Tensor-Invariant Magnetic Localization for 6-DoF Pose Estimation via Compact Magnetometer Arrays

Magnetic localization using compact sensor arrays is a key capability for miniature robotic systems operating in confined and occluded environments. This article proposes a two-stage tensor-based localization framework for estimating the full six-degree-of-freedom (6-DoF) pose of a rigid, compact array of magnetometers immersed in static magnetic fields generated by magnetic dipole sources. Under a first-order compact-array approximation, we first derive a structured estimator for the local magnetic gradient tensor from differential measurements for general compact array configurations. Building on this estimate, tensor-invariant structures are exploited to formulate a position estimation problem that is analytically independent of the unknown orientation, enabling explicit decoupling of position and orientation estimation and improving robustness to initialization and static magnetic disturbances. Orientation is subsequently recovered through a structured optimization on the SO(3) manifold, for which a proximal power iteration (PPI) algorithm is developed with convergence guarantees to a first-order stationary point. Simulation and hardware experiments demonstrate reliable 6-DoF pose recovery across different sensing geometries, with average runtimes of 7.2-8.6 ms across four sensor configurations, only modest degradation under simulated static magnetic disturbances, and a median rotation error of 0.065 rad in hardware path-following experiments.