{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T20:05:27Z","timestamp":1768680327039,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T00:00:00Z","timestamp":1655424000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program","award":["2020YFB1313200"],"award-info":[{"award-number":["2020YFB1313200"]}]},{"name":"National Key Research and Development Program","award":["2021YFC2803000"],"award-info":[{"award-number":["2021YFC2803000"]}]},{"name":"National Key Research and Development Program","award":["2020YFB1313200"],"award-info":[{"award-number":["2020YFB1313200"]}]},{"name":"National Key Research and Development Program","award":["2021YFC2803000"],"award-info":[{"award-number":["2021YFC2803000"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In order to solve the problem of inconsistent state estimation when multiple autonomous underwater vehicles (AUVs) are co-located, this paper proposes a method of multi-AUV co-location based on the consistent extended Kalman filter (EKF). Firstly, the dynamic model of cooperative positioning system follower AUV under two leaders alternately transmitting navigation information is established. Secondly, the observability of the standard linearization estimator based on the lead-follower multi-AUV cooperative positioning system is analyzed by comparing the subspace of the observable matrix of state estimation with that of an ideal observable matrix, it can be concluded that the estimation of state by standard EKF is inconsistent. Finally, aiming at the problem of inconsistent state estimation, a consistent EKF multi-AUV cooperative localization algorithm is designed. The algorithm corrects the linearized measurement values in the Jacobian matrix for cooperative positioning, ensuring that the linearized estimator can obtain accurate measurement values. The positioning results of the follower AUV under dead reckoning, standard EKF, and consistent EKF algorithms are simulated, analyzed, and compared with the real trajectory of the following AUV. The simulation results show that the follower AUV with a consistent EKF algorithm can keep synchronization with the leader AUV more stably.<\/jats:p>","DOI":"10.3390\/s22124563","type":"journal-article","created":{"date-parts":[[2022,6,19]],"date-time":"2022-06-19T21:19:26Z","timestamp":1655673566000},"page":"4563","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Consistent Extended Kalman Filter-Based Cooperative Localization of Multiple Autonomous Underwater Vehicles"],"prefix":"10.3390","volume":"22","author":[{"given":"Fubin","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Marine Science and Technology, Northwestern Polytechnical University, 127 West Youyi Road, Xi\u2019an 710072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4021-660X","authenticated-orcid":false,"given":"Xingqi","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Marine Science and Technology, Northwestern Polytechnical University, 127 West Youyi Road, Xi\u2019an 710072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peng","family":"Ma","sequence":"additional","affiliation":[{"name":"The 20th Research Institute of China Electronics Technology Group Corporation, 1 Baisha Road, Xi\u2019an 710075, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Li, D.L., and Du, L. (2021). AUV Trajectory Tracking Models and Control Strategies: A Review. J. Mar. Sci. Eng., 9.","DOI":"10.3390\/jmse9091020"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"16191","DOI":"10.3390\/s131216191","article-title":"An Observability Metric for Underwater Vehicle Localization Using Range Measurements","volume":"13","author":"Arrichiello","year":"2013","journal-title":"Sensors"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1109\/48.820738","article-title":"Underwater acoustic Sozernetworks","volume":"25","author":"Sozer","year":"2000","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Yang, S., Luo, Q.H., Liu, C., Yan, X.Z., and Yang, K.X. (2021). A multi-AUV cooperative navigation method. IOP Conf. Ser. Mater. Sci. Eng., 1207.","DOI":"10.1088\/1757-899X\/1207\/1\/012002"},{"key":"ref_5","unstructured":"Yuan, H., Wei, J.X., Shao, J.B., Fan, S.W., Yu, F., and Huang, W.J. (2021, January 22\u201324). Research on AUV Cooperative Positioning Technology Based on Improved-EKF with Error Estimation. Proceedings of the 2021 33rd Chinese Control and Decision Conference (CCDC), Kunming, China."},{"key":"ref_6","unstructured":"Wei, J.X., Chen, S.L., Wang, Y.Y., Wang, Q.X., Yu, F., and Huang, W.J. (2021, January 22\u201324). Research on AUV Cooperative Positioning Algorithm Based on Innovation Correction Method Based Central Differential Kalman Filter. Proceedings of the 2021 33rd Chinese Control and Decision Conference (CCDC), Kunming, China."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.ifacol.2021.10.096","article-title":"Cooperative single-beacon multiple AUV navigation under stringent communication bandwidth constraints","volume":"54","author":"Rego","year":"2021","journal-title":"IFAC-PapersOnLine"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Cario, G., Casavola, A., Gagliardi, G., Lupia, M., Severino, U., and Bruno, F. (2019, January 17\u201320). Analysis of error sources in underwater localization systems. Proceedings of the OCEANS, Marseille, France.","DOI":"10.1109\/OCEANSE.2019.8867536"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1109\/COMST.2021.3059998","article-title":"A Survey of Autonomous Underwater Vehicle Formation: Performance, Formation Control, and Communication Capability","volume":"23","author":"Yang","year":"2021","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Campos, R., Gracias, N., and Ridao, P. (2016). Underwater Multi-Vehicle Trajectory Alignment and Mapping Using Acoustic and Optical Constraints. Sensors, 3.","DOI":"10.3390\/s16030387"},{"key":"ref_11","first-page":"735","article-title":"Multiple-AUV cooperative navigation based on two-leader alternated navigation","volume":"35","author":"Gao","year":"2014","journal-title":"J. Harbin Eng. Univ."