{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T09:54:12Z","timestamp":1766138052600,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2022,7,15]],"date-time":"2022-07-15T00:00:00Z","timestamp":1657843200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100006222","name":"United States India Educational Foundation","doi-asserted-by":"publisher","award":["2482\/fnpdr\/2019"],"award-info":[{"award-number":["2482\/fnpdr\/2019"]}],"id":[{"id":"10.13039\/100006222","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Fixed-lag smoothing has been used across different disciplines for offline analysis in many applications. With rising computational power and parallel processing architectures, fixed-lag smoothers are increasingly integrated into online processing system with small delays. This delay is directly related to the lag-length used in system design, which needs to be chosen appropriately. In this work, an adaptive approach is devised to choose an appropriate lag-length that provides a good trade-off between accuracy and computational requirements. The analysis shown in this paper for the error dynamics of the fixed-lag smoother over the lags helps in understanding its saturation over increasing lags. In order to provide the empirical results, simulations are carried out over a second-order Newtonian system, single-axis attitude estimation, Van der Pol\u2019s oscillator, and three-axis attitude estimation. The simulation results demonstrate the performance achieved with an adaptive-lag smoother as compared to a fixed-lag smoother with very high lag-length.<\/jats:p>","DOI":"10.3390\/s22145310","type":"journal-article","created":{"date-parts":[[2022,7,18]],"date-time":"2022-07-18T01:53:22Z","timestamp":1658109202000},"page":"5310","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Adaptive Lag Smoother for State Estimation"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9661-8224","authenticated-orcid":false,"given":"Shashi","family":"Poddar","sequence":"first","affiliation":[{"name":"Department of Intelligent Machines & Communication Systems, CSIR\u2014Central Scientific Instruments Organisation, Chandigarh 160030, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8092-569X","authenticated-orcid":false,"given":"John L.","family":"Crassidis","sequence":"additional","affiliation":[{"name":"Department of Aerospace and Mechanical Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14260-4400, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mungu\u00eda, R., Urzua, S., and Grau, A. (2019). EKF-based parameter identification of multi-rotor unmanned aerial vehiclesmodels. Sensors, 19.","DOI":"10.3390\/s19194174"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Zhang, K.W., Hao, G., and Sun, S.L. (2018). Weighted Measurement Fusion Particle Filter for Nonlinear Systems with Correlated Noises. Sensors, 18.","DOI":"10.3390\/s18103242"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2082","DOI":"10.1109\/TSMC.2017.2778269","article-title":"Robust Kalman filters based on Gaussian scale mixture distributions with application to target tracking","volume":"49","author":"Huang","year":"2017","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Goodman, J.M., Wilkerson, S.A., Eggleton, C., and Gadsden, S.A. (2018, January 16\u201319). A multiple model adaptive SVSF-KF estimation strategy. Proceedings of the Signal Processing, Sensor\/Information Fusion, and Target Recognition XXVII, Orlando, FL, USA.","DOI":"10.1117\/12.2520018"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1109\/JPROC.2007.893255","article-title":"The smooth variable structure filter","volume":"95","author":"Habibi","year":"2007","journal-title":"Proc. IEEE"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Crassidis, J.L., and Junkins, J.L. (2012). Optimal Estimation of Dynamic Systems, CRC Press.","DOI":"10.1201\/b11154"},{"key":"ref_7","first-page":"16","article-title":"Fixed-lag smoothing results for linear dynamical systems","volume":"7","author":"Moore","year":"1973","journal-title":"Aust. Telecommun. Res."},{"key":"ref_8","unstructured":"Meditch, J.S. (1969). Stochastic Optimal Linear Estimation and Control, McGraw-Hill."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1109\/TIT.1974.1055170","article-title":"Stable fixed-lag smoothing of continuous time processes","volume":"20","author":"Chirarattananon","year":"1974","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1016\/j.measurement.2012.09.021","article-title":"A computationally efficient fixed-lag smoother using recent finite measurements","volume":"46","author":"Kim","year":"2013","journal-title":"Measurement"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chen, Y., Xu, L., Yan, B., and Li, C. (2020). A novel smooth variable structure smoother for robust estimation. Sensors, 20.","DOI":"10.3390\/s20061781"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5571","DOI":"10.1109\/TSP.2019.2941066","article-title":"Particle-Based Adaptive-Lag Online Marginal Smoothing in General State-Space Models","volume":"67","author":"Olsson","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1313","DOI":"10.1080\/00207728508926754","article-title":"Fixed-lag smoothing in the identification of time-varying systems with unknown dead time","volume":"16","author":"Fkirin","year":"1985","journal-title":"Int. J. Syst. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Leanza, A., Reina, G., and Blanco-Claraco, J.L. (2021). A Factor-Graph-Based Approach to Vehicle Sideslip Angle Estimation. Sensors, 21.","DOI":"10.3390\/s21165409"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1134\/S2075108711030023","article-title":"Suboptimal smoothing filter for the marine gravimeter GT-2M","volume":"2","author":"Bolotin","year":"2011","journal-title":"Gyroscopy Navig."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1134\/S2075108710010049","article-title":"Analysis of filtering and smoothing techniques as applied to aerogravimetry","volume":"1","author":"Stepanov","year":"2010","journal-title":"Gyroscopy Navig."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1109\/36.175324","article-title":"Application of Kalman filtering to real-time preprocessing of geophysical data","volume":"30","author":"Noriega","year":"1992","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1716","DOI":"10.1109\/JIOT.2020.3015351","article-title":"Tightly Coupled Integration of INS and UWB Using Fixed-Lag Extended UFIR Smoothing for Quadrotor Localization","volume":"8","author":"Xu","year":"2021","journal-title":"IEEE Internet Things J."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Hsiung, J., Hsiao, M., Westman, E., Valencia, R., and Kaess, M. (2018, January 1\u20135). Information Sparsification in Visual-Inertial Odometry. Proceedings of the 2018 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, Spain.","DOI":"10.1109\/IROS.2018.8594007"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Fetzer, T., Ebner, F., Deinzer, F., K\u00f6ping, L., and Grzegorzek, M. (2016, January 4\u20137). On Monte Carlo smoothing in multi sensor indoor localisation. Proceedings of the 2016 International Conference on Indoor Positioning and Indoor Navigation (IPIN), Madrid, Spain.","DOI":"10.1109\/IPIN.2016.7743670"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zhuang, Y., Wang, Q., Li, Y., Gao, Z., Zhou, B., Qi, L., Yang, J., Chen, R., and El-Sheimy, N. (2019). The Integration of Photodiode and Camera for Visible Light Positioning by Using Fixed-Lag Ensemble Kalman Smoother. Remote Sens., 11.","DOI":"10.3390\/rs11111387"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kim, P.S. (2019). Finite Memory Structure Filtering and Smoothing for Target Tracking in Wireless Network Environments. Appl. Sci., 9.","DOI":"10.3390\/app9142872"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Ullah, I., Qureshi, M.B., Khan, U., Memon, S.A., Shi, Y., and Peng, D. (2018). Multisensor-based target-tracking algorithm with out-of-sequence-measurements in cluttered environments. Sensors, 18.","DOI":"10.3390\/s18114043"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s11633-006-0425-x","article-title":"A new smoothing approach with diverse fixed-lags based on target motion model","volume":"3","author":"Li","year":"2006","journal-title":"Int. J. Autom. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Duong, T.T., Chiang, K.W., and Le, D.T. (2019). On-line smoothing and error modelling for integration of GNSS and visual odometry. Sensors, 19.","DOI":"10.3390\/s19235259"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Hartikainen, J., and S\u00e4rkk\u00e4, S. (September, January 29). Kalman filtering and smoothing solutions to temporal Gaussian process regression models. Proceedings of the 2010 IEEE International Workshop on Machine Learning for Signal Processing, Kittila, Finland.","DOI":"10.1109\/MLSP.2010.5589113"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.inffus.2021.03.002","article-title":"A joint introduction to Gaussian Processes and Relevance Vector Machines with connections to Kalman filtering and other kernel smoothers","volume":"74","author":"Martino","year":"2021","journal-title":"Inf. Fusion"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Simon, D. (2006). Optimal State Estimation: Kalman, H-Infinity, and Nonlinear Approaches, John Wiley & Sons.","DOI":"10.1002\/0470045345"},{"key":"ref_29","first-page":"240295","article-title":"Online Stochastic Convergence Analysis of the Kalman Filter","volume":"2013","author":"Rhudy","year":"2013","journal-title":"Int. J. Stoch. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1109\/9.754809","article-title":"Stochastic stability of the discrete-time extended Kalman filter","volume":"44","author":"Reif","year":"1999","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_31","first-page":"439","article-title":"A Survey of Attitude Representations","volume":"41","author":"Shuster","year":"1993","journal-title":"J. Astronaut. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"417","DOI":"10.2514\/3.56190","article-title":"Kalman Filtering for Spacecraft Attitude Estimation","volume":"5","author":"Lefferts","year":"1982","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1614","DOI":"10.2514\/1.G001025","article-title":"Attitude estimation employing common frame error representations","volume":"38","author":"Andrle","year":"2015","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Markley, F.L., and Crassidis, J.L. (2014). Fundamentals of Spacecraft Attitude Dynamics and Control, Springer.","DOI":"10.1007\/978-1-4939-0802-8"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/14\/5310\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:51:35Z","timestamp":1760140295000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/14\/5310"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,15]]},"references-count":34,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["s22145310"],"URL":"https:\/\/doi.org\/10.3390\/s22145310","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2022,7,15]]}}}