{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T01:41:04Z","timestamp":1772847664339,"version":"3.50.1"},"reference-count":42,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T00:00:00Z","timestamp":1766448000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotica"],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>This paper proposes an energy-efficient walking generation method utilizing limit cycles generated by nonlinear model predictive control (NMPC). Conventional limit cycle walking methods rely on strong feedback, such as output zeroing control, to attract the robot\u2019s state toward a predefined periodic trajectory. However, we hypothesize that employing feedback control that better leverages the robot\u2019s dynamics can improve energy efficiency during walking. Our previous work confirmed that using limit cycles generated by NMPC can produce energy-efficient walking patterns. This study builds upon this foundation and proposes a new method for generating walking in a general five-link bipedal robot. Through extensive numerical simulations, we demonstrate that the proposed method achieves highly energy-efficient walking while exhibiting excellent convergence to periodic trajectories.<\/jats:p>","DOI":"10.1017\/s0263574725103020","type":"journal-article","created":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T08:23:43Z","timestamp":1766478223000},"page":"1-36","source":"Crossref","is-referenced-by-count":0,"title":["Limit cycles generation for energy-efficient bipedal walking via nonlinear model predictive control"],"prefix":"10.1017","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1161-1101","authenticated-orcid":false,"given":"Yuta","family":"Hanazawa","sequence":"first","affiliation":[{"name":"Kyushu Institute of Technology"}]},{"given":"Yuhi","family":"Uchino","sequence":"additional","affiliation":[{"name":"Kyushu Institute of Technology"}]},{"given":"Shinichi","family":"Sagara","sequence":"additional","affiliation":[{"name":"Kyushu Institute of Technology"}]}],"member":"56","published-online":{"date-parts":[[2025,12,23]]},"reference":[{"key":"S0263574725103020_ref34","doi-asserted-by":"crossref","unstructured":"[34] Ghansah, A. B. , Kim, J. , Li, K. and Ames, A. D. , \u201cDynamic Walking on Highly Underactuated Point Foot Humanoids: Closing the Loop Between HZD and HLIP,\u201d IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS) (2024) pp. 12686\u201312693.","DOI":"10.1109\/IROS58592.2024.10802542"},{"key":"S0263574725103020_ref11","doi-asserted-by":"crossref","unstructured":"[11] Shigemi, S. , Goswami, A. and Vadakkepat, P. , \u201cAsimo and Humanoid Robot Research at Honda,\u201d In: Humanoid Robotics: A Reference (2018) pp. 55\u201390.","DOI":"10.1007\/978-94-007-6046-2_9"},{"key":"S0263574725103020_ref26","doi-asserted-by":"publisher","DOI":"10.1115\/1.4055770"},{"key":"S0263574725103020_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/j.robot.2021.103900"},{"key":"S0263574725103020_ref29","doi-asserted-by":"crossref","unstructured":"[29] Hereid, A. , Cousineau, E. A. , Hubicki, C. M. and Ames, A. D. , \u201c3D Dynamic Walking with Underactuated Humanoid Robots: A Direct Collocation Framework for Optimizing Hybrid Zero Dynamics,\u201d IEEE International Conference on Robotics and Automation (ICRA) (2016) pp. 1447\u20131454.","DOI":"10.1109\/ICRA.2016.7487279"},{"key":"S0263574725103020_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/S0263574721001363"},{"key":"S0263574725103020_ref31","doi-asserted-by":"crossref","unstructured":"[31] Takano, R. , Chang, J. and Yamakita, M. , \u201cPeriodic Trajectory Planning and Robust Output Zeroing Control for Underactuated Bipedal Robots with Predicted Disturbances,\u201d IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS) (2019) pp. 1415\u20131420.","DOI":"10.1109\/IROS40897.2019.8967765"},{"key":"S0263574725103020_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S026357472200162X"},{"key":"S0263574725103020_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/s10514-015-9476-6"},{"key":"S0263574725103020_ref16","doi-asserted-by":"publisher","DOI":"10.1177\/02783649241285161"},{"key":"S0263574725103020_ref36","doi-asserted-by":"publisher","DOI":"10.1017\/S0263574723001522"},{"key":"S0263574725103020_ref12","doi-asserted-by":"crossref","unstructured":"[12] Rodriguez, D. and Behnke, S. , \u201cDeepwalk: Omnidirectional Bipedal Gait by Deep Reinforcement Learning,\u201d 2021 IEEE international conference on robotics and automation (ICRA) (2021) pp. 3033\u20133039. 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J. , Dyck, A. , Zirkel, M. , Zentner, L. and Fidlin, A. , \u201cHybrid zero dynamics control for bipedal walking with a non-instantaneous double support phase,\u201d arXiv preprint arXiv:2303.05165 (2023)."},{"key":"S0263574725103020_ref42","doi-asserted-by":"publisher","DOI":"10.1080\/00207179508921561"},{"key":"S0263574725103020_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s11044-024-09991-3"},{"key":"S0263574725103020_ref9","doi-asserted-by":"crossref","unstructured":"[9] Kajita, S. , Kanehiro, F. , Kaneko, K. , Fujiwara, K. , Harada, K. , Yokoi, K. and Hirukawa, H. , \u201cBiped Walking Pattern Generation by Using Preview Control of Zero-Moment Point,\u201d IEEE International Conference on Robotics and Automation (ICRA), vol. 2 (2003) pp. 1620\u20131626.","DOI":"10.1109\/ROBOT.2003.1241826"},{"key":"S0263574725103020_ref1","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2022.3143227"},{"key":"S0263574725103020_ref6","doi-asserted-by":"crossref","unstructured":"[6] Csomay-Shanklin, N. , Compton, W. D. , Rodriguez, I. D. J. , Ambrose, E. R. , Yue, Y. and Ames, A. D. , \u201cRobust Agility via Learned Zero Dynamics Policies,\u201d IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS) (2024) pp. 9116\u20139123.","DOI":"10.1109\/IROS58592.2024.10801322"},{"key":"S0263574725103020_ref35","doi-asserted-by":"publisher","DOI":"10.1177\/027836499000900206"},{"key":"S0263574725103020_ref41","doi-asserted-by":"crossref","unstructured":"[41] Westervelt, E. R. , Grizzle, J. W. , Chevallereau, C. , Choi, J. H. and Morris, B. , \u201cFeedback control of dynamic bipedal robot locomotion\u201d (2018).10.1201\/9781420053739","DOI":"10.1201\/9781420053739"},{"key":"S0263574725103020_ref10","doi-asserted-by":"crossref","unstructured":"[10] Nishiwaki, K. and Kagami, S. , \u201cStrategies for Aadjusting the ZMP Reference Trajectory for Maintaining Balance in Humanoid Walking,\u201d IEEE International Conference on Robotics and Automation (ICRA) (2010) pp. 4230\u20134236.","DOI":"10.1109\/ROBOT.2010.5510002"},{"key":"S0263574725103020_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S0263574723000097"},{"key":"S0263574725103020_ref21","unstructured":"[21] Hobbelen, D. 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