Source-linked AI summary
Data-Driven Localization of Magneto-Mechanical Resonators from Time Signals
Artyom Tsanda, Konrad Scheffler, Sarah Reiss, Jonas Faltinath, Janik Bach, Florian Thieben, Pascal Stagge, Fabian Mohn, Tobias Knopp
TL;DR
Estimating sensor parameters from measured signals requires a computationally intensive nonlinear inverse problem, while fast estimation is needed for real-time applications such as catheter tracking. A deep-learning regressor maps measured coil-array signals to MMR position, trained on simulated data with noise recorded from the real system. 7.26 mm median Euclidean distance error across experimental frames and grid points, with a 23.7 mm 95th-percentile error.
Problem
Estimating sensor parameters from measured signals requires a computationally intensive nonlinear inverse problem, while fast estimation is needed for real-time applications such as catheter tracking.
Source: §I · p. 1Method
A deep-learning regressor maps measured coil-array signals to MMR position, trained on simulated data with noise recorded from the real system.
Source: §I · p. 1, B. Data Generation · p. 2, C. Network Architecture and Training · p. 2Results
7.26 mm median Euclidean distance error across experimental frames and grid points, with a 23.7 mm 95th-percentile error.
Source: §III · p. 3Takeaways & Limitations
Accuracy is comparable to the size of the MMR sensor itself and corresponds to 6% of a single coil size; inference requires 13 ms per batch, a small fraction of the 0.88 s frame duration.
Source: §IV · p. 3The approach assumes simulated data can adequately capture the distribution of real measurements; more accurate system-dynamics and noise modeling could improve it.
Source: §IV · p. 3
Abstract
from arXiv · showhide
Magneto-mechanical resonators (MMRs) are a recently introduced sensor technology for tracking and sensing applications that enables effective miniaturization down to the millimeter regime. However, estimating sensor parameters such as position from raw signals remains computationally challenging, because it requires solving a nonlinear inverse problem with computationally demanding objective function evaluations. In this work, we show how a deep learning method can accelerate this process by directly learning the inverse mapping from the measured signals. We consider the localization problem as an example and show that a convolutional-transformer regressor trained on simulated data with recorded noise can generalize to experimental data. The method achieves a median Euclidean distance error of 7 mm, which is comparable to the size of the sensor and corresponds to 6% of the size of a single coil. These results suggest that deep learning is a promising approach for rapid estimation of MMR parameters.
I. INTRODUCTION
MMRs are proposed for compact tracking and sensing, but estimating sensor parameters from measured signals requires a computationally demanding nonlinear inverse problem. This work introduces deep learning to estimate MMR position directly from coil-array time signals.
I. INTRODUCTION
- An MMR uses antiparallel magnets, with a torsionally oscillating rotor whose induced voltage signals in receive coils support position and orientation inference.Sensor frequency can also encode temperature, pressure, and pH.Source: §I · p. 1
- Parameter estimation from measured signals requires a computationally intensive nonlinear inverse problem, motivating faster methods for applications such as real-time catheter tracking.The objective function relies on numerical ODE solvers, numerical integration, and the Biot-Savart law [6].Source: §I · p. 1
- The proof-of-concept approach estimates MMR position directly from inductive coil-array time signals using deep learning.The study makes its source code and data publicly available.Source: §I · p. 1
A. Signal Model
The signal model describes the rotor’s free torsional oscillation and the voltages induced in the receive coils. It assumes gravity is negligible and the rotor has only a torsional degree of freedom.
A. Signal Model
- The rotor’s free oscillation is modeled as a second-order ordinary differential equation under negligible gravity and a single torsional degree of freedom.The rotor returns toward its stable equilibrium alignment when external magnetic forces are absent.Source: A. Signal Model · pp. 1–2
- The rotor’s deflection angle, natural frequency, maximum deflection and angular velocity, and initial phase characterize its oscillation.The initial phase ψstart lies in [0, 2π).Source: A. Signal Model · p. 2
- The receive-coil voltages depend on the coil sensitivity map, rotor position and orientation, the time derivative of its magnetic moment, and measurement noise.The signal is induced in L receive coils according to Faraday’s law.Source: A. Signal Model · p. 2
B. Data Generation
Training and validation data are simulated from the signal model using randomly sampled parameters spanning expected experimental ranges. Recorded system noise is added to the simulated signals.
