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After introducing a definition of hybrid averageability sufficient to recover the classical result, we illustrate its applicability by analysis of first a one-legged and then a two-legged hopping model. These abstract systems prepare the ground for the analysis of a significantly more complicated two legged model: a new template for quadrupedal running to be analyzed and implemented on a physical robot in a companion paper. We conclude with some rather more speculative remarks concerning the prospects for further extension and generalization of these ideas. <\/jats:p>","DOI":"10.1177\/0278364918756498","type":"journal-article","created":{"date-parts":[[2018,3,8]],"date-time":"2018-03-08T13:49:35Z","timestamp":1520516975000},"page":"266-286","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":13,"title":["A hybrid dynamical extension of averaging and its application to the analysis of legged gait stability"],"prefix":"10.1177","volume":"37","author":[{"given":"Avik","family":"De","sequence":"first","affiliation":[{"name":"Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA"}]},{"given":"Samuel A","family":"Burden","sequence":"additional","affiliation":[{"name":"Electrical Engineering, University of Washington, Seattle, WA, USA"}]},{"given":"Daniel E","family":"Koditschek","sequence":"additional","affiliation":[{"name":"Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA"}]}],"member":"179","published-online":{"date-parts":[[2018,3,8]]},"reference":[{"key":"bibr1-0278364918756498","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1029-0"},{"key":"bibr2-0278364918756498","doi-asserted-by":"publisher","DOI":"10.1177\/0278364904047389"},{"key":"bibr3-0278364918756498","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2014.2299335"},{"key":"bibr4-0278364918756498","doi-asserted-by":"publisher","DOI":"10.1109\/ACC.2006.1656621."},{"key":"bibr5-0278364918756498","doi-asserted-by":"publisher","DOI":"10.1007\/BF00197760"},{"key":"bibr6-0278364918756498","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2015.2411971"},{"key":"bibr7-0278364918756498","doi-asserted-by":"publisher","DOI":"10.1137\/15M1016588"},{"key":"bibr8-0278364918756498","unstructured":"De A (2017) Modular Hopping and Running via Parallel Composition. 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