{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T16:24:06Z","timestamp":1777652646212,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,3,30]],"date-time":"2024-03-30T00:00:00Z","timestamp":1711756800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China (NSFC) Joint Fund Project","award":["U2241201"],"award-info":[{"award-number":["U2241201"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Currently, magnetic gradient tensor-based localization methods face challenges such as significant errors in geomagnetic field estimation, susceptibility to local optima in optimization algorithms, and inefficient performance. In addressing these issues, this article propose a two-point localization method under the constraint of overlaying geometric invariants. This method initially establishes the relationship between the target position and the magnetic gradient tensor by substituting an intermediate variable for the magnetic moment. Exploiting the property of the eigenvector corresponding to the minimum absolute eigenvalue being perpendicular to the target position vector, this constraint is superimposed to formulate a nonlinear system of equations of the target\u2019s position. In the process of determining the target position, the Nara method is employed for obtaining the initial values, followed by the utilization of the Levenberg\u2013Marquardt algorithm to derive a precise solution. Experimental validation through both simulations and experiments confirms the effectiveness of the proposed method. The results demonstrate its capability to overcome the challenges faced by a single-point localization method in the presence of some errors in geomagnetic field estimation. In comparison to traditional two-point localization methods, the proposed method exhibits the highest precision. The localization outcomes under different noise conditions underscore the robust noise resistance and resilience of the proposed method.<\/jats:p>","DOI":"10.3390\/s24072224","type":"journal-article","created":{"date-parts":[[2024,3,31]],"date-time":"2024-03-31T13:32:56Z","timestamp":1711891976000},"page":"2224","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Two-Point Localization Algorithm of a Magnetic Target Based on Tensor Geometric Invariant"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9056-0498","authenticated-orcid":false,"given":"Cheng","family":"Chi","sequence":"first","affiliation":[{"name":"Institute of Remote Sensing, Navy Submarine Academy, Qingdao 266000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dan","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Remote Sensing, Navy Submarine Academy, Qingdao 266000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ronghua","family":"Tao","sequence":"additional","affiliation":[{"name":"Institute of Remote Sensing, Navy Submarine Academy, Qingdao 266000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianwei","family":"Li","sequence":"additional","affiliation":[{"name":"Institute of Remote Sensing, Navy Submarine Academy, Qingdao 266000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ye","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Remote Sensing, Navy Submarine Academy, Qingdao 266000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhentao","family":"Yu","sequence":"additional","affiliation":[{"name":"Institute of Remote Sensing, Navy Submarine Academy, Qingdao 266000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lu","family":"Yu","sequence":"additional","affiliation":[{"name":"Institute of Remote Sensing, Navy Submarine Academy, Qingdao 266000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,30]]},"reference":[{"key":"ref_1","first-page":"354","article-title":"Airborne tensor magnetic gradiometry-the latest progress of airborne magnetometric technology","volume":"3","author":"Zhang","year":"2006","journal-title":"Chin. J. Eng. Geophys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1071\/EG12020","article-title":"New methods for interpretation of magnetic vector and gradient tensor data I: Eigenvector analysis and the normalised source strength","volume":"43","author":"Clark","year":"2012","journal-title":"Explor. Geophys."},{"key":"ref_3","unstructured":"Wynn, W. (1972). Naval Ship Research and Development Laboratory Informal Report NSRDL\/PC 3493, Naval Ship Research and Development Laboratory."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1071\/EG14013","article-title":"Methods for determining remanent and total magnetizations of magnetic sources\u2014A review","volume":"45","author":"Clark","year":"2014","journal-title":"Explor. Geophys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7434","DOI":"10.1109\/JSEN.2020.3046120","article-title":"Inversion of Magnetic Dipole Parameters Using a Scalar Field Gradiometer","volume":"21","author":"Birsan","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3451","DOI":"10.1088\/0957-0233\/18\/11\/027","article-title":"Localization and magnetic moment estimation of a ferromagnetic target by simulated annealing","volume":"18","author":"Sheinker","year":"2007","journal-title":"Meas. Sci. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"23788","DOI":"10.3390\/s150923788","article-title":"A Dedicated Genetic Algorithm for Localization of Moving Magnetic Objects","volume":"15","author":"Alimi","year":"2015","journal-title":"Sensors"},{"key":"ref_8","first-page":"292","article-title":"Study of a hybrid algorithm for localization of mobile magnetic target by a single fluxgate","volume":"56","author":"Gao","year":"2016","journal-title":"J. Dalian Univ. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1109\/TMAG.2008.2006635","article-title":"Magnetic anomaly detection using a three-axis magnetometer","volume":"45","author":"Sheinker","year":"2009","journal-title":"IEEE Trans. Magn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3291","DOI":"10.1109\/TMAG.2006.879151","article-title":"A closed form formula for magnetic dipole localization by measurement of its magnetic field and spatial gradients","volume":"42","author":"Nara","year":"2006","journal-title":"IEEE Trans. Magn."