B. Data Generation
- 100,000 training and 10,000 validation signals are simulated, with parameters sampled randomly across expected real-world ranges.The parameter distributions are summarized in Table I; its caption identifies the table as simulation parameters and their distributions.Source: B. Data Generation · p. 2, Table I · p. 2
- Because the system operates in an unshielded environment, noise n(t) is difficult to model with a predefined distribution, so real-system noise samples are recorded and added to simulated signals.Source: B. Data Generation · p. 2
- Each simulated signal uses a fixed sampling frequency of 61,035 Hz and 53,700 sampled points, and the initial value problem is solved with Tsitouras 5/4 Runge-Kutta.Coil sensitivity is computed by numerical integration of the Biot-Savart law for the experimental array geometry.Source: B. Data Generation · p. 2
C. Network Architecture and Training
The model regresses rotor position from measured signals using convolutional feature extraction followed by temporal attention and a linear output layer. Training uses normalized, downsampled signals and MSE loss.
C. Network Architecture and Training
- The regressor maps coil signals to 3D rotor position and is trained by minimizing mean squared error over the training-data distribution.The model is denoted fθ and maps signals S to position r.Source: C. Network Architecture and Training · p. 2
- Two 1D convolutions extract features, a transformer encoder models temporal dependencies, and global average pooling feeds a linear coordinate-regression output.The embedding dimension is 128, and the multi-head attention uses 8 heads.Source: C. Network Architecture and Training · p. 2
- Signals are downsampled by 20 and normalized by their maximum column norm before training; target positions are scaled by 5.Training runs for 100 epochs with AdamW, while scaled measured-noise samples are added dynamically.Source: C. Network Architecture and Training · p. 2
D. Experiment
The experiment used a 16-coil array and a robot-positioned MMR to collect measurements across a three-dimensional grid. Figure 2 presents estimated and ground-truth positions in 3D and orthogonal projections.
D. Experiment
- A 16-coil array on a 4×4 grid received signals from an MMR built from two cylindrical magnets in a 3D-printed housing.The coils also transmitted signals through a transmit-receive circuit.Source: D. Experiment · p. 2
- Figure 2 shows estimated and ground-truth positions in 3D with receive-coil geometry and in x-y, x-z, and z-y projections, with coordinates in millimeters.Source: Figure 2 · p. 3
- The robot positioned the MMR on a 5×5×3 grid; at each position, 25 static frames were acquired while orientation remained fixed.Grid spacing was 87.25 mm, 90 mm, and 69 mm in x, y, and z; each frame had a 0.88 s readout.Source: D. Experiment · pp. 2–3
E. Evaluation
The evaluation compares estimated positions with robot-recorded ground truth using Euclidean distance across experimental frames and grid points. It also reports uncertainty for error quantiles and benchmarks batch inference time.
E. Evaluation
- Localization error is the Euclidean distance between estimated positions and robot-recorded ground truth across all frames and grid points.Source: E. Evaluation · p. 3
- Because predictions contain outliers, the evaluation reports the 50th and 95th error percentiles with confidence intervals estimated by percentile bootstrap.Source: E. Evaluation · p. 3
- Batch inference time was benchmarked on an AMD EPYC 9124 CPU over 1,000 calls after 10 warm-ups, reporting the minimum time.Source: E. Evaluation · p. 3
III. RESULTS
The method recovered the experimental grid structure, with a 7.26 mm median Euclidean error across frames and grid points. Variability was greater near the coil surface, where z-direction uncertainty dominated.
III. RESULTS
- 7.26 mm median Euclidean distance error and 23.7 mm 95th-percentile error were measured across all frames and grid points.The respective 95% confidence intervals were 7.12–7.46 mm and 23.0–24.6 mm.Source: §III · p. 3
- The inferred positions correctly recovered the grid structure, while errors varied more near the coil surface and were dominated there by z-direction uncertainty.Source: §III · p. 3
- Minimum inference time was 13 ms for a single batch.Source: §III · p. 3
IV. DISCUSSION AND CONCLUSION
The reported localization accuracy is comparable to the MMR's size and 6% of a single coil size, with 13 ms batch inference. The authors identify simulated-data fidelity as a key assumption and outline extensions beyond localization.
IV. DISCUSSION AND CONCLUSION
- Localization accuracy is comparable to the MMR sensor's size and corresponds to 6% of a single coil size; inference takes 13 ms per batch.The authors describe the inference time as a small fraction of the 0.88 s frame duration and the method as suitable for real-time applications.Source: §IV · p. 3
- The approach assumes simulated data adequately capture real measurements, and more accurate system-dynamics and noise modeling could improve the method.Source: §IV · p. 3
- The approach focuses on localization but could be extended to estimate orientation and sensing-related quantities using separate models or a multi-output model.The authors also identify alternative normalization strategies and model architectures as future directions.Source: §IV · p. 3