{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T20:31:04Z","timestamp":1778877064844,"version":"3.51.4"},"reference-count":38,"publisher":"MDPI AG","issue":"20","license":[{"start":{"date-parts":[[2022,10,19]],"date-time":"2022-10-19T00:00:00Z","timestamp":1666137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001381","name":"National Research Foundation (NRF) Singapore, Office for Space Technology &amp; Industry (OSTin)","doi-asserted-by":"publisher","award":["S21-19009-STDP"],"award-info":[{"award-number":["S21-19009-STDP"]}],"id":[{"id":"10.13039\/501100001381","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>For a small satellite, the processor onboard the attitude determination and control system (ADCS) is required to monitor, communicate, and control all the sensors and actuators. In addition, the processor is required to consistently communicate with the satellite bus. Consequently, the processor is unable to ensure all the sensors and actuators will immediately respond to the data acquisition request, which leads to asynchronous data problems. The extended Kalman filter (EKF) is commonly used in the attitude determination process, but it assumes fully synchronous data. The asynchronous data problem would greatly degrade the attitude determination accuracy by EKF. To minimize the attitude estimation accuracy loss due to asynchronous data while ensuring a reasonable computational complexity for small satellite applications, this paper proposes the simplex-back-propagation Kalman filter (SBPKF). The proposed SBPKF incorporates the time delay, gyro instability, and navigation error into both the measurement and covariance estimation during the Kalman update process. The performance of SBPKF has been compared with EKF, modified adaptive EKF (MAEKF), and moving\u2013covariance Kalman filter (MC-KF). Simulation results show that the attitude estimation error of SBPKF is at least 30% better than EKF and MC-KF. In addition, the SBPKF\u2019s computational complexity is 17% lower than MAEKF and 29% lower than MC-KF.<\/jats:p>","DOI":"10.3390\/s22207970","type":"journal-article","created":{"date-parts":[[2022,10,19]],"date-time":"2022-10-19T22:19:53Z","timestamp":1666217993000},"page":"7970","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Simplex Back Propagation Estimation Method for Out-of-Sequence Attitude Sensor Measurements"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7073-5462","authenticated-orcid":false,"given":"Shu Ting","family":"Goh","sequence":"first","affiliation":[{"name":"Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117583, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6269-0313","authenticated-orcid":false,"given":"M. S. C.","family":"Tissera","sequence":"additional","affiliation":[{"name":"Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117583, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"RongDe Darius","family":"Tan","sequence":"additional","affiliation":[{"name":"Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117583, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ankit","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117583, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6017-6943","authenticated-orcid":false,"given":"Kay-Soon","family":"Low","sequence":"additional","affiliation":[{"name":"Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117583, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lip San","family":"Lim","sequence":"additional","affiliation":[{"name":"Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117583, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,19]]},"reference":[{"key":"ref_1","unstructured":"BryceTech (2022). Smallsats by the Numbers 2022, BryceTech. Available online: https:\/\/www.brycetech.com\/reports\/report-documents\/Bryce_Smallsats_2022.pdf."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Tissera, M.S.C., Low, K.S., and Goh, S.T. (2021, January 6\u201313). On-orbit Gyroscope Bias Compensation to Improve Satellite Attitude Control Performance. Proceedings of the IEEE Aerospace Conference, Big Sky, MT, USA.","DOI":"10.1109\/AERO50100.2021.9438500"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1038","DOI":"10.1016\/j.asr.2018.10.003","article-title":"Nonlinear filtering for sequential spacecraft attitude estimation with real data: Cubature Kalman Filter, Unscented Kalman Filter and Extended Kalman Filter","volume":"63","author":"Garcia","year":"2019","journal-title":"Adv. Space Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.2514\/1.G003221","article-title":"Fully Multiplicative Unscented Kalman Filter for Attitude Estimation","volume":"41","author":"Zanetti","year":"2018","journal-title":"J.Guid.Control Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"536","DOI":"10.2514\/2.5102","article-title":"Unscented Filtering for Spacecraft Attitude Estimation","volume":"26","author":"Crassdis","year":"2003","journal-title":"J.Guid.Control Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"806","DOI":"10.1016\/j.cja.2018.01.023","article-title":"Adaptive robust cubature Kalman filtering for satellite attitude estimation","volume":"31","author":"QIU","year":"2018","journal-title":"Chin. J. Aeronaut."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.2514\/1.47236","article-title":"Particle Filtering for Attitude Estimation Using a Minimal Local-Error Representation","volume":"33","author":"Cheng","year":"2010","journal-title":"J.Guid.Control Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1458","DOI":"10.2514\/1.43119","article-title":"Norm-Constrained Kalman Filtering","volume":"32","author":"Zanetti","year":"2009","journal-title":"J.Guid.Control Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2546","DOI":"10.1016\/j.automatica.2014.08.007","article-title":"Continuous-time norm-constrained Kalman filtering","volume":"50","author":"Forbes","year":"2014","journal-title":"Automatica"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1109\/TAES.2014.130204","article-title":"Gain-scheduled extended kalman filter for nanosatellite attitude determination system","volume":"51","author":"Pham","year":"2015","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.actaastro.2021.11.008","article-title":"Attitude estimation and control based on modified unscented Kalman filter for gyro-less satellite with faulty sensors","volume":"191","author":"Pourtakdoust","year":"2022","journal-title":"Acta Astronaut."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1088\/1361-6501\/ac6c75","article-title":"Discrete-time complementary filter for attitude estimation based on MARG sensor","volume":"33","author":"Li","year":"2022","journal-title":"Meas. Sci. Technol."