Modelling and controllability of the motion of a slender, flexible micro-swimmer

TL;DR

This paper introduces a novel kinematic model for a slender, flexible micro-swimmer operating at low Reynolds numbers, combining rigid and flexible components, and analyzes its controllability and motion using advanced geometric methods.

Contribution

It develops a new modeling approach that incorporates flexible tail dynamics and provides controllability analysis for such micro-swimmers.

Findings

Model captures the shape-dependent force gradients using Cox theory.

Derived velocities expressed in Lie algebra of SE(2).

Simulation demonstrates velocity variation with shape changes.

Abstract

The mechanism of swimming at very low Reynolds number conditions is a topic of interest to biologists and engineering community. We develop a novel kinematic model of a slender flexible swimmer which locomotes in a low Reynolds number regime. In contrast to existing techniques that model such systems as a connected set of straight, rigid links, the novelty of our technique stems from the fact that we model the swimmer with two components - one is a straight, rigid body (the head) and the other is a flexible member (the tail). Using Cox theory we model the gradient of the forces as a function of the instantaneous shape of the swimmer and its velocity. By virtue of the low inertia conditions, an expression for the translational and rotational velocity of the head is obtained for the planar motion in the form of a Lie algebra of the Special Euclidean group. We explain the principal fiber bundle structure of the configuration space of the swimmer and use that to show a weak controllability result for a type of slender flexible swimmer where the shape space is the space of all continuous curves of a given length. A set of simulation results is presented showing the variation of the swimmer head velocity for a bump function moving along the swimmer length.