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Gaussian Mixture Approximation of Angles-Only Initial Orbit Determination Likelihood Function

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A method is developed to approximate the angles-only orbit determination likelihood function using a Gaussian mixture to incorporate information about an admissible region. The resulting probability density function can provide the a priori information for a Gaussian mixture orbit determination filter. This admissible region approach can compensate for the effects of measurement uncertainty over a short track of angles-only data better than deterministic approaches can. The new technique starts with a nonlinear batch least-squares solution. The solution enforces soft constraints on an admissible region defined in terms of minimum periapsis and maximum apoapsis. Although this soft-constrained solution lies in or near the admissible region, it does not characterize that region well. It provides a starting point to develop a Gaussian mixture approximation of the batch least-squares likelihood function as modified through multiplication by a finite-support function that is zero outside the admissible region and 1 inside that region. This Gaussian mixture is optimized to fit the resulting probability density in the two-dimensional subspace of position/velocity space that has the most uncertainty. The procedure starts by approximating the finite-support distribution using a single optimized Gaussian mixand. It automatically increases the number of mixands and reoptimizes until the resulting mixture fits the finite-support distribution to a prespecified precision. This optimal fitting procedure allows the Gaussian mixture to use a low number of mixands while fitting the finite-support probability density function well. By approximating the product of a finite-support function and the original likelihood function, the new method gains the capability to transition smoothly between regimes where the admissibility constraints dominate, i.e., high-altitude/short-measurement-arc cases, and those where they are less important, i.e., low-altitude/long-measurement-arc cases.


  • [1] Schaeperkoetter A., “A Comprehensive Comparison Between Angles-Only Initial Orbit Determination Techniques,” M.S. Thesis, Aerospace Engineering, Texas A&M Univ., College Station, TX, Dec. 2011. Google Scholar

  • [2] Weisman R. and Jah M., “Uncertainty Quantification for Angles-Only Initial Orbit Determination,” Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, AAS Paper  14-434, San Diego, CA, 2014. Google Scholar

  • [3] DeMars K. and Jah M., “Probabilistic Initial Orbit Determination Using Gaussian Mixture Models,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 5, 2013, pp. 1324–1335. doi: JGCODS 0731-5090 LinkGoogle Scholar

  • [4] Milani A., Gronchi G., Vitturi M. and Knežević Z., “Orbit Determination with Very Short Arcs: I. Admissible Regions,” Celestial Mechanics and Dynamical Astronomy, Vol. 90, Nos. 1–2, 2004, pp. 57–85. doi: CrossrefGoogle Scholar

  • [5] Tommei G., Milani A. and Rossi A., “Orbit Determination of Space Debris: Admissible Regions,” Celestial Mechanics and Dynamical Astronomy, Vol. 97, No. 4, 2007, pp. 289–304. doi: CrossrefGoogle Scholar

  • [6] Maruskin J., Scheeres D. and Alfriend K., “Correlation of Optical Observations of Objects in Earth Orbit,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 194–209. doi: JGCODS 0731-5090 LinkGoogle Scholar

  • [7] Farnocchia D., Tommei G., Milani A. and Rossi A., “Innovative Methods of Correlation and Orbit Determination for Space Debris,” Celestial Mechanics and Dynamical Astronomy, Vol. 107, Nos. 1–2, 2010, pp. 169–185. doi: CrossrefGoogle Scholar

  • [8] Fujimoto K. and Scheeres D., “Correlation of Optical Observations of Earth-Orbiting Objects and Initial Orbit Determination,” Journal of Guidance, Control, and Dynamics, Vol. 35, No. 1, 2012, pp. 208–221. doi: JGCODS 0731-5090 LinkGoogle Scholar

  • [9] Sorenson H. and Alspach D., “Recursive Bayesian Estimation Using Gaussian Sums,” Automatica, Vol. 7, No. 4, 1971, pp. 465–479. doi: ATCAA9 0005-1098 CrossrefGoogle Scholar

  • [10] Alspach D. and Sorenson H., “Nonlinear Bayesian Estimation Using Gaussian Sum Approximations,” IEEE Transactions on Automatic Control, Vol. 17, No. 4, 1972, pp. 439–448. doi: IETAA9 0018-9286 CrossrefGoogle Scholar

  • [11] DeMars K., Bishop R. and Jah M., “Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 4, 2013, pp. 1047–1057. doi: JGCODS 0731-5090 LinkGoogle Scholar

  • [12] Psiaki M., Schoenberg J. and Miller I., “Gaussian Sum Reapproximation for Use in a Nonlinear Filter,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 2, 2015, pp. 292–303. doi: JGCODS 0731-5090 LinkGoogle Scholar

  • [13] Psiaki M., “Gaussian Mixture Nonlinear Filtering with Resampling for Mixand Narrowing,” IEEE Transactions on Signal Processing, Vol. 64, No. 21, 2016, pp. 5499–5512. doi: ITPRED 1053-587X CrossrefGoogle Scholar

  • [14] Maruskin J., “On the Dynamical Propagation of Subvolumes and on the Geometry and Variational Principles of Nonholonomic Systems,” Ph.D. Dissertation, Applied and Interdisciplinary Mathematics, Univ. of Michigan, Ann Arbor, MI, 2008. Google Scholar

  • [15] Broucke R. and Cefola P., “On the Equinoctial Orbit Elements,” Celestial Mechanics, Vol. 5, No. 3, 1972, pp. 303–310. doi: CLMCAV 0008-8714 CrossrefGoogle Scholar

  • [16] Gill P., Murray W. and Wright M., Practical Optimization, Academic Press, New York, 1981, pp. 136–137. Google Scholar

  • [17] Bierman G., Factorization Methods for Discrete Sequential Estimation, Academic Press, New York, 1977, pp. 69–75, 115–121. Google Scholar

  • [18] Clancy D., “Associating Data In System of Systems Using Measures of Information,” Proceedings of InfoTech@Aerospace, AIAA Paper  2011-1540, 2011. LinkGoogle Scholar

  • [19] Williams J. and Maybeck P., “Cost-Function-Based Gaussian Mixture Reduction for Target Tracking,” Proceedings of the 6th International Conference on Information Fusion, IEEE Publ., Piscataway, NJ, 2003, pp. 1047–1054. Google Scholar