However, magnetometer data often suffer from disturbances caused by satellite dynamics, onboard instrument interference, and environmental noise. For instance, changes in satellite orientation can lead to anomalies in magnetic field measurements due to interference from electric currents within the satellite's instruments. These disturbances necessitate careful data correction to ensure the accuracy and reliability of measurements. Traditional correction methods rely heavily on human expertise and are rooted in well established physical and mathematical principles. While these methods have proven effective, they are inherently limited by their long processing times and delays in real time prediction [7] [6] [4] [2] [1]. In contrast, machine learning models, though rarely applied in this field, offer strong predictive capabilities and the potential for faster computations. This study seeks to address these limitations by combining the strengths of traditional correction methods with the adaptability and efficiency of machine learning models, thereby improving timeliness while ensuring both physical consistency and improved real time performance. This study bridges the gap between data driven models and physics based understanding by integrating Maxwell's equations into the neural network architecture as physical information. The key innovations are: 1 arXiv:2501.00020v3

We present a lightweight magnetic field simultaneous localisation and mapping (SLAM) approach for drift correction in odometry paths, where the interest is purely in the odometry and not in map building. We represent the past magnetic field readings as a one-dimensional trajectory against which the current magnetic field observations are matched. This approach boils down to sequential loop-closure detection and decision-making, based on the current pose state estimate and the magnetic field. We combine this setup with a path estimation framework using an extended Kalman smoother which fuses the odometry increments with the detected loop-closure timings. We demonstrate the practical applicability of the model with several different real-world examples from a handheld iPad moving in indoor scenes.

Indoor magnetic fields are a combination of Earth's magnetic field and disruptions induced by ferromagnetic objects, such as steel structural components in buildings. As a result of these disruptions, pervasive in indoor spaces, magnetic field data is often omitted from navigation algorithms in indoor environments. This paper leverages the spatially-varying disruptions to Earth's magnetic field to extract positional information for use in indoor navigation algorithms. The algorithm uses a rate gyro and an array of four magnetometers to estimate the robot's pose. Additionally, the magnetometer array is used to compute attitude-invariant measurements associated with the magnetic field and its gradient. These measurements are used to detect loop closure points. Experimental results indicate that the proposed approach can estimate the pose of a ground robot in an indoor environment within meter accuracy.

Electrolysis is crucial for eco-friendly hydrogen production, but gas bubbles generated during the process hinder reactions, reduce cell efficiency, and increase energy consumption. Additionally, these gas bubbles cause changes in the conductivity inside the cell, resulting in corresponding variations in the induced magnetic field around the cell. Therefore, measuring these gas bubble-induced magnetic field fluctuations using external magnetic sensors and solving the inverse problem of Biot-Savart's Law allows for estimating the conductivity in the cell and, thus, bubble size and location. However, determining high-resolution conductivity maps from only a few induced magnetic field measurements is an ill-posed inverse problem. To overcome this, we exploit Invertible Neural Networks (INNs) to reconstruct the conductivity field. Our qualitative results and quantitative evaluation using random error diffusion show that INN achieves far superior performance compared to Tikhonov regularization.

Localization of magnetically actuated medical robots is essential for accurate actuation, closed loop control and delivery of functionality. Despite extensive progress in the use of magnetic field and inertial measurements for pose estimation, these have been either under single external permanent magnet actuation or coil systems. With the advent of new magnetic actuation systems comprised of multiple external permanent magnets for increased control and manipulability, new localization techniques are necessary to account for and leverage the additional magnetic field sources. In this letter, we introduce a novel magnetic localization technique in the Special Euclidean Group SE(3) for multiple external permanent magnetic field actuation and control systems. The method relies on a milli-meter scale three-dimensional accelerometer and a three-dimensional magnetic field sensor and is able to estimate the full 6 degree-of-freedom pose without any prior pose information. We demonstrated the localization system with two external permanent magnets and achieved localization errors of 8.5 ? 2.4 mm in position norm and 3.7 ? 3.6? in orientation, across a cubic workspace with 20 cm length.

This work presents a number of techniques to improve the ability to create magnetic field maps on a UAV which can be used to quickly and reliably gather magnetic field observations at multiple altitudes in a workspace. Unfortunately, the electronics on the UAV can introduce their own magnetic fields, distorting the resultant magnetic field map. We show methods of reducing and working with UAV-induced noise to better enable magnetic fields as a sensing modality for indoor navigation. First, some gains in our flight controller create high-frequency motor commands that introduce large noise in the measured magnetic field. Next, we implement a common noise reduction method of distancing the magnetometer from other components on our UAV. Finally, we introduce what we call a compromise GPR (Gaussian process regression) map that can be trained on multiple flight tests to learn any flight-by-flight variations between UAV observation tests. We investigate the spatial density of observations used to train a GPR map then use the compromise map to define a consistency test that can indicate whether or not the magnetometer data and corresponding GPR map are appropriate to use for state estimation. The interventions we introduce in this work facilitate indoor position localization of a UAV whose estimates we found to be quite sensitive to noise generated by the UAV.