{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T10:56:46Z","timestamp":1777633006520,"version":"3.51.4"},"reference-count":31,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,7]],"date-time":"2024-11-07T00:00:00Z","timestamp":1730937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Major Research Plan of the National Natural Science Foundation of China","award":["92048301"],"award-info":[{"award-number":["92048301"]}]},{"name":"Major Research Plan of the National Natural Science Foundation of China","award":["52025054"],"award-info":[{"award-number":["52025054"]}]},{"name":"National Outstanding Youth Science Fund Project of National Natural Science Foundation of China","award":["92048301"],"award-info":[{"award-number":["92048301"]}]},{"name":"National Outstanding Youth Science Fund Project of National Natural Science Foundation of China","award":["52025054"],"award-info":[{"award-number":["52025054"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Biomimetics"],"abstract":"<jats:p>This article introduces a novel perspective on designing a stepping controller for bipedal robots. Typically, designing a state-feedback controller to stabilize a bipedal robot to a periodic orbit of step-to-step (S2S) dynamics based on a reduced-order model (ROM) can achieve stable walking. However, the model discrepancies between the ROM and the full-order dynamic system are often ignored. We introduce the latest results from behavioral systems theory by directly constructing a robust stepping controller using input-state data collected during flat-ground walking with a nominal controller in the simulation. The model uncertainty discrepancies are equivalently represented as bounded noise and over-approximated by bounded energy ellipsoids. We conducted extensive walking experiments in a simulation on a 22-degrees-of-freedom small humanoid robot, verifying that it demonstrates superior robustness in handling uncertain loads, various sloped terrains, and push recovery compared to the nominal S2S controller.<\/jats:p>","DOI":"10.3390\/biomimetics9110681","type":"journal-article","created":{"date-parts":[[2024,11,7]],"date-time":"2024-11-07T07:47:16Z","timestamp":1730965636000},"page":"681","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Bipedal Stepping Controller Design Considering Model Uncertainty: A Data-Driven Perspective"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-3637-7964","authenticated-orcid":false,"given":"Chao","family":"Song","sequence":"first","affiliation":[{"name":"State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5124-3999","authenticated-orcid":false,"given":"Xizhe","family":"Zang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boyang","family":"Chen","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5329-1823","authenticated-orcid":false,"given":"Shuai","family":"Heng","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changle","family":"Li","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanhe","family":"Zhu","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jie","family":"Zhao","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1177\/0278364917708248","article-title":"3LP: A linear 3D-walking model including torso and swing dynamics","volume":"36","author":"Faraji","year":"2017","journal-title":"Int. J. Robot. Res."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kajita, S., Morisawa, M., Miura, K., Nakaoka, S., Harada, K., Kaneko, K., Kanehiro, F., and Yokoi, K. (2010, January 18\u201322). Biped walking stabilization based on linear inverted pendulum tracking. Proceedings of the 2010 IEEE\/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan.","DOI":"10.1109\/IROS.2010.5651082"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1109\/TRO.2020.2982584","article-title":"Walking control based on step timing adaptation","volume":"36","author":"Khadiv","year":"2020","journal-title":"IEEE Trans. Robot."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"110959","DOI":"10.1016\/j.automatica.2023.110959","article-title":"Robust adaptive model predictive control with persistent excitation conditions","volume":"152","author":"Lu","year":"2023","journal-title":"Automatica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/s10514-013-9341-4","article-title":"Centroidal dynamics of a humanoid robot","volume":"35","author":"Orin","year":"2013","journal-title":"Auton. Robot."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1279","DOI":"10.1109\/LCSYS.2023.3234769","article-title":"Dynamic walking of bipedal robots on uneven stepping stones via adaptive-frequency MPC","volume":"7","author":"Li","year":"2023","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3414","DOI":"10.1109\/TRO.2022.3183785","article-title":"Dynamic complementarity conditions and whole-body trajectory optimization for humanoid robot locomotion","volume":"38","author":"Dafarra","year":"2022","journal-title":"IEEE Trans. Robot."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1109\/TRO.2019.2923971","article-title":"Capturability-Based Pattern Generation for Walking with Variable Height","volume":"36","author":"Caron","year":"2020","journal-title":"IEEE Trans. Robot."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2012","DOI":"10.1109\/TMECH.2023.3281463","article-title":"Balance Gait Controller for a Bipedal Robotic Walker with Foot Slip","volume":"28","author":"Mihalec","year":"2023","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2089","DOI":"10.1109\/TMECH.2022.3173805","article-title":"Robust Locomotion Exploiting Multiple Balance Strategies: An Observer-Based Cascaded Model Predictive Control Approach","volume":"27","author":"Ding","year":"2022","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1109\/TRO.2019.2926487","article-title":"A Robust Walking Controller Based on Online Optimization of Ankle, Hip, and Stepping Strategies","volume":"35","author":"Jeong","year":"2019","journal-title":"IEEE Trans. Robot."