{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T16:13:55Z","timestamp":1764000835996,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,6,7]],"date-time":"2024-06-07T00:00:00Z","timestamp":1717718400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["51979047","2022CFB865","YQ2021E011"],"award-info":[{"award-number":["51979047","2022CFB865","YQ2021E011"]}]},{"name":"Natural Science Foundation of Hubei Province of China","award":["51979047","2022CFB865","YQ2021E011"],"award-info":[{"award-number":["51979047","2022CFB865","YQ2021E011"]}]},{"name":"Natural Science Foundation of Heilongjiang Province of China","award":["51979047","2022CFB865","YQ2021E011"],"award-info":[{"award-number":["51979047","2022CFB865","YQ2021E011"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In order to reduce the position errors of the Global Positioning System\/Strapdown Inertial Navigation System (GPS\/SINS) integrated navigation system during GPS denial, this paper proposes a method based on the Particle Swarm Optimization\u2013Back Propagation Neural Network (PSO-BPNN) to replace the GPS for positioning. The model relates the position information, velocity information, attitude information output by the SINS, and the navigation time to the position errors between the position information output by the SINS and the actual position information. The performance of the model is compared with the BPNN through an actual ship experiment. The results show that the PSO-BPNN can obviously reduce the position errors in the case of GPS signal denial.<\/jats:p>","DOI":"10.3390\/s24123722","type":"journal-article","created":{"date-parts":[[2024,6,7]],"date-time":"2024-06-07T10:43:42Z","timestamp":1717757022000},"page":"3722","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["A Method for Predicting Inertial Navigation System Positioning Errors Using a Back Propagation Neural Network Based on a Particle Swarm Optimization Algorithm"],"prefix":"10.3390","volume":"24","author":[{"given":"Yabo","family":"Wang","sequence":"first","affiliation":[{"name":"Wuhan Second Ship Research and Design Institute, Wuhan 430205, China"}]},{"given":"Ruihan","family":"Jiao","sequence":"additional","affiliation":[{"name":"School of Intelligent Science and Engineer, Harbin Engineering University, Harbin 150001, China"}]},{"given":"Tingxiao","family":"Wei","sequence":"additional","affiliation":[{"name":"School of Intelligent Science and Engineer, Harbin Engineering University, Harbin 150001, China"}]},{"given":"Zhaoxing","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Intelligent Science and Engineer, Harbin Engineering University, Harbin 150001, China"}]},{"given":"Yueyang","family":"Ben","sequence":"additional","affiliation":[{"name":"School of Intelligent Science and Engineer, Harbin Engineering University, Harbin 150001, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7791","DOI":"10.1109\/TVT.2015.2497713","article-title":"Transversal strapdown INS based on reference ellipsoid for vehicle in the polar region","volume":"65","author":"Li","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Jiang, C., Chen, S., Chen, Y., Zhang, B., Feng, Z., Zhou, H., and Bo, Y. (2018). A MEMS IMU de-noising method using long short term memory recurrent neural networks (LSTM-RNN). Sensors, 18.","DOI":"10.3390\/s18103470"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"192","DOI":"10.5081\/jgps.4.1.192","article-title":"Comparative study of interpolation techniques for ultra-tight integration of GPS\/INS\/PL sensors","volume":"4","author":"Badu","year":"2005","journal-title":"J. Glob. Position. Syst."},{"key":"ref_4","first-page":"110","article-title":"Design of Strapdown Inertial Navigation System Based on MEMS","volume":"3","author":"Chen","year":"2020","journal-title":"Acad. J. Comput. Inf. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Shaukat, N., Ali, A., Javed Iqbal, M., Moinuddin, M., and Otero, P. (2021). Multi-sensor fusion for underwater vehicle localization by augmentation of RBF neural network and error-state Kalman filter. Sensors, 21.","DOI":"10.3390\/s21041149"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1109\/JSEN.2016.2642040","article-title":"Non-linear autoregressive delay-dependent INS\/GPS navigation system using neural networks","volume":"17","author":"Jaradat","year":"2017","journal-title":"IEEE Sens."