{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:59:29Z","timestamp":1760234369082,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,5,14]],"date-time":"2021-05-14T00:00:00Z","timestamp":1620950400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the project of National Key R&amp;D Program of China","award":["2018YFC2001300"],"award-info":[{"award-number":["2018YFC2001300"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["91848204","91948302"],"award-info":[{"award-number":["91848204","91948302"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Natural Science Foundation of Chin","award":["51675222"],"award-info":[{"award-number":["51675222"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Ankle push-off occurs when muscle\u2013tendon units about the ankle joint generate a burst of positive power at the end of stance phase in human walking. Ankle push-off mainly contributes to both leg swing and center of mass (CoM) acceleration. Humans use the amount of ankle push-off to induce speed changes. Thus, this study focuses on determining the faster walking speed and the lowest energy efficiency of biped robots by using ankle push-off. The real-time-space trajectory method is used to provide reference positions for the hip and knee joints. The torque curve during ankle push-off, composed of three quintic polynomial curves, is applied to the ankle joint. With the walking distance and the mechanical cost of transport (MCOT) as the optimization goals, the genetic algorithm (GA) is used to obtain the optimal torque curve during ankle push-off. The results show that the biped robot achieved a maximum speed of 1.3 m\/s, and the ankle push-off occurs at 41.27\u221248.34% of the gait cycle. The MCOT of the bipedal robot corresponding to the high economy gait is 0.70, and the walking speed is 0.54 m\/s. This study may further prompt the design of the ankle joint and identify the important implications of ankle push-off for biped robots.<\/jats:p>","DOI":"10.3390\/s21103435","type":"journal-article","created":{"date-parts":[[2021,5,17]],"date-time":"2021-05-17T02:31:34Z","timestamp":1621218694000},"page":"3435","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Torque Curve Optimization of Ankle Push-Off in Walking Bipedal Robots Using Genetic Algorithm"],"prefix":"10.3390","volume":"21","author":[{"given":"Qiaoli","family":"Ji","sequence":"first","affiliation":[{"name":"Key Laboratory of Bionic Engineering, Jilin University, Changchun 130022, China"}]},{"given":"Zhihui","family":"Qian","sequence":"additional","affiliation":[{"name":"Key Laboratory of Bionic Engineering, Jilin University, Changchun 130022, China"}]},{"given":"Lei","family":"Ren","sequence":"additional","affiliation":[{"name":"Key Laboratory of Bionic Engineering, Jilin University, Changchun 130022, China"},{"name":"School of Mechanical, Aerospace and Civil Engineering, University of Manchester, Manchester M13 9PL, UK"}]},{"given":"Luquan","family":"Ren","sequence":"additional","affiliation":[{"name":"Key Laboratory of Bionic Engineering, Jilin University, Changchun 130022, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1585","DOI":"10.1177\/0278364913498122","article-title":"Foot\u2013terrain interaction mechanics for legged robots: Modeling and experimental validation","volume":"32","author":"Ding","year":"2013","journal-title":"Int. J. Robot. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1177\/02783649922066376","article-title":"Postural stability of biped robots and the foot-rotation indicator (FRI) point","volume":"18","author":"Ambarish","year":"1999","journal-title":"Int. J. Robot. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1142\/S0219843604000083","article-title":"Zero moment point\u2013Thirty five years of its life","volume":"1","author":"Vukobratovic","year":"2004","journal-title":"Int. J. Hum. Robot."},{"unstructured":"Sakagami, Y., Watanabe, R., Aoyama, C., Matsunaga, S., Higaki, N., and Fujimura, K. (October, January 30). The intelligent ASIMO: System overview and integration. Proceedings of the 2002 IEEE\/RSJ Conference on Intelligent Systems and Robots (IROS), Lausanne, Switzerland.","key":"ref_4"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1177\/027836499000900206","article-title":"Passive dynamic walking","volume":"9","author":"McGeer","year":"1990","journal-title":"Int. J. Robot. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1530","DOI":"10.1177\/0278364914532391","article-title":"The effects of foot geometric properties on the gait of planar bipeds walking under HZD-based control","volume":"33","author":"Martin","year":"2014","journal-title":"Int. J. Robot. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1017\/S0263574708005006","article-title":"Effect of Circular Arc Feet on a Control Law for a Biped","volume":"27","author":"Kinugasa","year":"2009","journal-title":"Robotica"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1177\/0278364914527485","article-title":"Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge","volume":"33","author":"Bhounsule","year":"2014","journal-title":"Int. J. Robot. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"505","DOI":"10.3182\/20100901-3-IT-2016.00302","article-title":"3D bipedal robotic walking: Models, feedback control, and open problems","volume":"43","author":"Grizzle","year":"2010","journal-title":"IFAC Proc. Vol."},{"doi-asserted-by":"crossref","unstructured":"Fevre, M., Lin, H., and Schmiedeler, J.P. (2019, January 3\u20138). Stability and gait switching of underactuated biped walkers. Proceedings of the 2019 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Macau, China.","key":"ref_10","DOI":"10.1109\/IROS40897.2019.8967673"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1109\/MCS.2003.1234651","article-title":"RABBIT: A testbed for advanced control theory","volume":"23","author":"Chevallereau","year":"2003","journal-title":"IEEE Contr. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1170","DOI":"10.1177\/0278364910379882","article-title":"A compliant hybrid zero dynamics controller for stable, efficient and fast bipedal walking on MABEL","volume":"30","author":"Sreenath","year":"2011","journal-title":"Int. J. Robot. