{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T21:12:28Z","timestamp":1768338748002,"version":"3.49.0"},"reference-count":132,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","license":[{"start":{"date-parts":[[2024,6,1]],"date-time":"2024-06-01T00:00:00Z","timestamp":1717200000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/ieeexplore.ieee.org\/Xplorehelp\/downloads\/license-information\/IEEE.html"},{"start":{"date-parts":[[2024,6,1]],"date-time":"2024-06-01T00:00:00Z","timestamp":1717200000000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/ieeexplore.ieee.org\/Xplorehelp\/downloads\/license-information\/IEEE.html"},{"start":{"date-parts":[[2024,6,1]],"date-time":"2024-06-01T00:00:00Z","timestamp":1717200000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2024,6,1]],"date-time":"2024-06-01T00:00:00Z","timestamp":1717200000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"}],"funder":[{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-19-1-0080"],"award-info":[{"award-number":["W911NF-19-1-0080"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Science Foundation","award":["ECCS-1924723"],"award-info":[{"award-number":["ECCS-1924723"]}]},{"name":"National Science Foundation","award":["CMMI-1943900"],"award-info":[{"award-number":["CMMI-1943900"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. IEEE"],"published-print":{"date-parts":[[2024,6]]},"DOI":"10.1109\/jproc.2024.3440211","type":"journal-article","created":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T17:30:57Z","timestamp":1724088657000},"page":"585-608","source":"Crossref","is-referenced-by-count":13,"title":["Saltation Matrices: The Essential Tool for Linearizing Hybrid Dynamical Systems"],"prefix":"10.1109","volume":"112","author":[{"given":"Nathan J.","family":"Kong","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9083-2094","authenticated-orcid":false,"given":"J.","family":"Joe Payne","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7539-0782","authenticated-orcid":false,"given":"James","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1629-8329","authenticated-orcid":false,"given":"Aaron M.","family":"Johnson","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA"}]}],"member":"263","reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-57318-6_32"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2002.806650"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1109\/MCS.2008.931718"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1177\/0278364916639380"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1109\/IROS51168.2021.9636094"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1109\/ICAMechS.2014.6911592"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.21236\/ada225713"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8928(58)90033-9"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-7793-9"},{"key":"ref10","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-44398-8","volume-title":"Dynamics and Bifurcations of Non-Smooth Mechanical Systems","author":"Leine","year":"2004"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-8928(98)00087-2"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2021.109752"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1109\/CDC45484.2021.9683506"},{"key":"ref14","article-title":"Hybrid iLQR model predictive control for contact implicit stabilization on legged robots","author":"Kong","year":"2022","journal-title":"arXiv:2207.04591"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1109\/IROS47612.2022.9981218"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1109\/ICRA46639.2022.9811782"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2015.7402602"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2019.2961996"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2014.7039861"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112064001173"},{"key":"ref21","article-title":"Robust bipedal locomotion: Leveraging saltation matrices for gait optimization","author":"Tucker","year":"2022","journal-title":"arXiv:2209.10452"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2008.916443"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1049\/cp.2010.0191"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1017\/S0956792518000050"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1140\/epjst\/e2018-800033-y"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1115\/DETC99\/VIB-8034"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1016\/S0895-7177(02)00124-3"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1016\/j.euromechsol.2006.04.004"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1137\/050625060"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.3182\/20110828-6-IT-1002.00752"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1109\/81.828574"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-45501-9_3"},{"key":"ref33","first-page":"1","article-title":"Control of bifurcations in power electronic DC\u2013DC converters through manipulation of the saltation