{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,9]],"date-time":"2026-09-09T14:11:56Z","timestamp":1788963116613,"version":"build-2803163510"},"reference-count":34,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2022,8,29]],"date-time":"2022-08-29T00:00:00Z","timestamp":1661731200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Robot. AI"],"abstract":"<jats:p>Quadruped robots are widely applied in real-world environments where they have to face the challenges of walking on unknown rough terrains. This paper presents a control pipeline that generates robust and compliant legged locomotion for torque-controlled quadruped robots on uneven terrains. The Cartesian motion planner is designed to be reactive to unexpected early and late contacts using the estimated contact forces. Moreover, we present a novel scheme of optimal stiffness modulation that aims to coordinate desired compliance and tracking performance. It optimizes joint stiffness and contact forces coordinately in a quadratic programming (QP) formulation, where the constraints of non-slipping contacts and torque limits are imposed as well. In addition, the issue of stability under variable stiffness control is solved by imposing a tank-based passivity constraint explicitly. We finally validate the proposed control pipeline on our quadruped robot CENTAURO in experiments on uneven terrains and, through comparative tests, demonstrate the improvements of the variable stiffness locomotion.<\/jats:p>","DOI":"10.3389\/frobt.2022.874290","type":"journal-article","created":{"date-parts":[[2022,8,29]],"date-time":"2022-08-29T07:31:23Z","timestamp":1661758283000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":11,"title":["Variable stiffness locomotion with guaranteed stability for quadruped robots traversing uneven terrains"],"prefix":"10.3389","volume":"9","author":[{"given":"Xinyuan","family":"Zhao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuqiang","family":"Wu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yangwei","family":"You","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arturo","family":"Laurenzi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nikos","family":"Tsagarakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1965","published-online":{"date-parts":[[2022,8,29]]},"reference":[{"key":"B1","first-page":"3295","article-title":"Soft robotics: What cartesian stiffness can we obtain with passively compliant, uncoupled joints?","author":"Albu-Sch\u00e4ffer","year":"2004"},{"key":"B2","first-page":"6028","article-title":"Online optimal impedance planning for legged robots","author":"Angelini","year":"2019"},{"key":"B3","doi-asserted-by":"publisher","first-page":"1311","DOI":"10.1002\/rob.21839","article-title":"Advances in real-world applications for legged robots","volume":"35","author":"Bellicoso","year":"2018","journal-title":"J. Field Robot."},{"key":"B4","first-page":"2245","article-title":"Mit cheetah 3: Design and control of a robust, dynamic quadruped robot","author":"Bledt","year":""},{"key":"B5","first-page":"4399","article-title":"Contact model fusion for event-based locomotion in unstructured terrains","author":"Bledt","year":""},{"key":"B6","doi-asserted-by":"publisher","first-page":"1023","DOI":"10.1109\/lra.2017.2652491","article-title":"Probabilistic contact estimation and impact detection for state estimation of quadruped robots","volume":"2","author":"Camurri","year":"2017","journal-title":"IEEE Robot. Autom. Lett."},{"key":"B7","first-page":"4948","article-title":"A tank-based approach to impedance control with variable stiffness","author":"Ferraguti","year":"2013"},{"key":"B8","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s10514-016-9573-1","article-title":"High-slope terrain locomotion for torque-controlled quadruped robots","volume":"41","author":"Focchi","year":"2017","journal-title":"Auton. Robots"},{"key":"B9","first-page":"3287","article-title":"Control of dynamic gaits for a quadrupedal robot","author":"Gehring","year":"2013"},{"key":"B10","first-page":"339","article-title":"Learning optimal gait parameters and impedance profiles for legged locomotion","author":"Heijmink","year":"2017"},{"key":"B11","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1115\/1.3140701","article-title":"Impedance control: An approach to manipulation: Part III\u2014Applications","volume":"107","author":"Hogan","year":"1985","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"B12","doi-asserted-by":"publisher","first-page":"1595","DOI":"10.1109\/lra.2019.2896758","article-title":"Centauro: A hybrid locomotion and high power resilient manipulation platform","volume":"4","author":"Kashiri","year":"2019","journal-title":"IEEE Robot. Autom. Lett."