{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T01:31:50Z","timestamp":1778635910563,"version":"3.51.4"},"reference-count":36,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,7,20]],"date-time":"2018-07-20T00:00:00Z","timestamp":1532044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In this paper, we propose a robust adaptive cubature Kalman filter (CKF) to deal with the problem of an inaccurately known system model and noise statistics. In order to overcome the kinematic model error, we introduce an adaptive factor to adjust the covariance matrix of state prediction, and process the influence introduced by dynamic disturbance error. Aiming at overcoming the abnormality error, we propose the robust estimation theory to adjust the CKF algorithm online. The proposed adaptive CKF can detect the degree of gross error and subsequently process it, so the influence produced by the abnormality error can be solved. The paper also studies a typical application system for the proposed method, which is the ultra-tightly coupled navigation system of a hypersonic vehicle. Highly dynamical scene experimental results show that the proposed method can effectively process errors aroused by the abnormality data and inaccurate model, and has better tracking performance than UKF and CKF tracking methods. Simultaneously, the proposed method is superior to the tracing method based on a single-modulating loop in the tracking performance. Thus, the stable and high-precision tracking for GPS satellite signals are preferably achieved and the applicability of the system is promoted under the circumstance of high dynamics and weak signals. The effectiveness of the proposed method is verified by a highly dynamical scene experiment.<\/jats:p>","DOI":"10.3390\/s18072352","type":"journal-article","created":{"date-parts":[[2018,7,20]],"date-time":"2018-07-20T02:10:11Z","timestamp":1532052611000},"page":"2352","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Robust Adaptive Cubature Kalman Filter and Its Application to Ultra-Tightly Coupled SINS\/GPS Navigation System"],"prefix":"10.3390","volume":"18","author":[{"given":"Xin","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Nuclear Engineering, Rocket Force University of Engineering, Xi\u2019an 710025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6295-4908","authenticated-orcid":false,"given":"Jianli","family":"Li","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Opto-electc Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xunliang","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Astronautics, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shaowen","family":"Ji","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Opto-electc Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/978-3-319-78458-8_14","article-title":"An extended Kalman filter for time delay inspired by a fractional order model","volume":"496","author":"Haus","year":"2018","journal-title":"Lect. Notes Electr. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1109\/TMECH.2018.2792321","article-title":"An extended Kalman filter as an observer in a control structure for health monitoring of a matal-polymer hybrid soft actuator","volume":"23","author":"Schimmack","year":"2018","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.ifacol.2016.07.711","article-title":"Development and online validation of an UKF-based navigation algorithm for AUVs","volume":"49","author":"Allotta","year":"2016","journal-title":"IFAC-PaperOnLine"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1002\/asjc.1537","article-title":"A Seventh-degree Cubature Kalman Filter","volume":"20","author":"Meng","year":"2018","journal-title":"Asian J. Control"},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1109\/JPROC.2003.823141","article-title":"Unscented filtering and nonlinear estimation","volume":"92","author":"Julier","year":"2004","journal-title":"Proc. IEEE"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1109\/78.978387","article-title":"A particle algorithm for sequential Bayesian parameter estimation and model selection","volume":"50","author":"Lee","year":"2002","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1049\/ip-rsn:20030741","article-title":"Interacting multiple model particle filter","volume":"150","author":"Boers","year":"2003","journal-title":"IEEE Proc. Radar Sonar Navig."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.dsp.2016.09.011","article-title":"Cooperative Parallel Particle Filters for on-Line Model Selection and Applications to Urban Mobility","volume":"60","author":"Martino","year":"2017","journal-title":"Digit. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1049\/iet-smt.2016.0374","article-title":"Robust gaussian particle filter based on modified likelihood function","volume":"12","author":"Li","year":"2018","journal-title":"IET Sci. Meas. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1109\/TAC.2009.2019800","article-title":"Cubature Kalman Filters","volume":"54","author":"Ienkaran","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Pesonen, H., and Piche, R. (2010, January 11\u201312). Cubature-based Kalman filters for positioning. Proceedings of the 7th Workshop on Positioning Navigation and Communication, Dresden, Germany.","DOI":"10.1109\/WPNC.2010.5653829"},{"key":"ref_13","first-page":"1032","article-title":"INS\/GPS integrated navigation filter algorithm based on cubature Kalman filter","volume":"27","author":"Sun","year":"2012","journal-title":"Control Decis."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.sigpro.2015.07.014","article-title":"Perfomance of Cubature Kalman filter in a GPS\/IMU tightly-coupled navigation system","volume":"119","author":"Zhao","year":"2016","journal-title":"Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s10291-015-0510-0","article-title":"Analysis of a variational Bayesian adaptive cubature Kalman filter tracking loop for high dynamic conditions","volume":"21","author":"Miao","year":"2017","journal-title":"GPS Solut."