{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T15:58:31Z","timestamp":1774454311011,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T00:00:00Z","timestamp":1715644800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shanxi Provincial Postgraduate Scientific Research Innovation Project","award":["2023KY572"],"award-info":[{"award-number":["2023KY572"]}]},{"name":"Shanxi Provincial Postgraduate Scientific Research Innovation Project","award":["20231903"],"award-info":[{"award-number":["20231903"]}]},{"name":"Graduate Science and Technology Project of North University of China","award":["2023KY572"],"award-info":[{"award-number":["2023KY572"]}]},{"name":"Graduate Science and Technology Project of North University of China","award":["20231903"],"award-info":[{"award-number":["20231903"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Underwater mobile acoustic source target localization encounters several challenges, including the unknown propagation speed of the source signal, uncertainty in the observation platform\u2019s position and velocity (i.e., platform systematic errors), and economic costs. This paper proposes a new two-step closed-form localization algorithm that jointly estimates the angle of arrival (AOA), time difference of arrival (TDOA), and frequency difference of arrival (FDOA) to address these challenges. The algorithm initially introduces auxiliary variables to construct pseudo-linear equations to obtain the initial solution. It then exploits the relationship between the unknown and auxiliary variables to derive the exact solution comprising solely the unknown variables. Both theoretical analyses and simulation experiments demonstrate that the proposed method accurately estimates the position, velocity, and speed of the sound source even with an unknown sound speed and platform systematic errors. It achieves asymptotic optimality within a reasonable error range to approach the Cram\u00e9r\u2013Rao lower bound (CRLB). Furthermore, the algorithm exhibits low complexity, reduces the number of required localization platforms, and decreases the economic costs. Additionally, the simulation experiments validate the effectiveness of the proposed localization method across various scenarios, outperforming other comparative algorithms.<\/jats:p>","DOI":"10.3390\/s24103127","type":"journal-article","created":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T10:26:36Z","timestamp":1715682396000},"page":"3127","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Co-Localization Algorithm for Underwater Moving Targets with an Unknown Constant Signal Propagation Speed and Platform Errors"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-8443-9444","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"first","affiliation":[{"name":"College of Mechatronics Engineering, North University of China, Taiyuan 030051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Long","family":"He","sequence":"additional","affiliation":[{"name":"College of Mechatronics Engineering, North University of China, Taiyuan 030051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8982-9349","authenticated-orcid":false,"given":"Gang","family":"Fan","sequence":"additional","affiliation":[{"name":"College of Mechatronics Engineering, North University of China, Taiyuan 030051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xue","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ya","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mechatronics Engineering, North University of China, Taiyuan 030051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"181662","DOI":"10.1109\/ACCESS.2020.3028608","article-title":"A Hybrid Localization Algorithm Based on Doppler Shift and AOA for an Underwater Mobile Node","volume":"8","author":"Hao","year":"2020","journal-title":"IEEE Access"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"16051","DOI":"10.1109\/JSEN.2023.3272565","article-title":"A Semidefinite Relaxation Approach for Mobile Target Localization Based on TOA and Doppler Frequency Shift Measurements","volume":"23","author":"Meng","year":"2023","journal-title":"IEEE Sens. 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