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Certified Coil Geometry Learning for Short-Range Magnetic Actuation and Spacecraft Docking Application

Yuta Takahashi, Hayate Tajima, Shin-ichiro Sakai

TL;DR

This work tackles the problem of real-time, high-fidelity magnetic-field modeling for short-range spacecraft docking, where exact Biot–Savart calculations are too costly and Far-Field dipole approximations fail near contacts. It presents a learning-based surrogate that captures coil-geometry interactions through a multilayer perceptron with certified error bounds, enabling accurate near-field predictions while dramatically reducing compute. The method introduces a coil-geometry information vector, decentralized AC current modulation, and a robust data workflow that generalizes across coil radii; it also provides a theoretical bound on learning error and demonstrates this via extensive offline training, numerical docking simulations, and experimental validation on a docking testbed. The results show stable proximity control and significant speedups, supporting practical fuel-free docking, formation control, and multi-satellite collaboration with high assurance in prediction accuracy. The approach advances near-field magnetic actuation for aerospace and related domains by delivering reliable, fast, and scalable magnetic interaction models with provable accuracy.

Abstract

This paper presents a learning-based framework for approximating an exact magnetic-field interaction model, supported by both numerical and experimental validation. High-fidelity magnetic-field interaction modeling is essential for achieving exceptional accuracy and responsiveness across a wide range of fields, including transportation, energy systems, medicine, biomedical robotics, and aerospace robotics. In aerospace engineering, magnetic actuation has been investigated as a fuel-free solution for multi-satellite attitude and formation control. Although the exact magnetic field can be computed from the Biot-Savart law, the associated computational cost is prohibitive, and prior studies have therefore relied on dipole approximations to improve efficiency. However, these approximations lose accuracy during proximity operations, leading to unstable behavior and even collisions. To address this limitation, we develop a learning-based approximation framework that faithfully reproduces the exact field while dramatically reducing computational cost. The proposed method additionally provides a certified error bound, derived from the number of training samples, ensuring reliable prediction accuracy. The learned model can also accommodate interactions between coils of different sizes through appropriate geometric transformations, without retraining. To verify the effectiveness of the proposed framework under challenging conditions, a spacecraft docking scenario is examined through both numerical simulations and experimental validation.

Certified Coil Geometry Learning for Short-Range Magnetic Actuation and Spacecraft Docking Application

TL;DR

This work tackles the problem of real-time, high-fidelity magnetic-field modeling for short-range spacecraft docking, where exact Biot–Savart calculations are too costly and Far-Field dipole approximations fail near contacts. It presents a learning-based surrogate that captures coil-geometry interactions through a multilayer perceptron with certified error bounds, enabling accurate near-field predictions while dramatically reducing compute. The method introduces a coil-geometry information vector, decentralized AC current modulation, and a robust data workflow that generalizes across coil radii; it also provides a theoretical bound on learning error and demonstrates this via extensive offline training, numerical docking simulations, and experimental validation on a docking testbed. The results show stable proximity control and significant speedups, supporting practical fuel-free docking, formation control, and multi-satellite collaboration with high assurance in prediction accuracy. The approach advances near-field magnetic actuation for aerospace and related domains by delivering reliable, fast, and scalable magnetic interaction models with provable accuracy.

Abstract

This paper presents a learning-based framework for approximating an exact magnetic-field interaction model, supported by both numerical and experimental validation. High-fidelity magnetic-field interaction modeling is essential for achieving exceptional accuracy and responsiveness across a wide range of fields, including transportation, energy systems, medicine, biomedical robotics, and aerospace robotics. In aerospace engineering, magnetic actuation has been investigated as a fuel-free solution for multi-satellite attitude and formation control. Although the exact magnetic field can be computed from the Biot-Savart law, the associated computational cost is prohibitive, and prior studies have therefore relied on dipole approximations to improve efficiency. However, these approximations lose accuracy during proximity operations, leading to unstable behavior and even collisions. To address this limitation, we develop a learning-based approximation framework that faithfully reproduces the exact field while dramatically reducing computational cost. The proposed method additionally provides a certified error bound, derived from the number of training samples, ensuring reliable prediction accuracy. The learned model can also accommodate interactions between coils of different sizes through appropriate geometric transformations, without retraining. To verify the effectiveness of the proposed framework under challenging conditions, a spacecraft docking scenario is examined through both numerical simulations and experimental validation.

Paper Structure

This paper contains 19 sections, 2 theorems, 33 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multilayer Perceptron Model and Error Bound
  4. Electromagnetic Interaction Model of One-Axis Coil
  5. "Far-Field" Approximation Model
  6. Coil Geometry Information and Decentralized Dipole Allocation
  7. Three-Axis Coil Geometry Information with Coil Offset
  8. Decentralized Dipole Allocation Using Alternating Current
  9. Learning-based Coil Geometry Approximation
  10. Minimum-Dimensional Sample Collection and Learning
  11. Generalization for Arbitrary Radius Coil Inference
  12. Certified Error Bounds of Coil Geometry Information
  13. Validation by Satellite Docking Application
  14. Offline Training of Coil Geometry Model
  15. Numerical Validation for Spacecraft Docking
  16. ...and 4 more sections

Key Result

Theorem 4.1

For given MLP model $\mathfrak{g}{(\bm{X}, \bm{\theta})}=\mathfrak{g}{\left([\bm{r},\bm{\phi}], \bm{\theta}\right)}$ learned coil geometry information for coil radius $a_{\mathrm{NN}}$ in $\mathcal{S}_{a_{\mathrm{NN}}}$, consider the coil radius $a_{\mathrm{inf}}\triangleq \gamma a_{\mathrm{NN}}$ an Then, $\mathfrak{g}$ derives ${{g}}^a_{j_{v}\leftarrow k_w}({r}^{a}_{j\leftarrow k},\sigma_j,\sigma

Figures (5)

  • Figure 1: Coil geometry learning to predict the magnetic interaction.
  • Figure 2: The definition of circulant integration in Eq. (\ref{['circulant_integration_term']}) and the coil model of target and chaser with offset in subsection \ref{['Three_Axis_Coil_Geometry_Information_with_Coil_Offset']}.
  • Figure 3: Numerical results of two‐satellite docking control using the far‐field magnetic field approximation in Appendix \ref{['appendix_far_field_approximation']} (Dashed lines) and the exact magnetic field model in Eq. (\ref{['near_field_electromagnetic_interaction_model']}) (solid lines). The target and chaser satellite are depicted in gray and red, respectively. The exact model results are based on 30 random initial conditions.
  • Figure 4: The learned model-based docking control result. The time evolution of the distance and time comparison of direct computation (Biot–Savart double integral) and prediction using our model.
  • Figure 5: Experimental results of two‐satellite docking control using the learned magnetic interaction model on a microgravity testbed. With the constant $k=\gamma_{\mu/c}^2\ {\mu_0}/{(8\pi A^2)}\approx 2.205e^{-7}$ [H/m] applied, the y-axis values in Figs \ref{['fig:experimental_prediction_position_result']} and \ref{['fig:experimental_prediction_position_result']} indicate the coefficients relating the product of the two coil currents to the resulting force and torque. $\gamma_{\mu/c}$ [m$^2$] is the design ratio of experimental coils between the maximum dipole and maximum current ($\gamma_{\mu/c}\approx 2.1$takahashi2025noda_mmh).

Theorems & Definitions (5)

  • Theorem 4.1
  • proof
  • Theorem 4.2: Certified Error Bounds
  • proof
  • proof