Optimal Control of Microswimmers for Trajectory Tracking Using Bayesian Optimization

TL;DR

This work addresses trajectory tracking for microswimmers in the Stokes regime by formulating it as an optimal-control problem and solving it with Scalable Constrained Bayesian Optimization (SCBO) using B-spline parameterizations. It demonstrates the approach on two fidelity levels: an ODE-based elastic flagellated $N$-link swimmer with resistive-force theory and a PDE-based three-sphere swimmer, including wall interactions. The main contributions are (i) a unified BO-enabled pipeline for high-cost, gradient-free control of microswimmers, (ii) evidence that optimized, nontrivial stroke patterns (often non-sinusoidal) achieve superior tracking and handle wall-induced hydrodynamics, and (iii) validation of robustness and generality across models of different fidelity. The framework has practical significance for designing robust microrobotic propulsion in constrained fluidic environments with complex fluid–structure interactions.

Abstract

Trajectory tracking for microswimmers remains a key challenge in microrobotics, where low-Reynolds-number dynamics make control design particularly complex. In this work, we formulate the trajectory tracking problem as an optimal control problem and solve it using a combination of B-spline parametrization with Bayesian optimization, allowing the treatment of high computational costs without requiring complex gradient computations. Applied to a flagellated magnetic swimmer, the proposed method reproduces a variety of target trajectories, including biologically inspired paths observed in experimental studies. We further evaluate the approach on a three-sphere swimmer model, demonstrating that it can adapt to and partially compensate for wall-induced hydrodynamic effects. The proposed optimization strategy can be applied consistently across models of different fidelity, from low-dimensional ODE-based models to high-fidelity PDE-based simulations, showing its robustness and generality. These results highlight the potential of Bayesian optimization as a versatile tool for optimal control strategies in microscale locomotion under complex fluid-structure interactions.

Figures (16)

• Figure 1: 3D $N$-links model. The swimmer's head frame is defined as $\mathscr{R}^{h} = (\boldsymbol{X}, \boldsymbol{e}_1^h, \boldsymbol{e}_2^h, \boldsymbol{e}_3^h)$. Each link $i$ of length $l$ is oriented along the unit vector $\boldsymbol{e}_1^i$ from $\boldsymbol{X}^i$. Taken from palazzolo2025b
• Figure 2: Illustration of the three-sphere swimmer. The left sphere is denoted by $\mathscr{B}_1$, the right sphere by $\mathscr{B}_2$, and the central reference sphere by $\mathscr{B}_3$. The swimmer propels itself by varying the arm lengths $u_L$ and $u_R$ at speeds $\dot u_1$ and $\dot u_2$, respectively.
• Figure 3: Illustration of the boundedness property of B-splines of degree $3$ with $\mathscr{T}=\{0, 0,0, 0, 1, 2, 5,5,5,5\}$ (respected) compared to cubic splines (not respected).
• Figure 4: (a) Trajectory error in the $L^2$ norm between an $N$-link swimmer and an $(N+1)$-link swimmer under the magnetic field \ref{['eq:sin_ctrl']}, with $B=0.01$, $f=0.5$, and $T=4/f$. (b) Mean displacement $\Delta x$ per period for various frequencies using the magnetic field defined in \ref{['eq:sin_ctrl']}.
• Figure 5: Illustration of the angles $\theta_{\text{head}}$ and $\theta_{\text{tail}}$ in the $N$-links model, used to characterize the geometrical deformation of the swimming stroke in the planar case ($z=0$ plane).