Calculus of Variations and Geometric Measure Theory

R. Marchello - M. Morandotti - H. Shum - M. Zoppello

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

created by morandott on 10 Dec 2019
modified on 13 May 2022


Published Paper

Inserted: 10 dec 2019
Last Updated: 13 may 2022

Journal: Acta Applicandae Mathematicae
Volume: 178
Number: 6
Year: 2022
Doi: 10.1007/s10440-022-00480-3

ArXiv: 1912.04998 PDF


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.

Keywords: optimal control problems, motion in viscous fluids, micro-swimmers, resistive force theory, controllability