{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T04:05:49Z","timestamp":1768968349216,"version":"3.49.0"},"reference-count":45,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2023,9,22]],"date-time":"2023-09-22T00:00:00Z","timestamp":1695340800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Technology Innovation of Shanghai Institute of Technical Physics, Chinese Academy of Sciences","award":["CX-212"],"award-info":[{"award-number":["CX-212"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The number of resident space objects (RSOs) has been steadily increasing over time, posing significant risks to the safe operation of on-orbit assets. The accurate prediction of potential collision events and implementation of effective and nonredundant avoidance maneuvers require the precise estimation of the orbit positions of objects of interest and propagation of their associated uncertainties. Previous research mainly focuses on striking a balance between accurate propagation and efficient computation. A recently proposed approach that integrates uncertainty propagation with different coordinate representations has the potential to achieve such a balance. This paper proposes combining the generalized equinoctial orbital elements (GEqOE) representation with an adaptive Gaussian mixture model (GMM) for uncertainty propagation. Specifically, we implement a reformulation for the orbital dynamics so that the underlying state and the moment feature of the GMM are propagated under the GEqOE coordinates. Starting from an initial Gaussian probability distribution function (PDF), the algorithm iteratively propagates the uncertainty distribution using a detection-splitting module. A differential entropy-based nonlinear detector and a splitting library are utilized to adjust the number of GMM components dynamically. Component splitting is triggered when a predefined threshold of differential entropy is violated, generating several GMM components. The final probability density function (PDF) is obtained by a weighted summation of the component distributions at the target time. Benefiting from the nonlinearity reduction caused by the GEqOE representation, the number of triggered events largely decreases, causing the necessary number of components to maintain uncertainty realism also to decrease, which enables the proposed approach to achieve good performance with much more efficiency. As demonstrated by the results of propagation in three scenarios with different degrees of complexity, compared with the Cartesian-based approach, the proposed approach achieves comparable accuracy to the Monte Carlo method while largely reducing the number of components generated during propagation. Our results confirm that a judicious choice of coordinate representation can significantly improve the performance of uncertainty propagation methods in terms of accuracy and computational efficiency.<\/jats:p>","DOI":"10.3390\/rs15194652","type":"journal-article","created":{"date-parts":[[2023,9,24]],"date-time":"2023-09-24T10:46:21Z","timestamp":1695552381000},"page":"4652","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Orbital Uncertainty Propagation Based on Adaptive Gaussian Mixture Model under Generalized Equinoctial Orbital Elements"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7873-9399","authenticated-orcid":false,"given":"Hui","family":"Xie","sequence":"first","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China"},{"name":"Key Laboratory of Intelligent Infrared Perception, Chinese Academy of Sciences, Shanghai 200083, China"}]},{"given":"Tianru","family":"Xue","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"given":"Wenjun","family":"Xu","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"Key Laboratory of Intelligent Infrared Perception, Chinese Academy of Sciences, Shanghai 200083, China"}]},{"given":"Gaorui","family":"Liu","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"Key Laboratory of Intelligent Infrared Perception, Chinese Academy of Sciences, Shanghai 200083, China"}]},{"given":"Haibin","family":"Sun","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"Key Laboratory of Intelligent Infrared Perception, Chinese Academy of Sciences, Shanghai 200083, China"}]},{"given":"Shengli","family":"Sun","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China"},{"name":"Key Laboratory of Intelligent Infrared Perception, Chinese Academy of Sciences, Shanghai 200083, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s42064-019-0049-x","article-title":"Spacecraft formation reconfiguration with multi-obstacle avoidance under navigation and control uncertainties using adaptive artificial potential function method","volume":"4","author":"Wang","year":"2020","journal-title":"Astrodynamics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/s42064-022-0157-x","article-title":"Analytical configuration uncertainty propagation of geocentric interferometric detection constellation","volume":"7","author":"Qiao","year":"2023","journal-title":"Astrodynamics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s10569-015-9618-3","article-title":"Propagation of large uncertainty sets in orbital dynamics by automatic domain splitting","volume":"122","author":"Wittig","year":"2015","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2163","DOI":"10.2514\/1.G001610","article-title":"Space Object Collision Probability Using Multidirectional Gaussian Mixture Models","volume":"39","author":"Vittaldev","year":"2016","journal-title":"J. Guid. Control Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1063\/1.1477058","article-title":"A Fokker-Planck model for a two-body problem","volume":"617","author":"Bierbaum","year":"2002","journal-title":"Bayesian Inference Maximum Entropy Methods Sci. