Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Dynamic Data-Driven Uncertainty Quantification via Polynomial Chaos for Space Situational Awareness

  • Chapter
  • First Online:
Handbook of Dynamic Data Driven Applications Systems

Abstract

Knowledge of all space objects in orbit and the space environment is collected and maintained by the Space Surveillance Network (SSN). This task is becoming more difficult as the number of objects currently tracked increases due to breakup events and improvements in sensor detection capabilities. The SSN is tasked with maintaining information on over 22,000 objects, 1,100 of which are active. In particular, low-Earth orbiting satellites are heavily influenced by atmospheric drag which is difficult to model due to fluctuations in the upper atmospheric density. These fluctuations are caused by variations in the Solar energy flux which heats Earth’s atmosphere causing it to expand. This research uses probabilistic models to characterize and account for the fluctuations in the Earth’s atmosphere. By correctly estimating the fluctuations, our work contributes to improving the ability to determine the likelihood of satellite collisions in space.

The main focus of this chapter is the application of a new Polynomial Chaos based Uncertainty Quantification (UQ) approach for Space Situational Awareness (SSA). The challenge of applying UQ to SSA is the long-term integration problem, where simulations are used to forecast physics over long temporal and/or spatial extrapolation intervals. This chapter applies a Polynomial Chaos (PC) expansion and Gaussian Mixture Models (GMMs) in a hybrid fashion for UQ applied to satellite tracking. This chapter uses the GMM-PC approach for orbital UQ and the PC approach for atmospheric density UQ. Two different application examples are shown. The first example demonstrates the GMM-PC approach for orbital UQ for a low Earth orbit satellite under the influence of atmospheric perturbations. The second example demonstrates the PC approach for atmospheric density UQ, where a physics-based model is used to capture the uncertainty of the atmospheric density under uncertain Solar conditions. These two examples are not combined under this work but the tools developed provide a framework for an unified understanding of UQ for low Earth orbiting satellites.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    This number is problem dependent.

  2. 2.

    Multidimensional quadratures.

References

  1. R. Oliva, E. Blasch, R. Ogan, Applying aerospace technologies to current issues using systems engineering: 3rd AESS chapter summit. IEEE Aerosp. Electron. Syst. Mag. 28(2), 34–41 (2013)

    Article  Google Scholar 

  2. D.S. Bernstein, A. Ridley, J. Cutler, A. Cohn, Transformative advances in DDDAS with application to space weather monitoring. Technical report, Department of Aerospace Engineering, University of Michigan, Ann Arbor, 2015

    Google Scholar 

  3. H. Chen, G. Chen, E. Blasch, K. Pham, Comparison of several space target tracking filters. Proc. SPIE 7330, 73300I (2009)

    Article  Google Scholar 

  4. N. Weiner, The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)

    Article  MathSciNet  Google Scholar 

  5. V. Vittaldev, R.P. Russell, R. Linares, Spacecraft uncertainty propagation using gaussian mixture models and polynomial chaos expansions. J. Guid. Control. Dyn. 39, 2615–2626 (2016)

    Article  Google Scholar 

  6. A. Hedin, A revised thermospheric model based on mass spectrometer and incoherent scatter data: Msis-83. J. Geophys. Res. 88, 10170–10188 (1983)

    Article  Google Scholar 

  7. J. Picone, A. Hedin, D. Drob, A. Aikin, NRLMSISE-00 empirical model of the atmosphere: statistical comparisons and scientific issues. J. Geophys. Res. 107(A12), 1468 (2002)

    Article  Google Scholar 

  8. M.F. Storz, B.R. Bowman, M.J.I. Branson, S.J. Casali, W.K. Tobiska, High accuracy satellite drag model (HASDM). Adv. Space Res. 36(12), 2497–2505 (2005)

    Article  Google Scholar 

  9. A. Ridley, Y. Deng, G. Toth, The global ionosphere–thermosphere model. J. Atmos. Sol.-Terr. Phys. 68, 839–864 (2006)

    Article  Google Scholar 

  10. R. Bellmam, Dynamic Programming (Princeton University Press, Princeton, 1957)

    Google Scholar 

  11. C. Sabol, C. Binz, A. Segerman, K. Roe, P.W. Schumacher, Probability of collision with special perturbations dynamics using the monte carlo method, paper AAS 11-435, in AAS/AIAA Astrodynamics Specialist Conference, Girdwood, 31 Jul–4 Aug 2011

    Google Scholar 

  12. R.W. Ghrist, D. Plakalovic, Impact of non-gaussian error volumes on conjunction assessment risk analysis, paper AIAA 2012-4965, in AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, 13–16 Aug, 2012

    Google Scholar 

  13. N. Arora, V. Vittaldev, R.P. Russell, Parallel computation of trajectories using graphics processing units and interpolated gravity models. J. Guid. Control. Dyn. 38, 1345–1355 (2015). Accepted for Publication

