{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T12:06:03Z","timestamp":1784289963760,"version":"3.55.0"},"reference-count":42,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,5,28]],"date-time":"2024-05-28T00:00:00Z","timestamp":1716854400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["42176195"],"award-info":[{"award-number":["42176195"]}]},{"name":"National Natural Science Foundation of China","award":["42074010"],"award-info":[{"award-number":["42074010"]}]},{"name":"National Natural Science Foundation of China","award":["42174051"],"award-info":[{"award-number":["42174051"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>When magnetic matching aided navigation is applied to an underwater vehicle, the magnetometer must be installed inside the vehicle, considering the navigation safety and concealment of the underwater vehicle. Then, the interference magnetic field will seriously affect the accuracy of geomagnetic field measurement, which directly affects the accuracy of geomagnetic matching aided navigation. Therefore, improving the accuracy of geomagnetic measurements inside the vehicle through error compensation has become one of the most difficult problems that requires an urgent solution in geomagnetic matching aided navigation. In order to solve this problem, this paper establishes the calculation model of the internal magnetic field of the underwater vehicle and the geomagnetic measurement error model of the ship-borne magnetometer. Then, a compensation method for the geomagnetic measurement error of the ship-borne magnetometer, based on the constrained total least square method, is proposed. To verify the effectiveness of the method proposed in this paper, a simulation experiment of geomagnetic measurement and compensation of a ship-borne three-axis magnetometer was constructed. Among them, to be closer to the real situation, a combination of the geomagnetism model, the elliptic shell model and the magnetic dipole model was used to simulate the internal magnetic field of the underwater vehicle. The experimental results indicated that the root mean square error of geomagnetic measurement in an underwater vehicle was less than 5 nT after compensation, and the accuracy of geomagnetic measurement met the requirements of geomagnetic matching aided navigation.<\/jats:p>","DOI":"10.3390\/s24113478","type":"journal-article","created":{"date-parts":[[2024,5,28]],"date-time":"2024-05-28T07:38:39Z","timestamp":1716881919000},"page":"3478","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Compensation Method for the Geomagnetic Measurement Error of an Underwater Ship-Borne Magnetometer Based on Constrained Total Least Squares"],"prefix":"10.3390","volume":"24","author":[{"given":"Yude","family":"Tong","sequence":"first","affiliation":[{"name":"College of Electrical Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoying","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Operational Research and Programing, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yongbing","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenkui","family":"Li","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Feng","family":"Zha","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,28]]},"reference":[{"key":"ref_1","first-page":"44","article-title":"Research on error of underwater terrain aided navigation based on TERCOM algorithm","volume":"32","author":"Zhang","year":"2020","journal-title":"J. Nav. Univ. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"20866","DOI":"10.1109\/JSEN.2022.3208114","article-title":"Improved Particle Swarm Optimization screening iterative algorithm in gravity matching navigation","volume":"22","author":"Zou","year":"2022","journal-title":"IEEE Sens. J."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Gao, S., Cai, T., and Fang, K. (2022). Gravity-matching algorithm based on K-Nearest neighbor. Sensors, 22.","DOI":"10.3390\/s22124454"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"104501","DOI":"10.1063\/5.0019305","article-title":"The robust residual-based adaptive estimation Kalman filter method for strap-down inertial and geomagnetic tightly integrated navigation system","volume":"91","author":"Zhai","year":"2020","journal-title":"Rev. Sci. Instruments"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"226064","DOI":"10.1109\/ACCESS.2020.3043794","article-title":"Improved particle swarm optimization geomagnetic matching algorithm based on simulated annealing","volume":"8","author":"Ji","year":"2020","journal-title":"IEEE