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      An Optimized Two-Step Magnetic Correction Strategy by Means of a Lagrange Multiplier Estimator with an Ellipsoid Constraint

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          Abstract

          The geomagnetic field is as fundamental a constituent of passive navigation as Earth’s gravity. In cases where no other external attitude reference is available, for the direct heading angle estimation by a typical magnetic compass, a two-step optimized correction algorithm is proposed to correct the model coefficients caused by hard and soft iron nearby. Specifically, in Step 1, a Levenberg-Marquardt (L-M) fitting estimator with an ellipsoid constraint is applied to solve the hard magnetic coefficients. In Step 2, a Lagrange multiplier estimator is used to deal with the soft magnetic iron circumstance. The essential attribute of “the two-step” lies in its eliminating the coupling effects of hard and soft magnetic fields, and their mutual interferences on the pure geomagnetic field. Under the conditions of non-deterministic magnetic interference sources with noise, the numerical simulation by referring to International Geomagnetic Reference Field (IGRF), and the laboratory tests based upon the turntable experiments with Honeywell HMR3000 compass (Honeywell, Morristown, NJ, USA) conducted, the experimental results indicate that, in the presence of the variation of multi-magnetic interferences, the RMSE (Root Mean Square Error) value of the estimated total magnetic flux density by the proposed two-step estimator falls to 0.125 μT from its initial 2.503 μT, and the mean values of the heading angle error estimates are less than 1°. The proposed solution therefore, exhibits ideal convergent properties, fairly meeting the accuracy requirements of non-tactical level navigation applications.

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          Extension of a two-step calibration methodology to include nonorthogonal sensor axes

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            A Novel Calibration Method of Magnetic Compass Based on Ellipsoid Fitting

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              Novel Calibration Algorithm for a Three-Axis Strapdown Magnetometer

              A complete error calibration model with 12 independent parameters is established by analyzing the three-axis magnetometer error mechanism. The said model conforms to an ellipsoid restriction, the parameters of the ellipsoid equation are estimated, and the ellipsoid coefficient matrix is derived. However, the calibration matrix cannot be determined completely, as there are fewer ellipsoid parameters than calibration model parameters. Mathematically, the calibration matrix derived from the ellipsoid coefficient matrix by a different matrix decomposition method is not unique, and there exists an unknown rotation matrix R between them. This paper puts forward a constant intersection angle method (angles between the geomagnetic field and gravitational field are fixed) to estimate R. The Tikhonov method is adopted to solve the problem that rounding errors or other errors may seriously affect the calculation results of R when the condition number of the matrix is very large. The geomagnetic field vector and heading error are further corrected by R. The constant intersection angle method is convenient and practical, as it is free from any additional calibration procedure or coordinate transformation. In addition, the simulation experiment indicates that the heading error declines from ±1° calibrated by classical ellipsoid fitting to ±0.2° calibrated by a constant intersection angle method, and the signal-to-noise ratio is 50 dB. The actual experiment exhibits that the heading error is further corrected from ±0.8° calibrated by the classical ellipsoid fitting to ±0.3° calibrated by a constant intersection angle method.
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                Author and article information

                Journal
                Sensors (Basel)
                Sensors (Basel)
                sensors
                Sensors (Basel, Switzerland)
                MDPI
                1424-8220
                29 September 2018
                October 2018
                : 18
                : 10
                : 3284
                Affiliations
                [1 ]School of Automation Engineering, Northeast Electric Power University, Jilin 132012, China; jingtong0611@ 123456163.com (J.G.); yanghanrui1208@ 123456163.com (H.Y.)
                [2 ]School of Electrical and Information Engineering, The University of Sydney, Camperdown, NSW 2006, Australia; 4490980@ 123456gmail.com
                [3 ]Department of Engineering Management, Shenzhen Yuhu Power Co., Ltd., Shenzhen 518111, China; fuzhaolong_0406@ 123456126.com
                [4 ]School of Computer Science, Northeast Electric Power University, Jilin 132012, China; mengbo_nannan@ 123456163.com
                Author notes
                [* ]Correspondence: xiall521@ 123456neepu.edu.cn ; Tel.: +86-0432-6480-6483
                Article
                sensors-18-03284
                10.3390/s18103284
                6210750
                30274309
                f8453db9-fbf4-4a5a-96d0-b38efbabc5c1
                © 2018 by the authors.

                Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

                History
                : 17 July 2018
                : 26 September 2018
                Categories
                Article

                Biomedical engineering
                two-step optimized correction,ellipsoid constraint,non-deterministic magnetic interferences,l-m fitting,lagrange multiplier estimator

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