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A parameterized family of continuous-time controllers is assumed so that (1) a periodic orbit is induced for the hybrid system, and (2) the orbit is invariant under the choice of controller parameters. Properties of the Poincar\u00e9 map and its first- and second-order derivatives are used to translate the problem of exponential stabilization of the periodic orbit into a set of bilinear matrix inequalities (BMIs). A BMI optimization problem is then set up to tune the parameters of the continuous-time controller so that the Jacobian of the Poincar\u00e9 map has its eigenvalues in the unit circle. It is also shown how robustness against uncertainty in the switching condition of the hybrid system can be incorporated into the design problem. The power of this approach is illustrated by finding robust and stabilizing continuous-time feedback laws for walking gaits of two underactuated 3D bipedal robots.<\/jats:p>","DOI":"10.1177\/0278364915593400","type":"journal-article","created":{"date-parts":[[2015,8,21]],"date-time":"2015-08-21T21:50:24Z","timestamp":1440193824000},"page":"977-999","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":61,"title":["Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems: Application to bipedal locomotion with ground height variations"],"prefix":"10.1177","volume":"35","author":[{"given":"Kaveh Akbari","family":"Hamed","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, San Diego State University, San Diego, CA, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Brian G.","family":"Buss","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University Michigan, Ann Arbor, MI, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jessy W.","family":"Grizzle","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, University Michigan, Ann Arbor, MI, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2015,8,21]]},"reference":[{"key":"bibr1-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1093\/qjmam\/11.4.385"},{"key":"bibr2-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2014.7039613"},{"key":"bibr3-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2013.2287831"},{"key":"bibr4-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1109\/TSMCA.2011.2169246"},{"key":"bibr5-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1177\/0278364904047389"},{"key":"bibr6-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2014.2299342"},{"key":"bibr7-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2014.2299335"},{"key":"bibr8-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2007.4434880"},{"key":"bibr9-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-00602-9_2"},{"key":"bibr10-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1975.1100846"},{"key":"bibr11-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1007\/s10514-010-9216-x"},{"key":"bibr12-0278364915593400","volume-title":"Geometrical Methods in the Theory of Ordinary Differential Equations","author":"Arnold V","year":"1996","edition":"2"},{"key":"bibr13-0278364915593400","unstructured":"ATRIAS (2013) https:\/\/sites.google.com\/site\/atrias21\/."},{"key":"bibr14-0278364915593400","volume-title":"Systems With Impulse Effect: Stability, Theory and Applications","author":"Bainov D","year":"1989"},{"key":"bibr15-0278364915593400","doi-asserted-by":"publisher","DOI":"10.1177\/027836499401300201"},{"key":"bibr16-0278364915593400","doi-asserted-by":"crossref","unstructured":"Burden S, Revzen S, Sastry S (2015) Model reduction near periodic orbits of hybrid dynamical systems. 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