{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T06:15:26Z","timestamp":1776320126551,"version":"3.50.1"},"reference-count":39,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2013,10,21]],"date-time":"2013-10-21T00:00:00Z","timestamp":1382313600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["The International Journal of Robotics Research"],"published-print":{"date-parts":[[2014,1]]},"abstract":"<jats:p> Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Many critical tasks, such as locomotion and manipulation, often involve impacting the ground or objects in the environment. Most state-of-the-art techniques treat the discontinuous dynamics that result from impacts as discrete modes and restrict the search for a complete path to a specified sequence through these modes. Here we present a novel method for trajectory planning of rigid-body systems that contact their environment through inelastic impacts and Coulomb friction. This method eliminates the requirement for a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints. We leverage Sequential Quadratic Programming to naturally resolve contact constraint forces while simultaneously optimizing a trajectory that satisfies the complementarity constraints. The method scales well to high-dimensional systems with large numbers of possible modes. We demonstrate the approach on four increasingly complex systems: rotating a pinned object with a finger, simple grasping and manipulation, planar walking with the Spring Flamingo robot, and high-speed bipedal running on the FastRunner platform. <\/jats:p>","DOI":"10.1177\/0278364913506757","type":"journal-article","created":{"date-parts":[[2013,10,22]],"date-time":"2013-10-22T01:24:25Z","timestamp":1382405065000},"page":"69-81","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":477,"title":["A direct method for trajectory optimization of rigid bodies through contact"],"prefix":"10.1177","volume":"33","author":[{"given":"Michael","family":"Posa","sequence":"first","affiliation":[{"name":"Massachusetts Institute of Technology, USA"}]},{"given":"Cecilia","family":"Cantu","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, USA"}]},{"given":"Russ","family":"Tedrake","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, USA"}]}],"member":"179","published-online":{"date-parts":[[2013,10,21]]},"reference":[{"key":"bibr1-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-75392-6"},{"key":"bibr2-0278364913506757","unstructured":"Acary V, Brogliato B, Daniilidis A, Lemar\u00e9chal C (2004) On the equivalence between complementarity systems, projected systems and unilateral differential inclusions. Report for Institut National de Recherche en Informatique et en Automatique INRIA. Report no. 5107, January 2004."},{"key":"bibr3-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623402401221"},{"key":"bibr4-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008292328909"},{"key":"bibr5-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1115\/1.4001820"},{"key":"bibr6-0278364913506757","doi-asserted-by":"publisher","DOI":"10.2514\/2.4231"},{"key":"bibr7-0278364913506757","volume-title":"Practical Methods for Optimal Control Using Nonlinear Programming (SIAM Advances in Design and Control)","author":"Betts JT","year":"2001"},{"key":"bibr8-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-0557-2"},{"key":"bibr9-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2008.4543376"},{"key":"bibr10-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1007\/BF00249052"},{"key":"bibr11-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1126\/science.1107799"},{"key":"bibr12-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/ICRA.2012.6225250"},{"key":"bibr13-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/IROS.2011.6095035"},{"key":"bibr14-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/IROS.2012.6386181"},{"key":"bibr15-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1080\/02331939208843795"},{"key":"bibr16-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623402407382"},{"key":"bibr17-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018359900133"},{"key":"bibr18-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144504446096"},{"key":"bibr19-0278364913506757","doi-asserted-by":"publisher","DOI":"10.2514\/3.20223"},{"key":"bibr20-0278364913506757","volume-title":"Differential Dynamic Programming","author":"Jacobson DH","year":"1970"},{"key":"bibr21-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/IROS.2004.1389727"},{"key":"bibr22-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511983658"},{"key":"bibr23-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2003.1241930"},{"key":"bibr24-0278364913506757","first-page":"137","volume-title":"Proceedings of the ACM SIGGRAPH\/Eurographics symposium on computer animation","author":"Mordatch I","year":"2012"},{"key":"bibr25-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185539"},{"key":"bibr26-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2010.5509728"},{"key":"bibr27-0278364913506757","first-page":"527","volume-title":"Proceedings of the 10th workshop on the algorithmic foundations of robotics (WAFR)","author":"Posa M","year":"2012"},{"key":"bibr28-0278364913506757","volume-title":"Exploiting inherent robustness and natural dynamics in the control of bipedal walking robots","author":"Pratt J","year":"2000"},{"key":"bibr29-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.1998.680611"},{"key":"bibr30-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/TMECH.2009.2035112"},{"key":"bibr31-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1177\/0278364910388315"},{"key":"bibr32-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1038\/nature04113"},{"key":"bibr33-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144599360110"},{"key":"bibr34-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19960815)39:15<2673::AID-NME972>3.0.CO;2-I"},{"key":"bibr35-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1109\/IROS.2012.6386025"},{"key":"bibr36-0278364913506757","doi-asserted-by":"publisher","DOI":"10.15607\/RSS.2010.VI.022"},{"key":"bibr37-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.19970770411"},{"key":"bibr38-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1145\/1576246.1531366"},{"key":"bibr39-0278364913506757","doi-asserted-by":"publisher","DOI":"10.1201\/9781420053739"}],"updated-by":[{"DOI":"10.1177\/0278364914533878","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2014,5,28]],"date-time":"2014-05-28T00:00:00Z","timestamp":1401235200000}}],"container-title":["The International Journal of Robotics Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/0278364913506757","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.1177\/0278364913506757","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/0278364913506757","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,4]],"date-time":"2025-03-04T00:24:07Z","timestamp":1741047847000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/0278364913506757"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,21]]},"references-count":39,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["10.1177\/0278364913506757"],"URL":"https:\/\/doi.org\/10.1177\/0278364913506757","relation":{"correction":[{"id-type":"doi","id":"10.1177\/0278364914533878","asserted-by":"object"}]},"ISSN":["0278-3649","1741-3176"],"issn-type":[{"value":"0278-3649","type":"print"},{"value":"1741-3176","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,10,21]]}}}