{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T02:40:57Z","timestamp":1771468857932,"version":"3.50.1"},"reference-count":49,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,2,8]],"date-time":"2021-02-08T00:00:00Z","timestamp":1612742400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007847","name":"Natural Science Foundation of Jilin Province","doi-asserted-by":"publisher","award":["20200201170JC"],"award-info":[{"award-number":["20200201170JC"]}],"id":[{"id":"10.13039\/100007847","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In applications such as carrier attitude control and mobile device navigation, a micro-electro-mechanical-system (MEMS) gyroscope will inevitably be affected by random vibration, which significantly affects the performance of the MEMS gyroscope. In order to solve the degradation of MEMS gyroscope performance in random vibration environments, in this paper, a combined method of a long short-term memory (LSTM) network and Kalman filter (KF) is proposed for error compensation, where Kalman filter parameters are iteratively optimized using the Kalman smoother and expectation-maximization (EM) algorithm. In order to verify the effectiveness of the proposed method, we performed a linear random vibration test to acquire MEMS gyroscope data. Subsequently, an analysis of the effects of input data step size and network topology on gyroscope error compensation performance is presented. Furthermore, the autoregressive moving average-Kalman filter (ARMA-KF) model, which is commonly used in gyroscope error compensation, was also combined with the LSTM network as a comparison method. The results show that, for the x-axis data, the proposed combined method reduces the standard deviation (STD) by 51.58% and 31.92% compared to the bidirectional LSTM (BiLSTM) network, and EM-KF method, respectively. For the z-axis data, the proposed combined method reduces the standard deviation by 29.19% and 12.75% compared to the BiLSTM network and EM-KF method, respectively. Furthermore, for x-axis data and z-axis data, the proposed combined method reduces the standard deviation by 46.54% and 22.30% compared to the BiLSTM-ARMA-KF method, respectively, and the output is smoother, proving the effectiveness of the proposed method.<\/jats:p>","DOI":"10.3390\/s21041181","type":"journal-article","created":{"date-parts":[[2021,2,10]],"date-time":"2021-02-10T04:33:46Z","timestamp":1612931626000},"page":"1181","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["A Combined Method for MEMS Gyroscope Error Compensation Using a Long Short-Term Memory Network and Kalman Filter in Random Vibration Environments"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5784-0527","authenticated-orcid":false,"given":"Chenhao","family":"Zhu","sequence":"first","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"given":"Sheng","family":"Cai","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}]},{"given":"Yifan","family":"Yang","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3756-4913","authenticated-orcid":false,"given":"Wei","family":"Xu","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}]},{"given":"Honghai","family":"Shen","sequence":"additional","affiliation":[{"name":"Key Laboratory of Airborne Optical Imaging and Measurement, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}]},{"given":"Hairong","family":"Chu","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1109\/MAES.2005.1514768","article-title":"Gps\/ins uses low-cost mems imu","volume":"20","author":"Brown","year":"2005","journal-title":"IEEE Aerosp. Electron. Syst. Mag."},{"key":"ref_2","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":"2008","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chia, J., Low, K., Goh, S., and Xing, Y. (2016, January 5\u201312). In A low complexity Kalman filter for improving MEMS based gyroscope performance. Proceedings of the 2016 IEEE Aerospace Conference, Big Sky, MO, USA.","DOI":"10.1109\/AERO.2016.7500795"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/j.measurement.2016.03.013","article-title":"Investigating the effects of quadrature error in parametrically and harmonically excited MEMS rate gyroscopes","volume":"87","author":"Mohammadi","year":"2016","journal-title":"Measurement"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"015107","DOI":"10.1088\/0957-0233\/21\/1\/015107","article-title":"Improved multi-position calibration for inertial measurement units","volume":"21","author":"Zhang","year":"2009","journal-title":"Meas. Sci. Technol."