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arXiv · 1904.00517

Existence and stability of a limit cycle in the model of a planar passive biped walking down a slope

Oleg Makarenkov
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Abstract

We consider the simplest model of a passive biped walking down a slope given by the equations of switched coupled pendula (McGeer, 1990). Following the fundamental work by Garcia et al (1998), we view the slope of the ground as a small parameter $\gamma\ge 0$. When $\gamma=0$ the system can be solved in closed form and the existence of a family of limit cycles (i.e. potential walking cycles) can be established explicitly. As observed in Garcia et al (1998), the family of limit cycles disappears when $\gamma$ increases and only isolated asymptotically stable cycles (walking cycles) persist. However, no rigorous proofs of such a bifurcation (often referred to as Melnikov bifurcation) have ever been reported. The present paper fills in this gap in the field and offers the required proof.

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Oleg Makarenkov. 2019-04-01. Existence and stability of a limit cycle in the model of a planar passive biped walking down a slope. https://doi.org/10.1098/rspa.2019.0450

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