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4977","DOI":"10.1109\/TSP.2010.2056923","article-title":"Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations","volume":"58","author":"Ienkaran","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"727","DOI":"10.4028\/www.scientific.net\/AMR.219-220.727","article-title":"Square Root Cubature Particle Filter","volume":"219\u2013220","author":"Mu","year":"2011","journal-title":"Adv. Mater. Res."},{"key":"ref_18","first-page":"1454","article-title":"Iterated cubature Kalman filter and its application","volume":"33","author":"Mu","year":"2011","journal-title":"Syst. Eng. Electron."},{"key":"ref_19","first-page":"451","article-title":"An improved cubature Kalman filters based on strong tracking","volume":"41","author":"Sun","year":"2013","journal-title":"J. Huazhong Univ. Sci. Technol."},{"key":"ref_20","first-page":"2394","article-title":"Strong tracking adaptive square-root cubature Kalman filter algorithm","volume":"42","author":"Xu","year":"2014","journal-title":"Acta Electron. Sin."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1016\/j.amc.2014.12.036","article-title":"Design of adaptive robust square-root cubature Kalman filter with noise statistic estimator","volume":"256","author":"Zhao","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_22","first-page":"3885","article-title":"Application of adaptive high-degree cubature Kalman filter in target tracking","volume":"36","author":"Cui","year":"2015","journal-title":"Acta Aeronaut. Astronaut. Sin."},{"key":"ref_23","first-page":"572","article-title":"Robust Cubature Kalman filter based on Huber M estimator","volume":"29","author":"Huang","year":"2014","journal-title":"Control Decis."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"0218401","DOI":"10.7498\/aps.64.218401","article-title":"Robust cubature Kalman filter target tracking algorithm based on genernalized M-estiamtion","volume":"64","author":"Wu","year":"2015","journal-title":"Acta Phys. Sin."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1177\/0954410014548698","article-title":"Robust square-root cubature Kalman filter based on Huber\u2019s M-estimation methodology","volume":"229","author":"Li","year":"2015","journal-title":"Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. June"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Zhang, C.L., Zhi, R.R., Li, T.C., and Corchado, J.M. (2016, January 22\u201323). Adaptive M-estimation for Robust Cubature Kalman Filtering. Proceedings of the Sensor Signal Processing for Defence (SSPD), Edinburgh, UK.","DOI":"10.1109\/SSPD.2016.7590586"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.isatra.2016.09.010","article-title":"Performance analysis of improved iterated cubature Kalman filter and its application to GNSS\/INS","volume":"66","author":"Cui","year":"2017","journal-title":"ISA Trans."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4334","DOI":"10.1109\/TSP.2012.2196697","article-title":"Likelihood Consensus and Its Application to Distributed Particle Filtering","volume":"60","author":"Hlinka","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"64","DOI":"10.14429\/dsj.66.8326","article-title":"Ultra-tight GPS\/IMU integration based long-range rocket projectile navigation","volume":"66","author":"Zhao","year":"2016","journal-title":"Def. Sci. J."},{"key":"ref_30","unstructured":"Wang, Q.T. (2010). The Theory and Application Research of Adaptive-Robust UKF for Satellite Integrated Navigation System. [Ph.D. Thesis, Huazhong University of Science and Technology]."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1007\/s001900050182","article-title":"Robust biased estimation and its application in geodetic adjustments","volume":"72","author":"Gui","year":"1998","journal-title":"J. Geod."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1179\/003962610X12572516251646","article-title":"An extended adaptive Kalman filtering in tight coupled GPS\/INS integration","volume":"42","author":"Wu","year":"2010","journal-title":"Surv. Rev."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"72","DOI":"10.3724\/SP.J.1004.2008.00072","article-title":"An adaptive UKF algorithm for the state and parameter estimations of a mobile robot","volume":"34","author":"Song","year":"2008","journal-title":"Acta Autom. Sin."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1109\/MAES.2011.5746183","article-title":"Unified Approach to Ultra-Tightly-Coupled GPS\/INS Integrated Navigation System","volume":"3","author":"Hwang","year":"2011","journal-title":"IEEE Aerosp. Electron. Syst. Mag."},{"key":"ref_35","first-page":"2895","article-title":"Research progress and prospects of GNSS\/INS deep integration","volume":"37","author":"Niu","year":"2016","journal-title":"Acta Aeronaut. Astronaut. Sin."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"8082","DOI":"10.1016\/j.ijleo.2016.06.009","article-title":"Acquisition and loop control of ultra-tight INS\/BeiDou integration system","volume":"127","author":"Zeng","year":"2016","journal-title":"Optik"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/7\/2352\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:13:12Z","timestamp":1760195592000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/7\/2352"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,20]]},"references-count":36,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2018,7]]}},"alternative-id":["s18072352"],"URL":"https:\/\/doi.org\/10.3390\/s18072352","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,20]]}}}