Airborne Magnetic Anomaly Navigation (MagNav) provides a jamming-resistant and robust alternative to satellite navigation but requires the real-time compensation of the aircraft platform's large and dynamic magnetic interference. State-of-the-art solutions often rely on extensive offline calibration flights or pre-training, creating a logistical barrier to operational deployment. We present a fully adaptive MagNav architecture featuring a "cold-start" capability that identifies and compensates for the aircraft's magnetic signature entirely in-flight. The proposed method utilizes an extended Kalman filter with an augmented state vector that simultaneously estimates the aircraft's kinematic states as well as the coefficients of the physics-based Tolles-Lawson calibration model and the parameters of a Neural Network to model aircraft interferences. The Kalman filter update is mathematically equivalent to an online Natural Gradient descent, integrating superior convergence and data efficiency of state-of-the-art second-order optimization directly into the navigation filter. To enhance operational robustness, the neural network is constrained to a residual learning role, modeling only the nonlinearities uncorrected by the explainable physics-based calibration baseline. Validated on the MagNav Challenge dataset, our framework effectively bounds inertial drift using a magnetometer-only feature set. The results demonstrate navigation accuracy comparable to state-of-the-art models trained offline, without requiring prior calibration flights or dedicated maneuvers.
Figures & tables
Figure 1: Hybrid MagNav Architecture with NN -augmented calibration model.
Figure 2: Mean position error distribution and average runtimes from 100 Monte Carlo simulations for input parameter sets ϕm , ϕmv , and ϕall for different numbers of hidden neurons.
Figure 3: Magnetic-Anomaly aided odometry with NN -online-learning based platform interference compensation for four trajectories ( ϕ=ϕm , Nh=8 ).
Figure 4: NN-EKF performance on the lawnmower trajectory ( ϕm ) for Nh=2 and Nh=8 showing position RMSE improvement compared to odometry in (a) and platform interference approximation in (b).
Figure 5: EKF-based online adaptation of the NN parameters xNN for ϕ=ϕm , Nh=8 for the (a) Figure Eight, (b) Irregular, (c) Lawnmower, and (d) Spiral trajectories.
Figure 6: GNSS trajectory of flight line 1007.06.
Figure 7: Magnetic aiding performance results using magnetometer 1 to 5 and different hidden layer sizes Nh .
Figure 9: Evolution of the NN parameter variance PNN using Magnetometer 2 ( Nh=5 ), comparing (a) cold start and (b) warm start.
Figure 10: Evolution of NN (top) and TL (bottom) model parameters using Magnetometer 2 ( Nh=5 ), comparing (a), (c) cold start and (b), (d) warm start.
Figure 11: Comparison between the (a) TL model compensation components (including the bias state SCB ) and the (b) learned NN output using Magnetometer 2 in the cold start scenario.
Appendix figures & tables21 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 12: CRLBs along the lawnmower-trajectory for (a) small Nh=4 and (b) large Nh=32 .
Figure 13: Evolution of the positioning accuracy in cold and warm start scenarios using Magnetometer 1 ( Nh=5 ). (a) Our hybrid approach combining TL and NN calibration ( DRMS cold-start 17m / warm-start 17m ) (b) TL online calibration ( DRMS cold-start 17m / warm-start 17m ).
Figure 14: Evolution of NN (top) and TL model (bottom) parameters using Magnetometer 1 ( Nh=5 ), comparing (a), (c) cold start and (b), (d) warm start.
Figure 15: Comparison between the (a) TL model compensation components (including the bias state SCB ) and the (b) learned NN output using Magnetometer 1 in the cold start scenario.
Figure 16: Comparison of INS and MagNav position errors (north, east, and down) for Magnetometer 1 in (a) our hybrid TL + NN cold-start and (b) the TL -only cold-start scenarios.
Figure 17: Comparison of INS and MagNav position errors (north, east, and down) for Magnetometer 2 in (a) our hybrid TL + NN cold-start and (b) the TL -only cold-start scenarios.
Figure 18: Evolution of the positioning accuracy in cold and warm start scenarios using Magnetometer 3 ( Nh=2 ). (a) Hybrid approach combining TL and NN calibration ( DRMS cold-start 42m / warm-start 36m ) (b) TL online calibration ( DRMS cold-start 46m / warm-start 45m ).
Figure 19: Evolution of NN (top) and TL model (bottom) parameters using Magnetometer 3 ( Nh=5 ), comparing (a), (c) cold start and (b), (d) warm start.
Figure 20: Comparison between the (a) TL model compensation components (including the bias state SCB ) and the (b) learned NN output using Magnetometer 3 in the cold start scenario.
Figure 21: Comparison of INS and MagNav position errors (north, east, and down) for Magnetometer 3 in (a) our hybrid TL + NN cold-start and (b) the TL -only cold-start scenarios.
Figure 22: Evolution of the positioning accuracy in cold and warm start scenarios using Magnetometer 4 ( Nh=5 ). (a) Hybrid approach combining TL and NN calibration ( DRMS cold-start 37m / warm-start 46m ) (b) TL online calibration ( DRMS cold-start 58m / warm-start 54m ).
Figure 23: Evolution of NN (top) and TL model (bottom) parameters using Magnetometer 4 ( Nh=5 ), comparing (a), (c) cold start and (b), (d) warm start.
Figure 24: Comparison between the (a) TL model compensation components (including the bias state SCB ) and the (b) learned NN output using Magnetometer 4 in the cold start scenario.