},{"key":"ref_7","first-page":"222","article-title":"Model Predictive Forward Neural Network Algorithm and Its Application in Integrated Navigation","volume":"4","author":"Yang","year":"2014","journal-title":"J. Inert. Technol. China"},{"key":"ref_8","first-page":"2231","article-title":"Research on the Error Feedback Correction Method of MEMS-SINS Based on Neural Network Prediction During GPS Signal Loss","volume":"30","author":"Cao","year":"2009","journal-title":"J. Astronaut."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1017\/S0373463318000760","article-title":"A hybrid intelligent algorithm DGP-MLP for GNSS\/INS integration during GNSS outages","volume":"72","author":"Zhang","year":"2019","journal-title":"Navigation"},{"key":"ref_10","unstructured":"Wang, Q., and Li, H. (2016). Research on Indoor Positioning Technology Based on Machine Learning and Inertial Navigation. Electron. Meas. Technol., 6."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/S0925-2312(01)00702-0","article-title":"Time series forecasting using a hybrid ARIMA and neural network model","volume":"50","author":"Zhang","year":"2003","journal-title":"Neurocomputing"},{"key":"ref_12","first-page":"489","article-title":"Application of BP Neural Network in the Transfer Alignment of Inertial Navigation System","volume":"5","author":"Wang","year":"2008","journal-title":"J. Nav. Aeronaut. Eng. Coll."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"82134","DOI":"10.1109\/ACCESS.2019.2922212","article-title":"A Novel BPNN-Based Method to Overcome the GPS Outages for INS\/GPS System","volume":"7","author":"Wang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"012189","DOI":"10.1088\/1742-6596\/1732\/1\/012189","article-title":"Research on master-slave filtering of Celestial Navigation System\/Inertial Navigation System","volume":"1732","author":"Yan","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ma, S., Liu, S., and Meng, X. (2020). Optimized BP Neural Network Algorithm for Predicting Ship Trajectory, IEEE.","DOI":"10.1109\/ITNEC48623.2020.9085154"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Fan, Y., Wang, Z., Lin, Y., and Tan, H. (2020, January 24\u201325). Enhance the Performance of Navigation: A Two-Stage Machine Learning Approach. Proceedings of the 2020 6th International Conference on Big Data Computing and Communications (BIGCOM), Deqing, China.","DOI":"10.1109\/BigCom51056.2020.00036"},{"key":"ref_17","first-page":"1","article-title":"Integrated navigation on vehicle based on low-cost SINS\/GNSS using Deep Learning","volume":"126","author":"Liu","year":"2021","journal-title":"Wirel. Pers. Commun."},{"key":"ref_18","unstructured":"Yu, X. (2022). Research on Gyroscope Random Error Modeling and Filtering Technology Based on LSTM Neural Network. [Ph.D. Thesis, Huazhong University of Science and Technology]."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"115","DOI":"10.3390\/aerospace7080115","article-title":"Unsupervised anomaly detection in flight data using convolutional variational auto-encoder","volume":"7","author":"Milad","year":"2020","journal-title":"Aerospace"},{"key":"ref_20","unstructured":"Chen, Z. (1986). Principles of Strapdown Inertial Navigation Systems. [Master\u2019s Thesis, Beijing Aerospace Press]."},{"key":"ref_21","unstructured":"Wang, X., and Shi, F. (2013). MATLAB Neural Network: Analysis of 43 Cases. [Master\u2019s Thesis, Beihang University Press]."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/12\/3722\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:55:27Z","timestamp":1760108127000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/12\/3722"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,7]]},"references-count":21,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["s24123722"],"URL":"https:\/\/doi.org\/10.3390\/s24123722","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2024,6,7]]}}}