Full Magnetometer and Gyroscope Bias Estimation using Angular Rates: Theory and Experimental Evaluation of a Factor Graph-Based Approach

TL;DR

MEMS AHRS suffer from magnetometer biases (hard-iron $m_b$ and soft-iron $A$) and gyroscope bias, limiting attitude accuracy. MAGYC estimates full magnetometer calibration and gyroscope bias from magnetometer and angular-rate measurements using two factor-graph formulations (batch and online incremental). The key innovations include a Cholesky-based parametrization of $C = A^{-1}$ with unit-volume constraint, a residual that eliminates the need for attitude or local magnetic-field magnitude, and seamless integration with smoothing and mapping frameworks via $GTSAM$. Validation on Monte Carlo simulations and field data from an underwater mapping mission shows dramatic reductions in heading errors and dead-reckoning drift, e.g., from about 10% to 0.5% of traveled distance.

Abstract

Despite their widespread use in determining system attitude, Micro-Electro-Mechanical Systems (MEMS) Attitude and Heading Reference Systems (AHRS) are limited by sensor measurement biases. This paper introduces a method called MAgnetometer and GYroscope Calibration (MAGYC), leveraging three-axis angular rate measurements from an angular rate gyroscope to estimate both the hard- and soft-iron biases of magnetometers as well as the bias of gyroscopes. We present two implementation methods of this approach based on batch and online incremental factor graphs. Our method imposes fewer restrictions on instrument movements required for calibration, eliminates the need for knowledge of the local magnetic field magnitude or instrument's attitude, and facilitates integration into factor graph algorithms for Smoothing and Mapping frameworks. We validate the proposed methods through numerical simulations and in-field experimental evaluations with a sensor onboard an underwater vehicle. By implementing the proposed method in field data of a seafloor mapping dive, the dead reckoning-based position estimation error of the underwater vehicle was reduced from 10% to 0.5% of the distance traveled.

Figures (9)

• Figure 1: Diagram illustrating an underwater vehicle's dead reckoning position using different magnetic field sources for the heading estimation with the corresponding trajectories represented with dashed lines.
• Figure 2: A factor graph representation, depicting the residual factor, $L_{Ri}$, and the state, $x$.
• Figure 3: Simulated magnetometer data for three datasets: (a) wam, (b) mam, and (c) lam. The 3D plots show blue dots for magnetometer data, gray spheres for the true magnetic field, and orange ellipsoids for the distorted magnetic field.
• Figure 4: Performance comparison of seven calibration methods on three simulated datasets. The geodesic soft-iron error $\log{(||A^{-1/2} A^* A^{-1/2}||_F)}$, hard-iron error $|m_b - m_b^*|$, and gyroscope bias error $|w_b - w_b^*|$ are analyzed for the wam (green), mam (yellow), and lam (orange) datasets. Gray-shaded zones show the raw data value.
• Figure 5: In-field magnetic data for two datasets: (a) EXP1-14 and (b) EXP2-14. The 3D plots show blue dots for magnetometer data, gray spheres for true magnetic field based on the magnetic model magnitude, and the orange sphere representing the non-calibrated magnetic field.