The $N$-link swimmer in three dimensions: controllability and optimality results

TL;DR

This paper analyzes the controllability and optimal control strategies of a three-dimensional N-link swimmer in low Reynolds number fluid, demonstrating full controllability and exploring optimality in time and power consumption.

Contribution

It establishes controllability for the 2-link swimmer using geometric control theory and extends the results to N-link swimmers, also addressing optimal control strategies.

Findings

The 2-link swimmer can achieve all configurations via shape changes.

Controllability extends to N-link swimmers with similar properties.

Optimal control strategies for minimal time and power are qualitatively described.

Abstract

The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal $2$-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the $2$-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all six linearly independent directions in the configuration space; every direction and orientation can be achieved by operating on the two shape variables. The result is subsequently extended to the $N$-link swimmer. Finally, the minimal time optimal control problem and the minimisation of the power expended are addressed and a qualitative description of the optimal strategies is provided.