Effects of Differential Magnetic Field/Tensor and Redundant Measurements on Multi-DOF Motion Estimation of a Magnetic Sensing System

This article presents a multi-DOF motion sensing system consisting of a permanent magnet (PM) and a magnetic tensor sensor (MTS) comprising a $3\times $ 3 array of three-axis digital magnetic flux density (MFD) sensors, and the methods to measure their relative MTS-PM position/orientation in a 3-D space. The design enables redundant differential measurements of MFD vectors and gradient tensor components to account for the singularities due to matrix inversion, providing a basis to explore different methods for multi-DOF estimation of the PM position/pose. Formulated as a two-stage linear least-square (LS) problem to take advantage of the dipole simplicity and exploit its physics revealed by its inverse model to guide the design of a fully connected artificial neural network (ANN) to account for the MTS measurement noise and un-modeled factors, a prototype multi-MTS system capable of 5-DOF motion measurements is developed and evaluated experimentally along with a study analyzing the parametric effects on the estimation accuracy; both stationary and moving sensor scenarios are considered. Enhanced with a fully connected ANN, an accuracy within a root-mean-square error (RMSE) of $40 ~\mu \text{m}$ spatial position and 0.1° pose can be uniquely obtained without subtracting a predetermined geomagnetic field in both fixed and moving multi-MTS scenarios, representing a significant improvement over the (0.5 mm, $1^{\circ }{)}$ RMSE of the single-MTS.