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Measurement Model Nonlinearity in Estimation of Dynamical Systems

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Abstract

The role of nonlinearity of the measurement model and its interactions with the uncertainty of measurements and geometry of the problem is studied in this paper. An examination of the transformations of the probability density function in various coordinate systems is presented for several astrodynamics applications. Smooth and analytic nonlinear functions are considered for the studies on the exact transformation of uncertainty. Special emphasis is given to understanding the role of change of variables in the calculus of random variables. The transformation of probability density functions through mappings is shown to provide insight in to understanding the evolution of uncertainty in nonlinear systems. Examples are presented to highlight salient aspects of the discussion. A sequential orbit determination problem is analyzed, where the transformation formula provides useful insights for making the choice of coordinates for estimation of dynamic systems.

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References

  1. GAUSS, C. F. Theoria Motus, (English translation by Davis, C. H.) Little Brown and Company, Boston MA, 1857.

  2. JUNKINS, J. L. and SINGLA, P. “How Nonlinear Is It? - A Tutorial on Nonlinearity of Orbit and Attitude Dynamics,” The Journal of Astronautical Sciences, Vol 52, No. 1–2, Keynote Paper, 2004, pp. 7–60.

  3. PARK, R. S. and SCHEERES, D. J. “Nonlinear Mapping of Gaussian State Uncertainties: Theory and Applications to Spacecraft Control and Navigation,” Journal of Guidance, Control, and Dynamics, 2006, Vol. 29, No. 6, pp. 1367–1375.

    Article  Google Scholar 

  4. TEREJANU, G., SINGLA, P., SINGH, T., and SCOTT, P. D. “Uncertainty Propagation for Nonlinear Dynamical Systems Using Gaussian Mixture Models,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 6, Nov. 2008, pp. 1623–1633.

    Article  Google Scholar 

  5. KUMAR, M., CHAKRAVORTY, S., SINGLA, P., and JUNKINS, J. L. “The Partition of Unity Finite Element Approach with H-P Refinement to the Stationary Fokker-Planck Equation,” Journal of Sound and Vibration, Vol. 327, Issues 1–2, Oct. 2009, pp. 144–162.

    Article  Google Scholar 

  6. PARK, R. S. and SCHEERES, D. J. “Nonlinear Semi-analytic Methods for Trajectory Estimation,” Journal of Guidance, Control and Dynamics, 2007, Vol. 30, No. 6, pp. 1668 -1676.

    Article  Google Scholar 

  7. MAJJI, M., JUNKINS, J., and TURNER, J. D. “High Order Methods for Estimation of Dynamical Systems,” The Journal of the Astronautical Sciences, Vol. 52, 2009.

  8. MAJJI, M., JUNKINS, J. L., and TURNER, J. D. “A Perturbation Method for Estimation of Dynamic Systems,” Nonlinear Dynamics, Vol. 60, No.3, DOI 10.1007/s11071-009-9597-6, 2010.

  9. JULIER, S. and UHLMANN, J. “Unscented Filtering and Nonlinear Estimation,” Proceedings of the IEEE, 2002, Vol. 92, No. 3, pp. 401–422.

    Article  Google Scholar 

  10. JUNKINS, J. L., AKELLA, M. R., and ALFRIEND, K. T. “Non-Gaussian Error Propagation in Orbital Mechanics,” Proceedings of the 19th Annual AAS Guidance and Control Conference, Breckenridge, CO, Feb, 1996, Journal of the Astronautical Sciences, Vol. 44, No. 4, 1996, pp. 541–564.

  11. JUNKINS, J. L. “Adventures on the Interface of Dynamics and Control,” invited Theodore von Karman Lecture, AIAA Aerospace Sciences Conference, Reno, NV, January, 1997, Journal of Guidance, Control, and Dynamic s, Vol. 20, No. 6, Nov-Dec, 1997, pp. 1058–1072.

  12. SORENSON, H. W. Parameter Estimation: Principles and Problems, Control and Systems Theory Series, Marcel Dekker Publications, New York, 1980.

  13. RAUCH, H. E., TUNG, F., and STRIEBEL, T. “Maximum Likelihood Estimates of Linear Dynamic Systems,” AIAA Journal, Vol. 3., No. 8, August 1965, pp. 1445–1450.