},{"key":"ref_12","first-page":"1898","article-title":"A cooperative navigation algorithm for two leader AUVs based on range measurements","volume":"36","author":"Zhang","year":"2016","journal-title":"Syst. Eng. Theory Pract."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ullah, I., Gao, M.S., Kamal, M.M., and Khan, Z. (2017, January 8\u201310). A survey on underwater localization, localization techniques and its algorithms. Proceedings of the 3rd Annual International Conference on Electronics, Electrical Engineering and Information Science (EEEIS), Guangdong, China.","DOI":"10.2991\/eeeis-17.2017.35"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Li, J., and Zhang, J. (2016, January 7\u201310). Research on the algorithm of multi-autonomous underwater vehicles navigation and localization based on the Extended Kalman Filter. Proceedings of the 2016 IEEE International Conference on Mechatronics and Automation, Harbin, China.","DOI":"10.1109\/ICMA.2016.7558951"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Bresciani, M., Costanzi, R., Manzari, V., Peralta, G., Terracciano, D.S., and Caiti, A. (2020, January 5\u201330). Comparative analysis of EKF and Particle Filter performance for an acoustic tracking system for AUVs exploiting bearing-only measurements. Proceedings of the Global Oceans 2020: Singapore\u2013U.S. Gulf Coast, Biloxi, MS, USA.","DOI":"10.1109\/IEEECONF38699.2020.9389288"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1017\/S037346331400085X","article-title":"Moving Horizon Estimation for Cooperative Localisation with Communication Delay","volume":"68","author":"Gao","year":"2015","journal-title":"J. Navig."},{"key":"ref_17","first-page":"2229","article-title":"Single Beacon-Based Localization With Constraints and Unknown Initial Poses","volume":"63","author":"Wang","year":"2016","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bar-Shalom, Y., Li, X.R., and Kirubarajan, T. (2002). Estimation with Applications to Tracking and Navigation, New York John Wiley and Sons.","DOI":"10.1002\/0471221279"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Bahr, A., Walter, M.R., and Leonard, J.J. (2009, January 12\u201317). Consistent cooperative localization. Proceedings of the 2009 IEEE International Conference on Robotics and Automation, Kobe, Japan.","DOI":"10.1109\/ROBOT.2009.5152859"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/s10514-010-9207-y","article-title":"Observability-based consistent EKF estimators for multi-robot cooperative localization","volume":"30","author":"Huang","year":"2011","journal-title":"Auton. Robot."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1177\/0278364909353640","article-title":"Observability-based rules for designing consistent EKF SLAM estimators","volume":"29","author":"Huang","year":"2010","journal-title":"Int. J. Robot. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1177\/0278364913509675","article-title":"Camera-IMU-based localization: Observability analysis and consistency improvenment","volume":"30","author":"Hesch","year":"2014","journal-title":"Int. J. Robot. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.oceaneng.2015.12.058","article-title":"A new AUV navigation system exploiting unscented Kalman filter","volume":"113","author":"Allotta","year":"2016","journal-title":"Ocean. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Sun, C., Zhang, Y., Wang, G., and Gao, W. (2018). A New Variational Bayesian Adaptive Extended Kalman Filter for Cooperative Navigation. Sensors, 18.","DOI":"10.3390\/s18082538"},{"key":"ref_25","unstructured":"Gadre, A.S. (2007). Observability Analysis in Navigation Systems with an Underwater Vehicle Application. [Ph.D. Thesis, Virginia Polytechnic Institute and State University]."},{"key":"ref_26","unstructured":"Chen, Z., Jiang, K., and Hung, J.C. (1990, January 27\u201330). Local observability matrix and its application to observability analyses. Proceedings of the 16th Annual Conference of IEEE Industrial Electronics Society, Pacific Grove, CA, USA."},{"key":"ref_27","unstructured":"Huang, G. (2013). Improving the Consistency of Nonlinear Estimator: Analysis, Algorithms, and Application. [Ph.D. Thesis, University of Minnesota]."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1581","DOI":"10.1016\/j.robot.2014.05.004","article-title":"An optimization based moving horizon estimation with application to localization of autonomous underwater vehicles","volume":"62","author":"Wang","year":"2014","journal-title":"Robot. Auton. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hesch, J.A., and Roumeliotis, S.I. (2012, January 16\u201321). Consistency analysis and improvement for single-camera localization. Proceedings of the 2012 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, Providence, RI, USA.","DOI":"10.1109\/CVPRW.2012.6239190"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1109\/TRO.2013.2277549","article-title":"Consistency Analysis and Improvement of Vision-aided Inertial Navigation","volume":"30","author":"Hesch","year":"2014","journal-title":"IEEE Trans. Robot."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Zhang, L., Wang, T., Zhang, F., and Xu, D. (2017). Cooperative Localization for Multi-AUVs Based on GM-PHD Filters and Information Entropy Theory. Sensors, 17.","DOI":"10.3390\/s17102286"},{"key":"ref_32","unstructured":"Huang, G.P., and Roumeliotis, S.I. (2008). An Observability Constrained UKF for Improving SLAM Consistency, MARS Lab, University of Minnesota. Technical Report."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/12\/4563\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:33:34Z","timestamp":1760139214000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/12\/4563"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,17]]},"references-count":32,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["s22124563"],"URL":"https:\/\/doi.org\/10.3390\/s22124563","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,17]]}}}