{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T21:10:21Z","timestamp":1779225021871,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T00:00:00Z","timestamp":1627257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61803015"],"award-info":[{"award-number":["61803015"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In the field of high accuracy strapdown inertial navigation system (SINS), the inertial measurement unit (IMU) biases can severely affect the navigation accuracy. Traditionally we use Kalman filter (KF) to estimate those biases. However, KF is an unbiased estimation method based on the assumption of Gaussian white noise (GWN) while IMU sensors noise is irregular. Kalman filtering will no longer be accurate when the sensor\u2019s noise is irregular. In order to obtain the optimal solution of the IMU biases, this paper proposes a novel method for the calibration of IMU biases utilizing the KF-based AdaGrad algorithm to solve this problem. Three improvements were made as the following: (1) The adaptive subgradient method (AdaGrad) is proposed to overcome the difficulty of setting step size. (2) A KF-based AdaGrad numerical function is derived and (3) a KF-based AdaGrad calibration algorithm is proposed in this paper. Experimental results show that the method proposed in this paper can effectively improve the accuracy of IMU biases in both static tests and car-mounted field tests.<\/jats:p>","DOI":"10.3390\/s21155055","type":"journal-article","created":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T22:22:46Z","timestamp":1627338166000},"page":"5055","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["An Improved Calibration Method for the IMU Biases Utilizing KF-Based AdaGrad Algorithm"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8694-6021","authenticated-orcid":false,"given":"Zeyang","family":"Wen","sequence":"first","affiliation":[{"name":"School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gongliu","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9956-4740","authenticated-orcid":false,"given":"Qingzhong","family":"Cai","sequence":"additional","affiliation":[{"name":"School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"14391","DOI":"10.1109\/ACCESS.2017.2726519","article-title":"A new process uncertainty robust student\u2019s t based Kalman filter for SINS\/GPS integration","volume":"5","author":"Huang","year":"2017","journal-title":"IEEE Access"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Qi, N. (2013, January 21\u201323). Research on initial alignment of SINS for marching vehicle. Proceedings of the 2013 Third International Conference on Instrumentation, Measurement, Computer, Communication and Control, Shenyang, China.","DOI":"10.1109\/IMCCC.2013.106"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Tedaldi, D., Pretto, A., and Menegatti, E. (June, January 31). A robust and easy to implement method for IMU calibration without external equipments. Proceedings of the 2014 IEEE International Conference on Robotics and Automation (ICRA), Hong Kong, China.","DOI":"10.1109\/ICRA.2014.6907297"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"085202","DOI":"10.1088\/0957-0233\/19\/8\/085202","article-title":"Methods for in-field user calibration of an inertial measurement unit without external equipment","volume":"19","author":"Fong","year":"2008","journal-title":"Meas. Sci. Technol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1809","DOI":"10.1016\/j.automatica.2006.05.005","article-title":"Observability analysis of rotation estimation by fusing inertial and line-based visual information: A revisit","volume":"42","author":"Wu","year":"2006","journal-title":"Automatica"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3058","DOI":"10.1109\/TIM.2012.2202186","article-title":"Error analysis and gyro-bias calibration of analytic coarse alignment for airborne POS","volume":"61","author":"Li","year":"2012","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5588","DOI":"10.1016\/j.ijleo.2013.03.151","article-title":"A new method for bias calibration of laser gyros using a single-axis turning table","volume":"124","author":"Han","year":"2013","journal-title":"Optik"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.sna.2015.05.002","article-title":"An eight-position self-calibration method for a dual-axis rotational Inertial Navigation System","volume":"232","author":"Zheng","year":"2015","journal-title":"Sens. Actuators A Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.neucom.2016.10.026","article-title":"An adaptive Kalman filter estimating process noise covariance","volume":"223","author":"Wang","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.ast.2017.08.020","article-title":"Adaptive unscented Kalman filter based on maximum posterior and random weighting","volume":"71","author":"Gao","year":"2017","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/j.sna.2007.05.008","article-title":"Calibration and data fusion solution for the miniature attitude and heading reference system","volume":"138","author":"Jurman","year":"2007","journal-title":"Sens. Actuators A Phys."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Munguia, R., and Grau, A. (2011, January 27\u201330). Attitude and Heading System based on EKF total state configuration. Proceedings of the 2011 IEEE International Symposium on Industrial Electronics, Gdansk, Poland.","DOI":"10.1109\/ISIE.2011.5984493"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1017\/S0373463312000331","article-title":"Effective adaptive kalman filter for MEMS-IMU\/magnetometers integrated attitude and heading reference systems","volume":"66","author":"Li","year":"2013","journal-title":"J. Navig."},{"key":"ref_14","first-page":"503814","article-title":"A DCM Based Attitude Estimation Algorithm for Low-Cost MEMS IMUs","volume":"2015","author":"Hyyti","year":"2015","journal-title":"Int. J. Navig. Obs."