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When the desired walking gait is defined as virtual constraints, i.e., as functions of a phasing variable and not on time, for the evolution of the swing foot and the vertical oscillation of the CoM, the walk will asymptotically converge to the periodic motion under disturbance with proper choice of the virtual constraints, thus a self-stabilization is obtained. It is shown that the vertical CoM oscillation, positions of the swing foot and the choice of the switching condition play crucial roles in stability. Moreover, a PI controller of the CoM velocity along the sagittal axis is also proposed such that the walking speed of the robot can converge to another periodic motion with a different walking speed. In this way, a natural walking gait is illustrated as well as the possibility of velocity adaptation as observed in human walking.<\/jats:p>","DOI":"10.1142\/s0219843619500403","type":"journal-article","created":{"date-parts":[[2019,12,2]],"date-time":"2019-12-02T16:04:43Z","timestamp":1575302683000},"page":"1950040","source":"Crossref","is-referenced-by-count":7,"title":["Walking Stability of a Variable Length Inverted Pendulum Controlled with Virtual Constraints"],"prefix":"10.1142","volume":"16","author":[{"given":"Qiuyue","family":"Luo","sequence":"first","affiliation":[{"name":"Laboratoire des Sciences du Num\u00e9rique de Nantes (LS2N), CNRS, Centrale Nantes, Universit\u00e9 de Nantes, 1 rue de la No\u00eb, Nantes, 44321, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christine","family":"Chevallereau","sequence":"additional","affiliation":[{"name":"Laboratoire des Sciences du Num\u00e9rique de Nantes (LS2N), CNRS, Centrale Nantes, Universit\u00e9 de Nantes, 1 rue de la No\u00eb, Nantes, 44321, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yannick","family":"Aoustin","sequence":"additional","affiliation":[{"name":"Laboratoire des Sciences du Num\u00e9rique de Nantes (LS2N), CNRS, Centrale Nantes, Universit\u00e9 de Nantes, 1 rue de la No\u00eb, Nantes, 44321, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2020,1,9]]},"reference":[{"key":"S0219843619500403BIB001","doi-asserted-by":"publisher","DOI":"10.1109\/IROS.2001.973365"},{"key":"S0219843619500403BIB002","doi-asserted-by":"crossref","first-page":"2935","DOI":"10.1242\/jeb.201.21.2935","volume":"201","author":"Lee C. 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