Figure 25: Comparison of INS and MagNav position errors (north, east, and down) for Magnetometer 4 in (a) our hybrid TL + NN cold-start and (b) the TL -only cold-start scenarios.
Figure 26: Evolution of the positioning accuracy in cold and warm start scenarios using Magnetometer 5 ( Nh=5 ). (a) Hybrid approach combining TL and NN calibration ( DRMS cold-start 14m / warm-start 15m ) (b) TL online calibration ( DRMS cold-start 15m / warm-start 15m ).
Figure 27: Evolution of NN (top) and TL model (bottom) parameters using Magnetometer 5 ( Nh=5 ), comparing (a), (c) cold start and (b), (d) warm start.
Figure 28: Comparison between the (a) TL model compensation components (including the bias state SCB ) and the (b) learned NN output using Magnetometer 5 in the cold start scenario.
Figure 29: Comparison of INS and MagNav position errors (north, east, and down) for Magnetometer 5 in (a) our hybrid TL + NN cold-start and (b) the TL -only cold-start scenarios.
Figure 30: Evolution of the innovation using ( Nh=5 ), comparing (a) Magnetometer 4 (C), (b) Magnetometer 4 (W), (c) Magnetometer 5 (C), and (d) Magnetometer 5 (W).
Figure 31: GNSS trajectory of all segments of flight line 1003.01 to 1003.09.
Figure 32: Evolution of the positioning accuracy for our hybrid approach combining TL and NN calibration using Magnetometer 3 ( Nh=2 ) in the (a) cold-start ( DRMS 142m ) and (b) warm-start ( DRMS 70m ) scenarios.
Autonomous platforms require accurate positioning to complete their tasks. To this end, a Kalman filter-based algorithms, such as the extended Kalman filter or invariant Kalman filter, utilizing inertial and external sensor fusion are applied. To cope with real-world scenarios, adaptive noise estimation methods have been developed primarily for classical Euclidean formulations. However, these methods remain largely unexplored in the tangent Lie space, despite it provides a principled geometric framework with favorable error dynamics on Lie groups. To fill this gap, we combine invariant filtering theory with neural-aided adaptive noise estimation in real-world settings. To this end, we derive a novel theoretical extension of classical innovation-based process noise adaptation formulated directly within the Lie-group framework. We further propose a lightweight neural network that estimates the process noise covariance parameters directly from raw inertial data. Trained entirely in a sim2real framework via domain adaptation, the network captures motion-dependent and sensor-dependent noise characteristics without requiring labeled real-world data. To examine our proposed neural-aided adaptive invariant Kalman filter, we focus on the challenging real-world scenario of autonomous underwater navigation. Experimental results demonstrate superior performance compared to existing methods in terms of position root mean square error. These results validate our sim2real pipeline and further confirm that geometric invariance significantly enhances learning-based adaptation and that adaptive noise estimation in the tangent Lie space offers a powerful mechanism for improving navigation accuracy in nonlinear autonomous platforms.
Accurate state estimation of nonlinear dynamical systems is fundamental to modern aerospace operations across air, sea, and space domains. Online tracking of adversarial unmanned aerial vehicles (UAVs) is especially challenging due to agile nonlinear motion, noisy and sparse sensor measurements, and unknown control inputs; conditions that violate key assumptions of classical Kalman filter variants and degrade estimation performance. Neural networks (NNs) can learn complex nonlinear relationships from data, but lack principled uncertainty quantification, which is critical for state estimation tasks where confidence bounds drive downstream decisions. We address this with Bayesian Neural Networks (BNNs), which model uncertainty through distributions over network weights and produce predictive means and uncertainties via Monte Carlo sampling. Building on this, we propose the Bayesian Neural Kalman Filter (BNKF): a hybrid framework coupling a trained BNN with a Kalman correction step for robust online UAV state estimation. Unlike related neural Kalman approaches, BNKF produces full state predictions and incorporates Bayesian uncertainty directly into covariance propagation, improving robustness under high noise conditions. We evaluate BNKF under varying radar noise levels and sampling rates using synthetic nonlinear UAV flight data. Five fold cross validation demonstrates that BNKF outperforms Extended and Unscented Kalman Filters in accuracy, precision, and truth containment under degraded sensing. An ensemble variant (BNKFe) further improves precision in high-noise edge cases at a slight accuracy tradeoff. Runtime analysis confirms minimal inference overhead, supporting real-time deployment feasibility.
Akhil Gupta, Erhan Guven
Whiting School of Engineering EP Program, Johns Hopkins University, Baltimore, Maryland, USA
Navigation in GPS-denied environments is commonly based on magnetic maps, which are used to correct trajectory drift and enable relocalization. However, such navigation systems suffer from a strong dependence on magnetic maps that are complex, heavy, and costly to construct. Bio-inspired magnetic navigation (BIMN) does not rely on magnetic maps, making it simpler and more frugal. In this paper, we present a gradient descent BIMN method based on the Rao-Blackwellized particle filter (RBPF) for local gradient estimation. We show that magnetic navigation can be performed without magnetic maps, as gradient descent naturally attracts the system toward the destination, resulting in a simple and memory-efficient approach.
Zayed Herma, Andrea Vassilev, Philippe Cardin +1
Univ. Grenoble Alpes, CEA, Leti, 38000 Grenoble, France · Univ. Grenoble Alpes, CNRS, ISTerre, 38000 Grenoble, France