{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T02:28:36Z","timestamp":1773800916829,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T00:00:00Z","timestamp":1648684800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program","doi-asserted-by":"publisher","award":["2018YFB0505200"],"award-info":[{"award-number":["2018YFB0505200"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Action Plan Project of the Beijing University of Posts and Telecommunications supported by the Fundamental Research Funds for the Central Universities","award":["2019XD-A06"],"award-info":[{"award-number":["2019XD-A06"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61872046"],"award-info":[{"award-number":["61872046"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62002026"],"award-info":[{"award-number":["62002026"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Joint Research Fund for Beijing Natural Science Foundation and Haidian Original Innovation","award":["L192004"],"award-info":[{"award-number":["L192004"]}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["4212024"],"award-info":[{"award-number":["4212024"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["4222034"],"award-info":[{"award-number":["4222034"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Research and Development Project from Hebei Province","award":["19210404D"],"award-info":[{"award-number":["19210404D"]}]},{"name":"Science and Technology Plan Project of Inner Mongolia Autonomous Regio","award":["2019GG328"],"award-info":[{"award-number":["2019GG328"]}]},{"name":"Key Research and Development Project from Hebei Province:","award":["21310102D"],"award-info":[{"award-number":["21310102D"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The tightly coupled navigation system is commonly used in UAV products and land vehicles. It adopts the Kalman filter to combine raw satellite observations, including the pseudorange, pseudorange rate and Doppler frequency, with the inertial measurements to achieve high navigational accuracy in GNSS-challenged environments. The accurate estimation of measurement noise covariance can ensure the quick convergence of the Kalman filter and the accuracy of the navigation results. Existing tightly coupled integrated navigation systems employ either constant noise covariance or simple noise covariance updating methods, which cannot accurately reflect the dynamic measurement noises. In this article, we propose an adaptive measurement noise estimation algorithm using a transformer and residual denoising autoencoder (RDAE), which can dynamically estimate the covariance of measurement noise. The residual module is used to solve the gradient degradation problem. The DAE is adopted to learn the essential characteristics from the noisy ephemeris data. By introducing the attention mechanism, the transformer can effectively learn the time and space dependency of long-term ephemeris data, and thus dynamically adjusts the noise covariance with the predicted factors. Extensive experimental results demonstrate that our method can achieve sub-meter positioning accuracy in the outdoor open environment. In a GNSS-degraded environment, our proposed method can still obtain about 3 m positioning accuracy. Another test on a new dataset also confirms that our proposed method has reasonable robustness and adaptability.<\/jats:p>","DOI":"10.3390\/rs14071691","type":"journal-article","created":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T21:34:29Z","timestamp":1648762469000},"page":"1691","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Towards Predicting the Measurement Noise Covariance with a Transformer and Residual Denoising Autoencoder for GNSS\/INS Tightly-Coupled Integrated Navigation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6031-7892","authenticated-orcid":false,"given":"Hongfu","family":"Xu","sequence":"first","affiliation":[{"name":"School of Computer Science, National Demonstration Software Institute, Beijing University of Posts and Telecommunications, Beijing 100876, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6827-4225","authenticated-orcid":false,"given":"Haiyong","family":"Luo","sequence":"additional","affiliation":[{"name":"Research Center for Ubiquitous Computing Systems, Institute of Computing Technology Chinese Academy of Sciences, Beijing 100080, China"}]},{"given":"Zijian","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Computer Science, National Demonstration Software Institute, Beijing University of Posts and Telecommunications, Beijing 100876, China"}]},{"given":"Fan","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Computer Science, National Demonstration Software Institute, Beijing University of Posts and Telecommunications, Beijing 100876, China"}]},{"given":"Linfeng","family":"Bao","sequence":"additional","affiliation":[{"name":"Research Center for Ubiquitous Computing Systems, Institute of Computing Technology Chinese Academy of Sciences, Beijing 100080, China"}]},{"given":"Fang","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Computer Science, National Demonstration Software Institute, Beijing University of Posts and Telecommunications, Beijing 100876, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Titterton, D., and Weston, J. (2004). Strapdown Inertial Navigation Technology, Institution of Engineering and Technology.","DOI":"10.1049\/PBRA017E"},{"key":"ref_2","unstructured":"(2007). GNSS\/INS Integration. Global Positioning Systems, Inertial Navigation, and Integration, John Wiley and Sons."