matrix","volume-title":"Proc. PhysCon","author":"Maity"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1109\/ECCTD.2011.6043387"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2011.2161371"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1109\/PEDES.2012.6484468"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1002\/cta.1864"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1109\/ECCTD.2013.6662284"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1109\/ECCE44975.2020.9235870"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.79.037201"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1117\/12.2178007"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1137\/140959031"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-016-3011-y"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2017.02.001"},{"key":"ref45","article-title":"Stability analysis of hybrid systems with higher order transverse discontinuity mapping","author":"Chawla","year":"2022","journal-title":"arXiv:2203.13222"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1021\/ie970738y"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1007\/s10827-013-0448-6"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1515\/jaiscr-2015-0023"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-017-01511-y"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1017\/S0956792518000128"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsv.2003.10.051"},{"key":"ref52","volume":"163","author":"Bernardo","year":"2008","journal-title":"Piecewise-Smooth Dyn. Systems: Theory and Applications"},{"issue":"1","key":"ref53","first-page":"87","article-title":"On the bifurcation of periodic orbits in discontinuous systems","volume":"8","author":"Fe\u010dkan","year":"2010","journal-title":"Commun. Math. Anal."},{"key":"ref54","first-page":"1","article-title":"A new software for dynamical analysis of nonsmooth systems","volume-title":"Proc. Eur. Nonlinear Oscillations Conf.","author":"Mandal"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1587\/transfun.E101.A.904"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.97.032213"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.12783\/dtcse\/ammso2019\/30101"},{"key":"ref58","article-title":"Master stability function for piecewise smooth networks","author":"Dieci","year":"2021","journal-title":"arXiv:10.48550"},{"key":"ref59","doi-asserted-by":"publisher","DOI":"10.1109\/JETCAS.2015.2462013"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1109\/ICIT.2006.372582"},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1049\/cp:20080548"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1109\/ISCAS.2008.4541478"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.1002\/cta.505"},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1109\/ISCAS.2010.5536979"},{"key":"ref65","first-page":"1","article-title":"Control of bifurcation of DC\/DC buck converters controlled by double-edged PWM waveform","volume-title":"Proc. Eur. Nonlinear Oscillations Conf.","author":"Elbkosh"},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.1109\/ISCAS.2011.5938164"},{"key":"ref67","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2011.2167273"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.4316\/AECE.2011.04015"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2012.2188953"},{"key":"ref70","first-page":"1","article-title":"On the co-occurrence of slaw-scale bifurcation in period-doubled orbits in a high order system","volume-title":"Proc. World Congr. Eng. Comput. Sci.","volume":"2","author":"Daho"},{"key":"ref71","volume-title":"Analysis and control of nonlinear characteristics in DC\/DC converters","author":"Imrayed","year":"2012"},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1109\/COMPEL.2013.6626474"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1109\/IECON.2013.6700539"},{"key":"ref74","volume-title":"Dynamical analysis of resonant DC\u2013DC converters","author":"Mandal","year":"2013"},{"key":"ref75","doi-asserted-by":"publisher","DOI":"10.1109\/TII.2013.2277942"},{"key":"ref76","doi-asserted-by":"publisher","DOI":"10.1109\/MCAS.2014.2360803"},{"key":"ref77","doi-asserted-by":"publisher","DOI":"10.1109\/APEC.2014.6803704"},{"key":"ref78","doi-asserted-by":"publisher","DOI":"10.1109\/TPEL.2015.2402281"},{"key":"ref79","doi-asserted-by":"publisher","DOI":"10.1049\/iet-pel.2017.0147"},{"key":"ref80","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2017.2780318"},{"key":"ref81","doi-asserted-by":"publisher","DOI":"10.1109\/ISCAS.2017.8050578"},{"key":"ref82","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2018.2883159"},{"key":"ref83","first-page":"1","article-title":"Border collisions in interleaved multi-output DC\u2013DC boost converters","volume-title":"Proc. Int. Symp. Nonlinear Theory Appl.","author":"Gkizas"},{"key":"ref84","doi-asserted-by":"publisher","DOI":"10.1109\/TIE.2018.2826451"},{"key":"ref85","doi-asserted-by":"publisher","DOI":"10.1142\/S0219455419410049"},{"issue":"1","key":"ref86","first-page":"1","article-title":"Dynamic behavior of PWM controlled DC