{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T07:59:24Z","timestamp":1777276764135,"version":"3.51.4"},"reference-count":58,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,4,17]],"date-time":"2021-04-17T00:00:00Z","timestamp":1618617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2019YFB1309502"],"award-info":[{"award-number":["2019YFB1309502"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>There has been a rising interest in compliant legged locomotion to improve the adaptability and energy efficiency of robots. However, few approaches can be generalized to soft ground due to the lack of consideration of the ground surface. When a robot locomotes on soft ground, the elastic robot legs and compressible ground surface are connected in series. The combined compliance of the leg and surface determines the natural dynamics of the whole system and affects the stability and efficiency of the robot. This paper proposes a bio-inspired leg compliance planning and implementation method with consideration of the ground surface. The ground stiffness is estimated based on analysis of ground reaction forces in the frequency domain, and the leg compliance is actively regulated during locomotion, adapting them to achieve harmonic oscillation. The leg compliance is planned on the condition of resonant movement which agrees with natural dynamics and facilitates rhythmicity and efficiency. The proposed method has been implemented on a hydraulic quadruped robot. The simulations and experimental results verified the effectiveness of our method.<\/jats:p>","DOI":"10.3390\/s21082838","type":"journal-article","created":{"date-parts":[[2021,4,19]],"date-time":"2021-04-19T21:59:49Z","timestamp":1618869589000},"page":"2838","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Bio-Inspired Compliance Planning and Implementation Method for Hydraulically Actuated Quadruped Robots with Consideration of Ground Stiffness"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9774-2462","authenticated-orcid":false,"given":"Xiaoxing","family":"Zhang","sequence":"first","affiliation":[{"name":"State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haoyuan","family":"Yi","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junjun","family":"Liu","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qi","family":"Li","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin","family":"Luo","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Raibert, M.H. (1986). Legged Robots that Balance, MIT Press.","DOI":"10.1109\/MEX.1986.4307016"},{"key":"ref_2","unstructured":"Boston Dynamics (2021, February 01). Robots. Available online: https:\/\/www.bostondynamics.com\/robots."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.1177\/0278364914532150","article-title":"High speed trot-running: Implementation of a hierarchical controller using proprioceptive impedance control on the MIT Cheetah","volume":"33","author":"Hyun","year":"2014","journal-title":"Int. J. Rob. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1177\/0278364917694244","article-title":"High-speed bounding with the MIT Cheetah 2: Control design and experiments","volume":"36","author":"Park","year":"2017","journal-title":"Int. J. Rob. Res."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Di Carlo, J., Wensing, P.M., Katz, B., Bledt, G., Kim, S., and Kim, S. (2018, January 1\u20135). Dynamic locomotion in the mit cheetah 3 through convex model-predictive control. Proceedings of the 2018 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, Spain.","DOI":"10.1109\/IROS.2018.8594448"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Katz, B., Carlo, J.D., and Kim, S. (2019, January 20\u201324). Mini Cheetah: A Platform for Pushing the Limits of Dynamic Quadruped Control. Proceedings of the 2019 International Conference on Robotics and Automation (ICRA), Montreal, QC, Canada.","DOI":"10.1109\/ICRA.2019.8793865"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.1177\/0278364916640102","article-title":"Implementation of trot-to-gallop transition and subsequent gallop on the MIT Cheetah I","volume":"35","author":"Hyun","year":"2016","journal-title":"Int. J. Rob. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1177\/0278364915578839","article-title":"Towards versatile legged robots through active impedance control","volume":"34","author":"Semini","year":"2015","journal-title":"Int. J. Rob. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1324","DOI":"10.1109\/TRO.2015.2482061","article-title":"Model-Based Hydraulic Impedance Control for Dynamic Robots","volume":"31","author":"Boaventura","year":"2015","journal-title":"IEEE Trans. Rob."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1080\/01691864.2017.1378591","article-title":"ANYmal\u2014Toward legged robots for harsh environments","volume":"31","author":"Hutter","year":"2017","journal-title":"Adv. Rob."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4170","DOI":"10.1109\/LRA.2019.2931284","article-title":"Dynamic Locomotion on Slippery Ground","volume":"4","author":"Jenelten","year":"2019","journal-title":"IEEE Rob. Autom. Lett."},{"key":"ref_12","unstructured":"Unitree Robotics (2021, April 10). Aliengo. Available online: https:\/\/www.unitree.com\/products\/aliengo."},{"key":"ref_13","unstructured":"Deep Robotics (2021, April 10). Jueying Robots. Available online: http:\/\/deeprobotics.cn\/en\/products.html."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"eabb2174","DOI":"10.1126\/scirobotics.abb2174","article-title":"Multi-expert learning of adaptive legged locomotion","volume":"5","author":"Yang","year":"2020","journal-title":"Sci. Rob."