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"25277","DOI":"10.3390\/s151025277","article-title":"Auto regressive moving average (ARMA) modeling method for Gyro random noise using a robust Kalman filter","volume":"15","author":"Huang","year":"2015","journal-title":"Sensors"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.sna.2016.09.036","article-title":"ARMA model based adaptive unscented fading Kalman filter for reducing drift of fiber optic gyroscope","volume":"251","author":"Narasimhappa","year":"2016","journal-title":"Sens. Actuators"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1109\/7.826331","article-title":"Equivalent ARMA model representation for RLG random errors","volume":"36","author":"Seong","year":"2000","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Quinchia, A.G., Ferrer, C., Falco, G., Falletti, E., and Dovis, F. (2012, January 25\u201327). In Analysis and modelling of MEMS inertial measurement unit. Proceedings of the 2012 International Conference on Localization and GNSS, Starnberg, Germany.","DOI":"10.1109\/ICL-GNSS.2012.6253129"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"371","DOI":"10.5139\/IJASS.2011.12.4.371","article-title":"Improvement of a low cost MEMS inertial-GPS integrated system using wavelet denoising techniques","volume":"12","author":"Kang","year":"2011","journal-title":"Int. J. Aeronaut. Space Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1016\/j.measurement.2012.02.001","article-title":"Modeling and compensation of MEMS gyroscope output data based on support vector machine","volume":"45","author":"Zhang","year":"2012","journal-title":"Measurement"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"9448","DOI":"10.3390\/s120709448","article-title":"An enhanced mems error modeling approach based on nu-support vector regression","volume":"12","author":"Bhatt","year":"2012","journal-title":"Sensors"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1017\/S0373463304002875","article-title":"An efficient neural network model for de-noising of MEMS-based inertial data","volume":"57","year":"2004","journal-title":"J. Navig."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Jiang, C., Chen, S., Chen, Y., Zhang, B., Feng, Z., Zhou, H., and Bo, Y. (2018). A MEMS IMU de-noising method using long short term memory recurrent neural networks (LSTM-RNN). Sensors, 18.","DOI":"10.3390\/s18103470"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Jiang, C., Chen, S., Chen, Y., Bo, Y., Han, L., Guo, J., Feng, Z., and Zhou, H. (2018). Performance analysis of a deep simple recurrent unit recurrent neural network (SRU-RNN) in MEMS gyroscope de-noising. Sensors, 18.","DOI":"10.3390\/s18124471"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Jiang, C., Chen, Y., Chen, S., Bo, Y., Li, W., Tian, W., and Guo, J. (2019). A mixed deep recurrent neural network for MEMS gyroscope noise suppressing. Electronics, 8.","DOI":"10.3390\/electronics8020181"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5491243","DOI":"10.1155\/2019\/5491243","article-title":"A MEMS Gyroscope Noise Suppressing Method Using Neural Architecture Search Neural Network","volume":"2019","author":"Zhu","year":"2019","journal-title":"Math. Probl. Eng."},{"key":"ref_19","unstructured":"Barbour, N.M. (2010). Inertial Navigation Sensors, Charles Stark Draper Lab Inc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2019","DOI":"10.1109\/JSEN.2014.2306912","article-title":"PEM stochastic modeling for MEMS inertial sensors in conventional and redundant IMUs","volume":"14","author":"Jafari","year":"2014","journal-title":"IEEE Sens. J."},{"key":"ref_21","unstructured":"Cho, J.Y. (2012). High-Performance Micromachined Vibratory Rate and Rate-Integrating Gyroscopes. [Ph.D. Thesis, University of Michigan]."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"115006","DOI":"10.1088\/0960-1317\/25\/11\/115006","article-title":"Analysis of compensation for a g-sensitivity scale-factor error for a MEMS vibratory gyroscope","volume":"25","author":"Park","year":"2015","journal-title":"J. Micromech. Microeng."},{"key":"ref_23","unstructured":"Dean, R., Flowers, G., Hodel, S., MacAllister, K., and Horvath, R. (2002). In Vibration Isolation of MEMS Sensors for Aerospace Applications. Proceedings of the SPIE Proceedings Series, SPIE."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1109\/58.660162","article-title":"A micromachined vibration isolation system for reducing the vibration sensitivity of surface transverse wave resonators","volume":"45","author":"Reid","year":"1998","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_25","unstructured":"Reid, J.R., Bright, V.M., Stewart, J.T., and Kosinski, J.A. (1996, January 5\u20137). In Reducing the normal acceleration sensitivity of surface transverse wave resonators using micromachined isolation systems. Proceedings of the 1996 IEEE International Frequency Control Symposium, Honolulu, HI, USA."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Dean, R., Flowers, G., Sanders, N., Horvath, R., Johnson, W., Kranz, M., and Whitley, M. (2006). Experimental validation and testing of components for active damping control for micromachined mechanical vibration isolation filters using electrostatic actuation. Smart Structures and Materials 2006: Smart Electronics, MEMS, BioMEMS, and Nanotechnology, International Society for Optics and Photonics.","DOI":"10.1117\/12.658091"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"555","DOI":"10.5139\/IJASS.2017.18.3.555","article-title":"Vibration-Robust Attitude and Heading Reference System Using Windowed Measurement Error Covariance","volume":"18","author":"Kim","year":"2017","journal-title":"Int. J. Aeronaut. Space Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1049\/el.2013.0422","article-title":"De-noising MEMS inertial sensors for low-cost vehicular attitude estimation based on singular spectrum analysis and independent component analysis","volume":"49","author":"Wu","year":"2013","journal-title":"Electron. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hao, X.Y., Li, M., Han, X.F., and Jia, H.G. (2012). In Analysis on the influence of random vibration on MEMS gyro precision and error compensation. Applied Mechanics and Materials, Trans Tech Publications.","DOI":"10.4028\/www.scientific.net\/AMM.130-134.4164"},{"key":"ref_30","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_31","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1109\/72.279181","article-title":"Learning long-term dependencies with gradient descent is difficult","volume":"5","author":"Bengio","year":"1994","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Graves, A., and Schmidhuber, J. (August, January 31). In Framewise phoneme classification with bidirectional LSTM networks. Proceedings of the 2005 IEEE International Joint Conference on Neural Networks, Montreal, QC, Canada.","DOI":"10.1016\/j.neunet.2005.06.042"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1115\/1.3662552","article-title":"A new approach to linear filtering and prediction problems","volume":"82","author":"Kalman","year":"1960","journal-title":"J. Basic Eng. Mar"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2258","DOI":"10.1109\/LCOMM.2015.2495212","article-title":"Handoff decision using a Kalman filter and fuzzy logic in heterogeneous wireless networks","volume":"19","author":"Kustiawan","year":"2015","journal-title":"IEEE Commun. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Hosseinyalamdary, S. (2018). Deep Kalman filter: Simultaneous multi-sensor integration and modelling; A GNSS\/IMU case study. Sensors, 18.","DOI":"10.20944\/preprints201803.0121.v1"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Peng, C., Mou, D., Li, M., and Quan, W. (2018). An adaptive filtering approach based on the dynamic variance model for reducing MEMS gyroscope random error. Sensors, 18.","DOI":"10.3390\/s18113943"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Yan, Y., Guo, P., and Liu, L. (2014, January 3\u20136). In A novel hybridization of artificial neural networks and ARIMA models for forecasting resource consumption in an IIS web server. Proceedings of the 2014 IEEE International Symposium on Software Reliability Engineering Workshops, Naples, Italy.","DOI":"10.1109\/ISSREW.2014.27"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.measurement.2015.05.034","article-title":"Identification of error sources in high precision weight measurements of gyroscopes","volume":"73","author":"Tajmar","year":"2015","journal-title":"Measurement"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1111\/j.1467-9892.1982.tb00349.x","article-title":"An approach to time series smoothing and forecasting using the EM algorithm","volume":"3","author":"Shumway","year":"1982","journal-title":"J. Time Ser. Anal."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1214\/aos\/1176346060","article-title":"On the convergence properties of the EM algorithm","volume":"11","author":"Wu","year":"1983","journal-title":"Ann. Stat."},{"key":"ref_41","unstructured":"Andrieu, C., and Doucet, A. (2003, January 6\u201310). In Online expectation-maximization type algorithms for parameter estimation in general state space models. Proceedings of the 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, Hong Kong, China."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3763","DOI":"10.1002\/aic.13776","article-title":"Data-based linear Gaussian state-space model for dynamic process monitoring","volume":"58","author":"Wen","year":"2012","journal-title":"AIChE J."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s00521-010-0344-1","article-title":"Nonlinear maximum likelihood estimation of electricity spot prices using recurrent neural networks","volume":"20","author":"Mirikitani","year":"2011","journal-title":"Neural Comput. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"895","DOI":"10.1109\/TSP.2007.907814","article-title":"Maximum-likelihood estimation, the Cram\u00e9r\u2013Rao bound, and the method of scoring with parameter constraints","volume":"56","author":"Moore","year":"2008","journal-title":"IEEE Trans. Signal. Process."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1109\/JBHI.2020.2982935","article-title":"An Adaptive Kalman Filter Bank for ECG Denoising","volume":"25","author":"Hesar","year":"2020","journal-title":"IEEE J. Biomed. Health Inform."},{"key":"ref_46","unstructured":"Hartikainen, J., Solin, A., and S\u00e4rkk\u00e4, S. (2011). Optimal Filtering with Kalman Filters and Smoothers\u2014A Manual for MATLAB Toolbox EKF\/UKF, Aalto University School of Science."},{"key":"ref_47","unstructured":"Rousseeuw, P.J., and Leroy, A.M. (2005). Robust Regression and Outlier Detection, John Wiley & Sons."},{"key":"ref_48","first-page":"1929","article-title":"Dropout: A simple way to prevent neural networks from overfitting","volume":"15","author":"Srivastava","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"ref_49","unstructured":"Kingma, D.P., and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/4\/1181\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:21:15Z","timestamp":1760160075000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/4\/1181"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,8]]},"references-count":49,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["s21041181"],"URL":"https:\/\/doi.org\/10.3390\/s21041181","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,8]]}}}