Divergence form of Euler´s equation for magnetic dipole localization

This paper presents a novel algorithm and sensor for estimating the position of a magnetic dipole irrespective of its posture in 3D space. We transform the so-called Euler equation that is the linear equation relating the dipole position to the magnetic field and its gradients into the equation that the divergence of a certain vector field is zero. Then, we derive its weak form using the Gauss theorem. As a result, the dipole positions can be obtained as the solution to linear equations in which the coefficients to be measured are the differences of the surface integrals of the magnetic field, not the gradients of the magnetic field as in Euler´s equation. A cubic sensor consisting of 18 coils for measuring those quantities is developed. The method is verified experimentally.