Access"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Li, J. (2013). Study on the Characteristics and Compensation Method of Vehicle Interferential Magnetic Field in Geomagnetic Measurement. [Ph.D. Thesis, National University of Defense Technology].","DOI":"10.1177\/0142331213497620"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Henkel, P. (2017). Calibration of Magnetometers with GNSS Receivers and Magnetometer-Aided GNSS Ambiguity Fixing. J. Sens., 17.","DOI":"10.3390\/s17061324"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Henkel, P., Berthold, P., and Kiam, J.J. (2014, January 12\u201313). Calibration of Magnetic Field Sensors with Two Mass-Market GNSS Receivers. Proceedings of the IEEE Workshop on Positioning, Navigation and Communication (WPNC), Dresden, Germany.","DOI":"10.1109\/WPNC.2014.6843306"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"115","DOI":"10.2514\/1.6278","article-title":"Real-time attitude-independent three-axis magnetometer calibration","volume":"28","author":"Crassidis","year":"2005","journal-title":"J. Guid. Control Dyn."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Alimi, R., Fisher, E., and Nahir, K. (2023). In Situ Underwater Localization of Magnetic Sensors Using Natural Computing Algorithms. J. Sens., 23.","DOI":"10.3390\/s23041797"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"6403161","DOI":"10.1155\/2020\/6403161","article-title":"A Review of Underwater Localization Techniques, Algorithms, and Challenges","volume":"2020","author":"Su","year":"2020","journal-title":"J. Sens."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Wang, H., Wang, S., Bu, R., and Zhang, E. (2017). A Novel Cross-Layer Routing Protocol Based on Network Coding for Underwater Sensor Networks. Sensors, 17.","DOI":"10.3390\/s17081821"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"709318","DOI":"10.1155\/2010\/709318","article-title":"Silent Localization of Underwater Sensors Using Magnetometers","volume":"2010","author":"Callmer","year":"2010","journal-title":"EURASIP J. Adv. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/BF03546249","article-title":"Complete Linear Attitude-Independent Magnetometer Calibration","volume":"50","author":"Alonso","year":"2003","journal-title":"J. Astronaut. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1061\/(ASCE)0893-1321(2006)19:2(87)","article-title":"Calibration of Strapdown Magnetometers in Magnetic Field Domain","volume":"19","author":"Elkaim","year":"2006","journal-title":"J. Aerosp. Eng."},{"key":"ref_16","first-page":"1707","article-title":"Component Calibration of Three-Axis Magnetometer with Two Step method","volume":"31","author":"Zhang","year":"2018","journal-title":"Chin. J. Sens. Actuators"},{"key":"ref_17","first-page":"763","article-title":"Truncated total least squares algorithm in restraining random error of geomagnetic navigation magnetometer","volume":"23","author":"Wang","year":"2015","journal-title":"J. Chin. Inert. Technol."},{"key":"ref_18","first-page":"215","article-title":"Scalar Calibration of Tri-Axial Magnetometer with Linearized Parameter Model","volume":"25","author":"Zhang","year":"2012","journal-title":"Chin. J. Sens. Actuators"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.sna.2009.10.008","article-title":"Calibration methods for inertial and magnetic sensors","volume":"156","author":"Bonnet","year":"2009","journal-title":"Sens. Actuators A Phys."},{"key":"ref_20","first-page":"371","article-title":"Error Calibration of Fluxgate Magnetometers in Arbitrary Attitude Situation","volume":"25","author":"Pang","year":"2011","journal-title":"J. Test Meas. Technol."},{"key":"ref_21","first-page":"1202","article-title":"Calibration of Strapdown Three-axis Magnetometer and Measurement Error Compensation of Geomagnetic Field Based on Total Least Squares","volume":"33","author":"Wu","year":"2012","journal-title":"Acta Armamentarii"},{"key":"ref_22","first-page":"427","article-title":"Calibration of Vehicular Three-axis Magnetometer via Truncated Total Least Squares Algorithm","volume":"36","author":"Zhang","year":"2015","journal-title":"Acta Armamentarii"},{"key":"ref_23","first-page":"149","article-title":"Integrated Compensation Method of Three-axis Magnetometer Based on EKF Algorithm","volume":"39","author":"Sun","year":"2019","journal-title":"J. Proj. Rocket. Missiles Guid."