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.actaastro.2013.04.027","article-title":"State propagation in an uncertain asteroid gravity field","volume":"91","author":"Melman","year":"2013","journal-title":"Acta Astronaut."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.paerosci.2016.12.002","article-title":"A review of uncertainty propagation in orbital mechanics","volume":"89","author":"Luo","year":"2017","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"497","DOI":"10.2514\/1.54385","article-title":"Analytical Nonlinear Propagation of Uncertainty in the Two-Body Problem","volume":"35","author":"Fujimoto","year":"2012","journal-title":"J. Guid. Control Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.2514\/1.G003897","article-title":"Exact Computation of High-Order State Transition Tensors for Perturbed Orbital Motion","volume":"42","author":"Younes","year":"2019","journal-title":"J. Guid. Control Dyn."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Foss\u00e0, A., Armellin, R., Delande, E., Losacco, M., and Sanfedino, F. (2022, January 3\u20137). Multifidelity Orbit Uncertainty Propagation using Taylor Polynomials. Proceedings of the AIAA SCITECH 2022 Forum, San Diego, CA, USA.","DOI":"10.2514\/6.2022-0859"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1109\/JPROC.2003.823141","article-title":"Unscented filtering and nonlinear estimation","volume":"92","author":"Julier","year":"2004","journal-title":"Proc. IEEE"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1109\/9.855552","article-title":"Gaussian filters for nonlinear filtering problems","volume":"45","author":"Ito","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1109\/TAC.2009.2019800","article-title":"Cubature Kalman Filters","volume":"54","author":"Arasaratnam","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"683","DOI":"10.2514\/1.A33686","article-title":"Orbital Uncertainty Propagation Using Positive Weighted Compact Quadrature Rule","volume":"54","author":"Jia","year":"2017","journal-title":"J. Spacecr. Rockets"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Jia, B., Xin, M., and Cheng, Y. (2012, January 10\u201313). The high-degree cubature Kalman filter. Proceedings of the 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), Maui, HI, USA.","DOI":"10.1109\/CDC.2012.6426413"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1137\/1015023","article-title":"Approximate Calculation of Multiple Integrals (A. H. Stroud)","volume":"15","author":"Stenger","year":"1973","journal-title":"SIAM Rev."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Adurthi, N., Singla, P., and Singh, T. (2012, January 27\u201329). The Conjugate Unscented Transform\u2014An approach to evaluate multi-dimensional expectation integrals. Proceedings of the 2012 American Control Conference (ACC), Montreal, QC, Canada.","DOI":"10.1109\/ACC.2012.6314970"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"430","DOI":"10.2514\/1.57599","article-title":"Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos","volume":"36","author":"Jones","year":"2013","journal-title":"J. Guid. Control Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1839","DOI":"10.2514\/1.53793","article-title":"Gaussian Sum Filters for Space Surveillance: Theory and Simulations","volume":"34","author":"Horwood","year":"2011","journal-title":"J. Guid. Control Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1109\/TAC.1972.1100034","article-title":"Nonlinear Bayesian estimation using Gaussian sum approximations","volume":"17","author":"Alspach","year":"1972","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Giza, D., Singla, P., and Jah, M. (2009, January 10\u201313). An Approach for Nonlinear Uncertainty Propagation: Application to Orbital Mechanics. Proceedings of the AIAA Guidance, Navigation, and Control Conference, Chicago, IL, USA.","DOI":"10.2514\/6.2009-6082"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.2514\/1.G000472","article-title":"Nonlinear Uncertainty Propagation for Perturbed Two-Body Orbits","volume":"37","author":"Vishwajeet","year":"2014","journal-title":"J. Guid. Control Dyn."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"888","DOI":"10.2514\/1.G003071","article-title":"Nonlinear Analytical Uncertainty Propagation for Relative Motion near J2-Perturbed Elliptic Orbits","volume":"41","author":"Yang","year":"2018","journal-title":"J. Guid. Control Dyn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s42064-018-0036-7","article-title":"Nonlinear semi-analytical uncertainty propagation of trajectory under impulsive maneuvers","volume":"3","author":"Yang","year":"2019","journal-title":"Astrodynamics"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"34511","DOI":"10.1007\/s11433-018-9267-6","article-title":"Nonlinear orbital uncertainty propagation with differential algebra and Gaussian mixture model","volume":"62","author":"Sun","year":"2018","journal-title":"Sci. China Phys. Mech. Astron."