Abstract
Multi-fidelity approaches to orbit-state probability density prediction reduce computation time, but introduce a systematic error in the single-point prediction of a spacecraft state. An estimate of the systematic error may be quantified using cross-validation. Credibilistic filters based on Outer Probability Measures (OPMs) enable a principled and unified representation of random and systematic errors in object tracking. The quantified error of the multi-fidelity approach defines an OPM-based transition kernel, which is used in a credibilistic filter to account for the systematic error in the orbit determination process. An approach based on automatic domain splitting is proposed to reduce the error beyond what is normally achievable with multi-fidelity methods. A proof-of-concept for the approach is demonstrated for a simulated scenario tracking a newly-detected space object in low-Earth orbit via two ground stations generating radar measurements. An OPM-based definition of the admissible region combined with the multi-fidelity credibilistic filter establishes custody of the object.
Similar content being viewed by others
Notes
If the system is deterministic, i.e., has no random uncertainty, then its probability measure (and associated PDF) concentrate all their mass to a single element of the state space, representing the true state of the system.
References
Armellin, R., Di Lizia, P., Bernelli-Zazzera, F., Berz, M.: Asteroid close encounters characterization using differential algebra: the case of Apophis. Celest. Mech. Dyn. Astron. 107(4), 451–470 (2010)
Balch, M.S., Martin, R., Ferson, S.: Satellite conjunction analysis and the false confidence theorem. arXiv:1706.08565 (2018)
Delande, E.D., Houssineau, J., Jah, M.K.: A new representation of uncertainty for data fusion in SSA detection and tracking problems. In: Proceedings of the 21st International Conference on Information Fusion (2018)
Delande, E., Houssineau, J., Jah, M.: Physics and human-based information fusion for improved resident space object tracking. Adv. Space Res. 62 (7), 1800–1812 (2018)
Delande, E.D., Jah, M.K., Jones, B.A.: A new representation of uncertainty for collision assessment. In: AAS/AIAA Space Flight Mechanics Meeting, Kaanapali, HI (2019)
DeMars, K.J., Cheng, Y, Bishop, R.H., Jah, M.K.: Methods for splitting gaussian distributions and applications within the AEGIS filter. In: 2012 AAS/AIAA Space Flight Mechanics Meeting, Charleston, SC (2012)
DeMars K.J., Jah, M.K.: Probabilistic initial orbit determination using Gaussian mixture models. J. Guid. Control Dyn. 36(5), 1324–1335 (2013)
Dormand, J.R., Prince, R.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)
Farnocchia, D., Tommei, G., Milani, A., Rossi, A.: Innovative methods of correlation and orbit determination for space debris. Celest. Mech. Dyn. Astron. 107(1-2), 169–185 (2010)
Folkner W.M., Williams, J.G., Boggs, D.H., Park, R.S., Kuchynka, P.: The planetary and lunar ephemerides DE430 and DE431. IPN Progress Report 42-196, Jet Propulsion Laboratory, California Institute of Technology. http://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf (2009)
Ghanem, R.G., Higdon, D., Owhadi, H., eds: Handbook of Uncertainty Quantification. Springer International Publishing, Switzerland (2017)
Hampton, J., Fairbanks, H.R., Narayan, A., Doostan, A.: Practical error bounds for a non-intrusive bi-fidelity approach to parametric/stochastic model reduction. J. Comput. Phys. 368, 315–332 (2018)
Horwood J.T., Aragon, N.D., Poore, A.B.: Gaussian sum filters for space surveillance: Theory and simulations. J. Guida. Control Dyn. 34(6), 1839–1851 (2011)
Houssineau, J., Bishop, A.N.: Smoothing and filtering with a class of outer measures. SIAM/ASA J. Uncertain. Quantif. 6(2), 845–866 (2018)
Jones, B.A.: Multi-fidelity methods for orbit determination. In: Proceedings of the 2018 Advanced Maui Optical and Space Surveillance Technologies Conference, Wailea, Maui, Hawaii (2018)
Jones, B.A., Delande, E.D., Zucchelli, E.M., Jah, M.K.: Multi-fidelity orbit uncertainty propagation with systematic errors. In: Proceedings of the 2019 Advanced Maui Optical and Space Surveillance Technologies Conference, Wailea, Maui, Hawaii (2019)
Jones, B.A., Doostan, A., Born, G.H.: Nonlinear propagation of orbit uncertainty using non-intrusive polynomial chaos. J. Guid. Control Dyn. 36(2), 430–444 (2013)
Jones, B.A., Weisman, R.: Multi-fidelity orbit uncertainty propagation. Acta Astronaut. 155, 406–417 (2019)
Julier, S., Uhlmann, J.K.: Unscented filtering and nonlinear estimation. Proc. IEEE 92(3), 401–422 (2004)
Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME–J. Basic Eng. 82(Series D), 35–45 (1960)
Narayan, A., Gittelson, C., Xiu, D.: A stochastic collocation algorithm with multifidelity models. SIAM J. Sci. Comput. 36(2), A495–A521 (2014)
Park, I., Fujimoto, K., Scheeres, D.J.: Effect of dynamical accuracy for uncertainty propagation of perturbed keplerian motion. J. Guid. Control Dyn. 38(12), 2287–2300 (2015)
Park, I., Scheeres, D.J.: Optimization of hybrid method for uncertainty propagation of non-keplerian motion. In: AIAA/AAS Astrodynamics Specialist Conference, pp 2016–5630. AIAA, Long Beach California (2016)
Petit, G., Luzum, B.: IERS conventions (2010) IERS Technical Note 36, International Earth Rotation and Reference Systems Service (IERS), Frankfurt am Main, Germany (2010)
Pirovano, L., Santeramo, D.A, Armellin, R., Lizia, P., Wittig, A.: Probabilistic data association: the orbit set. Celest. Mech. Dyn. Astron. 132(2), 1–27 (2020)
Poore, A.B., Aristoff, J.M., Horwood, J.T., Armellin, R., Cerven, W.T., Cheng, Y., Cox, C.M., Erwin, R.S., Frisbee, J.H., Hejduk, M.D, Jones, B.A., Pierluigi, D.L., Scheeres, D.J., Vallado, D.A., Weisman, R: Covariance and uncertainty realism in space surveillance and tracking. Technical Report AD1020892, Air Force Space Command Astrodynamics Innovation Committee (2016)
Standish, E.M., Newhall, X.X., Williams, J.G., Folkner, W.M.: JPL planetary and lunar ephemerides, DE403/LE403. Interoffice memorandum IOM 314.10-127, Jet Propulsion Laboratory (1995)
Tapley, B.D., Ries, J.C., Bettadpur, S.V., Chambers, D., Cheng, M., Condi, F., Poole, S.: The GGM03 mean earth gravity model from GRACE. In: American Geophysical Union, Fall Meeting, Abstract No. G42A-03 (2007)
Tapley, BD., Schutz, B.E., Born, G.H.: Statistical Orbit Determination, 1st edn. Elsevier Academic Press, Burlington, MA (2004)
Tapley, B.D., Watkins, M.M., Ries, J.C., Davis, G.W., Eanes, R.J., Poole, S.R., Rim, H.J., Schutz, B.E., Shum, C.K., Steven Nerem, R., Lerch, F.J., Marshall, J.A., Klosko, S.M., Pavlis, N.K., Williamson, R.W.: The joint gravity model 3. J. Geophys. Res. 101(B12), 28029–28050 (1996)
Tuggle, K., Zanetti, R.: Automated splitting gaussian mixture nonlinear measurement update. J. Guid. Control Dyn. 41(3), 725–734 (2018)
Vittaldev, V., Russell, R.P.: Multidirectional Gaussian mixture models for nonlinear uncertainty propagation. Comput. Model. Eng. Sci. 111(1), 83–117 (2016)
Worthy, J.L., Holzinger, M.J.: Incorporating uncertainty in admissible regions for uncorrelated detections. J. Guid. Control Dyn. 38(9), 1673–1689 (2015)
Worthy, IIIJ.L., Holzinger, M.J.: Use of uninformative priors to initialize state estimation for dynamical systems. Adv. Space Res. 60(7), 1373–1388 (2017)
Zhu, X., Narayan, A., Xiu, D.: Computational aspects of stochastic collocation with multifidelity models. SIAM/ASA J. Uncertain. Quantif. 2(1), 444–463 (2014)
Funding
This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-19-1-0404.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interests
The authors declare that they have no conflict of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article belongs to the Topical Collection: Advanced Maui Optical and Space Surveillance Technologies (AMOS 2018 & 2019) Guest Editors: James M. Frith, Lauchie Scott, Islam Hussein
Rights and permissions
About this article
Cite this article
Zucchelli, E.M., Delande, E.D., Jones, B.A. et al. Multi-Fidelity Orbit Determination with Systematic Errors. J Astronaut Sci 68, 695–727 (2021). https://doi.org/10.1007/s40295-021-00267-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40295-021-00267-y