Properties of the Linear Equations Derived From Euler's Equation and Its Application to Magnetic Dipole Localization

This paper presents a novel algorithm and sensor for estimating the position of a magnetic dipole. By transforming Euler's equation of degree -3 into an integral form, we have linear equations relating the dipole position to the surface integrals of the magnetic flux densities on a cube. To measure all the quantities required in the linear equations, we develop a cubic sensor with a side length of 50 mm which consists of 18 coils. We show that the coefficient matrix of the linear equations is symmetric and traceless, which can be used to improve localization accuracy. By performing the nonlinear least squares method with the initial solution given by the proposed method, the average and maximum error are 8.3 and 17.5 mm, respectively, in the range of 500 mm.