{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T17:26:38Z","timestamp":1783013198222,"version":"3.54.6"},"reference-count":37,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2015,9,17]],"date-time":"2015-09-17T00:00:00Z","timestamp":1442448000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotica"],"published-print":{"date-parts":[[2017,3]]},"abstract":"<jats:title>SUMMARY<\/jats:title><jats:p>Swing-leg retraction in walking is the slowing or reversal of the forward rotation of the swing leg at the end of the swing phase prior to ground contact. For retraction, a hip torque is often applied to the swing leg at about the same time as stance-leg push-off. Due to mechanical coupling, the push-off force affects leg swing, and hip torque affects the stance-leg extension. This coupling makes the energetic costs of retraction and push-off depend on their relative timing. Here, we find the energy-optimal relative timing of these actions. We first use a simplified walking model with non-regenerative actuators, a work-based energetic-cost, and impulsive actuations. Depending on whether the late-swing hip torque is retracting or extending (pushing the leg forward), we find that the optimum is obtained by applying the impulsive hip torque either following or prior to the impulsive push-off force, respectively. These trends extend to other bipedal models and to aperiodic gaits, and are independent of step lengths and walking speeds. In one simulation, the cost of a walking step is increased by 17.6% if retraction torque comes before push-off. To consider non-impulsive actuation and the cost of force production, we add a force-squared (<jats:italic>F<\/jats:italic><jats:sup>2<\/jats:sup>) term to the work cost. We show that this cost promotes simultaneous push-off force and retracting torque, but does not change the result that any extending torque should come prior to push-off. A high-fidelity optimization of the Cornell Ranger robot is consistent with the swing-retraction trends from the models above.<\/jats:p>","DOI":"10.1017\/s0263574715000764","type":"journal-article","created":{"date-parts":[[2015,9,17]],"date-time":"2015-09-17T03:03:09Z","timestamp":1442458989000},"page":"654-686","source":"Crossref","is-referenced-by-count":14,"title":["Energy-optimal relative timing of stance-leg push-off and swing-leg retraction in walking"],"prefix":"10.1017","volume":"35","author":[{"given":"S. Javad","family":"Hasaneini","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"John E. A.","family":"Bertram","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chris J. B.","family":"Macnab","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2015,9,17]]},"reference":[{"key":"S0263574715000764_ref36","volume-title":"Robot Modeling and Control","author":"Spong","year":"2006"},{"key":"S0263574715000764_ref19","doi-asserted-by":"publisher","DOI":"10.1080\/01691864.2013.791656"},{"key":"S0263574715000764_ref16","doi-asserted-by":"publisher","DOI":"10.1177\/027836499000900206"},{"key":"S0263574715000764_ref28","volume-title":"Biomechanics and Energetics of Muscular Exercise","author":"Margaria","year":"1976"},{"key":"S0263574715000764_ref20","doi-asserted-by":"publisher","DOI":"10.1098\/rsif.2009.0544"},{"key":"S0263574715000764_ref6","doi-asserted-by":"crossref","unstructured":"M. Wisse , C. G. Atkeson and D. K. Kloimwieder , \u201cSwing Leg Retraction Helps Biped Walking Stability,\u201d Proceedings 5th IEEE-RAS International Conference on Humanoid Robots, Tsukuba, Japan (Dec. 2005) pp. 295\u2013300.","DOI":"10.1109\/ICHR.2005.1573583"},{"key":"S0263574715000764_ref15","doi-asserted-by":"crossref","unstructured":"S. J. Hasaneini , C. J. Macnab , J. E. Bertram and H. Leung , \u201cSwing-Leg Retraction Efficiency in Bipedal Walking,\u201d Intelligent Robots and Systems (IROS 2014), 2014 IEEE\/RSJ International Conference on, IEEE (Chicago: IL, 2014) pp. 2515\u20132522.","DOI":"10.1109\/IROS.2014.6942905"},{"key":"S0263574715000764_ref5","unstructured":"S. J. Hasaneini , Energy Efficient Bipedal Locomotion Ph.D. dissertation (University of Calgary, Calgary, Alberta, Canada, Jan. 2014)."