Geometric Controllability of The Purcell's Swimmer and its Symmetrized Cousin

TL;DR

This paper investigates the controllability of the 3-link Purcell's swimmer and its symmetric variant using geometric methods, extending classical theorems to analyze low Reynolds number locomotion.

Contribution

It introduces a geometric framework utilizing principal fiber bundles and extends Chow's theorem to analyze controllability of these swimmers.

Findings

Established controllability conditions for the swimmer models.

Derived the connection form for the symmetric swimmer.

Analyzed the influence of the structure group's Abelian nature.

Abstract

We analyse weak and strong controllability notions for the locomotion of the 3-link Purcell's swimmer, the simplest possible swimmer at low Reynolds number from a geometric framework. After revisiting a purely kinematic form of the equations, we apply an extension of Chow's theorem to analyze controllability in the strong and weak sense. Further, the connection form for the symmetric version of the Purcell's swimmer is derived, based on which, the controllability analysis utilizing the Abelian nature of the structure group is presented. The novelty in our approach is the usage of geometry and the principal fiber bundle structure of the configuration manifold of the system to arrive at strong and weak controllability notions.