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Near-Linear Orbit Uncertainty Propagation Using the Generalized Equinoctial Orbital Elements

Published Online:22 Dec 2022https://doi.org/10.2514/1.G006864

Abstract

This paper addresses the problem of minimizing the impact of nonlinearities when dealing with uncertainty propagation in the perturbed two-body problem. The recently introduced generalized equinoctial orbital elements set is employed as a means to reduce nonlinear effects stemming from J2 and higher-order gravity field harmonics. The uncertainty propagation performance of the proposed set of elements in different Earth orbit scenarios, including low-thrust orbit raising, is evaluated using a Cramér–von Mises test on the Mahalanobis distance of the uncertainty distribution. A considerable improvement compared to all sets of elements proposed so far is obtained.

References

  • [1] Maybeck P. S., Stochastic Models, Estimation, and Control Volume 2, Vol. 141-2 of Mathematics in Science and Engineering, Academic Press, New York, 1982, pp. 192–202. Google Scholar

  • [2] Armellin R., Di Lizia P., Bernelli-Zazzera F. and Berz M., “Asteroid Close Encounters Characterization Using Differential Algebra: The Case of Apophis,” Celestial Mechanics and Dynamical Astronomy, Vol. 107, No. 4, 2010, pp. 451–470. https://doi.org/10.1007/s10569-010-9283-5 CrossrefGoogle Scholar

  • [3] Park R. S. and Scheeres D. J., “Nonlinear Mapping of Gaussian Statistics: Theory and Applications to Spacecraft Trajectory Design,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 6, 2006, pp. 1367–1375. https://doi.org/10.2514/1.2017 LinkGoogle Scholar

  • [4] Roa J. and Park R. S., “Reduced Nonlinear Model for Orbit Uncertainty Propagation and Estimation,” Journal of Guidance, Control, and Dynamics, Vol. 44, No. 9, 2021, pp. 1–15. https://doi.org/10.2514/1.G005519 Google Scholar

  • [5] Giza D., Singla P. and Jah M., “An Approach for Nonlinear Uncertainty Propagation: Application to Orbital Mechanics,” AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2009-6082, 2009. https://doi.org/10.2514/6.2009-6082 Google Scholar

  • [6] Terejanu G., Singla P., Singh T. and Scott P. D., “Uncertainty Propagation for Nonlinear Dynamic Systems Using Gaussian Mixture Models,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 6, 2008, pp. 1623–1633. https://doi.org/10.2514/1.36247 LinkGoogle Scholar

  • [7] Julier S. J. and Uhlmann J. K., “New Extension of the Kalman Filter to Nonlinear Systems,” Signal Processing, Sensor Fusion, and Target Recognition VI, Vol. 3068, International Soc. for Optics and Photonics, Bellingham, WA, 1997, pp. 182–193. https://doi.org/10.1117/12.280797 Google Scholar

  • [8] Jones B. A., Doostan A. and Born G. H., “Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 2, 2013, pp. 430–444. https://doi.org/10.2514/1.57599 LinkGoogle Scholar

  • [9] Milani A., Sansaturio M. E., Tommei G., Arratia O. and Chesley S. R., “Multiple Solutions for Asteroid Orbits: Computational Procedure and Applications,” Astronomy & Astrophysics, Vol. 431, No. 2, 2005, pp. 729–746. https://doi.org/10.1051/0004-6361:20041737 Google Scholar

  • [10] Tardioli C., Kubicek M., Vasile M., Minisci E. and Riccardi A., “Comparison of Non-Intrusive Approaches to Uncertainty Propagation in Orbital Mechanics,” AAS/AIAA Astrodynamics Specialist Conference 2015, AAS Paper 2015-545, Vail, CA, 2015, pp. 3979–3992. Google Scholar

  • [11] Horwood J. T. and Poore A. B., “Gauss von Mises Distribution for Improved Uncertainty Realism in Space Situational Awareness,” SIAM/ASA Journal on Uncertainty Quantification, Vol. 2, No. 1, 2014, pp. 276–304. https://doi.org/10.1137/130917296 CrossrefGoogle Scholar