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129","DOI":"10.3389\/frobt.2018.00129","article-title":"An Overview on Principles for Energy Efficient Robot Locomotion","volume":"5","author":"Kashiri","year":"2018","journal-title":"Front. Rob. AI"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1126\/science.288.5463.100","article-title":"How animals move: An integrative view","volume":"288","author":"Dickinson","year":"2000","journal-title":"Science"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Biewener, A.A., and Patek, S.N. (2018). Animal Locomotion, Oxford University Press.","DOI":"10.1093\/oso\/9780198743156.001.0001"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3325","DOI":"10.1242\/jeb.202.23.3325","article-title":"Templates and anchors: Neuromechanical hypotheses of legged locomotion on land","volume":"202","author":"Full","year":"1999","journal-title":"J. Exp. Biol."},{"key":"ref_19","first-page":"1","article-title":"Toward a Unified Approximate Analytical Representation for Spatially Running Spring-Loaded Inverted Pendulum Model","volume":"37","author":"Yu","year":"2020","journal-title":"IEEE Trans. Rob."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Alexander, R.M. (2003). Principles of Animal Locomotion, Princeton University Press.","DOI":"10.1515\/9781400849512"},{"key":"ref_21","first-page":"4252","article-title":"Elastic ankle muscle-tendon interactions are adjusted to produce acceleration during walking in humans","volume":"220","author":"Farris","year":"2017","journal-title":"J. Exp. Biol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1152\/jappl.1997.82.1.15","article-title":"Interaction of leg stiffness and surface stiffness during human hopping","volume":"82","author":"Ferris","year":"1997","journal-title":"J. Appl. Physiol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1016\/S0021-9290(99)00078-0","article-title":"Runners adjust leg stiffness for their first step on a new running surface","volume":"32","author":"Ferris","year":"1999","journal-title":"J. Biomech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1016\/j.jbiomech.2011.02.072","article-title":"Leg stiffness increases with speed to modulate gait frequency and propulsion energy","volume":"44","author":"Kim","year":"2011","journal-title":"J. Biomech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1016\/j.jbiomech.2015.01.051","article-title":"Running with a load increases leg stiffness","volume":"48","author":"Silder","year":"2015","journal-title":"J. Biomech."},{"key":"ref_26","first-page":"3276","article-title":"Running, hopping and trotting: Tuning step frequency to the resonant frequency of the bouncing system favors larger animals","volume":"218","author":"Cavagna","year":"2015","journal-title":"J. Exp. Biol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1038\/265114a0","article-title":"Storage of elastic strain energy in muscle and other tissues","volume":"265","author":"Alexander","year":"1977","journal-title":"Nature"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"446","DOI":"10.1038\/35106566","article-title":"The central nervous system stabilizes unstable dynamics by learning optimal impedance","volume":"414","author":"Burdet","year":"2001","journal-title":"Nature"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1078\/0944-2006-00057","article-title":"Walking and running at resonance","volume":"105","author":"Ahlborn","year":"2002","journal-title":"Zoology"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"678","DOI":"10.1007\/s004240050451","article-title":"The resonant step frequency in human running","volume":"434","author":"Cavagna","year":"1997","journal-title":"Pflug. Arch. Eur. J. Physiol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1098\/rsta.2006.1911","article-title":"Intelligence by mechanics","volume":"365","author":"Blickhan","year":"2007","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1177\/027836498400300207","article-title":"Experiments in balance with a 3D one-legged hopping machine","volume":"3","author":"Raibert","year":"1984","journal-title":"Int. J. Robot. Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1109\/JRA.1986.1087044","article-title":"Running on four legs as though they were one","volume":"2","author":"Raibert","year":"1986","journal-title":"Robot. Autom. IEEE J."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"7983","DOI":"10.1016\/0021-9290(90)90043-3","article-title":"Trotting, pacing and bounding by a quadruped robot","volume":"23","author":"Raibert","year":"1990","journal-title":"J. Biomech."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Park, H.-W., Chuah, M.Y., and Kim, S. (2014, January 14\u201318). Quadruped bounding control with variable duty cycle via vertical impulse scaling. Proceedings of the 2014 IEEE\/RSJ International Conference on Intelligent Robots and Systems, Chicago, IL, USA.","DOI":"10.1109\/IROS.2014.6943013"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Park, H.-W., Park, S., and Kim, S. (2015, January 26\u201330). Variable-speed quadrupedal bounding using impulse planning: Untethered high-speed 3d running of mit cheetah 2. Proceedings of the 2015 IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, USA.","DOI":"10.1109\/ICRA.2015.7139918"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"025003","DOI":"10.1088\/1748-3190\/10\/2\/025003","article-title":"Quadrupedal galloping control for a wide range of speed via vertical impulse scaling","volume":"10","author":"Park","year":"2015","journal-title":"Bioinspir. Biomim."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Valenzuela, A.K., and Kim, S. (2012, January 14\u201318). Optimally scaled hip-force planning: A control approach for quadrupedal running. Proceedings of the 2012 IEEE International Conference on Robotics and Automation, Saint Paul, MN, USA.","DOI":"10.1109\/ICRA.2012.6225251"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1109\/TRO.2016.2640183","article-title":"Proprioceptive Actuator Design in the MIT Cheetah: Impact Mitigation and High-Bandwidth Physical Interaction for Dynamic Legged Robots","volume":"33","author":"Wensing","year":"2017","journal-title":"IEEE Trans. Rob."