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1109\/TVT.2008.926076","article-title":"Performance Enhancement of MEMS-Based INS\/GPS Integration for Low-Cost Navigation Applications","volume":"58","author":"Noureldin","year":"2009","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/TITS.2008.2011712","article-title":"In-Car Positioning and Navigation Technologies\u2014A Survey","volume":"10","author":"Skog","year":"2009","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4256","DOI":"10.1109\/TVT.2010.2070850","article-title":"Two-Filter Smoothing for Accurate INS\/GPS Land-Vehicle Navigation in Urban Centers","volume":"59","author":"Liu","year":"2010","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_6","first-page":"191","article-title":"Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems\u2014Second Edition","volume":"67","author":"Groves","year":"2013","journal-title":"J. Navig."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3694","DOI":"10.1109\/TIM.2010.2050981","article-title":"A Robust Solution to High-Accuracy Geolocation: Quadruple Integration of GPS, IMU, Pseudolite, and Terrestrial Laser Scanning","volume":"60","author":"Toth","year":"2011","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.isatra.2014.10.006","article-title":"A derivative UKF for tightly coupled INS\/GPS integrated navigation","volume":"56","author":"Hu","year":"2015","journal-title":"ISA Trans"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1016\/j.cja.2017.12.011","article-title":"GPS\/BDS\/INS tightly coupled integration accuracy improvement using an improved adaptive interacting multiple model with classified measurement update","volume":"31","author":"Han","year":"2018","journal-title":"Chin. J. Aeronaut."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"829","DOI":"10.3390\/s130100829","article-title":"About Non-Line-Of-Sight satellite detection and exclusion in a 3D map-aided localization algorithm","volume":"13","author":"Peyraud","year":"2013","journal-title":"Sensors"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Dhital, A., Lachapelle, G., and Bancroft, J.B. (2015, January 13\u201316). Bancroft. Improving the Reliability of Personal Navigation Devices in Harsh Environments. Proceedings of the 2015 International Conference on Indoor Positioning and Indoor Navigation (IPIN), Banff, AB, Canada.","DOI":"10.1109\/IPIN.2015.7346950"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1109\/TAC.2017.2730480","article-title":"A Novel Adaptive Kalman Filter with Inaccurate Process and Measurement Noise Covariance Matrices","volume":"63","author":"Huang","year":"2018","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/nature14539","article-title":"Deep learning","volume":"521","author":"LeCun","year":"2015","journal-title":"Nature"},{"key":"ref_14","first-page":"112","article-title":"Relative Study of Measurement Noise Covariance R and Process Noise Covariance Q of the Kalman Filter in Estimation","volume":"10","year":"2015","journal-title":"IOSR J. Electr. Electron. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.actaastro.2015.12.014","article-title":"Covariance matching based adaptive unscented Kalman filter for direct filtering in INS\/GNSS integration","volume":"120","author":"Meng","year":"2016","journal-title":"Acta Astronaut."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1132","DOI":"10.1109\/TIM.2006.877718","article-title":"Robust Positioning Technique in Low-Cost DR\/GPS for Land Navigation","volume":"55","author":"Cho","year":"2006","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4315","DOI":"10.1109\/TIE.2012.2193854","article-title":"Seam Tracking Monitoring Based on Adaptive Kalman Filter Embedded Elman Neural Network During High-Power Fiber Laser Welding","volume":"59","author":"Gao","year":"2012","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Qiusheng., H., Wei, C., and Xu., Y. (2017, January 20\u201322). An Improved Adaptive Kalman Filtering Algorithm for balancing vehicle. Proceedings of the 2017 Chinese Automation Congress (CAC), Jinan, China.","DOI":"10.1109\/CAC.2017.8243804"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.jprocont.2011.01.001","article-title":"Identification of process and measurement noise covariance for state and parameter estimation using extended Kalman filter","volume":"21","author":"Bavdekar","year":"2011","journal-title":"J. Process Control"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1017\/S0373463302002151","article-title":"Adaptive Kalman Filtering for Low-cost INS\/GPS","volume":"56","author":"Hide","year":"2003","journal-title":"J. Navig."},{"key":"ref_21","first-page":"696","article-title":"Measurement-based adaptive Kalman filtering algorithm for GPS\/INS integrated navigation system","volume":"18","author":"Zhang","year":"2010","journal-title":"J. Chin. Inert. