{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T10:17:45Z","timestamp":1768558665103,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T00:00:00Z","timestamp":1579132800000},"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":["No.51575500"],"award-info":[{"award-number":["No.51575500"]}],"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":["No.51705477"],"award-info":[{"award-number":["No.51705477"]}],"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":["No.61973280"],"award-info":[{"award-number":["No.61973280"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the National Natural Science Funds for Distinguished Young Scholars","award":["No.51225504"],"award-info":[{"award-number":["No.51225504"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The optimization-based alignment (OBA) methods, which are implemented by the optimal attitude estimation using vector observations\u2014also called double-vectors\u2014have proven to be effective at solving the in-flight alignment (IFA) problem. However, the traditional OBA methods are not applicable for the low-cost strap-down inertial navigation system (SINS) since the error of double-vectors will be accumulated over time due to the substantial drift of micro-electronic- mechanical system (MEMS) gyroscope. Moreover, the existing optimal estimation method is subject to a large computation burden, which results in a low alignment speed. To address these issues, in this article we propose a new fast IFA method based on modified double-vectors construction and the gradient descent method. To be specific, the modified construction method is implemented by reducing the integration interval and identifying the gyroscope bias during the construction procedure, which improves the accuracy of double-vectors and IFA; the gradient descent scheme is adopted to estimate the optimal attitude of alignment without complex matrix operation, which results in the improvement of alignment speed. The effect of different sizes of mini-batch on the performance of the gradient descent method is also discussed. Extensive simulations and vehicle experiments demonstrate that the proposed method has better accuracy and faster alignment speed than the related traditional methods for the low-cost SINS\/global positioning system (GPS) integrated navigation system<\/jats:p>","DOI":"10.3390\/s20020512","type":"journal-article","created":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T07:39:02Z","timestamp":1579246742000},"page":"512","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["A New In-Flight Alignment Method with an Application to the Low-Cost SINS\/GPS Integrated Navigation System"],"prefix":"10.3390","volume":"20","author":[{"given":"Zhenglong","family":"Lu","sequence":"first","affiliation":[{"name":"National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan 030051, China"}]},{"given":"Jie","family":"Li","sequence":"additional","affiliation":[{"name":"National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan 030051, China"},{"name":"Key Laboratory of Instrumentation Science &amp; Dynamic Measurement, North University of China, Taiyuan 030051, China"}]},{"given":"Xi","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Electrical Control Engineering, North University of China, Taiyuan 030051, China"}]},{"given":"Kaiqiang","family":"Feng","sequence":"additional","affiliation":[{"name":"National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan 030051, China"}]},{"given":"Xiaokai","family":"Wei","sequence":"additional","affiliation":[{"name":"National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan 030051, China"}]},{"given":"Debiao","family":"Zhang","sequence":"additional","affiliation":[{"name":"National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan 030051, China"}]},{"given":"Jing","family":"Mi","sequence":"additional","affiliation":[{"name":"National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan 030051, China"}]},{"given":"Yang","family":"Liu","sequence":"additional","affiliation":[{"name":"National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan 030051, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Yu, F., Gao, W., and Wang, Y. (2018). An improved strapdown inertial navigation system initial alignment algorithm for unmanned vehicles. Sensors, 18.","DOI":"10.3390\/s18103297"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Chang, L., Qin, F., and Jiang, S. (2019). Strapdown Inertial Navigation System Initial Alignment based on Modified Process Model. IEEE Sens. J.","DOI":"10.1109\/JSEN.2019.2910213"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1108\/SR-07-2016-0125","article-title":"Pedestrian navigation system using MEMS sensors for heading drift and altitude error correction","volume":"37","author":"Tian","year":"2017","journal-title":"Sens. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hu, H., and Zhang, J. (2018, January 24\u201326). Application of Hybrid Filtering Algorithm Based on Neural Network in INS\/GPS Integrated Navigation System. Proceedings of the 2018 IEEE 4th International Conference on Control Science and Systems Engineering (ICCSSE), Wuhan, China.","DOI":"10.1109\/CCSSE.2018.8724815"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1016\/j.ymssp.2017.07.051","article-title":"An innovative information fusion method with adaptive Kalman filter for integrated INS\/GPS navigation of autonomous vehicles","volume":"100","author":"Liu","year":"2018","journal-title":"Mech. Syst. Sig. Process."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Zhang, H., Li, T., Yin, L., Liu, D., Zhou, Y., Zhang, J., and Pan, F. (2019). A Novel KGP Algorithm for Improving INS\/GPS Integrated Navigation Positioning Accuracy. Sensors, 19.","DOI":"10.3390\/s19071623"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"61296","DOI":"10.1109\/ACCESS.2019.2911025","article-title":"A Fusion Methodology to Bridge GPS Outages for INS\/GPS Integrated Navigation System","volume":"7","author":"Zhang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3346","DOI":"10.1109\/TIE.2017.2752137","article-title":"Robust navigational system for a transporter using GPS\/INS fusion","volume":"65","author":"Kim","year":"2018","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhang, G., Lu, C., and Li, Y. (2019, January 15\u201317). Research on Initial Alignment Method of SINS with Improved CKF. Proceedings of the 2019 IEEE 3rd Information Technology, Networking, Electronic and Automation Control Conference (ITNEC), Sichuan, China.","DOI":"10.1109\/ITNEC.2019.8729514"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1017\/S0373463314000198","article-title":"An Improved Optimal Method For Initial Alignment","volume":"67","author":"Li","year":"2014","journal-title":"J. Navig."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1109\/TIM.2014.2355652","article-title":"Initial Alignment by Attitude Estimation for Strapdown Inertial Navigation Systems","volume":"64","author":"Chang","year":"2014","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1017\/S037346331700090X","article-title":"Fast fine initial self-alignment of INS in erecting process on stationary base","volume":"71","author":"Li","year":"2018","journal-title":"J. Navig."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1049\/iet-smt.2017.0535","article-title":"Analytic coarse alignment and calibration for inertial navigation system on swaying base assisted by star sensor","volume":"12","author":"Lu","year":"2018","journal-title":"IET Sci. Meas. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zhang, W., Peng, G., Yuan, B., Wang, P., Huo, Z., and Yang, Z. (2019, January 27\u201330). Improved Maximum Likelihood Filter Based on UD Decomposition Algorithm and its Application in Transfer Alignment. Proceedings of the 2019 Chinese Control Conference (CCC), Guangzhou, China.","DOI":"10.23919\/ChiCC.2019.8866004"},{"key":"ref_15","unstructured":"Wei, X., Huang, G.R., Lu, H., Peng, Z.Y., Hao, S.Y., and Xu, M.Q. (2018, January 7\u20139). Marginal Reduced High-degree CKF and its Application in Nonlinear Rapid Transfer Alignment. Proceedings of the 2018 International Conference on Computer Information Science and Application Technology, Daqing, China."},{"key":"ref_16","first-page":"612","article-title":"Initial Self-Alignment Method for Inertial Platform on a Stationary Base","volume":"38","author":"Ding","year":"2017","journal-title":"J. Astronaut."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Xu, Y., and Zhou, T. (2019). Research on In-Flight Alignment for Micro Inertial Navigation System Based on Changing Acceleration using Exponential Function. Micromachines, 10.","DOI":"10.3390\/mi10010024"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.ast.2016.11.014","article-title":"In-flight Initial Alignment for Small UAV MEMS-based Navigation via Adaptive Unscented Kalman Filtering approach","volume":"61","author":"Wang","year":"2017","journal-title":"Aerosol Sci. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"603","DOI":"10.24425\/123904","article-title":"Estimation of small uav position and attitude with reliable in-flight initial alignment for MEMS inertial sensors","volume":"25","author":"Wang","year":"2018","journal-title":"Metrol. Meas. Syst."},{"key":"ref_20","first-page":"439","article-title":"A survey of attitude representations","volume":"8","author":"Shuster","year":"1993","journal-title":"Navigation"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Pei, F., Wei, X., and Liang, Q. (2017, January 26\u201327). A Fast Alignment Algorithm Based on Adaptive Quaternion Kalman Filter. Proceedings of the 2017 9th International Conference on Intelligent Human-Machine Systems and Cybernetics (IHMSC), Hangzhou, China.","DOI":"10.1109\/IHMSC.2017.78"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1550147719844929","DOI":"10.1177\/1550147719844929","article-title":"In-flight alignment method of navigation system based on microelectromechanical systems sensor measurement","volume":"15","author":"Liu","year":"2019","journal-title":"Int. J. Distrib. Sens. Netw."