    Article  Google Scholar 

  14. H. Shen, V. Vittaldev, C.D. Karlgaard, R.P. Russell, E. Pellegrini, Parallelized sigma point and particle filters for navigation problems, paper AAS 13-034, in 36th Annual AAS Guidance and Control Conference, 1–6 Feb, Breckenridge, 2013

    Google Scholar 

  15. N. Nakhjiri, B.F. Villac, An algorithm for trajectory propagation and uncertainty mapping on GPU, paper AAS 13-376, in 23rd AAS/AIAA Space Flight Mechanics Meeting, Kauai, 2013

    Google Scholar 

  16. S.-Z. Ueng, M. Lathara, S.S. Baghsorkhi, W.-M. W. Hwu, Languages and Compilers for Parallel Computing, ch. CUDA-Lite: Reducing GPU Programming Complexity (Springer, Berlin/Heidelberg, 2008), pp. 1–15

    Book  Google Scholar 

  17. A.B. Poore, Propagation of uncertainty in support of SSA missions, in 25th AAS/AIAA Space Flight Mechanics Meeting, Williamsburg, 2015

    Google Scholar 

  18. D.L. Alspach, H.W. Sorenson, Nonlinear Bayesian estimation using gaussian sum approximations. IEEE Trans. Autom. Control 17(4), 439–448 (1972)

    Article  Google Scholar 

  19. R.S. Park, D.J. Scheeres, Nonlinear mapping of gaussian statistics: theory and applications to spacecraft trajectory design. J. Guid. Control. Dyn. 29(6), 1367–1375 (2006)

    Article  Google Scholar 

  20. S. Julier, J.K. Uhlmann, Unscented filtering and nonlinear estimation. Proc. IEEE 92, 401–402 (2004)

    Article  Google Scholar 

  21. M. Norgaard, N.K. Poulsen, O. Ravn, New developments in state estimation for nonlinear systems. Automatica 36(11), 1627–1638 (2000)

    Article  MathSciNet  Google Scholar 

  22. I. Arasaratnam, S. Haykin, Cubature Kalman filters. IEEE Trans. Autom. Control 54(6), 1254–1269 (2009)

    Article  MathSciNet  Google Scholar 

  23. D.F. Crouse, On measurement-based light-time corrections for bistatic orbital debris tracking. IEEE Trans. Aerosp. Electron. Syst. 51(3), 2502–2518 (2015)

    Article  Google Scholar 

  24. B. Jia, K.D. Pham, E. Blasch, D. Shen, Z. Wang, G. Chen, Cooperative space object tracking using space-based optical sensors via consensus-based filters. IEEE Trans. Aerosp. Electron. Syst. 52(4), 1908–1936 (2016)

    Article  Google Scholar 

  25. K.J. DeMars, R.H. Bishop, M.K. Jah, Entropy-based approach for uncertainty propagation of nonlinear dynamical systems. J. Guid. Control. Dyn. 36(4), 1047–1057 (2013)

    Article  Google Scholar 

  26. K. Vishwajeet, P. Singla, M. Jah, Nonlinear uncertainty propagation for perturbed two-body orbits. J. Guid. Control. Dyn. 37(5), 1415–1425 (2014)

    Article  Google Scholar 

  27. J.T. Horwood, N.D. Aragon, A.B. Poore, Gaussian sum filters for space surveillance: theory and simulations. J. Guid. Control. Dyn. 34(6), 1839–1851 (2011)

    Article  Google Scholar 

  28. K.J. DeMars, M.K. Jah, A probabilistic approach to initial orbit determination via gaussian mixture models. J. Guid. Control. Dyn. 36(5), 1324–1335 (2013)

    Article  Google Scholar 

  29. K.J. DeMars, Y. Cheng, M.K. Jah, Collision probability with Gaussian mixture orbit uncertainty. J. Guid. Control. Dyn. 37(3), 979–985 (2014)

    Article  Google Scholar 

  30. V. Vittaldev, R.P. Russell, Collision probability for resident space objects using Gaussian mixture models, paper AAS 13-351, in 23rd AAS/AIAA Spaceflight Mechanics Meeting, Kauai, 2013

    Google Scholar 

  31. D. Xiu, G.E. Karniadakis, The wiener-askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)

    Article  MathSciNet  Google Scholar 

  32. S. Oladyshkin, W. Nowak, Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliab. Eng. Syst. Saf. 106, 179–190 (2012)

    Article  Google Scholar 

  33. D.M. Luchtenburga, S.L. Bruntonc, C.W. Rowleyb, Long-time uncertainty propagation using generalized polynomial chaos and flow map composition. J. Comput. Phys. 274, 783–802 (2014)

    Article  MathSciNet  Google Scholar 

  34. X. Li, P.B. Nair, Z. Zhang, L. Gao, C. Gao, Aircraft robust trajectory optimization using nonintrusive polynomial chaos. J. Aircr. 51(5), 1592–1603 (2014)

    Article  Google Scholar 

  35. L. Mathelin, M.Y. Hussaini, T.A. Zang, Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algorithms 38(1–3), 209–236 (2005)

    Article  MathSciNet  Google Scholar 

  36. M. Dodson, G.T. Parks, Robust aerodynamic design optimization using polynomial chaos. J. Aircr. 46(2), 635–646 (2009)