},{"key":"ref_15","first-page":"1018415","article-title":"Online calibration technique for LDV in SINS\/LDV integrated navigation systems","volume":"10184","author":"Wang","year":"2017","journal-title":"Sens. Command. Control. Commun. Intell. Technol. Homel. Secur. Defense Law Enforc. Appl. XVI"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1016\/j.measurement.2018.07.065","article-title":"Online self-calibration research of single-axis rotational inertial navigation system","volume":"129","author":"Li","year":"2018","journal-title":"Meas. J. Int. Meas. Confed."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"28755","DOI":"10.1109\/ACCESS.2018.2833290","article-title":"Attitude Estimation Fusing Quasi-Newton and Cubature Kalman Filtering for Inertial Navigation System Aided with Magnetic Sensors","volume":"6","author":"Huang","year":"2018","journal-title":"IEEE Access"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"55788","DOI":"10.1109\/ACCESS.2019.2912871","article-title":"Particle Swarm Optimization-Based Gyro Drift Estimation Method for Inertial Navigation System","volume":"7","author":"He","year":"2019","journal-title":"IEEE Access"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"66923","DOI":"10.1109\/ACCESS.2018.2878756","article-title":"Study on Installation Error Analysis and Calibration of Acoustic Transceiver Array Based on SINS\/USBL Integrated System","volume":"6","author":"Jinwu","year":"2018","journal-title":"IEEE Access"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.automatica.2008.05.032","article-title":"Estimation of the disturbance structure from data using semidefinite programming and optimal weighting","volume":"45","author":"Rajamani","year":"2009","journal-title":"Automatica"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Sheimy, N.E.I., Hou, H., and Niu, X. (2008). Analysis and Modeling of Inertial Sensors Using AV. IEEE Trans. Instrum. Meas., 56\u201357.","DOI":"10.1109\/TIM.2007.908635"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"980","DOI":"10.2514\/2.4666","article-title":"On-line estimation of Allan variance parameters","volume":"23","author":"Ford","year":"2000","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2750","DOI":"10.3390\/s130302750","article-title":"Systematic angle random walk estimation of the constant rate biased ring laser Gyro","volume":"13","author":"Yu","year":"2013","journal-title":"Sensors"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s10957-007-9305-y","article-title":"Gradient descent approach to optimal mode scheduling in hybrid dynamical systems","volume":"136","author":"Axelsson","year":"2008","journal-title":"J. Optim. Theory Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s001860000043","article-title":"Steepest descent methods for multicriteria optimization","volume":"51","author":"Fliege","year":"2000","journal-title":"Math. Methods Oper. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1007\/s10107-016-1017-3","article-title":"Stochastic compositional gradient descent: Algorithms for minimizing compositions of expected-value functions","volume":"161","author":"Wang","year":"2017","journal-title":"Math. Program."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.ins.2017.05.005","article-title":"Multi-step heuristic dynamic programming for optimal control of nonlinear discrete-time systems","volume":"411","author":"Luo","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"47288","DOI":"10.1109\/ACCESS.2018.2866818","article-title":"System-level calibration for the star sensor installation error in the stellar-inertial navigation system on a swaying base","volume":"6","author":"Zhang","year":"2018","journal-title":"IEEE Access"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"101019","DOI":"10.1109\/ACCESS.2019.2930988","article-title":"Analysis and Compensation of Installation Errors for Rotating Semi-Strapdown Inertial Navigation System","volume":"7","author":"Zhang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6381","DOI":"10.1109\/JSEN.2019.2910213","article-title":"Strapdown Inertial Navigation System Initial Alignment Based on Modified Process Model","volume":"19","author":"Chang","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"8998","DOI":"10.1109\/JSEN.2016.2616356","article-title":"A Self-Calibration Method for Non-Orthogonal Angles of Gimbals in Tri-Axis Rotational Inertial Navigation System","volume":"16","author":"Gao","year":"2016","journal-title":"IEEE Sens. J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"7243","DOI":"10.1109\/JSEN.2019.2916067","article-title":"Calibration, alignment, and dynamic tilt maintenance method based on vehicular hybrid measurement unit","volume":"19","author":"Lu","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_33","first-page":"1","article-title":"Modeling and Calibration of the Gyro- Accelerometer Asynchronous Time in Dual-Axis RINS","volume":"70","author":"Wen","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_34","first-page":"1","article-title":"A Robust In-Motion Alignment Method with Inertial Sensors and Doppler Velocity Log","volume":"70","author":"Xu","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3816","DOI":"10.1109\/TVT.2020.2974524","article-title":"A Fast Robust In-Motion Alignment Method for SINS with DVL Aided","volume":"69","author":"Xu","year":"2020","journal-title":"IEEE Trans. Veh. Technol."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/15\/5055\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:35:09Z","timestamp":1760164509000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/15\/5055"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,26]]},"references-count":35,"journal-issue":{"issue":"15","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["s21155055"],"URL":"https:\/\/doi.org\/10.3390\/s21155055","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,26]]}}}