},{"key":"S0263574715000764_ref8","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2008.917002"},{"key":"S0263574715000764_ref35","doi-asserted-by":"publisher","DOI":"10.1242\/jeb.01498"},{"key":"S0263574715000764_ref9","doi-asserted-by":"crossref","unstructured":"J. G. D. Karssen , M. Haberland , M. Wisse and S. Kim , \u201cThe optimal swing-leg retraction rate for running,\u201d Proceedings 2011 IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China (May 2011) pp. 4000\u20134006.","DOI":"10.1109\/ICRA.2011.5980168"},{"key":"S0263574715000764_ref24","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(80)90070-X"},{"key":"S0263574715000764_ref18","doi-asserted-by":"publisher","DOI":"10.1038\/nature04113"},{"key":"S0263574715000764_ref10","unstructured":"S. J. Hasaneini , C. J. Macnab , J. E. Bertram and A. Ruina , \u201cSeven Reasons to Brake Leg swing Just Before Heel Strike,\u201d Online Proceedings of Dynamic Walking Conference (Pittsburgh, P.A., 2013)."},{"key":"S0263574715000764_ref37","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)AS.1943-5525.0000017"},{"key":"S0263574715000764_ref34","doi-asserted-by":"publisher","DOI":"10.1006\/jtbi.2001.2279"},{"key":"S0263574715000764_ref2","doi-asserted-by":"crossref","unstructured":"K. L. Poggensee , M. A. Sharbafi and A. Seyfarth , \u201cCharacterizing Swing-Leg Retraction in Human Locomotion,\u201d Proceedings of 17th International Conference on Climbing and Walking Robots, Poznan, Poland (Jul. 2014).","DOI":"10.1142\/9789814623353_0044"},{"key":"S0263574715000764_ref26","doi-asserted-by":"crossref","DOI":"10.1515\/9780691221540","volume-title":"Muscles, Reflexes, and Locomotion","author":"McMahon","year":"1984"},{"key":"S0263574715000764_ref7","doi-asserted-by":"publisher","DOI":"10.1242\/jeb.00463"},{"key":"S0263574715000764_ref23","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":"S0263574715000764_ref31","doi-asserted-by":"publisher","DOI":"10.1145\/2558904"},{"key":"S0263574715000764_ref1","volume-title":"Human Walking","author":"Inman","year":"1981"},{"key":"S0263574715000764_ref11","doi-asserted-by":"publisher","DOI":"10.1177\/0278364914527485"},{"key":"S0263574715000764_ref13","first-page":"113","article-title":"Energetics of actively powered locomotion using the simplest walking model","volume":"124","author":"Kuo","year":"2002","journal-title":"J. Biomed. Eng."},{"key":"S0263574715000764_ref29","doi-asserted-by":"publisher","DOI":"10.1016\/0167-9457(94)90010-8"},{"key":"S0263574715000764_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9290(01)00169-5"},{"key":"S0263574715000764_ref32","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623499350013"},{"key":"S0263574715000764_ref25","first-page":"297","article-title":"The human bicycle","volume":"5","author":"Basmajian","year":"1976","journal-title":"Biomechanics VA"},{"key":"S0263574715000764_ref21","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0117384"},{"key":"S0263574715000764_ref4","volume-title":"Human Walking","author":"Rose","year":"1994"},{"key":"S0263574715000764_ref30","unstructured":"P. A. Bhounsolue , A controller design framework for bipedal robots: Trajectory optimization and event-based stabilization Ph.D. dissertation (Cornell University, Ithaca, NY, USA, May 2012)."},{"key":"S0263574715000764_ref17","doi-asserted-by":"crossref","unstructured":"S. J. Hasaneini , C. J. Macnab , J. E. Bertram and H. Leung , \u201cOptimal Relative Timing of Stance Push-Off and swing leg retraction,\u201d Intelligent Robots and Systems (IROS), 2013 IEEE\/RSJ International Conference on. 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