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1177\/0278364912473344","article-title":"Embedding active force control within the compliant hybrid zero dynamics to achieve stable, fast running on MABEL","volume":"32","author":"Sreenath","year":"2013","journal-title":"Int. J. Robot. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1497","DOI":"10.1177\/0278364916648388","article-title":"ATRIAS: Design and validation of a tether-free 3D-capable spring-mass bipedal robot","volume":"35","author":"Hubicki","year":"2016","journal-title":"Int. J. Robot. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3676","DOI":"10.1242\/jeb.140376","article-title":"A unified perspective on ankle push-off in human walking","volume":"219","author":"Zelik","year":"2016","journal-title":"J. Exp. Biol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1242\/jeb.115451","article-title":"Six degree-of-freedom analysis of hip, knee, ankle and foot provides updated understanding of biomechanical work during human walking","volume":"218","author":"Zelik","year":"2015","journal-title":"J. Exp. Biol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.clinbiomech.2003.12.001","article-title":"Roll-over shapes of human locomotor systems: Effect of walking speed","volume":"19","author":"Hansen","year":"2004","journal-title":"Clin. Biomech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3953","DOI":"10.1242\/jeb.02455","article-title":"The advantages of a rolling foot in human walking","volume":"209","author":"Adamczyk","year":"2006","journal-title":"J. Exp. Biol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1177\/02783640122067561","article-title":"A three-dimensional passive-dynamic walking robot with two legs and knees","volume":"20","author":"Collins","year":"2001","journal-title":"Int. J. Robot. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1082","DOI":"10.1126\/science.1107799","article-title":"Efficient bipedal robots based on passive-dynamic walkers","volume":"307","author":"Collins","year":"2005","journal-title":"Science"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1177\/0278364908095005","article-title":"Controlling the walking speed in limit cycle walking","volume":"27","author":"Hobbelen","year":"2008","journal-title":"Int. J. Robot. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1115\/1.1427703","article-title":"Energetics of actively powered locomotion using the simplest walking model","volume":"124","author":"Kuo","year":"2002","journal-title":"J. Biomech. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1109\/TRO.2016.2636297","article-title":"Once-per-step control of ankle push-off work improves balance in a three-dimensional simulation of bipedal walking","volume":"33","author":"Kim","year":"2017","journal-title":"IEEE Trans. Robot."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"036005","DOI":"10.1088\/1748-3182\/7\/3\/036005","article-title":"Robots in human biomechanics\u2014a study on ankle push-off in walking","volume":"7","author":"Renjewski","year":"2012","journal-title":"Bioinspir. Biomim."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1098\/rsif.2008.0415","article-title":"Elastic coupling of limb joints enables faster bipedal walking","volume":"6","author":"Dean","year":"2009","journal-title":"J. R. Soc. Interface"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1109\/TRO.2014.2298058","article-title":"Torso inclination enables faster walking in a planar biped robot with passive ankles","volume":"30","author":"Geng","year":"2004","journal-title":"IEEE Trans. Robot."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1218","DOI":"10.1242\/jeb.107391","article-title":"Impulsive ankle push-off powers leg swing in human walking","volume":"217","author":"Lipfert","year":"2014","journal-title":"J. Exp. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"101972","DOI":"10.1016\/j.simpat.2019.101972","article-title":"Modeling of a biped robot for investigating foot drop using MATLAB\/Simulink","volume":"98","author":"Eldirdiry","year":"2020","journal-title":"Simul. Model. Pract. Th."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1109\/MRA.2007.380653","article-title":"Choosing your steps carefully","volume":"14","author":"Kuo","year":"2007","journal-title":"IEEE Robot. Autom. Mag."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3717","DOI":"10.1242\/jeb.205.23.3717","article-title":"Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of human walking","volume":"205","author":"Donelan","year":"2002","journal-title":"J. Exp. Biol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1177\/0278364906063822","article-title":"Fast biped walking with a sensor-driven neuronal controller and real-time online learning","volume":"25","author":"Geng","year":"2006","journal-title":"Int. J. Robot. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"20180197","DOI":"10.1098\/rsif.2018.0197","article-title":"Contributions of metabolic and temporal costs to human gait selection","volume":"15","author":"Summerside","year":"2018","journal-title":"J. R. Soc. Interface"},{"key":"ref_33","first-page":"277","article-title":"Energy-speed relation and optimal speed during level walking","volume":"17","author":"Ralston","year":"1958","journal-title":"Int. Z. Angew. Physiol. Einschl. Arb."},{"unstructured":"Collins, S.H., and Ruina, A. (2005, January 18\u201322). A Bipedal Walking Robot with Efficient and Human-Like Gait. Proceedings of the 2005 IEEE\/RSJ International Conference on Robotics and Automation (ICRA), Barcelona, Spain.","key":"ref_34"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/10\/3435\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:01:00Z","timestamp":1760162460000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/10\/3435"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,14]]},"references-count":34,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["s21103435"],"URL":"https:\/\/doi.org\/10.3390\/s21103435","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2021,5,14]]}}}