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2615","DOI":"10.2514\/1.G001571","article-title":"Spacecraft Uncertainty Propagation Using Gaussian Mixture Models and Polynomial Chaos Expansions","volume":"39","author":"Vittaldev","year":"2016","journal-title":"J. Guid. Control Dyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4179","DOI":"10.1016\/j.asr.2022.03.041","article-title":"Kernel-based ensemble gaussian mixture filtering for orbit determination with sparse data","volume":"69","author":"Yun","year":"2022","journal-title":"Adv. Space Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1016\/j.actaastro.2018.10.023","article-title":"Multi-fidelity orbit uncertainty propagation","volume":"155","author":"Jones","year":"2019","journal-title":"Acta Astronaut."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Xu, T., Zhang, Z., and Han, H. (2023). Adaptive Gaussian Mixture Model for Uncertainty Propagation Using Virtual Sample Generation. Appl. Sci., 13.","DOI":"10.3390\/app13053069"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"D\u2019Ortenzio, A., and Manes, C. (2021, January 1\u20134). Composite Transportation Dissimilarity in Consistent Gaussian Mixture Reduction. Proceedings of the 2021 IEEE 24th International Conference on Information Fusion (FUSION), Sun City, South Africa.","DOI":"10.23919\/FUSION49465.2021.9627011"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1109\/MAES.2004.1263228","article-title":"Multiple hypothesis tracking for multiple target tracking","volume":"19","author":"Blackman","year":"2004","journal-title":"IEEE Aerosp. Electron. Syst. Mag."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"603","DOI":"10.2514\/1.G002801","article-title":"Adaptive Split\/Merge-Based Gaussian Mixture Model Approach for Uncertainty Propagation","volume":"41","author":"Vishwajeet","year":"2018","journal-title":"J. Guid. Control Dyn."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF01228432","article-title":"On the equinoctial orbit elements","volume":"5","author":"Broucke","year":"1972","journal-title":"Celest. Mech."},{"key":"ref_34","first-page":"90920F","article-title":"Beyond covariance realism: A new metric for uncertainty realism","volume":"Volume 9092","author":"Drummond","year":"2014","journal-title":"Proceedings of the Signal and Data Processing of Small Targets 2014"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/s10569-021-10004-0","article-title":"On a set of J2 equinoctial orbital elements and their use for uncertainty propagation","volume":"133","author":"Aristoff","year":"2021","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1007\/s10569-021-10049-1","article-title":"A generalization of the equinoctial orbital elements","volume":"133","author":"Bombardelli","year":"2021","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"654","DOI":"10.2514\/1.G006864","article-title":"Near-Linear Orbit Uncertainty Propagation Using the Generalized Equinoctial Orbital Elements","volume":"46","author":"Bombardelli","year":"2023","journal-title":"J. Guid. Control Dyn."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1109\/TAC.2011.2142610","article-title":"Adaptive Gaussian Sum Filters for Space Surveillance","volume":"56","author":"Horwood","year":"2011","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"73","DOI":"10.2514\/1.G005891","article-title":"Gaussian Mixture Filter for Angles-Only Orbit Determination in Modified Equinoctial Elements","volume":"45","author":"Psiaki","year":"2022","journal-title":"J. Guid. Control Dyn."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.2514\/1.58987","article-title":"Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems","volume":"36","author":"DeMars","year":"2013","journal-title":"J. Guid. Control Dyn."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"770","DOI":"10.2514\/1.G006696","article-title":"Hybrid Gaussian Mixture Splitting Techniques for Uncertainty Propagation in Nonlinear Dynamics","volume":"46","author":"Sun","year":"2023","journal-title":"J. Guid. Control Dyn."},{"key":"ref_42","first-page":"83","article-title":"Multidirectional Gaussian Mixture Models for Nonlinear Uncertainty Propagation","volume":"111","author":"Vittaldev","year":"2016","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_43","unstructured":"Shampine, L.F., and Gordon, M.K. (1975). Computer Solution of Ordinary Differential Equations: The Initial Value Problem, Freeman."},{"key":"ref_44","first-page":"G42A-03","article-title":"The GGM03 mean Earth gravity model from GRACE","volume":"2007","author":"Tapley","year":"2007","journal-title":"AGU Fall Meet. Abstr."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"105","DOI":"10.3847\/1538-3881\/abd414","article-title":"The JPL Planetary and Lunar Ephemerides DE440 and DE441","volume":"161","author":"Park","year":"2021","journal-title":"Astron. J."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/19\/4652\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:55:57Z","timestamp":1760129757000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/19\/4652"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,22]]},"references-count":45,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["rs15194652"],"URL":"https:\/\/doi.org\/10.3390\/rs15194652","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,22]]}}}