  • [12] Vittaldev V., Russell R. P. and Linares R., “Spacecraft Uncertainty Propagation Using Gaussian Mixture Models and Polynomial Chaos Expansions,” Journal of Guidance, Control, and Dynamics, Vol. 39, No. 12, 2016, pp. 2615–2626. https://doi.org/10.2514/1.G001571 LinkGoogle Scholar

  • [13] Junkins J. L., Akella M. R. and Alfriend K. T., “Non-Gaussian Error Propagation in Orbital Mechanics,” Journal of the Astronautical Sciences, Vol. 44, No. 4, 1996, pp. 541–563. Google Scholar

  • [14] Sabol C., Sukut T., Hill K., Alfriend K. T., Wright B., Li Y. and Schumacher P., “Linearized Orbit Covariance Generation and Propagation Analysis via Simple Monte Carlo Simulations,” AAS/AIAA Space Flight Mechanics Conference, AAS Paper 2010-134, Feb. 2010, pp. 14–17. Google Scholar

  • [15] Folcik Z., Lue A. and Vatsky J., “Reconciling Covariances with Reliable Orbital Uncertainty,” 12th Advanced Maui Optical and Space Surveillance Technologies Conference, Curran Associates, Inc., Red Hook, NY, 2011, pp. 309–318. Google Scholar

  • [16] Hernando-Ayuso J. and Bombardelli C., “Orbit Covariance Propagation via Quadratic-Order State Transition Matrix in Curvilinear Coordinates,” Celestial Mechanics and Dynamical Astronomy, Vol. 129, Sept. 2017, pp. 215–234. https://doi.org/10.1007/s10569-017-9773-9 CrossrefGoogle Scholar

  • [17] Coppola V. T. and Tanygin S., “Using Bent Ellipsoids to Represent Large Position Covariance in Orbit Propagation,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 9, 2015, pp. 1775–1784. https://doi.org/10.2514/1.G001011 LinkGoogle Scholar

  • [18] Melton R. G., “Time-Explicit Representation of Relative Motion Between Elliptical Orbits,” Journal of Guidance, Control, and Dynamics, Vol. 23, No. 4, 2000, pp. 604–610. https://doi.org/10.2514/2.4605 LinkGoogle Scholar

  • [19] Lane C. M. and Axelrad P., “Formation Design in Eccentric Orbits Using Linearized Equations of Relative Motion,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 1, 2006, pp. 146–160. https://doi.org/10.2514/1.13173 LinkGoogle Scholar

  • [20] Hill K., Alfriend K. and Sabol C., “Covariance-Based Uncorrelated Track Association,” AIAA/AAS Astrodynamics Specialist Conference, AIAA Paper 2008-7211, 2008. https://doi.org/10.2514/6.2008-7211 LinkGoogle Scholar

  • [21] Weis L. M., “Uncertainty in KS Space with Arbitrary Forces,” AIAA SciTech Forum, AIAA Paper 2018-0728, 2018. https://doi.org/10.2514/6.2018-0728 Google Scholar

  • [22] Roa Vicens J. and Peláez Álvarez J., “Efficient Trajectory Propagation for Orbit Determination Problems,” AAS/AIAA Astrodynamics Specialist Conference, AAS Paper 2015-730, Univelt, Vail, CO, Aug. 2015. Google Scholar

  • [23] Horwood J. T., Aristoff J. M., Singh N. and Poore A. B., “A Comparative Study of New Non-Linear Uncertainty Propagation Methods for Space Surveillance,” SPIE Proceedings: Signal and Data Processing of Small Targets, Vol. 9092, June 2014. https://doi.org/10.1117/12.2051353 Google Scholar

  • [24] Aristoff J. M., Horwood J. T. and Alfriend K. T., “On a Set of J2 Equinoctial Orbital Elements and Their Use for Uncertainty Propagation,” Celestial Mechanics and Dynamical Astronomy, Vol. 133, No. 9, 2021, pp. 1–19. https://doi.org/10.1007/s10569-021-10004-0 Google Scholar