},{"key":"B13","volume-title":"Nonlinear systems","author":"Khalil","year":"2002"},{"key":"B14","first-page":"1","article-title":"Supervised autonomous locomotion and manipulation for disaster response with a centaur-like robot","author":"Klamt","year":"2018"},{"key":"B15","doi-asserted-by":"publisher","first-page":"1298","DOI":"10.1109\/tro.2016.2593492","article-title":"Stability considerations for variable impedance control","volume":"32","author":"Kronander","year":"2016","journal-title":"IEEE Trans. Robot."},{"key":"B16","first-page":"591","article-title":"Cartesi\/o: A ros based real-time capable cartesian control framework","author":"Laurenzi","year":"2019"},{"key":"B17","doi-asserted-by":"publisher","first-page":"755","DOI":"10.1109\/tnsre.2014.2346927","article-title":"Time-varying ankle mechanical impedance during human locomotion","volume":"23","author":"Lee","year":"2014","journal-title":"IEEE Trans. Neural Syst. Rehabil. Eng."},{"key":"B18","doi-asserted-by":"publisher","first-page":"eabc5986","DOI":"10.1126\/scirobotics.abc5986","article-title":"Learning quadrupedal locomotion over challenging terrain","volume":"5","author":"Lee","year":"2020","journal-title":"Sci. Robot."},{"key":"B19","first-page":"1096","article-title":"Trajectory and foothold optimization using low-dimensional models for rough terrain locomotion","author":"Mastalli","year":"2017"},{"key":"B20","first-page":"3406","article-title":"Inverse dynamics control of floating base systems using orthogonal decomposition","author":"Mistry","year":"2010"},{"key":"B21","first-page":"77","article-title":"Xbotcore: A real-time cross-robot software platform","author":"Muratore","year":"2017"},{"key":"B22","doi-asserted-by":"publisher","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. Robots"},{"key":"B23","first-page":"26","article-title":"Posture and balance control for biped robots based on contact force optimization","author":"Ott","year":"2011"},{"key":"B24","first-page":"4180","article-title":"Cartesian impedance control for a variable stiffness robot arm","author":"Petit","year":"2011"},{"key":"B25","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1177\/0278364912469821","article-title":"Optimal distribution of contact forces with inverse-dynamics control","volume":"32","author":"Righetti","year":"2013","journal-title":"Int. J. Robotics Res."},{"key":"B26","first-page":"440","article-title":"Unified passivity-based cartesian force\/impedance control for rigid and flexible joint robots via task-energy tanks","author":"Schindlbeck","year":"2015"},{"key":"B27","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1109\/tro.2010.2043757","article-title":"Compliant control of multicontact and center-of-mass behaviors in humanoid robots","volume":"26","author":"Sentis","year":"2010","journal-title":"IEEE Trans. Robot."},{"key":"B28","volume-title":"Modelling, planning and control. Advanced textbooks in control and signal processing","author":"Siciliano","year":"2009"},{"key":"B29","first-page":"1248","article-title":"Dynamic balance force control for compliant humanoid robots","author":"Stephens","year":"2010"},{"key":"B30","doi-asserted-by":"publisher","first-page":"2201","DOI":"10.1109\/lra.2017.2723931","article-title":"Fast trajectory optimization for legged robots using vertex-based zmp constraints","volume":"2","author":"Winkler","year":"2017","journal-title":"IEEE Robot. Autom. Lett."},{"key":"B31","doi-asserted-by":"publisher","first-page":"5357","DOI":"10.3390\/s20185357","article-title":"An intuitive formulation of the human arm active endpoint stiffness","volume":"20","author":"Wu","year":"","journal-title":"Sensors"},{"key":"B32","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1109\/lra.2020.3033260","article-title":"A framework for autonomous impedance regulation of robots based on imitation learning and optimal control","volume":"6","author":"Wu","year":"","journal-title":"IEEE Robot. Autom. Lett."},{"key":"B33","first-page":"5028","article-title":"Locomotion adaptation in heavy payload transportation tasks with the quadruped robot centauro","author":"Zhao","year":"2021"},{"key":"B34","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/s0921-8890(03)00042-3","article-title":"An experimental study on compliance control for a redundant personal robot arm","volume":"44","author":"Zollo","year":"2003","journal-title":"Robotics Aut. Syst."}],"container-title":["Frontiers in Robotics and AI"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/frobt.2022.874290\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,29]],"date-time":"2022-08-29T07:31:33Z","timestamp":1661758293000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/frobt.2022.874290\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,29]]},"references-count":34,"alternative-id":["10.3389\/frobt.2022.874290"],"URL":"https:\/\/doi.org\/10.3389\/frobt.2022.874290","relation":{},"ISSN":["2296-9144"],"issn-type":[{"value":"2296-9144","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,29]]},"article-number":"874290"}}