},{"key":"ref_13","first-page":"769","article-title":"Update with out-of-sequence measurements in tracking: Exact solution","volume":"8","year":"2002","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_14","unstructured":"Mallick, M., Krant, J., and Bar-Shalom, Y. (2002, January 8\u201311). Multi-sensor multi-target tracking using out-of-sequence measurements. Proceedings of the Fifth International Conference on Information Fusion, Annapolis, MD, USA."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1518","DOI":"10.2514\/1.47984","article-title":"Estimation with multitemporal measurements","volume":"33","author":"Fosbury","year":"2010","journal-title":"J.Guid.Control Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"664","DOI":"10.1007\/s11432-010-4151-1","article-title":"Single-step-lag OOSM algorithm based on unscented transformation","volume":"54","author":"Chen","year":"2011","journal-title":"Sci. China Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Mallick, M., and Marrs, A. (2003, January 8\u201311). Comparison of the KF and particle filter based out-of-sequence measurement filtering algorithms. Proceedings of the Sixth International Conference of Information Fusion, QLD, Australia.","DOI":"10.1109\/ICIF.2003.177477"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.ymssp.2019.106421","article-title":"A method for the reduction of the computational cost associated with the implementation of particle-filter-based failure prognostic algorithms","volume":"135","author":"Rozas","year":"2020","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Tang, C., and Dou, L. (2020). An Improved Game Theory-Based Cooperative Localization Algorithm for Eliminating the Conflicting Information of Multi-Sensors. Sensors, 20.","DOI":"10.3390\/s20195579"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.ast.2012.11.012","article-title":"A Weighted Measurement Fusion Kalman Filter implementation for UAV navigation","volume":"28","author":"Goh","year":"2013","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Khosravian, A., Trumpf, J., Mahony, R., and Hamel, T. (2015, January 1\u20133). Recursive attitude estimation in the presence of multi-rate and multi-delay vector measurements. Proceedings of the American Control Conference (ACC), Chicago, IL, USA.","DOI":"10.1109\/ACC.2015.7171825"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.actaastro.2021.04.033","article-title":"Nano satellite attitude determination with randomly delayed measurements","volume":"185","author":"Fei","year":"2021","journal-title":"Acta Astronaut."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Pornsarayouth, S., and Wongsaisuwan, M. (2009, January 22\u201325). Sensor fusion of delay and non-delay signal using Kalman Filter with moving covariance. Proceedings of the IEEE International Conference on Robotics and Biomimetics, Bangkok, Thailand.","DOI":"10.1109\/ROBIO.2009.4913316"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1109\/TIM.2015.2450358","article-title":"Statistical Modeling of Random Walk Errors for Triaxial Rate Gyros","volume":"65","author":"Yuan","year":"2016","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Crassidis, J.L., and Junkins, J.L. (2004). Optimal Estimation of Dynamic Systems, Chapman & Hall\/CRC.","DOI":"10.1201\/9780203509128"},{"key":"ref_26","unstructured":"Zekavat, S.A.R., and Buehrer, R.M. (2019). An Introduction to Kalman Filtering Implementation for Localization and Tracking Applications. Handbook of Position Location: Theory, Practice, and Advances, Wiley-IEEE Press."},{"key":"ref_27","unstructured":"Riggins, M., and Humphrey, J. (2020). Sun. The Astronomical Almanac For The Year 2020, United States Naval Observatory (USNO)Nautical Almanac Office."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1007\/s40314-017-0502-5","article-title":"Kalman filter for attitude determination of a CubeSat using low-cost sensors","volume":"37","author":"Baroni","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.2514\/1.42849","article-title":"Deterministic Relative Attitude Determination of Three-Vehicle Formation","volume":"32","author":"Andrle","year":"2009","journal-title":"J.Guid.Control Dyn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2386","DOI":"10.2514\/1.G002718","article-title":"Simultaneous estimation of attitude and Markov-modeled rate corrections of gyroless spacecraft","volume":"40","author":"Challa","year":"2017","journal-title":"J.Guid.Control Dyn."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Casey, R.T., Karpenko, M., Curry, R., and Elkaim, G. (2013, January 19\u201322). Attitude Representation and Kinematic Propagation for Low-Cost UAVs. Proceedings of the AIAA Guidance, Navigation, and Control (GNC) Conference, Boston, MA, USA.","DOI":"10.2514\/6.2013-4615"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/S1566-2535(02)00070-2","article-title":"Some remarks on Kalman filters for the multisensor fusion","volume":"3","author":"Gao","year":"2002","journal-title":"Inf. Fusion"},{"key":"ref_33","unstructured":"Vallado, D.A. (2007). Coordinate and Time Systems. Fundamentals of Astrodynamics and Applications, Microcosm Press. [3rd ed.]."},{"key":"ref_34","unstructured":"R\u00f6nnb\u00e4ck, S. (2000). Development of a INS\/GPS navigation loop for an UAV. [Master\u2019s Thesis, Department of Computer Science and Electrical Engineering, Lule\u00e5 University of Technology]."},{"key":"ref_35","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 Dyn."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4257","DOI":"10.1007\/s12206-022-0743-0","article-title":"Identification method of nonlinear maneuver model for unmanned surface vehicle from sea trial data based on support vector machine","volume":"36","author":"Wu","year":"2022","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"387","DOI":"10.2514\/1.54864","article-title":"Constraint Estimation of Spacecraft Positions","volume":"35","author":"Goh","year":"2012","journal-title":"J.Guid.Control. Dyn."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Wertz, J.R. (1978). Magnetic Field Models. Spacecraft Attitude Determination and Control, Springer.","DOI":"10.1007\/978-94-009-9907-7"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/20\/7970\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:57:18Z","timestamp":1760144238000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/20\/7970"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,19]]},"references-count":38,"journal-issue":{"issue":"20","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["s22207970"],"URL":"https:\/\/doi.org\/10.3390\/s22207970","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,19]]}}}