Localization of Passive 3-D Coils as an Inverse Problem: Theoretical Analysis and a Numerical Method

Abstract

This article provides both a theoretical analysis and a numerical method for the inverse source problem of locating multiple passive 3-D coils based on measurements of their superposed magnetic fields. In our context, a 3-D coil consists of three concentric circular coils being mutually perpendicular, and the term "passive" means that these coils are not connected to an active power source. Instead, their current is induced externally by a low-frequency alternating magnetic field which is generated by a closed exciter wire. The underlying inductively coupled system is modeled with regard to the 3-D coils as magnetic dipoles and their localization is formulated as an inverse problem. Since its ill posedness mainly arises from strong sensitivity to observational noise, an approximate upper bound for the localization error is derived mathematically by linearization. The Levenberg-Marquardt algorithm is applied as a method for localization and modified for better performance as well as the ability to estimate the number of 3-D coils in the localization area. Our method is tested with simulated and real data in order to confirm its capability of locating up to three passive 3-D coils within the front of a wooden shelf surrounded by the exciter wire and eight rectangular loop antennas.

[Retracted] Modeling and Research on Human Capital Accumulation Complex System of High‐Tech Enterprises Based on Big Data

“…So we have a deeper understanding of the core human capital and its internal relationship with the competitiveness of high-tech enterprises. In the competition of the enterprise, only know the enemy and the friend can win the victory [13].…”

Effective Solution of the Problem of Electromagnetic Positioning Based on Two-Axial Radiator

“…In the cited works, the least squares method was not used, which allows us to hope to improve an accuracy for problems with the same input data if the least squares method is applied. The algorithm based on a three-axis radiator and the least squares method was proposed in [5] and developed in [6,7].Another approach [8] requires such an excessive number of electromagnetic field generators that the navigation problem is reduced to the usual measurement range problem [9].The above-mentioned works motivated to construct an algorithm of positioning a minimal configuration on the basis of two dipoles (a two-axis radiator) via the solution to the minimal system of size 6×6, which is done in the present paper. In fact, we propose a new electromagnetic positioning method on the basis of two magnetic dipoles with coincident centers.…”

“…In the cited works, the least squares method was not used, which allows us to hope to improve an accuracy for problems with the same input data if the least squares method is applied. The algorithm based on a three-axis radiator and the least squares method was proposed in [5] and developed in [6,7].…”