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

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.