{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T04:17:35Z","timestamp":1782361055257,"version":"3.54.5"},"reference-count":25,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2019,8,15]],"date-time":"2019-08-15T00:00:00Z","timestamp":1565827200000},"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":["61773116"],"award-info":[{"award-number":["61773116"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Navigation grade inertial measurement units (IMUs) should be calibrated after Inertial Navigation Systems (INSs) are assembled and be re-calibrated after certain periods of time. The multi-position calibration methods with advantage of not requiring high-precision equipment are widely discussed. However, the existing multi-position calibration methods for IMU are based on the model of linear scale factors. To improve the precision of INS, the nonlinear scale factors should be calibrated accurately. This paper proposes an optimized multi-position calibration method with nonlinear scale factor for IMU, and the optimal calibration motion of IMU has been designed based on the analysis of sensitivity of the cost function to the calibration parameters. Besides, in order to improve the accuracy and robustness of the optimization, an estimation method on initial values is presented to solve the problem of setting initial values for iterative methods. Simulations and experiments show that the proposed method outperforms the calibration method without nonlinear scale factors. The navigation accuracy of INS can be improved by up to 17% in lab conditions and 12% in the moving vehicle experiment, respectively.<\/jats:p>","DOI":"10.3390\/s19163568","type":"journal-article","created":{"date-parts":[[2019,8,15]],"date-time":"2019-08-15T11:11:00Z","timestamp":1565867460000},"page":"3568","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Optimized Multi-Position Calibration Method with Nonlinear Scale Factor for Inertial Measurement Units"],"prefix":"10.3390","volume":"19","author":[{"given":"Zihui","family":"Wang","sequence":"first","affiliation":[{"name":"School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China"},{"name":"Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of Education, Southeast University, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4002-2592","authenticated-orcid":false,"given":"Xianghong","family":"Cheng","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China"},{"name":"Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of Education, Southeast University, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinbo","family":"Fu","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,15]]},"reference":[{"key":"ref_1","unstructured":"Savage, P.G. (2000). Strapdown Analytics, Strapdown Associates. [2nd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1115\/1.4034419","article-title":"A comprehensive overview of inertial sensor calibration techniques","volume":"139","author":"Poddar","year":"2017","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"170","DOI":"10.5081\/jgps.7.2.170","article-title":"Calibration and stochastic modelling of inertial navigation sensor errors","volume":"7","author":"Pagiatakis","year":"2008","journal-title":"J. Global Positioning Syst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Dong, C., Ren, S., Chen, X., and Wang, Z. (2018). A Separated Calibration Method for Inertial Measurement Units Mounted on Three-Axis Turntables. Sensors, 18.","DOI":"10.3390\/s18092846"},{"key":"ref_5","first-page":"311","article-title":"Procedure for effortless in-field calibration of three-axis rate gyros and accelerometers","volume":"7","author":"Ferraris","year":"1995","journal-title":"Sens. Mater."},{"key":"ref_6","first-page":"1","article-title":"A new calibration method for strapdown inertial navigation systems","volume":"127","author":"Shin","year":"2002","journal-title":"Z. Vermess"},{"key":"ref_7","unstructured":"Skog, I., and H\u00e4ndel, P. (2006, January 17\u201322). Calibration of a MEMS inertial measurement unit. Proceedings of the XVII IMEKO World Congress, Rio de Janeiro, Brazil."},{"key":"ref_8","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"},{"key":"ref_9","first-page":"1","article-title":"Improved multi-position calibration for inertial measurement units","volume":"21","author":"Zhang","year":"2009","journal-title":"Meas. Sci. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Cheuk, C.M., Lau, T.K., Lin, K.W., and Liu, Y. (2012, January 5\u20137). Automatic calibration for inertial measurement unit. Proceedings of the 12th International Conference on Control Automation Robotics & Vision (ICARCV), Guangzhou, China.","DOI":"10.1109\/ICARCV.2012.6485340"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"8157","DOI":"10.3390\/s120608157","article-title":"Improved iterative calibration for triaxial accelerometers based on the optimal observation","volume":"12","author":"Yang","year":"2012","journal-title":"Sensors"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1088\/0957-0233\/24\/10\/105002","article-title":"Accelerometer calibration with nonlinear scale factor based on multi-position observation","volume":"24","author":"Cai","year":"2013","journal-title":"Meas. Sci. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3470","DOI":"10.1109\/JSEN.2017.2694488","article-title":"A multi-position calibration method for consumer-grade accelerometers, gyroscopes, and magnetometers to field conditions","volume":"17","author":"Nieminen","year":"2017","journal-title":"IEEE Sens. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.ast.2017.05.001","article-title":"A new Multi-position calibration method for gyroscope\u2019s drift coefficients on centrifuge","volume":"68","author":"Wang","year":"2017","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1090\/S0025-5718-2014-02847-0","article-title":"A dwindling filter line search method for unconstrained optimization","volume":"84","author":"Chen","year":"2015","journal-title":"Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1016\/j.ejor.2015.06.044","article-title":"A multi-layer line search method to improve the initialization of optimization algorithms","volume":"247","author":"Ivorra","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1371","DOI":"10.1007\/s11590-015-0848-9","article-title":"A nonmonotone line search method for noisy minimization","volume":"9","author":"Nikolovski","year":"2015","journal-title":"Optim. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s10107-015-0893-2","article-title":"Recent advances in trust region algorithms","volume":"151","author":"Yuan","year":"2015","journal-title":"Math. Program."},{"key":"ref_19","unstructured":"Wu, Y., Mansimov, E., Grosse, R.B., Liao, S., and Ba, J. (2017, January 4\u20139). Scalable trust-region method for deep reinforcement learning using kronecker-factored approximation. Proceedings of the 31st Conference on Neural Information Processing Systems (NIPS), Long Beach, CA, USA."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1007\/s10107-017-1141-8","article-title":"Stochastic optimization using a trust-region method and random models","volume":"169","author":"Chen","year":"2018","journal-title":"Math. Program."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.ins.2017.09.059","article-title":"Accelerated nonrigid image registration using improved Levenberg\u2013Marquardt method","volume":"423","author":"Dong","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2343","DOI":"10.1109\/TIT.2018.2881187","article-title":"Globally convergent Levenberg-Marquardt method for phase retrieval","volume":"65","author":"Ma","year":"2019","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apradiso.2017.07.053","article-title":"Fast adaptive particle spectrum fitting algorithm based on moment-estimated initial parameters","volume":"129","author":"Shi","year":"2017","journal-title":"Appl. Radiat. Isot."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1016\/j.ymssp.2017.06.030","article-title":"Fuzzy adaptive integration scheme for low-cost SINS\/GPS navigation system","volume":"99","author":"Nourmohammadi","year":"2018","journal-title":"Mech. Syst. Signal Proc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/dac.3296","article-title":"Levenberg-Marquardt optimization method for coverage and connectivity control in backbone-based wireless networks","volume":"30","author":"Khan","year":"2017","journal-title":"Int. J. Commun. Syst."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/16\/3568\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:11:28Z","timestamp":1760188288000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/16\/3568"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,15]]},"references-count":25,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2019,8]]}},"alternative-id":["s19163568"],"URL":"https:\/\/doi.org\/10.3390\/s19163568","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8,15]]}}}