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2405","DOI":"10.1109\/TRO.2022.3150219","article-title":"3-d underactuated bipedal walking via h-lip based gait synthesis and stepping stabilization","volume":"38","author":"Xiong","year":"2022","journal-title":"IEEE Trans. Robot."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1109\/TAC.2014.2299335","article-title":"Rapidly exponentially stabilizing control lyapunov functions and hybrid zero dynamics","volume":"59","author":"Ames","year":"2014","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Xiong, X., Chen, Y., and Ames, A.D. (2021, January 14\u201317). Robust disturbance rejection for robotic bipedal walking: System-level-synthesis with step-to-step dynamics approximation. Proceedings of the 2021 60th IEEE Conference on Decision and Control (CDC), Austin, TX, USA.","DOI":"10.1109\/CDC45484.2021.9683065"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hinojosa, E.H., and Bhounsule, P.A. (2023, January 12\u201314). Data-driven Identification of a Non-homogeneous Inverted Pendulum Model for Enhanced Humanoid Control. Proceedings of the 2023 IEEE-RAS 22nd International Conference on Humanoid Robots (Humanoids), Austin, TX, USA.","DOI":"10.1109\/Humanoids57100.2023.10375164"},{"key":"ref_16","first-page":"909","article-title":"Formulas for data-driven control: Stabilization, optimality, and robustness","volume":"65","author":"Tesi","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_17","unstructured":"Dai, M., Xiong, X., and Ames, A.D. (2022). Data-driven step-to-step dynamics based adaptive control for robust and versatile underactuated bipedal robotic walking. arXiv."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1016\/j.sysconle.2004.09.003","article-title":"A note on persistency of excitation","volume":"54","author":"Willems","year":"2005","journal-title":"Syst. Control Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.arcontrol.2021.09.005","article-title":"Behavioral systems theory in data-driven analysis, signal processing, and control","volume":"52","author":"Markovsky","year":"2021","journal-title":"Annu. Rev. Control"},{"key":"ref_20","first-page":"162","article-title":"From noisy data to feedback controllers: Nonconservative design via a matrix S-lemma","volume":"67","author":"Camlibel","year":"2020","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"110537","DOI":"10.1016\/j.automatica.2022.110537","article-title":"Data-driven control via Petersen\u2019s lemma","volume":"145","author":"Bisoffi","year":"2022","journal-title":"Automatica"},{"key":"ref_22","first-page":"4351","article-title":"Learning controllers for performance through LMI regions","volume":"68","author":"Bisoffi","year":"2022","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4618","DOI":"10.1109\/TAC.2022.3209342","article-title":"Combining prior knowledge and data for robust controller design","volume":"68","author":"Berberich","year":"2022","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Shen, J., Zhang, J., Liu, Y., and Hong, D. (2022, January 28\u201330). Implementation of a robust dynamic walking controller on a miniature bipedal robot with proprioceptive actuation. Proceedings of the 2022 IEEE-RAS 21st International Conference on Humanoid Robots (Humanoids), Ginowan, Japan.","DOI":"10.1109\/Humanoids53995.2022.10000075"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Gibson, G., Dosunmu-Ogunbi, O., Gong, Y., and Grizzle, J. (2022, January 23\u201327). Terrain-adaptive, alip-based bipedal locomotion controller via model predictive control and virtual constraints. Proceedings of the 2022 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Kyoto, Japan.","DOI":"10.1109\/IROS47612.2022.9981969"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ding, Y., Khazoom, C., Chignoli, M., and Kim, S. (2022, January 28\u201330). Orientation-aware model predictive control with footstep adaptation for dynamic humanoid walking. Proceedings of the 2022 IEEE-RAS 21st International Conference on Humanoid Robots (Humanoids), Ginowan, Japan.","DOI":"10.1109\/Humanoids53995.2022.10000244"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1080\/23307706.2017.1397554","article-title":"A rewriting system for convex optimization problems","volume":"5","author":"Agrawal","year":"2018","journal-title":"J. Control Decis."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s12532-020-00179-2","article-title":"OSQP: An operator splitting solver for quadratic programs","volume":"12","author":"Stellato","year":"2020","journal-title":"Math. Program. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"eadh5401","DOI":"10.1126\/scirobotics.adh5401","article-title":"Dtc: Deep tracking control","volume":"9","author":"Jenelten","year":"2024","journal-title":"Sci. Robot."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6082","DOI":"10.1109\/TAC.2023.3234889","article-title":"Learning controllers from data via approximate nonlinearity cancellation","volume":"68","author":"Rotulo","year":"2023","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_31","unstructured":"Zhao, F., D\u00f6rfler, F., Chiuso, A., and You, K. (2024). Data-Enabled Policy Optimization for Direct Adaptive Learning of the LQR. arXiv."}],"container-title":["Biomimetics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2313-7673\/9\/11\/681\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:27:57Z","timestamp":1760113677000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2313-7673\/9\/11\/681"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,7]]},"references-count":31,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["biomimetics9110681"],"URL":"https:\/\/doi.org\/10.3390\/biomimetics9110681","relation":{},"ISSN":["2313-7673"],"issn-type":[{"value":"2313-7673","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,7]]}}}