Pairwise controllability and motion primitives for micro-rotors in a bounded Stokes flow

Abstract

Micro-robots that can propel themselves in a low Reynolds number fluid flow by converting their rotational motion into translation have begun attracting much attention due to their ease of fabrication. The dynamics and controllability of the motion of such microswimmers are investigated in this paper. The microswimmers under consideration here are spinning spheres (or rotors) whose dynamics are approximated by rotlets, a singularity solution of the Stokes equations. While singularities of Stokes flows are commonly used as theoretical models for microswimmers and micro-robots, rotlet models of microswimmers have received less attention. While a rotlet alone cannot generate translation, a pair of rotlets can interact and execute net motion. Taking the control inputs to be the strengths of the micro rotors, the positions of a pair of rotors are not controllable in an unbounded planar fluid domain. However, in a bounded domain, which is often the case of practical interest, we show that the positions of the micro rotors are controllable. This is enabled by the interaction of the rotors with the boundaries of the domain. We show how control inputs can be constructed based on combinations of Lie brackets to move the rotors from one point to another in the domain. Another contribution of this paper is the creation of a framework for path planning and control of the motion of Stokes singularities that model the dynamics of microswimmers. This can be extended to microswimmers with other shapes moving in confined fluid domains with complex boundaries.