How do we walk? Using hybrid holonomy to approximate non-holonomic systems

TL;DR

This paper introduces a geometric framework to analyze how hybrid systems, like walking robots, achieve net movement through holonomy, by examining the effects of impact sequences and their limits.

Contribution

It develops a method to compute the holonomy group of hybrid systems and analyzes their behavior as impacts become infinitely frequent.

Findings

Holonomy explains net motion in walking systems.

Impact sequences influence the holonomy group.

Behavior of hybrid systems approaches a limit with infinite impacts.

Abstract

Why do we move forward when we walk? Our legs undergo periodic motion and thus possess no net change in position; however, our bodies do possess a net change in position and we are propelled forward. From a geometric perspective, this phenomenon of periodic input producing non-periodic output is holonomy. To obtain non-zero holonomy and propel forward, we must alternate which leg is in contact with the ground; a non-zero net motion can be obtained by concatenating arcs that would individually produce no net motion. We develop a framework for computing the holonomy group of hybrid systems and analyze their behavior in the limit as the number of impacts goes to infinity.