  • [25] Baù G., Hernando-Ayuso J. and Bombardelli C., “A Generalization of the Equinoctial Orbital Elements,” Celestial Mechanics and Dynamical Astronomy, Vol. 133, No. 50, 2021. https://doi.org/10.1007/s10569-021-10049-1 Google Scholar

  • [26] Hernando-Ayuso J., Bombardelli C., Baù G. and Martínez-Cacho A., “Near-Linear Orbit Uncertainty Propagation in Low-Earth Orbit,” 31st Workshop on JAXA Astrodynamics and Flight Mechanics, Sagamihara (Japan), Inst. of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA), July 2021. Google Scholar

  • [27] Arsenault J. L., Ford K. C. and Koskela P. E., “Orbit Determination Using Analytic Partial Derivatives of Perturbed Motion,” AIAA Journal, Vol. 8, No. 1, 1970, pp. 4–9. https://doi.org/10.2514/3.5597 LinkGoogle Scholar

  • [28] Broucke R. A. and Cefola P. J., “On the Equinoctial Orbit Elements,” Celestial Mechanics, Vol. 5, No. 3, May 1972, pp. 303–310. https://doi.org/10.1007/BF01228432 CrossrefGoogle Scholar

  • [29] Horwood J. T., Aragon N. D. and Poore A. B., “Gaussian Sum Filters for Space Surveillance: Theory and Simulations,” Journal of Guidance, Control, and Dynamics, Vol. 34, Nov. 2011, pp. 1839–1851. https://doi.org/10.2514/1.53793 LinkGoogle Scholar

  • [30] Battin R., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA, Reston, VA, 1999, pp. 483, 504. https://doi.org/10.2514/4.861543 LinkGoogle Scholar

  • [31] Brouwer D., “Solution of the Problem of Artificial Satellite Theory Without Drag,” Astronomical Journal, Vol. 64, Nov. 1959, p. 378. https://doi.org/10.1086/107958 CrossrefGoogle Scholar

  • [32] Csörgö S. and Faraway J. J., “The Exact and Asymptotic Distributions of Cramér-von Mises Statistics,” Journal of the Royal Statistical Society: Series B (Methodological), Vol. 58, No. 1, 1996, pp. 221–234. https://doi.org/10.1111/j.2517-6161.1996.tb02077.x Google Scholar

  • [33] Abbasrezaee P., Mirshams M. and Seyed-Zamani S., “Conceptual GEO Telecommunication All-Electric Satellite Design Based on Statistical Model,” 2019 9th International Conference on Recent Advances in Space Technologies (RAST), Inst. of Electrical and Electronics Engineers, New York, 2019, pp. 503–507. https://doi.org/10.1109/RAST.2019.8767854 Google Scholar

  • [34] Ries J., Bettadpur S., Eanes R., Kang Z., Ko U., McCullough C., Nagel P., Pie N., Poole S., Richter T. and et al., “Development and Evaluation of the Global Gravity Model GGM05-CSR-16-02,” TR CSR-16-02, Center for Space Research, Univ. of Texas at Austin, Austin, TX, 2016. https://doi.org/10.26153/tsw/1461 Google Scholar

  • [35] Cunningham L. E., “On the Computation of the Spherical Harmonic Terms Needed During the Numerical Integration of the Orbital Motion of an Artificial Satellite,” Celestial Mechanics, Vol. 2, No. 2, 1970, pp. 207–216. https://doi.org/10.1007/BF01229495 CrossrefGoogle Scholar

  • [36] Métris G., Xu J. and Wytrzyszczak I., “Derivatives of the Gravity Potential with Respect to Rectangular Coordinates,” Celestial Mechanics and Dynamical Astronomy, Vol. 71, No. 2, 1998, pp. 137–151. https://doi.org/10.1023/A:1008361202235 CrossrefGoogle Scholar

  • [37] Fantino E. and Casotto S., “Methods of Harmonic Synthesis for Global Geopotential Models and Their First-, Second- and Third-Order Gradients,” Journal of Geodesy, Vol. 83, No. 7, 2009, pp. 595–619. https://doi.org/10.1007/s00190-008-0275-0 CrossrefGoogle Scholar

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