Accurate and Computational-Efficient Analytical Solutions for the Extended Two-Step Magnetometer Calibration

One of the biggest challenges involving low-cost inertial and magnetic sensors, especially the latter, is the inherent distortion/corruption of their measurements by systematic errors. Such errors can be so compromising that using the corrupted measurements for navigation purposes becomes an impossible task. In addition to sensor fusion, which aims at improving performance, calibration techniques can precisely estimate these errors, which, then, can be compensated for. This article revisits the problem of in-field magnetometer calibration, focusing on the traditional and well-established ellipsoid fitting-based extended two-step (ETS) method. Despite being largely employed, especially for initializing more sophisticated and iterative optimization-based calibration techniques, ETS implementation is not straightforward, as the mapping between its estimated intermediate parameters and the sensor error parameters of interest (biases, scale factors, and misalignments) has not been clearly provided in the literature yet. In this article, hence, we carefully derive the latter, aiming at facilitating ETS numerical implementation. Withal, and also figuring as main contribution of this article, we provide analytical closed-form solutions for ETS, which are especially suitable for low-cost, real-time embedded applications. In order to validate their accuracy, insensitivity to local-minima convergence issues, and computational efficiency, we present simulations—including a Monte Carlo analysis—and hardware implementation, alongside a comprehensive comparison with state-of-the-art magnetometer calibration techniques.