Non-Locally Controllable but Trackable Magnetic Head Flagellated Swimmer

TL;DR

The paper investigates a magnetic-head microswimmer with an elastic tail in a Stokes flow, deriving a planar two-control-affine model via Resistive Force Theory. It proves that the swimmer is not small-time locally controllable in planar motion using Lie-bracket analysis and symbolic verification, while nonetheless enabling effective trajectory tracking through Bayesian optimization with B-spline controls. This highlights a fundamental limitation in local controllability for planar configurations, yet demonstrates practical trajectory guidance in a data-driven optimization framework. The results provide insight into control strategies for microrobots and motivate extending the analysis to fully three-dimensional dynamics and more complex fluid environments.

Abstract

Unlike macroscopic swimmers, microswimmers operate in a low-Reynolds-number regime dominated by viscous forces. This paper investigates the controllability of a magnetic microswimmer composed of a spherical magnetic head and an elastic, non-magnetic flagellum. The swimmer evolves in a Stokes flow and is modeled using the resistive force theory. We prove that, under planar motion, the system is not small-time locally controllable and numerically identify regions that remain inaccessible. Nevertheless, simulations show that trajectory tracking can still be achieved via Bayesian optimization, though it requires large-amplitude transverse deformations.

Figures (3)

• Figure 1: 3D $N$-link model of the swimmer. The head frame is $\mathscr{R}^h = (\boldsymbol{X}, \boldsymbol{e}_1^h, \boldsymbol{e}_2^h, \boldsymbol{e}_3^h)$; each link $i$ is oriented along $\boldsymbol{e}_1^i$ of length $l$.
• Figure 2: Trajectories around origin equilibrium for $N_{\text{MC}} = 2000$ realizations under the random oscillating control \ref{['eq:random_control']}. $a)$ Trajectories of the $2$-link swimmer. $b)$ Trajectories of the $10$-link swimmer. A zoom on the neighborhood of the origin is shown for both plots. The parameters are taken from \ref{['table:swimmerparameters']}, with $\varepsilon = 1$ and $T = 1$.
• Figure 3: Trajectories of $2$-link swimmer for portion of elliptical references. From left to right: results for $b = \frac{L}{2}$, $b = L$, and $b = \frac{3L}{2}$. Top row: optimal trajectory (blue), trajectory under sinusoidal magnetic field of frequency $0.7Hz$ with tangent alignment (dashed line), and reference trajectory \ref{['eq:ellipse_traj']} (black). Bottom row: optimized controls $u_1$ (red) and $u_2$ (green). The swimmer parameters are taken from \ref{['table:swimmerparameters']}.

Theorems & Definitions (14)

• Definition 1: beauchard2018, $W^{k,\infty}$-STLC
• Definition 2: moreau2024, B-STLC
• Theorem 1: moreau2024, Theorem 3.2
• Theorem 2: moreau2024, Theorem 3.8
• Theorem 2: moreau2024, Theorem 3.8
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof