{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,3]],"date-time":"2025-12-03T17:57:38Z","timestamp":1764784658490,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T00:00:00Z","timestamp":1621987200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the U.S. Air Force Office of Scientific Research, Air Force Material Command","award":["No. FA9550-18-C-0201"],"award-info":[{"award-number":["No. FA9550-18-C-0201"]}]},{"name":"CONICYT\/PIA Project","award":["AFB180004"],"award-info":[{"award-number":["AFB180004"]}]},{"name":"CONICYT FONDECYT Postdoctorado 2018","award":["number 3180319"],"award-info":[{"award-number":["number 3180319"]}]},{"name":"CONICYT FONDECYT project","award":["1190979"],"award-info":[{"award-number":["1190979"]}]},{"name":"Becas CONICYT Doctorado Nacional","award":["2012"],"award-info":[{"award-number":["2012"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper presents a Bayesian filter based solution to the Space Object (SO) tracking problem using simulated optical telescopic observations. The presented solution utilizes the Probabilistic Admissible Region (PAR) approach, which is an orbital admissible region that adheres to the assumption of independence between newborn targets and surviving SOs. These SOs obey physical energy constraints in terms of orbital semi-major axis length and eccentricity within a range of orbits of interest. In this article, Low Earth Orbit (LEO) SOs are considered. The solution also adopts the Partially Uniform Birth (PUB) intensity, which generates uniformly distributed births in the sensor field of view. The measurement update then generates a particle SO distribution. In this work, a Poisson Labeled Multi-Bernoulli (PLMB) multi-target tracking filter is proposed, using the PUB intensity model for the multi-target birth density, and a PAR for the spatial density to determine the initial orbits of SOs. Experiments are demonstrated using simulated SO trajectories created from real Two-Line Element data, with simulated measurements from twelve telescopes located in observatories, which form part of the Falcon telescope network. Optimal Sub-Pattern Assignment (OSPA) and CLEAR MOT metrics demonstrate encouraging multi-SO tracking results even under very low numbers of observations per SO pass.<\/jats:p>","DOI":"10.3390\/s21113684","type":"journal-article","created":{"date-parts":[[2021,5,25]],"date-time":"2021-05-25T22:02:23Z","timestamp":1621980143000},"page":"3684","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Space Debris Tracking with the Poisson Labeled Multi-Bernoulli Filter"],"prefix":"10.3390","volume":"21","author":[{"given":"Leonardo","family":"Cament","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Universidad de Chile, Av. Tupper 2007, Santiago 8370451, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1085-0506","authenticated-orcid":false,"given":"Martin","family":"Adams","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Universidad de Chile, Av. Tupper 2007, Santiago 8370451, Chile"}]},{"given":"Pablo","family":"Barrios","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Universidad de Chile, Av. Tupper 2007, Santiago 8370451, Chile"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1324","DOI":"10.2514\/1.59844","article-title":"Probabilistic initial orbit determination using gaussian mixture models","volume":"36","author":"DeMars","year":"2013","journal-title":"J. Guid. Control Dyn."},{"key":"ref_2","unstructured":"Delande, E.D., Frueh, C., Houssineau, J., and Clark, D.E. (2015, January 11\u201315). Multi-object filtering for space situational awareness. Proceedings of the AAS Space Flight Mechanics Meeting, Williamsburg, VA, USA."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Milani, A., Gronchi, G.F., Vitturi, M.D.M., and Kne\u017eevi\u0107, Z. (2004). Orbit determination with very short arcs. I admissible regions. Celestial Mechanics and Dynamical Astronomy, Universita di Pisa.","DOI":"10.1007\/s10569-004-6593-5"},{"key":"ref_4","unstructured":"Hussein, I., Roscoe, C., Wilkins, M., and Schumacher, P. (2014, January 9\u201312). Probabilistic Admissible Region for Short-Arc Angles-Only Observations. Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, HI, USA."},{"key":"ref_5","unstructured":"Bar-Shalom, Y., and Li, X.R. (1995). Multitarget-Multisensor Tracking: Principles and Techniques, YBs."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","unstructured":"Mahler, R.P.S. (2007). Statistical Multisource-multitarget Information Fusion, Artech House Publishers.","DOI":"10.1201\/9781420053098.ch16"},{"key":"ref_8","unstructured":"Mahler, R. (2014). Advances in Statistical Multisource-Multitarget Information Fusion, Artech House."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Jones, B.A., Vo, B.T., and Vo, B.N. (2016). Generalized Labeled Multi-Bernoulli Space-Object Tracking with Joint Prediction and Update. AIAA\/AAS Astrodynamics Specialist Conference, American Institute of Aeronautics and Astronautics.","DOI":"10.2514\/6.2016-5502"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1456","DOI":"10.1109\/TAES.2018.2793378","article-title":"CPHD filter birth modeling using the probabilistic admissible region","volume":"54","author":"Jones","year":"2018","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1016\/j.asr.2019.04.012","article-title":"A new multi-target tracking algorithm for a large number of orbiting objects","volume":"64","author":"Delande","year":"2019","journal-title":"Adv. Space Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"107714","DOI":"10.1016\/j.sigpro.2020.107714","article-title":"The Histogram Poisson, Labeled Multi-Bernoulli Multi-Target Tracking Filter","volume":"176","author":"Cament","year":"2020","journal-title":"Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1109\/TAES.2018.2805153","article-title":"Poisson multi-Bernoulli mixture filter: Direct derivation and implementation","volume":"54","author":"Williams","year":"2018","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Vo, B.N., and Vo, B.T. (2017, January 10\u201313). An implementation of the multi-sensor generalized labeled multi-Bernoulli filter via Gibbs sampling. Proceedings of the 2017 20th International Conference on Information Fusion (Fusion), Xi\u2019an, China.","DOI":"10.23919\/ICIF.2017.8009647"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Mahler, R. (2010, January 26\u201329). Linear-complexity CPHD filters. Proceedings of the 2010 13th International Conference on Information Fusion, Edinburgh, UK.","DOI":"10.1109\/ICIF.2010.5711919"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1664","DOI":"10.1109\/TAES.2015.130550","article-title":"Marginal multi-bernoulli filters\u2014RFS derivation of MHT, JIPDA, and association-based member","volume":"51","author":"Williams","year":"2015","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1287\/opre.16.3.682","article-title":"An algorithm for ranking all the assignments in order of increasing costs","volume":"16","author":"Murthy","year":"1968","journal-title":"Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"095003","DOI":"10.1088\/1538-3873\/aad03f","article-title":"A new global array of optical telescopes: The Falcon telescope network","volume":"130","author":"Chun","year":"2018","journal-title":"Publ. Astron. Soc. Pac."},{"key":"ref_19","unstructured":"Mahooti, M., and Sharifi, M. (2009, January 17). Variable and Fixed Step Integrations for Dynamic Orbit Determination of LEO Satellites. Proceedings of the 9th Conference of Iranian Aerospace Society, Tehran, Iran."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"119","DOI":"10.2514\/1.G002231","article-title":"Robust joint target detection and tracking for space situational awareness","volume":"41","author":"Pak","year":"2018","journal-title":"J. Guid. Control Dyn."},{"key":"ref_21","unstructured":"Hussein, I.I., Roscoe, C.W., Wilkins, M.P., and Schumacher, P.W. (2014, January 5\u20139). Probabilistic admissibility in angles-only initial orbit determination. Proceedings of the 24th International Symposium on Space Flight Dynamics, Maui, HI, USA."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Franco, J., Delande, E.D., Frueh, C., Houssineau, J., and Clark, D.E. (2016, January 5\u201312). A spherical co-ordinate space parameterisation for orbit estimation. Proceedings of the 2016 IEEE Aerospace Conference, Big Sky, MT, USA.","DOI":"10.1109\/AERO.2016.7500573"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Mahler, R. (2017, January 11\u201312). On CPHD filters with track labeling. Proceedings of the Signal Processing, Sensor\/Information Fusion, and Target Recognition XXVI. International Society for Optics and Photonics, Anaheim, CA, USA.","DOI":"10.1117\/12.2263508"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Reuter, S., Danzer, A., Stuebler, M., Scheel, A., and Granstr\u00f6m, K. (2017). A fast implementation of the Labeled Multi-Bernoulli filter using gibbs sampling. Intell. Veh. Symp., 765\u2013772.","DOI":"10.1109\/IVS.2017.7995809"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10569-015-9647-y","article-title":"DROMO propagator revisited","volume":"124","author":"Urrutxua","year":"2016","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.actaastro.2018.02.008","article-title":"SPOOK-A comprehensive Space Surveillance and Tracking analysis tool","volume":"158","author":"Fernandez","year":"2019","journal-title":"Acta Astronaut."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.asr.2019.10.013","article-title":"Propagation of grid-scale density model uncertainty to orbital uncertainties","volume":"65","author":"Schiemenz","year":"2020","journal-title":"Adv. Space Res."},{"key":"ref_28","unstructured":"Dundar, I.U. (2018). Improvement of a Space Surveillance Tracking Analysis Tool. [Master\u2019s Thesis, Lulea University of Technology]."},{"key":"ref_29","unstructured":"M\u00f6ckel, M., Bennett, J., Stoll, E., and Zhang, K. (2016, January 20\u201323). High performance orbital propagation using a generic software architecture. Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, HI, USA."},{"key":"ref_30","unstructured":"Berry, M.M. (2004). A Variable-Step Double-Integration Multi-Step Integrator. [Ph.D. Thesis, Virginia Tech]."},{"key":"ref_31","unstructured":"Mallick, M., Rubin, S., and Vo, B.N. (2013, January 9\u201312). An introduction to force and measurement modeling for space object tracking. Proceedings of the 16th International Conference on Information Fusion, Istanbul, Turkey."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1853","DOI":"10.1016\/S0273-1177(03)00171-6","article-title":"Orbit determination in satellite geodesy","volume":"31","author":"Beutler","year":"2003","journal-title":"Adv. Space Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1007\/s40328-016-0159-3","article-title":"Assessment of numerical integration methods in the context of low Earth orbits and inter-satellite observation analysis","volume":"51","author":"Papanikolaou","year":"2016","journal-title":"Acta Geod. Geophys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1800","DOI":"10.1016\/j.asr.2018.06.033","article-title":"Physics and human-based information fusion for improved resident space object tracking","volume":"62","author":"Delande","year":"2018","journal-title":"Adv. Space Res."},{"key":"ref_35","unstructured":"McCabe, J.S., DeMars, K.J., and Frueh, C. (2015, January 11\u201315). Integrated detection and tracking for multiple space objects. Proceedings of the AAS Space Flight Mechanics Meeting, Williamsburg, VA, USA."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2402","DOI":"10.1109\/TPAMI.2016.2522425","article-title":"EM algorithms for weighted-data clustering with application to audio-visual scene analysis","volume":"38","author":"Gebru","year":"2016","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_37","unstructured":"Sola, J., Deray, J., and Atchuthan, D. (2018). A micro Lie theory for state estimation in robotics. arXiv."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Calinon, S., and Jaquier, N. (2019). Gaussians on Riemannian manifolds for robot learning and adaptive control. arXiv.","DOI":"10.1109\/MRA.2020.2980548"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3447","DOI":"10.1109\/TSP.2008.920469","article-title":"A consistent metric for performance evaluation of multi-object filters","volume":"56","author":"Schuhmacher","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Beard, M., Vo, B.T., and Vo, B.N. (November, January 31). OSPA (2): Using the OSPA metric to evaluate multi-target tracking performance. Proceedings of the 2017 International Conference on Control, Automation and Information Sciences (ICCAIS), Chiang Mai, Thailand.","DOI":"10.1109\/ICCAIS.2017.8217598"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2008\/246309","article-title":"Evaluating multiple object tracking performance: The CLEAR MOT metrics","volume":"2008","author":"Bernardin","year":"2008","journal-title":"EURASIP J. Image Video Process."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"113856","DOI":"10.1016\/j.eswa.2020.113856","article-title":"Fuzzy C-Means clustering algorithm for data with unequal cluster sizes and contaminated with noise and outliers: Review and development","volume":"165","author":"Askari","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Granstr\u00f6m, K., Svensson, L., Xia, Y., Williams, J., and Garc\u00eda-Fem\u00e1ndez, \u00c1.F. (2018, January 10\u201313). Poisson multi-Bernoulli mixture trackers: Continuity through random finite sets of trajectories. Proceedings of the 2018 21st International Conference on Information Fusion (FUSION), Cambridge, UK.","DOI":"10.23919\/ICIF.2018.8455849"},{"key":"ref_44","unstructured":"Vallado, D.A., and McClain, W.D. (2013). Fundamentals of Astrodynamics and Applications, The Space Technology Library, Microcosm Press. [4th ed.]."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/11\/3684\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:07:51Z","timestamp":1760162871000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/11\/3684"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,26]]},"references-count":44,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["s21113684"],"URL":"https:\/\/doi.org\/10.3390\/s21113684","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2021,5,26]]}}}