{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,9]],"date-time":"2026-09-09T14:08:47Z","timestamp":1788962927015,"version":"build-2803163510"},"reference-count":41,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T00:00:00Z","timestamp":1736121600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["62303457"],"award-info":[{"award-number":["62303457"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["U21A20482"],"award-info":[{"award-number":["U21A20482"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"National Key Research and Development Program of China","award":["2022YFB3303800"],"award-info":[{"award-number":["2022YFB3303800"]}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"crossref","award":["2023M733737"],"award-info":[{"award-number":["2023M733737"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Beijing Municipal Natural Science Foundation, China","award":["4252053"],"award-info":[{"award-number":["4252053"]}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Biomimetics"],"abstract":"<jats:p>With advancements in bipedal locomotion for humanoid robots, a critical challenge lies in generating gaits that are bounded to ensure stable operation in complex environments. Traditional Model Predictive Control (MPC) methods based on Linear Inverted Pendulum (LIP) or Cart\u2013Table (C-T) methods are straightforward and linear but inadequate for robots with flexible joints and linkages. To overcome this limitation, we propose a Flexible MPC (FMPC) framework that incorporates joint dynamics modeling and emphasizes bounded gait control to enable humanoid robots to achieve stable motion in various conditions. The FMPC is based on an enhanced flexible C-T model as the motion model, featuring an elastic layer and an auxiliary second center of mass (CoM) to simulate joint systems. The flexible C-T model\u2019s inversion derivation allows it to be effectively transformed into the predictive equation for the FMPC, therefore enriching its flexible dynamic behavior representation. We further use the Zero Moment Point (ZMP) velocity as a control variable and integrate multiple constraints that emphasize CoM constraint, embed explicit bounded constraint, and integrate ZMP constraint, therefore enabling the control of model flexibility and enhancement of stability. Since all the above constraints are shown to be linear in the control variables, a quadratic programming (QP) problem is established that guarantees that the CoM trajectory is bounded. Lastly, simulations validate the effectiveness of the proposed method, emphasizing its capacity to generate bounded CoM\/ZMP trajectories across diverse conditions, underscoring its potential to enhance gait control. In addition, the validation of the simulation of real robot motion on the robots CASBOT and Openloong, in turn, demonstrates the effectiveness and robustness of our approach.<\/jats:p>","DOI":"10.3390\/biomimetics10010030","type":"journal-article","created":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T08:08:52Z","timestamp":1736150932000},"page":"30","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Flexible Model Predictive Control for Bounded Gait Generation in Humanoid Robots"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-6453-5790","authenticated-orcid":false,"given":"Tianbo","family":"Yang","sequence":"first","affiliation":[{"name":"Institute of Automation, Chinese Academy of Sciences, Beijing 100089, China"},{"name":"The School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuchuang","family":"Tong","sequence":"additional","affiliation":[{"name":"Institute of Automation, Chinese Academy of Sciences, Beijing 100089, China"},{"name":"The School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1659-7879","authenticated-orcid":false,"given":"Zhengtao","family":"Zhang","sequence":"additional","affiliation":[{"name":"Institute of Automation, Chinese Academy of Sciences, Beijing 100089, China"},{"name":"The School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1109\/MIE.2020.3008136","article-title":"Robots in industry: The past, present, and future of a growing collaboration with humans","volume":"15","author":"Grau","year":"2020","journal-title":"IEEE Ind. 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