    Article  Google Scholar 

  37. B.A. Jones, A. Doostan, G.H. Born, Nonlinear propagation of orbit uncertainty using non-intrusive polynomial chaos. J. Guid. Control. Dyn. 36(2), 415–425 (2013)

    Article  Google Scholar 

  38. V. Vittaldev, R. Russell, R. Linares, Spacecraft uncertainty propagation using gaussian mixture models and polynomial chaos expansions. J. Guid. Control Dyn. 39(12), 2615–2626 (2016)

    Article  Google Scholar 

  39. B.A. Jones, A. Doostan, Satellite collision probability estimation using polynomial chaos. Adv. Space Res. 52(11), 1860–1875 (2013)

    Article  Google Scholar 

  40. B.A. Jones, N. Parrish, A. Doostan, Postmaneuver collision probability estimation using sparse polynomial chaos expansions. J. Guid. Control. Dyn. 38(8), 1425–1437 (2015)

    Article  Google Scholar 

  41. G. Terejanu, P. Singla, T. Singh, P.D. Scott, Uncertainty propagation for nonlinear dynamic systems using gaussian mixture models. J. Guid. Control. Dyn. 31(6), 1623–1633 (2008)

    Article  Google Scholar 

  42. M.F. Huber, T. Bailey, H. Durrant-Whyte, U.D. Hanebeck, On entropy approximation for gaussian mixture random vectors, in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2008), 2008, pp. 181–188

    Google Scholar 

  43. V. Vittaldev, R.P. Russell, Multidirectional Gaussian mixtrure models for nonlinear uncertainty propagation. CMES 111(1), 83–117 (2016)

    Google Scholar 

  44. V. Vittaldev, R.P. Russell, Uncertainty propagation using gaussian mixture models. In SIAM Conference on Uncertainty Quatification, (Savannah, GA 2014)

    Google Scholar 

  45. J.M. Aristoff, J.T. Horwood, T. Singh, A.B. Poore, Nonlinear uncertainty propagation in orbital elements and transformation to cartesian space without loss of realism, in AIAA/AAS Astrodynamics Specialist Conference, San Diego, Aug 4–Aug 7, 2014

    Google Scholar 

  46. V. Vittaldev, R.P. Russell, Collision probability using multidirectional gaussian mixture models, in 25th AAS/AIAA Space Flight Mechanics Meeting, Williamsburg, 2015

    Google Scholar 

  47. R. Madankan, P. Singla, T. Singh, P.D. Scott, Polynomial-chaos-based Bayesian approach for state and parameter estimations. J. Guid. Control. Dyn. 36(4), 1058–1074 (2013)

    Article  Google Scholar 

  48. S. Hosder, R.W. Walter, R. Perez, A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations, in 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, 2006

    Google Scholar 

  49. S. Hosder, R.W. Walter, Non-intrusive polynomial chaos methods for uncertainty quantification in fluid dynamics, in 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, 2010

    Google Scholar 

  50. D. Xiu, G.E. Karniadakis, The wiener–askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

    Article  MathSciNet  Google Scholar 

  51. S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Doklady Akademii nauk SSSR, 1(4), 240–243 (1963)

    MATH  Google Scholar 

  52. K. Rawer, D. Bilitza, S. Ramakrishnan, Goals and status of the international reference ionosphere. Rev. Geophys. 16, 177 (1978)

    Article  Google Scholar 

  53. B.R. Bowman, W.K. Tobiska, F.A. Marcos, C.Y. Huang, C.S. Lin, W.J. Burke, A new empirical thermospheric density model jb2008 using new solar and geomagnetic indices, AIAA 2008–6483, in AIAA/AAS Astrodynamics Specialist Conference, Honolulu, 2008

    Google Scholar 

Download references

Acknowledgements

The first author wish to acknowledge support of this work by the Air Force’s Office of Scientic Research under Contract Number FA9550-18-1-0149 issued by Erik Blasch.

Author information

Authors and Affiliations

  1. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Richard Linares

  2. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX, USA

    Vivek Vittaldev

  3. Applied Mathematics Group, Los Alamos National Laboratory, Los Alamos, NM, USA

    Humberto C. Godinez

Authors
  1. Richard Linares
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vivek Vittaldev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Humberto C. Godinez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richard Linares .

Editor information

Editors and Affiliations

  1. Air Force Office of Scientific Research, Air Force Research Laboratory, Arlington, VA, USA

    Erik Blasch

  2. Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

    Sai Ravela

  3. Information Directorate, Air Force Research Laboratory, Rome, NY, USA

    Alex Aved

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Linares, R., Vittaldev, V., Godinez, H.C. (2018). Dynamic Data-Driven Uncertainty Quantification via Polynomial Chaos for Space Situational Awareness. In: Blasch, E., Ravela, S., Aved, A. (eds) Handbook of Dynamic Data Driven Applications Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-95504-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-95504-9_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-95503-2

  • Online ISBN: 978-3-319-95504-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation