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Optimal Transport Based Tracking of Space Objects in Cylindrical Manifolds

  • Niladri DasEmail author
  • Riddhi Pratim Ghosh
  • Nilabja Guha
  • Raktim Bhattacharya
  • Bani Mallick
Article
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Abstract

In this paper, we examine the performance of satellite state estimation algorithms in the modified equinoctial coordinate system, defined on the cylindrical manifold of \(\mathbb {R}^{5}\times \mathbb {S}\), where \(\mathbb {R}\) is the space of reals and \(\mathbb {S}\) denotes circular space. A comparison is made between an optimal transport based filter and ensemble Kalman filter algorithm in the context of satellite state estimation. The initial state joint probability density function is modeled in \(\mathbb {R}^{5}\times \mathbb {S}\) using the Gauss von Mises distribution. The sensor noise for optimal transport filtering is modeled in the same manifold. The ensemble Kalman filter, by definition, requires the sensor noise to be Gaussian and is modeled in \(\mathbb {R}^{6}\) for this problem. We observe that there is a clear advantage in using an optimal transport based filtering algorithm where we represent the initial condition uncertainty and sensor noise, in the cylindrical manifold. These two filtering algorithms are implemented on a simulated International Space Station orbit, with measurement at equal intervals. We compare two distinct scenarios in this simulation based study. In the first one, sensor noise characteristics are assumed to be known. For the second one, we present a new algorithm within the optimal transport framework with unknown sensor noise characteristics, a practical and relevant issue in the space-tracking problems. The optimal transport based algorithm provides more consistent and robust estimates compared to that of ensemble Kalman filter, in each of these scenarios.

Keywords

Space situational awareness Gauss von Mises distribution State estimation optimal transport 

Notes

Acknowledgments

This research was sponsored by AFOSR DDDAS grant FA9550-15-1-0071, with Dr. Erik Blasch as the program manager. We thank Professor Faming Liang, Department of Biostatistics, University of Florida, Gainesville and Jingnan Xue at Texas A&M University for their insightful comments. Nilabja Guha is supported by University of Massachusetts Lowell start-up grant.

References

  1. 1.
    Akhlaghi, S., Zhou, N., Huang, Z.: Adaptive Adjustment of Noise Covariance in Kalman Filter for Dynamic State Estimation. In: 2017 IEEE Power Energy Society General Meeting, pp. 1–5.  https://doi.org/10.1109/PESGM.2017.8273755 (2017)
  2. 2.
    Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics, Rev. Edn. AIAA Education Series. American Institute of Aeronautics and Astronautics. Reston, VA (1999)Google Scholar
  3. 3.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J., et al.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends®; Mach. Learn. 3(1), 1–122 (2011).  https://doi.org/10.1561/2200000016 zbMATHGoogle Scholar
  4. 4.
    Broucke, R.A., Cefola, P.J.: On the equinoctial orbit elements. Celest. Mech. 5(3), 303–310 (1972)zbMATHGoogle Scholar
  5. 5.
    Celestrack: Iridium 33/Cosmos 2251 Collision (2009) http://celestrak.com/events/collision/
  6. 6.
    Chen, M.Y., Wang, H.Y.: Nonlinear measurement update for recursive filtering based on the gauss von mises distribution. Procedia Comput. Sci. 92, 543–548 (2016).  https://doi.org/10.1016/j.procs.2016.07.380 Google Scholar
  7. 7.
    Corner, B., Narayanan, R., Reichenbach, S.: Noise estimation in remote sensing imagery using data masking. Int. J. Remote Sens. 24(4), 689–702 (2003).  https://doi.org/10.1080/01431160210164271 Google Scholar
  8. 8.
    Das, N., Deshpande, V., Bhattacharya, R.: Optimal Transport Based Tracking of Space Objects Using Range Data from a Single Ranging Station. Journal of Guidance, Control, and Dynamics (2018)Google Scholar
  9. 9.
    Daum, F., Huang, J.: Particle Flow for Nonlinear Filters with Log-Homotopy. In: Signal and Data Processing of Small Targets 2008, vol. 6969, pp. 696918. International Society for Optics and Photonics.  https://doi.org/10.1117/12.764909 (2008)
  10. 10.
    De Mars, K., Cheng, Y., Bishop, R., Jah, M.: Methods for splitting gaussian distributions and applications within the aegis filter. Adv. Astronaut. Sci. 143, 2379–2398 (2012)Google Scholar
  11. 11.
    Evensen, G.: The ensemble kalman filter: Theoretical formulation and practical implementation. Ocean Dyn. 53(4), 343–367 (2003).  https://doi.org/10.1007/s10236-003-0036-9 Google Scholar
  12. 12.
    Fujimoto, K., Scheeres, D.: Non-linear propagation of uncertainty with non-conservative effects. Adv. Astronaut. Sci. 143, 2409–2427 (2012)Google Scholar
  13. 13.
    Gillijns, S., Mendoza, O.B., Chandrasekar, J., De Moor, B., Bernstein, D., Ridley, A.: What is the Ensemble Kalman Filter and How Well Does It Work? In: American Control Conference, vol. 6. IEEE.  https://doi.org/10.1109/acc.2006.1657419 (2006)
  14. 14.
    Gordon, N., Salmond, D., Smith, A.: Novel approach to nonlinear/non-gaussian bayesian state estimation. IEEE Proc. F Radar Signal Process. 140(2), 107 (1993).  https://doi.org/10.1049/ip-f-2.1993.0015 Google Scholar
  15. 15.
    Hairer, E., Wanner, G.: Solving ordinary differential equations II. Springer, Berlin.  https://doi.org/10.1007/978-3-662-09947-6 (1991)
  16. 16.
    Hammersley, J., Handscomb, D., Weiss, G.: Monte carlo methods. Phys. Today 18, 55 (1965)Google Scholar
  17. 17.
    Horwood, J.T.: Methods and systems for updating a predicted location of an object in a multi-dimensional space (2014). US Patent 8,909,589Google Scholar
  18. 18.
    Horwood, J.T., Aragon, N.D., Poore, A.B.: Gaussian sum filters for space surveillance: Theory and simulations. J. Guid. Control Dyn. 34(6), 1839–1851 (2011).  https://doi.org/10.2514/1.53793 Google Scholar
  19. 19.
    Horwood, J.T., Poore, A.B.: Adaptive gaussian sum filters for space surveillance. IEEE Trans. Autom. Control 56(8), 1777–1790 (2011).  https://doi.org/10.1109/Tac.2011.2142610 MathSciNetzbMATHGoogle Scholar
  20. 20.
    Horwood, J.T., Poore, A.B.: Orbital state uncertainty realism. In: Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, pp. 356–365. Wailea (2012)Google Scholar
  21. 21.
    Horwood, J.T., Poore, A.B.: Gauss von mises distribution for improved uncertainty realism in space situational awareness. Siam-Asa J. Uncertain. Quantif. 2(1), 276–304 (2014).  https://doi.org/10.1137/130917296 MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82(1), 35 (1960).  https://doi.org/10.1115/1.3662552 Google Scholar
  23. 23.
    Krag, H., Serrano, M., Braun, V., Kuchynka, P., Catania, M., Siminski, J., Schimmerohn, M., Marc, X., Kuijper, D., Shurmer, I., O’Connell, A., Otten, M., Muñoz, I., Morales, J., Wermuth, M., McKissock, D.: A 1 cm space debris impact onto the sentinel-1a solar array. Acta Astronaut. 137, 434–443 (2017).  https://doi.org/10.1016/j.actaastro.2017.05.010 Google Scholar
  24. 24.
    McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 (2), 309–323 (1995).  https://doi.org/10.1215/S0012-7094-95-08013-2 MathSciNetzbMATHGoogle Scholar
  25. 25.
    Monge, G.: Mémoire sur la théorie des déblais et des remblais. De l’Imprimerie Royale (1781)Google Scholar
  26. 26.
    Moselhy, T.A.E., Marzouk, Y.M.: Bayesian inference with optimal maps. J. Comput. Phys. 231(23), 7815–7850 (2012).  https://doi.org/10.1016/j.jcp.2012.07.022 MathSciNetzbMATHGoogle Scholar
  27. 27.
    Park, R.S., Scheeres, D.J.: Nonlinear mapping of gaussian statistics: Theory and applications to spacecraft trajectory design. J. Guid. Control Dyn. 29(6), 1367–1375 (2006).  https://doi.org/10.2514/1.20177 Google Scholar
  28. 28.
    Park, R.S., Scheeres, D.J.: Nonlinear semi-analytic methods for trajectory estimation. J. Guid. Control Dyn. 30(6), 1668–1676 (2007).  https://doi.org/10.2514/1.29106 Google Scholar
  29. 29.
    Reich, S.: A nonparametric ensemble transform method for bayesian inference. Siam J. Sci. Comput. 35(4), A2013–A2024 (2013).  https://doi.org/10.1137/130907367 MathSciNetzbMATHGoogle Scholar
  30. 30.
    Ristic, B., Arulampalam, S., Gordon, N.: Beyond the kalman filter. IEEE Aerosp. Electron. Syst. Mag. 19, 37–38 (2004)zbMATHGoogle Scholar
  31. 31.
    Schaub, H., Junkins, J.: Analytical mechanics of space systems, second edition american institute of aeronautics and astronautics.  https://doi.org/10.2514/4.105210 (2009)
  32. 32.
    Tang, Y., Ambandan, J., Chen, D.: Nonlinear measurement function in the ensemble kalman filter. Adv. Atmos. Sci. 31(3), 551–558 (2014).  https://doi.org/10.1007/s00376-013-3117-9 Google Scholar
  33. 33.
    Terejanu, G., Singla, P., Singh, T., Scott, P.D.: Uncertainty propagation for nonlinear dynamic systems using gaussian mixture models. J. Guid. Control Dyn. 31(6), 1623–1633 (2008).  https://doi.org/10.2514/1.36247 Google Scholar
  34. 34.
    Vallado, D.A., McClain, W.D.: Fundamentals of Astrodynamics and Applications, 2nd edn. Space technology library. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  35. 35.
    Villani, C.: Optimal Transport: Old and New. Grundlehren Der Mathematischen Wissenschaften. Springer, Berlin (2009)Google Scholar
  36. 36.
    Walker, M.: A set of modified equinoctial orbit elements. Celest. Mech. Dyn. Astron. 38(4), 391–392 (1986)zbMATHGoogle Scholar
  37. 37.
    Walker, M., et al.: A set modified equinoctial orbit elements. Celest. Mech. Dyn. Astron. 36(4), 409–419 (1985)zbMATHGoogle Scholar
  38. 38.
    Welch, G.: Kalman Filter. Springer, US.  https://doi.org/10.1007/978-0-387-31439-6_716 (2014)

Copyright information

© American Astronautical Society 2019

Authors and Affiliations

  1. 1.Aerospace EngineeringTexas A, M UniversityCollege StationUSA
  2. 2.Deptartment of StatisticsTexas A, M UniversityCollege StationUSA
  3. 3.Mathematical SciencesUniversity of Massachusetts LowellLowellUSA

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