{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T03:48:14Z","timestamp":1777866494609,"version":"3.51.4"},"reference-count":25,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T00:00:00Z","timestamp":1777507200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Young Elite Scientists Sponsorship Program of CAST","award":["No. 2023QNRC001"],"award-info":[{"award-number":["No. 2023QNRC001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Anti-collision detection technologies primarily rely on optical, radar, or laser sensors; however, their performance often deteriorates severely under adverse weather conditions (e.g., rain, snow, fog) or in scenarios involving visual occlusion. By contrast, magnetic anomaly detection leverages perturbations in the geomagnetic field induced by target objects (e.g., vehicles, metallic obstacles), exhibiting intrinsic all-weather operability and strong anti-interference capability. Nevertheless, conventional magnetic anomaly detection methods suffer from the limited applicability of the magnetic dipole model, which only affords coarse positioning accuracy and is predominantly suited for long-range targets. To address this limitation, this paper proposes an Extended N-th-Pole Magnetic Dipole (E-NMD) model that improves accuracy by analyzing the Lagrangian cosine term and rigorously constraining truncation errors under specific operational conditions. Experimental results demonstrate that, for steel with a relative permeability of 200, the model achieves a fitting variance of 99.87%. Furthermore, to overcome the inversion difficulties arising when the strength of short-range magnetic anomalies is comparable to sensor noise, an Adaptive Iterative Extended Kalman Filter (AI-EKF) is developed to enable robust noise suppression and precise distance estimation. Results indicate that E-NMD outperforms the traditional N-th-Pole Magnetic Dipole (NMD) model in proximal state estimation, achieving a 39.62% reduction in Root Mean Square Error (RMSE). Finally, in light of parameter uncertainty in magnetic anomaly targets under real-world conditions, a Dual-Mode Pairwise Iterative Extended Kalman Filter (DI-EKF) is introduced to jointly estimate parameters and system states, yielding an 89% reduction in RMSE compared to AI-EKF.<\/jats:p>","DOI":"10.3390\/s26092792","type":"journal-article","created":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T08:46:26Z","timestamp":1777538786000},"page":"2792","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Magnetic Heterodyne Target Proximal Distance Estimate Using Extended N-th-Pole Magnetic Dipole Model via Iterative Extended Kalman Filter"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-1660-3582","authenticated-orcid":false,"given":"Xuyi","family":"Miao","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yipeng","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zumeng","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shaojie","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"He","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5729-798X","authenticated-orcid":false,"given":"Keren","family":"Dai","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,4,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Knoepfel, H. (2000). Magnetic Fields: A Comprehensive Theoretical Treatise for Practical Use, Wiley.","DOI":"10.1002\/9783527617418"},{"key":"ref_2","first-page":"100129","article-title":"Positioning system design of closed-loop magnetically driven laser steering manipulator for endoscopic microsurger","volume":"3","author":"Zhang","year":"2023","journal-title":"Biomim. Intell. Robot."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5935411","DOI":"10.1109\/TGRS.2024.3494868","article-title":"Predictive Model-Based Correction of Magnetic Sensor Array Sway Errors","volume":"62","author":"Lu","year":"2024","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"075205","DOI":"10.1088\/1361-6501\/aac747","article-title":"A LiDAR data-based camera self-calibration method","volume":"29","author":"Xu","year":"2018","journal-title":"Meas. Sci. Technol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"015205","DOI":"10.1088\/0957-0233\/27\/1\/015205","article-title":"A high success rate full-waveform lidar echo decomposition method","volume":"27","author":"Xu","year":"2016","journal-title":"Meas. Sci. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111719","DOI":"10.1016\/j.measurement.2022.111719","article-title":"High-resolution LiDAR using random modulated continuous wave with a low code rate","volume":"202","author":"Wang","year":"2022","journal-title":"Measurement"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1038\/s41378-022-00397-9","article-title":"Improved sampling scheme for LiDAR in Lissajous scanning mode","volume":"8","author":"Wang","year":"2022","journal-title":"Microsyst. Nanoeng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"25882","DOI":"10.1109\/JSEN.2023.3319475","article-title":"A Hybrid Method for Positioning a Moving Magnetic Target and Estimating Its Magnetic Moment","volume":"23","author":"Chi","year":"2023","journal-title":"IEEE Sens. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"045114","DOI":"10.1088\/1361-6501\/ad1dae","article-title":"Underwater target correlation detection and location method based on normalized magnetic momen","volume":"35","author":"Jin","year":"2024","journal-title":"Meas. Sci. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"126147","DOI":"10.1088\/1361-6501\/ad7a99","article-title":"An adaptive weighted magnetic anomaly detection based on full magnetic gradient orthonormal basis function","volume":"35","author":"Yan","year":"2024","journal-title":"Meas. Sci. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"085003","DOI":"10.1088\/0957-0233\/25\/8\/085003","article-title":"Deep coupling of star tracker and MEMS-gyro data under highly dynamic and long exposure conditions","volume":"25","author":"Sun","year":"2014","journal-title":"Meas. Sci. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"6187","DOI":"10.1364\/OE.23.006187","article-title":"A real-time detection and positioning method for small and weak targets using a 1D morphology-based approach in 2D images","volume":"23","author":"Li","year":"2015","journal-title":"Opt. Express"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6007808","DOI":"10.1109\/TIM.2021.3084283","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_14","doi-asserted-by":"crossref","first-page":"035005","DOI":"10.1088\/1361-6439\/aaf943","article-title":"UKF-based MEMS micromirror angle estimation for LiDAR","volume":"29","author":"Wang","year":"2019","journal-title":"J. Micromech. Microeng."},{"key":"ref_15","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. Mater."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1145960","DOI":"10.1155\/2022\/1145960","article-title":"Correction of Three Sensors in a Magnetic Field Gradiometer","volume":"2022","author":"Yang","year":"2022","journal-title":"J. Sens."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"065304","DOI":"10.1088\/1361-6501\/ac5a9a","article-title":"Adaptive surge detection of magnetic suspension centrifugal blower based on rotor radial displacement signal and SOGI-FLL with prefilter","volume":"33","author":"Guan","year":"2022","journal-title":"Meas. Sci. Technol."},{"key":"ref_18","first-page":"2120","article-title":"The Development of Vehicle Position Estimation Algorithms Based on the Use of AMR Sensors","volume":"13","author":"Taghvaeeyan","year":"2012","journal-title":"IEEE Trans. Intell. Transp."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4003609","DOI":"10.1109\/TMAG.2019.2910481","article-title":"A Real-Time Magnetic Dipole Localization Method Based on Cube Magnetometer Array","volume":"55","author":"Liu","year":"2019","journal-title":"IEEE Trans. Magn."},{"key":"ref_20","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":"Yin","year":"2014","journal-title":"J. Appl. Remote Sens."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"055107","DOI":"10.1088\/1361-6501\/acb608","article-title":"Adaptive filter entropy monitoring method for scalar magnetic detection using optically pumped magnetometers","volume":"34","author":"Qiao","year":"2023","journal-title":"Meas. Sci. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1109\/TMAG.1975.1058672","article-title":"Advanced superconducting gradiometer\/Magnetometer arrays and a novel signal processing technique","volume":"11","author":"Wynn","year":"1975","journal-title":"IEEE Trans. Magn."},{"key":"ref_23","first-page":"126","article-title":"Nonlinear opto-magnetic response of magnetic nanoparticles: Opto-magnetic particle spectroscopy","volume":"21","author":"Xu","year":"2025","journal-title":"Appl. Phys. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"943","DOI":"10.1109\/JSEN.2011.2166066","article-title":"Motion Measurement Using Inertial Sensors, Ultrasonic Sensors, and Magnetometers with Extended Kalman Filter for Data Fusion","volume":"12","author":"Zhao","year":"2012","journal-title":"IEEE Sens. J."},{"key":"ref_25","first-page":"5904913","article-title":"Estimating Locations and Moments of Multiple Dipole-Like Magnetic Sources from Magnetic Gradient Tensor Data Using Differential Evolution","volume":"60","author":"Ding","year":"2021","journal-title":"IEEE Trans. Geosci. Remote Sens."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/9\/2792\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T08:47:08Z","timestamp":1777538828000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/9\/2792"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,30]]},"references-count":25,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["s26092792"],"URL":"https:\/\/doi.org\/10.3390\/s26092792","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,30]]}}}