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ge, B., Zhang, H., Jiang, L., Li, Z., and Butt, M.M. (2019). Adaptive Unscented Kalman Filter for Target Tracking with Unknown Time-Varying Noise Covariance. Sensors, 19.","DOI":"10.3390\/s19061371"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Ge, B., Zhang, H., Fu, W., and Yang, J. (2020). Enhanced Redundant Measurement-Based Kalman Filter for Measurement Noise Covariance Estimation in INS\/GNSS Integration. Remote Sens., 12.","DOI":"10.3390\/rs12213500"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1007\/s10291-021-01213-z","article-title":"Pseudorange error prediction for adaptive tightly coupled GNSS\/IMU navigation in urban areas","volume":"26","author":"Sun","year":"2021","journal-title":"GPS Solut."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"9002643","DOI":"10.1155\/2021\/9002643","article-title":"A Bayesian Adaptive Unscented Kalman Filter for Aircraft Parameter and Noise Estimation","volume":"2021","author":"Ding","year":"2021","journal-title":"J. Sens."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Bao, X., Chen, H., and Li, J. (2021, January 22\u201324). Adaptive Tracking Algorithm with Radar Position Errors and Measurement Noise Covariance Matrix. Proceedings of the 2021 33rd Chinese Control and Decision Conference (CCDC), Kunming, China.","DOI":"10.1109\/CCDC52312.2021.9601990"},{"key":"ref_27","unstructured":"Haarnoja, T., Ajay, A., Levine, S., and Abbeel, P. (2016, January 5\u201310). Backprop KF: Learning Discriminative Deterministic State Estimators. Proceedings of the 30th International Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Liu, K., Ok, K., Vega-Brown, W., and Roy, N. (2018, January 21\u201325). Deep Inference for Covariance Estimation: Learning Gaussian Noise Models for State Estimation. Proceedings of the 2018 IEEE International Conference on Robotics and Automation (ICRA), Brisbane, Australia.","DOI":"10.1109\/ICRA.2018.8461047"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Coskun., H., Achilles., F., DiPietro., R., Navab, N., and Tombari, F. (2017, January 22\u201329). Long Short-Term Memory Kalman Filters: Recurrent Neural Estimators for Pose Regularization. Proceedings of the IEEE International Conference on Computer Vision (ICCV), Venice, Italy.","DOI":"10.1109\/ICCV.2017.589"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Gao, X., Luo, H., Ning, B., Zhao, F., Bao, L., Gong, Y., Xiao, Y., and Jiang, J. (2020). RL-AKF: An Adaptive Kalman Filter Navigation Algorithm Based on Reinforcement Learning for Ground Vehicles. Remote Sens., 12.","DOI":"10.3390\/rs12111704"},{"key":"ref_31","first-page":"2504013","article-title":"Predicting the Noise Covariance with a Multitask Learning Model for Kalman Filter-Based GNSS\/INS Integrated Navigation","volume":"70","author":"Wu","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Noureldin, A., Karamat., T.B., and Georgy, J. (2013). Global Positioning System. Fundamentals of Inertial Navigation, Satellite-Based Positioning and Their Integration, Spring.","DOI":"10.1007\/978-3-642-30466-8"},{"key":"ref_33","unstructured":"Misra, P., and Enge, P. (2001). Global Positioning System: Signals, Measurements and Performance, Ganga-Jamuna Press."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"10599","DOI":"10.3390\/s130810599","article-title":"The performance analysis of a real-time integrated INS\/GPS vehicle navigation system with abnormal GPS measurement elimination","volume":"13","author":"Chiang","year":"2013","journal-title":"Sensors"},{"key":"ref_35","unstructured":"Gelb, A. (1974). Applied Optimal Estimation, MIT Press."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Vincent, P., Larochelle, H., Bengio, Y., and Manzagol, P.-A. (2008, January 5\u20139). Extracting and composing robust features with denoising autoencoders. Proceedings of the 25th International Conference on Machine Learning(ICML), Helsinki, Finland.","DOI":"10.1145\/1390156.1390294"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","article-title":"Long short-term memory","volume":"9","author":"Hochreiter","year":"1997","journal-title":"Neural Comput."},{"key":"ref_38","first-page":"5998","article-title":"Attention Is All You Need","volume":"30","author":"Vaswani","year":"2017","journal-title":"Adv. Neural Inf. Processing Syst."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Akhlaghi, S., Zhou, N., and Huang, Z. (2017, January 16\u201320). Adaptive Adjustment of Noise Covariance in Kalman Filter for Dynamic State Estimation. Proceedings of the 2017 IEEE Power & Energy Society General Meeting, Chicago, IL, USA.","DOI":"10.1109\/PESGM.2017.8273755"}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/14\/7\/1691\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:47:50Z","timestamp":1760136470000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/14\/7\/1691"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,31]]},"references-count":39,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["rs14071691"],"URL":"https:\/\/doi.org\/10.3390\/rs14071691","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,31]]}}}