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

## TL;DR

This paper rigorously proves the existence and bifurcation behavior of stable walking limit cycles in a simplified passive biped model as the slope parameter varies.

## Contribution

It provides the first rigorous proof of the Melnikov bifurcation in a passive biped walking model with a small slope parameter.

## Key findings

- Existence of a family of limit cycles at zero slope.
- Disappearance of the family of limit cycles as slope increases.
- Persistence of isolated stable walking cycles for positive slope.

## 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.