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1024","DOI":"10.1109\/TAES.2013.6494396","article-title":"Velocity\/Position Integration Formula Part II: Application to Strapdown Inertial Navigation Computation","volume":"49","author":"Wu","year":"2013","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"70","DOI":"10.2514\/3.19717","article-title":"Three-axis attitude determination from vector observations","volume":"4","author":"Shuster","year":"1981","journal-title":"J. Guid. Control."},{"key":"ref_25","first-page":"245","article-title":"Attitude determination using vector observations and the singular value decomposition","volume":"36","author":"Markley","year":"1988","journal-title":"J. Astronaut. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"328","DOI":"10.2514\/2.4239","article-title":"Euler-q Algorithm for Attitude Determination from Vector Observations","volume":"21","author":"Mortari","year":"1998","journal-title":"J. Guid. Control Dyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ast.2010.05.004","article-title":"Optimization-based alignment for inertial navigation systems: Theory and algorithm","volume":"15","author":"Wu","year":"2011","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1697","DOI":"10.1109\/TAES.2016.130824","article-title":"Optimization-based alignment for strapdown inertial navigation system: Comparison and extension","volume":"52","author":"Chang","year":"2016","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1979","DOI":"10.1109\/TIM.2018.2805978","article-title":"In-Motion Filter-QUEST Alignment for Strapdown Inertial Navigation Systems","volume":"67","author":"Xu","year":"2018","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.1109\/TAES.2013.6494395","article-title":"Velocity\/Position Integration Formula Part I: Application to In-Flight Coarse Alignment","volume":"49","author":"Wu","year":"2013","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3930","DOI":"10.1109\/JSEN.2019.2896274","article-title":"In-Motion Coarse Alignment Method for SINS\/GPS Using Position Loci","volume":"19","author":"Xu","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1137\/1007077","article-title":"A least squares estimate of satellite attitude","volume":"7","author":"Wahba","year":"1965","journal-title":"SIAM Rev."},{"key":"ref_33","unstructured":"Lerner, G.M. (1978). Three-axis attitude determination. Spacecraft Attitude Determination and Control, Kluwer Academic."},{"key":"ref_34","unstructured":"Keat, J. (1977). Analysis of Least-Squares Attitude Determination Routine DOAOP, Computer Sciences Corporation Report CSC. Technical Report TM-77\/6034."},{"key":"ref_35","unstructured":"Shuster, M. (1973, January 20\u201322). Approximate algorithms for fast optimal attitude computation. Proceedings of the Guidance and Control Conference, Key Biscayne, FL, USA."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"19","DOI":"10.2514\/2.4228","article-title":"Strapdown inertial navigation integration algorithm design part 1: Attitude algorithms","volume":"21","author":"Savage","year":"1998","journal-title":"J. Guid. Control Dyn."},{"key":"ref_37","unstructured":"Xu, Y. (2016). Research on several key techniques of MINS\/GNSS integrated navigation system in the guided projectiles. [Ph.D. Thesis, Nanjing University of Science & Technology]."},{"key":"ref_38","first-page":"286","article-title":"Double-vector attitude determination algorithm improving coarse alignment accuracy of strapdown inertial navigation system for sea cucumber fishing device","volume":"33","author":"Bao","year":"2017","journal-title":"Trans. Chin. Soc. Agric. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Liu, M., Gao, Y., Li, G., Guang, X., and Li, S. (2016). An improved alignment method for the Strapdown Inertial Navigation System (SINS). Sensors, 16.","DOI":"10.3390\/s16050621"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1850","DOI":"10.1109\/TMECH.2017.2698639","article-title":"Indirect Kalman Filtering Based Attitude Estimation for Low-Cost Attitude and Heading Reference Systems","volume":"22","author":"Chang","year":"2017","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_41","unstructured":"Ruder, S. (2016). An overview of gradient descent optimization algorithms. arXiv."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/2\/512\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:26:43Z","timestamp":1760365603000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/2\/512"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,16]]},"references-count":41,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["s20020512"],"URL":"https:\/\/doi.org\/10.3390\/s20020512","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,16]]}}}