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1109\/TMECH.2014.2339013","article-title":"Design Principles for Energy-Efficient Legged Locomotion and Implementation on the MIT Cheetah Robot","volume":"20","author":"Seok","year":"2015","journal-title":"IEEE-ASME Trans. Mech."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Park, H.W., Wensing, P.M., and Kim, S. (2015, January 13\u201317). Online Planning for Autonomous Running Jumps Over Obstacles in High-Speed Quadrupeds. Proceedings of the 2015 Robotics: Science and Systems Conference (RSS), Rome, Italy.","DOI":"10.15607\/RSS.2015.XI.047"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Farshidian, F., Jelavi\u0107, E., Winkler, A.W., and Buchli, J. (2017, January 24\u201328). Robust whole-body motion control of legged robots. Proceedings of the 2017 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Vancouver, BC, Canada.","DOI":"10.1109\/IROS.2017.8206328"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2553","DOI":"10.1109\/LRA.2019.2908502","article-title":"Passive Whole-Body Control for Quadruped Robots: Experimental Validation Over Challenging Terrain","volume":"4","author":"Fahmi","year":"2019","journal-title":"IEEE Rob. Autom. Lett."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1109\/TRO.2019.2954670","article-title":"STANCE: Locomotion Adaptation Over Soft Terrain","volume":"36","author":"Fahmi","year":"2020","journal-title":"IEEE Trans. Rob."},{"key":"ref_45","first-page":"385","article-title":"Design and implementation of scalf, an advanced hydraulic quadruped robot","volume":"36","author":"Chai","year":"2014","journal-title":"Robot"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"10822","DOI":"10.3182\/20080706-5-KR-1001.01833","article-title":"Bigdog, the rough-terrain quadruped robot","volume":"41","author":"Raibert","year":"2008","journal-title":"IFAC Proc. Vol."},{"key":"ref_47","unstructured":"Semini, C. (2010). HyQ Design and Development of a Hydraulically Actuated Quadruped Robot, University of Genoa."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Hutter, M., Remy, C.D., Hoepflinger, M.A., and Siegwart, R. (2011, January 25\u201330). ScarlETH Design and control of a planar running robot. Proceedings of the 2011 IEEE\/RSJ International Conference on Intelligent Robots and Systems, San Francisco, CA, USA.","DOI":"10.1109\/IROS.2011.6048146"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Reher, J., Ma, W., and Ames, A.D. (2019, January 25\u201328). Dynamic Walking with Compliance on a Cassie Bipedal Robot. Proceedings of the 2019 18th European Control Conference (ECC), Naples, Italy.","DOI":"10.23919\/ECC.2019.8796090"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1601","DOI":"10.1016\/j.robot.2013.06.009","article-title":"Variable impedance actuators: A review","volume":"61","author":"Vanderborght","year":"2013","journal-title":"Rob. Auton. Syst."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2418","DOI":"10.1109\/TMECH.2015.2501019","article-title":"Variable Stiffness Actuators: Review on Design and Components","volume":"21","author":"Wolf","year":"2016","journal-title":"IEEE-ASME Trans. Mech."},{"key":"ref_52","first-page":"846","article-title":"Design of an Active Compliance Controller for a Bionic Hydraulic Quadruped Robot","volume":"10463","author":"Zhang","year":"2017","journal-title":"Comput. Vis."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.mechatronics.2017.06.011","article-title":"Stability enhancement of admittance control with acceleration feedback and friction compensation","volume":"45","author":"Aung","year":"2017","journal-title":"Mechatronics"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2568","DOI":"10.1109\/TCYB.2018.2828654","article-title":"Neural Networks Enhanced Adaptive Admittance Control of Optimized Robot-Environment Interaction","volume":"49","author":"Yang","year":"2019","journal-title":"IEEE Trans. Cybern."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1177\/02783640122067309","article-title":"Virtual model control: An intuitive approach for bipedal locomotion","volume":"20","author":"Pratt","year":"2001","journal-title":"Int. J. Rob. Res."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"026006","DOI":"10.1088\/1748-3182\/5\/2\/026006","article-title":"Swing leg control in human running","volume":"5","author":"Blum","year":"2010","journal-title":"Bioinspir. Biomim."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.mechatronics.2018.05.003","article-title":"Control-oriented friction modeling of hydraulic actuators based on hysteretic nonlinearity of lubricant film","volume":"53","author":"Pan","year":"2018","journal-title":"Mechatronics"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.euromechsol.2018.10.016","article-title":"Modelling and analysis of dynamic frictional interactions of vibro-driven capsule systems with viscoelastic property","volume":"74","author":"Liu","year":"2019","journal-title":"Eur. J. Mech. A-Solid"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/8\/2838\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:49:14Z","timestamp":1760161754000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/8\/2838"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,17]]},"references-count":58,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["s21082838"],"URL":"https:\/\/doi.org\/10.3390\/s21082838","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,17]]}}}