drive","volume":"1","author":"Chakrabarty","year":"2020","journal-title":"Sci. Voyage"},{"key":"ref87","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2020.2993793"},{"key":"ref88","doi-asserted-by":"publisher","DOI":"10.23939\/mmc2021.01.043"},{"key":"ref89","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2020.107080"},{"key":"ref90","doi-asserted-by":"publisher","DOI":"10.3390\/en15010352"},{"key":"ref91","doi-asserted-by":"publisher","DOI":"10.1016\/0960-0779(94)00170-U"},{"key":"ref92","doi-asserted-by":"publisher","DOI":"10.23919\/ACC.1991.4791666"},{"key":"ref93","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0103847"},{"key":"ref94","doi-asserted-by":"publisher","DOI":"10.1115\/DETC2005-84574"},{"key":"ref95","doi-asserted-by":"publisher","DOI":"10.1109\/ICRA57147.2024.10611641"},{"key":"ref96","doi-asserted-by":"publisher","DOI":"10.1016\/S0005-1098(98)00019-3"},{"key":"ref97","doi-asserted-by":"publisher","DOI":"10.1109\/CDC51059.2022.9992825"},{"key":"ref98","doi-asserted-by":"publisher","DOI":"10.1016\/j.nahs.2018.07.003"},{"key":"ref99","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(98)00125-1"},{"key":"ref100","doi-asserted-by":"publisher","DOI":"10.1016\/B978-1-4831-9822-4.50016-9"},{"key":"ref101","doi-asserted-by":"publisher","DOI":"10.1145\/643120.643122"},{"key":"ref102","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-0153-7_34"},{"key":"ref103","doi-asserted-by":"publisher","DOI":"10.1109\/ICRA.2017.7989472"},{"key":"ref104","first-page":"1","article-title":"Reference spreading trajectory tracking control: Experimental analysis on a one-degree-of-freedom setup","volume-title":"Proc. Eur. Nonlinear Oscillations Conf.","author":"Rijnen"},{"key":"ref105","volume-title":"User\u2019s guide for DIRCOL\u2014A direct collocation method for the numerical solution of optimal control problems","author":"Von Stryk","year":"1999"},{"key":"ref106","doi-asserted-by":"publisher","DOI":"10.1137\/16M1062569"},{"key":"ref107","doi-asserted-by":"publisher","DOI":"10.1109\/TMECH.2009.2035112"},{"key":"ref108","doi-asserted-by":"publisher","DOI":"10.1109\/ICRA.2016.7487270"},{"key":"ref109","doi-asserted-by":"publisher","DOI":"10.1177\/0278364913506757"},{"key":"ref110","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185539"},{"key":"ref111","doi-asserted-by":"publisher","DOI":"10.1109\/TCSII.2014.2356873"},{"key":"ref112","article-title":"Invariant filtering for bipedal walking on dynamic rigid surfaces with orientation-based measurement model","author":"Gao","year":"2021","journal-title":"arXiv:2109.01241"},{"key":"ref113","doi-asserted-by":"publisher","DOI":"10.1109\/TMECH.2022.3176015"},{"key":"ref114","first-page":"1","article-title":"Monitoring and diagnosis of hybrid systems using particle filtering methods","volume-title":"Proc. Int. Symp. Math. Theory Netw. Syst.","author":"Koutsoukos"},{"key":"ref115","doi-asserted-by":"publisher","DOI":"10.1115\/1.4054481"},{"key":"ref116","doi-asserted-by":"publisher","DOI":"10.1145\/3049797.3049807"},{"key":"ref117","doi-asserted-by":"publisher","DOI":"10.1137\/15M1016588"},{"key":"ref118","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4340-7"},{"key":"ref119","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(00)00243-7"},{"key":"ref120","doi-asserted-by":"publisher","DOI":"10.1177\/027836499000900206"},{"key":"ref121","volume-title":"Ordinary Differential Equations With Applications","author":"Chicone","year":"2006"},{"key":"ref122","volume-title":"Quadratic Forms Random Variables","author":"Mathai","year":"1992"},{"key":"ref123","volume-title":"An introduction to the Kalman filter","author":"Welch","year":"1995"},{"key":"ref124","doi-asserted-by":"publisher","DOI":"10.1109\/JPROC.2003.823141"},{"key":"ref125","volume":"1","author":"Bertsekas","year":"2012","journal-title":"Dynamic Programming and Optimal Control"},{"key":"ref126","doi-asserted-by":"publisher","DOI":"10.1002\/9783527618385"},{"key":"ref127","doi-asserted-by":"publisher","DOI":"10.1201\/9781315136370"},{"key":"ref128","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2013.2263192"},{"key":"ref129","article-title":"Impacts","author":"Johnson","year":"2021","journal-title":"Advanced Robot Dynamics"},{"key":"ref130","doi-asserted-by":"publisher","DOI":"10.1037\/0096-1523.27.5.1163"},{"key":"ref131","doi-asserted-by":"publisher","DOI":"10.1515\/9781400842643"},{"key":"ref132","volume-title":"Underactuated Robotics","author":"Tedrake","year":"2022"}],"container-title":["Proceedings of the IEEE"],"original-title":[],"link":[{"URL":"https:\/\/ieeexplore.ieee.org\/ielam\/5\/10654605\/10638633-aam.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/xplorestaging.ieee.org\/ielx8\/5\/10654605\/10638633.pdf?arnumber=10638633","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,1]],"date-time":"2024-09-01T04:14:02Z","timestamp":1725164042000},"score":1,"resource":{"primary":{"URL":"https:\/\/ieeexplore.ieee.org\/document\/10638633\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6]]},"references-count":132,"journal-issue":{"issue":"6"},"URL":"https:\/\/doi.org\/10.1109\/jproc.2024.3440211","relation":{},"ISSN":["0018-9219","1558-2256"],"issn-type":[{"value":"0018-9219","type":"print"},{"value":"1558-2256","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6]]}}}