Gait controllability of length-changing slender microswimmers

TL;DR

This work analyzes the gait controllability of length-changing two-link slender microswimmers within the framework of Geometric Control Theory and Resistive Force Theory. By modeling shape dynamics with a small number of controllable parameters and exploiting Lie brackets of the associated control vector fields, the authors derive sufficient conditions for gait controllability and deduce total controllability from gait controllability for four length-change mechanisms: stretching, sliding, growing, and telescopic links. They provide explicit Lie-bracket computations and covariance analyses showing independence of bracket directions, yielding gait controllability and, where feasible, total controllability via constant shape matrices. Numerical simulations illustrate the generation of geometric phases that produce translation and rotation, validating the theoretical criteria and demonstrating practical actuation strategies for microswimmer locomotion. The results offer a blueprint for minimal actuation schemes enabling robust locomotion at low Reynolds number and inform design principles for bio-inspired micro-robots with length-changing capabilities.

Abstract

Controllability results of four models of two-link microscale swimmers that are able to change the length of their links are obtained. The problems are formulated in the framework of Geometric Control Theory, within which the notions of fiber, total, and gait controllability are presented, together with sufficient conditions for the latter two. The dynamics of a general two-link swimmer is described by resorting to Resistive Force Theory and different mechanisms to produce a length-change in the links, namely, active deformation, a sliding hinge, growth at the tip, and telescopic links. Total controllability is proved via gait controllability in all four cases, and illustrated with the aid of numerical simulations.

Figures (8)

• Figure 1: The incompetent crawler of Example \ref{['ex:notFCtoPC']}.
• Figure 2: The parametrization of the two-link swimmer given in Section \ref{['sec:generalmodel']}.
• Figure 3: Illustration of the possible shape changes of the stretching links model of Section \ref{['sec:stretch']}. The red bullet and the yellow star represent the position of some exemplifying material points in consecutive steps.
• Figure 4: Illustration of the elongation-type shape changes of the sliding links model of Section \ref{['sec:sliding']}. The red bullet, white square and yellow star represent the position of some exemplifying material points in consecutive steps.
• Figure 5: Illustration of the elongation of the $+$-link in the growing links model of Section \ref{['sec:growing']} (left) and in the telescopic links model of Section \ref{['sec:telescopic']} (right). The red bullet, white square and yellow star represent the position of some exemplifying material points in consecutive steps. The appearance of additional material points in successive steps is due to the eversion/protusion mechanism.

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2: total controllability
• Definition 2.3: fiber controllability
• Definition 2.4: gait controllability
• Definition 2.5: gait controllability at $s^\star$
• Proposition 2.6
• Proposition 2.7
• proof
• Theorem 2.8
• Corollary 2.9