true,"given":"Ismail","family":"Uyanik","sequence":"additional","affiliation":[{"name":"Hacettepe University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3158-3961","authenticated-orcid":true,"given":"\u00d6mer","family":"Morg\u00fcl","sequence":"additional","affiliation":[{"name":"Bilkent University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"266","published-online":{"date-parts":[[2026,8,7]]},"reference":[{"key":"bbae7e2dbib1","doi-asserted-by":"publisher","first-page":"3325","DOI":"10.1242\/jeb.202.23.3325","type":"journal-article","article-title":"Templates and anchors: neuromechanical hypotheses of legged locomotion on land","volume":"202","author":"Full","year":"1999","journal-title":"J. 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