    Article  MathSciNet  Google Scholar 

  14. LIGHT, D. L. “Satellite Photogrammetry,” Manual of Photogrammetry, edited by C. C. Slama, Chap. 17, American Society of Photogrammetry, Falls Church, VA, 4th Ed, 1980.

  15. CRASSIDIS, J. and JUNKINS, J. Optimal Estimation of Dynamical Systems, Chapman- Hall/CRC Press, Boca Raton, FL, 2004.

    Book  Google Scholar 

  16. JAZWINSKI, A. H. Stochastic Processes and Filtering Theory, Dover Publications, Mineola, NY, 2007.

    MATH  Google Scholar 

  17. ESCOBAL, P. E. Methods of Orbit Determination, Krieger Publishing Company, Malabar, FL, 1965.

    Google Scholar 

  18. TAPLEY, B., SCHUTZ, B. E., and BORN, G. H., Statistical Orbit Determination, Elsevier Science and Technology Series, Elsevier Academic Publishers, Burlington, MA, 2004.

  19. VALLADO, D. A. Fundamentals of Astrodynamics and Applications, McGraw Hill, New York, NY, 1997.

    Google Scholar 

  20. ARULAMPALAM, M. S., MASKELL, S., GORDON, N., and CLAPP, T. “A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking,” IEEE Transactions on Signal Processing, Vol. 50, No. 2, February 2002, pp. 174–188.

    Article  Google Scholar 

  21. PAPOULIS, A. Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York, NY, 1991.

  22. JUNKINS, J. L. and MAJJI, M. “Small Body Proximity Sensing with a Novel HD3D LADAR System,” Proceedings of the AAS Guidance Navigation and Control Conference, Breckenridge, CO, February, 2011.

  23. LERRO, D. and BAR-SHALOM, Y. “Tracking with Debiased Consistent Converted Measurements Versus EKF,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 29, No. 3, July 1993, pp. 1015–1022.

    Article  Google Scholar 

  24. LONGBIN, M., XIAOQUAN, S., YIYU, Z., KANG, S., and BAR-SHALOM, Y. “Unbiased Converted Measurements for Tracking,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 34, No. 3., August, 2002, pp. 1023–1027.

    Article  Google Scholar 

  25. BATTIN, R. An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education Series, American Institute of Aeronautics and Astronautics, Washington, D. C., 1995.

  26. HOW, J. P., BREGER, L. S., MITCHELL, M., ALFRIEND, K. T., and CARPENTER, R. “Differential Semimajor Axis Estimation Performance Using Carrier Phase Differential Global Positioning System Measurements,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 2, March-April 2007, pp. 301–312.

    Article  Google Scholar 

  27. ALFRIEND, K. T. and CARPENTER, R. “Navigation Accuracy Guidelines for Orbital Formation Flying,” NASA GSFC Technical Report, No. 20040079829, 2004, pp. 1–18.

  28. CARPENTER, J. R. and SCHIESSER, E. R. “Semimajor Axis Knowledge and GPS Orbit Determination,” Navigation, Vol. 48, No. 1, Spring 2001, pp. 57–68.

    Google Scholar 

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Authors and Affiliations

  1. Research Associate, Aerospace Engineering Department, Texas A&M University, College Station, TX, 77840

    Manoranjan Majji

  2. Distinguished Professor, Regents Professor, Royce Wisenbaker Chair in Engineering, Aerospace Engineering Department, Texas A&M University, College Station, TX, 77843-3141, Fellow, AAS, Fellow, AIAA

    J. L. Junkins

  3. Research Professor, Aerospace Engineering Department, Texas A&M University, College Station, TX, 77843-3141, Associate Fellow, AAS

    J. D. Turner

Authors
  1. Manoranjan Majji
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  2. J. L. Junkins
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  3. J. D. Turner
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Correspondence to Manoranjan Majji.

Additional information

Dedicated to Professor Kyle T. Alfriend for his contributions in Astronautics.

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Majji, M., Junkins, J.L. & Turner, J.D. Measurement Model Nonlinearity in Estimation of Dynamical Systems. J of Astronaut Sci 59, 41–62 (2012). https://doi.org/10.1007/s40295-013-0005-6

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