Optimal Control of Microswimmers for Trajectory Tracking Using Bayesian Optimization

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.

💡 Research Summary

This paper addresses the challenging problem of trajectory tracking for microswimmers operating in low‑Reynolds‑number environments, where viscous forces dominate and conventional control design is hampered by highly nonlinear, computationally expensive dynamics. The authors formulate the tracking task as an optimal control problem: given a reference path p_ref(t) over a finite horizon, they minimize a cost consisting of the integrated squared tracking error and a terminal error term, subject to the swimmer’s governing equations and any physical constraints.

To make the problem tractable, the control signal is represented by a set of B‑spline basis functions. This reduces the infinite‑dimensional control function to a finite set of spline control points while preserving smoothness and local support, which is essential for realistic actuation (e.g., magnetic field components or link‑length variations). The resulting low‑dimensional parameter vector is then optimized using Scalable Constrained Bayesian Optimization (SCBO), implemented with the BoTorch library. SCBO builds a Gaussian‑process surrogate of the expensive black‑box objective, handles constraints probabilistically, and selects new candidates via an acquisition function (Expected Improvement). Because gradients of the objective are unavailable or costly to compute, the Bayesian approach is well‑suited for both ordinary‑differential‑equation (ODE) and partial‑differential‑equation (PDE) models.

Two microswimmer models are studied. The first is an N‑link flagellated swimmer with a magnetic head. The head is a sphere of radius r, and the flagellum is discretized into N rigid links. Hydrodynamic drag is approximated by Resistive Force Theory, elastic torques are modeled by torsional springs, and actuation is provided by a time‑varying magnetic field B(t) = (u₁(t), u₂(t), u₃(t))ᵀ acting on a magnetic moment aligned with the head. The resulting dynamics form a coupled ODE system for the head position, orientation (three Euler angles), and the internal joint angles (2N variables). Using the B‑spline parametrization of the magnetic field, SCBO finds control profiles that reproduce a variety of biologically inspired trajectories (circles, helices, S‑shapes) with only a few dozen simulation evaluations. The tracking error is on the order of 10⁻³, and the generated motions closely match experimentally observed sperm trajectories.

The second model is the classic three‑sphere swimmer, where three equal spheres are aligned collinearly and change their inter‑sphere distances to generate propulsion. Here the fluid–structure interaction is solved in full detail using the Feel++ finite‑element library, which discretizes the unsteady Stokes equations coupled with Newton’s laws for the spheres. The authors consider a bounded fluid domain to capture wall‑induced hydrodynamic effects that are common in microfluidic channels. By applying SCBO to the time‑varying link‑length functions (again represented by B‑splines), the optimizer discovers non‑trivial stroke patterns that compensate for wall drag: the optimal gait forms a double‑loop trajectory near the wall, markedly improving tracking accuracy compared with simple sinusoidal strokes. This demonstrates that Bayesian optimization can adapt control strategies to complex environmental constraints without explicit analytical insight.

Key contributions of the work include: (1) the first systematic application of Bayesian optimization to microswimmer control, (2) a flexible B‑spline control parametrization that reduces dimensionality while preserving actuation realism, (3) the use of SCBO to handle constraints (e.g., bounded magnetic fields, physical limits) in high‑cost simulations, (4) validation across models of vastly different fidelity—from low‑dimensional ODEs to high‑resolution PDE simulations—showing the method’s robustness, and (5) a demonstration that the approach can partially counteract wall‑induced hydrodynamic disturbances, a critical step toward practical micro‑robotic navigation in confined environments.

Limitations are also discussed. Although BO requires far fewer evaluations than exhaustive grid searches, each PDE simulation remains expensive, which may preclude real‑time control in highly detailed models. The performance of SCBO can degrade as the number of spline control points grows, potentially leading to scalability issues in very long‑duration or highly complex maneuvers. Moreover, the study is purely simulation‑based; experimental validation on actual microswimmers would be needed to assess robustness against sensor noise, actuation delays, and model mismatches.

Future directions suggested by the authors include integrating online BO for adaptive, real‑time updates, extending the framework to multi‑objective formulations (e.g., minimizing energy consumption while tracking), and testing the methodology on physical microswimmer platforms in microfluidic testbeds. Overall, the paper presents a compelling case for Bayesian optimization as a versatile, gradient‑free tool for optimal control of microscale locomotion under complex fluid‑structure interactions.