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"17E504","DOI":"10.1063\/1.4861675","article-title":"Moore\u2013Penrose generalized inverse of the gradient tensor in Euler\u2019s equation for locating a magnetic dipole","volume":"115","author":"Nara","year":"2014","journal-title":"J. Appl. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"166274","DOI":"10.1016\/j.jmmm.2019.166274","article-title":"A Closed-Form Formula for Magnetic Dipole Localization by Measurement of Its Magnetic Field Vector and Magnetic Gradient Tensor","volume":"499","author":"Yin","year":"2020","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_13","first-page":"3995","article-title":"Online magnetic target location method based on the magnetic gradient tensor of two points","volume":"60","author":"Liu","year":"2017","journal-title":"Chin. J. Geophys."},{"key":"ref_14","first-page":"44","article-title":"Two-point magnetic gradient tensor positioning method","volume":"40","author":"Dai","year":"2018","journal-title":"J. Detect. Control"},{"key":"ref_15","first-page":"1","article-title":"Magnetic Target Linear Location Method Using Two-Point Gradient Full Tensor","volume":"70","author":"Xu","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"083596","DOI":"10.1117\/1.JRS.8.083596","article-title":"Magnetic dipole localization based on magnetic gradient tensor data at a single point","volume":"8","author":"Gang","year":"2014","journal-title":"J. Appl. Remote Sens."},{"key":"ref_17","first-page":"1250","article-title":"Improved method of magnetic localization based on magnetic gradient tensor","volume":"36","author":"Yu","year":"2014","journal-title":"Syst. Eng. Electron."},{"key":"ref_18","first-page":"1","article-title":"Magnetic Tensor Sensor for Gradient-Based Localization of Ferrous Object in Geomagnetic Field","volume":"52","author":"Lee","year":"2016","journal-title":"IEEE Trans. Magn."},{"key":"ref_19","first-page":"21","article-title":"Localization Method of magnetic field gradient tensor under carriers moving parallelly","volume":"41","author":"Zhang","year":"2013","journal-title":"J. Huazhong Univ. Sci. Tech."},{"key":"ref_20","first-page":"9700410","article-title":"Magnetic Dipole Two-Point Tensor Positioning Based on Magnetic Moment Constraints","volume":"70","author":"Liu","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Wiegert, R., Lee, K., and Oeschger, J. (2008, January 15\u201318). Improved magnetic STAR methods for real-time, point-by-point localization of unexploded ordnance and buried mines. Proceedings of the OCEANS 2008, Quebec City, QC, Canada.","DOI":"10.1109\/OCEANS.2008.5152073"},{"key":"ref_22","first-page":"5","article-title":"Analysis of the magnetic gradient tensor","volume":"8","author":"Wilson","year":"1985","journal-title":"Tech. Memo."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmmm.2015.11.034","article-title":"Detection of ferromagnetic target based on mobile magnetic gradient tensor system","volume":"402","author":"Gang","year":"2016","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_24","first-page":"60","article-title":"Research on the asphericity error elimination of the invariant of magnetic gradient tensor","volume":"64","author":"Lv","year":"2015","journal-title":"Acta Phys. Sinica"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1109\/TEMC.2016.2615132","article-title":"A Software-Based Calibration Technique for Characterizing the Magnetic Signature of EUTs in Measuring Facilities","volume":"59","author":"Kakarakis","year":"2017","journal-title":"IEEE Trans. Electromagn. Compat."},{"key":"ref_26","first-page":"97","article-title":"Magnetic Field Sensor Calibration for Attitude Determination","volume":"8","author":"Klingbeil","year":"2014","journal-title":"J. Appl. Geod."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Hu, C., Meng, M.Q., and Mandal, M. (2006, January 9\u201315). The Calibration of 3-Axis Magnetic Sensor Array System for Tracking Wireless Capsule Endoscope. Proceedings of the 2006 IEEE\/RSJ International Conference on Intelligent Robots and Systems, Beijing, China.","DOI":"10.1109\/IROS.2006.282118"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"5000506","DOI":"10.1109\/TMAG.2019.2895355","article-title":"Calibration of Room Temperature Magnetic Sensor Array for Biomagnetic Measurement","volume":"55","author":"Adachi","year":"2019","journal-title":"IEEE Trans. Magn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1287\/mnsc.13.5.344","article-title":"Non-linear programming via penalty functions","volume":"13","author":"Zangwill","year":"1967","journal-title":"Manag. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Seber, G.A.F., and Wild, C.J. (2003). Nonlinear Regression, Wiley-Interscience.","DOI":"10.1002\/9780471722199"},{"key":"ref_31","first-page":"2415006","article-title":"Positioning Algorithm of Homonymous Object Points in Multi-Vision System Based on Weighted Levenberg-Marquardt","volume":"58","author":"Zhou","year":"2021","journal-title":"Laser Optoelectron. Prog."},{"key":"ref_32","first-page":"6688364","article-title":"Simulation Analysis of Magnetic Gradient Full-Tensor Measurement System","volume":"2021","author":"Xu","year":"2021","journal-title":"Math. Probl. Eng."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/7\/2224\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:21:24Z","timestamp":1760106084000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/7\/2224"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,30]]},"references-count":32,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["s24072224"],"URL":"https:\/\/doi.org\/10.3390\/s24072224","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,30]]}}}