Formation Mechanism of a Basin of Attraction for Passive Dynamic Walking Induced by Intrinsic Hyperbolicity

TL;DR

This paper investigates how intrinsic saddle hyperbolicity influences the shape of the basin of attraction in passive dynamic walking, revealing fundamental stability mechanisms through dynamical systems theory.

Contribution

It identifies the role of saddle hyperbolicity in shaping the basin of attraction, providing new insights into passive walking stability mechanisms.

Findings

The basin of attraction has a fractal-like, thin shape.

Intrinsic saddle hyperbolicity affects basin geometry.

Understanding this mechanism aids in stability analysis of passive walking.

Abstract

Passive dynamic walking is a useful model for investigating the mechanical functions of the body that produce energy-efficient walking. The basin of attraction is very small and thin, and it has a fractal-like shape; this explains the difficulty in producing stable passive dynamic walking. The underlying mechanism that produces these geometric characteristics was not known. In this paper, we consider this from the viewpoint of dynamical systems theory, and we use the simplest walking model to clarify the mechanism that forms the basin of attraction for passive dynamic walking. We show that the intrinsic saddle-type hyperbolicity of the upright equilibrium point in the governing dynamics plays an important role in the geometrical characteristics of the basin of attraction; this contributes to our understanding of the stability mechanism of bipedal walking.