},{"key":"ref_24","first-page":"650","article-title":"Integrated Error Compensation Method for Three-Axis Magnetometer in Geomagnetic Navigation. In Lecture Notes in Electrical Engineering","volume":"Volume 650","author":"Yang","year":"2020","journal-title":"Proceedings of the China Satellite Navigation Conference (CSNC) 2020 Proceedings: Volume I. CSNC 2020"},{"key":"ref_25","first-page":"31","article-title":"Calibration Method of Magnetometer Based on BP Neural Network","volume":"6","author":"Cheng","year":"2020","journal-title":"J. Comput. Commun."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"045106","DOI":"10.1088\/1361-6501\/abd1b4","article-title":"An improved neural network method for aeromagnetic compensation","volume":"32","author":"Yu","year":"2021","journal-title":"Meas. Sci. Technol."},{"key":"ref_27","first-page":"126","article-title":"Method on eliminating magnetic interference in shipboard geomagnetic field measurement","volume":"41","author":"Zhu","year":"2019","journal-title":"Ship Sci. Technol."},{"key":"ref_28","first-page":"2246","article-title":"Compensation of geomagnetic field measurement error based on damped particle swarm optimization","volume":"38","author":"Li","year":"2017","journal-title":"Chin. J. Sci. Instrum."},{"key":"ref_29","first-page":"170","article-title":"New Measuring and Calculate Method of Submarine Space Magnetic Field","volume":"4","author":"Zhang","year":"2010","journal-title":"Ship Electron. Eng."},{"key":"ref_30","first-page":"136","article-title":"Modeling Analysis and Application of Submarine Space Magnetic","volume":"38","author":"Zhang","year":"2018","journal-title":"Ship Electron. Eng."},{"key":"ref_31","first-page":"54","article-title":"Single-row Ellipsoid and Magnetic Dipole Mixed Model in Ship Magnetic Field Depth Conversion","volume":"3","author":"Lin","year":"1996","journal-title":"Mine Warf. Ship Prot."},{"key":"ref_32","first-page":"7","article-title":"Analysis and comparison on magnetic field modeling method of submarine","volume":"33","author":"Qu","year":"2011","journal-title":"Ship Sci. Technol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1186\/s40623-020-01288-x","article-title":"International Geomagnetic Reference Field: The thirteenth generation","volume":"73","author":"Alken","year":"2021","journal-title":"Earth Planets Space"},{"key":"ref_34","unstructured":"Zhou, Y., and Zhang, G. (2004). Ship Magnetic Field Analysis and Calculation, National Defence Industry Press."},{"key":"ref_35","first-page":"60","article-title":"Warship Magnetic Field Simulation Technology Based on Magnetic Dipoles","volume":"37","author":"Yang","year":"2017","journal-title":"Ship Electron. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1070","DOI":"10.1109\/TAES.2008.4655364","article-title":"Extension of a Two-Step Calibration Methodology to Include Nonorthogonal Sensor Axes","volume":"44","author":"Foster","year":"2008","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1109\/TAES.2011.5751259","article-title":"Geometric Approach to Strapdown Magnetometer Calibration in Sensor Frame","volume":"47","author":"Vasconcelos","year":"2011","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_38","first-page":"465","article-title":"Least squares methods","volume":"1","author":"Bjorck","year":"1990","journal-title":"Handb. Numer. Anal."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1070","DOI":"10.1109\/78.80955","article-title":"The constrained total least squares technique and its applications to harmonic superresolution","volume":"39","author":"Abatzoglou","year":"1991","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Golub, G.H., and Loan, C.V. (1980). An Analysis of the Total Least Squares Problem, Cornell University.","DOI":"10.1137\/0717073"},{"key":"ref_41","unstructured":"Huffel, S.V., and Vandewalle, J. (1991). The Total Least Squares Problem: Computational Aspects and Analysis, Society for Industrial and Applied Mathematics."},{"key":"ref_42","unstructured":"Wu, Z. (2013). Research on Geomagnetic Measurement Error Compensation for Underwater Geomagnetic Navigation. [Ph.D. Thesis, National University of Defense Technology]."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/11\/3478\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:49:30Z","timestamp":1760107770000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/11\/3478"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,28]]},"references-count":42,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["s24113478"],"URL":"https:\/\/doi.org\/10.3390\/s24113478","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,28]]}}}