Control and Path Planning Strategies of a Micro-swimmer – M1

[antrcy]

Control and Path Planning Strategies of a Micro-swimmer – M1

Student : Antoine Ruch
Supervisors : Lucas Palazzolo (INRIA) & Céline Van Landeghem (IRMA)
Organism : Institut de Recherche en Mathématiques Avancées (IRMA), Strabourg
Time period : June to August 2025

Context

This internship acts as the continuation of a semester project on path planning and numerical control strategies for micro-swimming. We first introduce a trajectory finding method with boundary distance constraints, which we used to test a point-to-point guiding procedure in complex 2D fluid environments. We then propose an implementation of the coupling between a Finite Element fluid solver (Feelpp) and a Bayesian optimization algorithm (SCBO) for which we provide simple test cases.

The distance function strategy

Let $F$ the distance field over the computational fluid domain $\text{𝓕}$:

$$ F(x) = \min_{y \in \partial \mathcal F} \lVert x - y \rVert_2 $$

We define a boundary distance constraint $d > 0$ with which we associate a level set of $F(x) - d = 0$, and we evaluate $F(x) - d$ over $\text{𝓕}_h$, the discretized fluid domain. We then invoke a simple criterion to decide whether a mesh element (triangle or tetrahedral) falls on this level set: if at least two of its vertices map to both negative and positive values, the element is selected. Taking their centroids we obtain a point cloud that approximates the level set.

Geometric graph & Dijkstra's algorithm

My contribution was suggesting to transform this point cloud data set into a geometric graph, with each point acting as vertex connected to its neighbor if their distance falls below a given threshold. We can then use Dijkstra's algorithm to find the optimal path between two points with boundary distance constraints. The resulting ordered sequence of points can then be used by a point-to-point guiding strategy, adjusting an external uniform magnetic field to produce a torque on the swimmer's magnetic body.

Numerical results

A simple Dijkstra test case.

We also have a method to interpolate check-points on the way. We simulate a micro-swimmer guided around a level set.

This time we find an optimal path (in red) inside a super-level set of the zebra-fish tail. The swimmer's trajectory is indicated with a dashed blue line.

Bayesian optimization of swimmer controls

Bayesian optimization uses stochastic processes modelling techniques to approach the behavior of black box, high dimensional functions in order to determine their global optimum in a limited amount of evaluations:

$$ x^* = \arg \min_{x \in \mathcal S} f(x), $$

with $\mathcal S$ a hypercube region. At each iteration of this method, a point or a batch of points is selected using an acquisition function, based on the posterior estimation of $f$ constructed from the available training data. SCBO is one instance of a Bayesian optimization algorithm that uses a trust region based approach to acquisition.

We assume the swimmer's trajectory $X_u$ is the solution of a control problem:

$$ \begin{cases} \dot X_u(t) = f(t, X_u(t), u(t)), \\ X_u(0) = X_0, \end{cases} $$

with $u(t)$ the control. Let $X_{\text{ref}}: [0, 1] \to \mathbb R^m$ a prescribed trajectory and $\gamma$ a parametrization, we aim to minimize the following objective function:

$$ C(u, \gamma) = \int_{0}^T \lVert X_{\text{ref}}(\gamma(t)) - X_u(t) \rVert_Q + \lVert u(t) \rVert_R dt + \lVert X_u(T) - X_{\text{ref}}(1) \rVert_S $$

The optimization spaces of $\gamma$ and $u$ are approached using B-Splines, i.e, linear combinations of piece-wise polynomial functions.

My contribution was to implement the coupling of a provided implementation of SCBO and the Feelpp micro-swimming solver. We provide test cases for the optimization of simple controls:

• $u_1(t)$ : the target linear velocity,
• $u_2(t)$ : the angular velocity of an external magnetic field.

We also set the energy cost $R=0$ to focus on trajectory accuracy.

Test case n°1 - Reaching a target point

$Q$	$R$	$n_{\text{iter}}$	$s_{\text{batch}}$	$C_{\text{final}}$
0	$I_3$	$250$	$10$	$7.46 \times 10^{-2}$

The swimmer must reach the final position of the reference path. We enforce periodic orientation.

Test case n°2 - Avoiding an obstacle

$Q$	$R$	$n_{\text{iter}}$	$s_{\text{batch}}$	$C_{\text{final}}$
$J_3$	$0.1 \times I_3$	$250$	$10$	$4.31$

An obstacle is placed in the way of the swimmer.

Test case n°3 - Following a curve

$Q$	$R$	$n_{\text{iter}}$	$s_{\text{batch}}$	$C_{\text{final}}$
$J_3$	$0$	$250$	$10$	$7.18 \times 10^{-2}$

The curve has the analytical expression of a sine wave.

Test case n°4 - Overcoming boundary effects

$Q$	$R$	$n_{\text{iter}}$	$s_{\text{batch}}$	$C_{\text{final}}$
$J_3$	$0$	$200$	$10$	$2.66 \times 10^{-1}$

The swimmer is placed closer to the bottom wall, which would normally attracts the swimmer.

Discussions

We obtained overall satisfying results for simple test cases, but we were limited by the computational burden of each simulation - this prevented us from conducting a full parameter space and performance study. Future work should focus on reducing the computational load, by parallelizing batch evaluations or pre-optimizing on low-fidelity solvers.

Conclusion

In this internship we explored two key aspects of micro-swimming simulations: trajectory planning, or deciding on feasible paths to follow, and trajectory control, i.e. implementing numerical strategies to ensure accurate tracking. On a personal perspective this internship allowed me to develop a strong intuition and methodological approach for computational fluid dynamics. I was also introduced to Bayesian optimization and surrogates, which are concepts I would gladly work with again in future projects.

Useful links

• 📄 Report
• 🎤 Presentation