Skip to main content
SpringerLink
Log in

A high order method for estimation of dynamic systems

  • Published:
The Journal of the Astronautical Sciences Aims and scope Submit manuscript

Abstract

An analytical approach is presented for developing an estimation framework (called the Jth Moment Extended Kalman Filter (JMEKF)). This forms an important component of a class of architectures under investigation to study the interplay of major issues in nonlinear estimation such as model nonlinearity, measurement sparsity and initial condition uncertainty in the presence of low process noise. Utilizing an automated nonlinear expansion of the model about the current best estimated trajectory, a Jth order approximate solution for the departure motion dynamics about a nominal trajectory is derived in the form of state transition tensors. This solution is utilized in evaluating the evolution of statistics of the departure motion as a function of the statistics of initial conditions. The statistics thus obtained are used in the determination of a state estimate assuming a Kalman update structure. Central to the state transition tensor integration about a nominal trajectory, is the high order sensitivity calculations of the nonlinear models (dynamics and measurement), being automated by OCEA (Object Oriented Coordinate Embedding Method), a computational tool generating the required various order partials of the system differential equations without user intervention. Working in tandem with an OCEA automation of the derivation of the state transition tensor differential equations is a vector matrix representation structure of tensors of arbitrary rank, facilitating faster and more accurate computations. High order moment update equations are derived to incorporate the statistical effects of the innovations process more rigorously, improving the effectiveness of the estimation scheme. Numerical simulations on an orbit estimation example investigate the gain obtained in using the proposed methodology in situations where the classical extended Kalman filter’s domain of convergence is smaller. The orbit estimation example presented examines a situation that requires us to determine the position and velocity state of the orbiter from range, azimuth and elevation measurements being made available sparsely.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. KALMAN, R. E. “A New Approach to Linear Filtering and Prediction Problems,” Transactions of the ASME Journal of Basic Engineering, 1960, Vol. 82, No. D, pp. 35–45.

    Article  Google Scholar 

  2. MYERS, K. A. and TAPLEY, B.D. “Dynamical Model Compensation for Near-Earth Satellite Orbit Determination,” AIAA Journal, 1975, Vol. 13, No. 3, pp. 343–349.

    Article  MATH  Google Scholar 

  3. MYERS, K. A. and TAPLEY, B.D. “Adaptive Sequential Estimation with Unknown Noise Statistics,” IEEE Transactions on Automatic Control, 1976, Vol. 21, No. 4, pp. 520–523.

    Article  MATH  Google Scholar 

  4. TAPLEY, B.D. and BORN, G. H. “Sequential Estimation of the State and the Observation-Error Covariance Matrix,” AIAA Journal, 1971, Vol. 9, No. 2, pp. 212–217.

    Article  Google Scholar 

  5. TAPLEY, B.D. and INGRAM, D. S. “Orbit Determination in the Presence of Unmodeled Accelerations,” IEEE Transactions on Automatic Control, 1973, Vol. 18, No. 4, pp. 364–373.

    Article  Google Scholar 

  6. TAPLEY, B. D. and SCHUTZ, B. E. “Estimation of Unmodeled Forces on a Lunar Satellite,” Celestial Mechanics and Dynamical Astronomy, 1975, Vol. 12, No. 4, pp. 409–424.

    Google Scholar 

  7. TAPLEY, B.D., SCHUTZ, B. E., and BORN, G. H. “Statistical Orbit Determination,” Elsevier Science and Technology Books, April, 2004, Burlington, MA, Elsevier Academic Press.

    Google Scholar 

  8. CRASSIDIS, J. L. and JUNKINS, J. L. Optimal Estimation of Dynamic Systems, Applied Mathematics and Nonlinear Science Series, 2004, Boca Raton, FL, Chapman and Hall/CRC Press.

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. GELB, A. ed. Applied Optimal Estimation, 1974, The M.I.T. Press, Reading, MA.

    Google Scholar 

  11. 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 

  12. ATHANS, M., WISHNER, R., and BERTOLINI, A. “Suboptimal State Estimation for Continuous-Time Nonlinear Systems from Discrete Noisy Measurements,” IEEE Transactions of Automatic Control, 1968, Vol. 13, No. 5, pp. 504–514.

    Article  Google Scholar 

  13. GUSTAFSSON, F., GUNNARSSON, F., BERGMAN, N., JANSSON, U. F. J., KARLSON, R., and NORDLUND, P. J. “Particle Filters for Positioning, Navigation and Tracking,” IEEE Transactions on Signal Processing, 2002, Vol. 50, No. 2, pp. 425–437.

    Article  Google Scholar 

  14. 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 

  15. 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, 2006, Vol. 29, No. 6, pp. 1367–1375.

    Article  Google Scholar 

  16. WISHNER, R. P., TABACZYNSKI, J. A., and Athans, M. “A Comparison of Three Nonlinear Filters,” Automatica, 1969, Vol. 5, pp. 487–496.

    Article  MATH  Google Scholar 

  17. NORGAARD, M., POULSEN, N.K., and RAVN, O. “New Developments in State Estimation of Nonlinear Systems,” Automatica, 2000, Vol. 36, pp. 1627–1638.

    Article  MathSciNet  Google Scholar 

  18. WIBERG, D.M. “The MIMO Wiberg Estimator,” Proceedings of the IEEE 28th Conference on Decision and Control, 1989, Tampa, FL.

    Google Scholar 

  19. GRIFFITH, D. T., TURNER, J. D., and JUNKINS, J. L. “An Embedded Function Tool for Modeling and Simulating Estimation Problems in Aerospace Engineering,” Proceedings of the 2004 AAS-AIAA Space Flight Mechanics Meeting, 2004, Advances in Astronautical Science, American Astronautical Society, Maui, Hawaii.

    Google Scholar 

  20. BADER, B.W. and KOLDA, T. G. “Algorithm 862: MATLAB Tensor Classes for Fast Algorithm Prototyping,” ACM Transactions on Mathematical Software, 2006, Vol. 32, No. 4, pp. 635–653.

    Article  MathSciNet  MATH  Google Scholar 

  21. LAWDEN, D. F. “Introduction to Tensor Calculus, Relativity and Cosmology,” 3rd ed., 2002, Mineola, NY, Dover Publications.

    Google Scholar 

  22. TURNER, J. D. “Automatic Generation of High-Order Partial Derivative Models,” AIAA Journal, 2003, Vol. 41, No. 8.

    Google Scholar 

  23. GRAHAM, A. “Kronecker Products and Matrix Calculus with Applications,” Ellis Horwood Series: Mathematics and its Applications, ed. G.M. Bell. 1981, New York, NY, John Wiley & Sons.

    Google Scholar 

  24. RAUHALA, U. A. “Array Algebra Expansion of Matrix and Tensor Calculus: Part 1,” SIAM Journal of Matrix Analysis and Applications, 2002, Vol. 24, No. 2, pp. 490–508.

    Article  MathSciNet  MATH  Google Scholar 

  25. GOLUB, G. H. and VANLOAN, C. F. Matrix Computations, Johns Hopkins Studies in Mathematical Sciences, 1996, Baltimore, MD, The Johns Hopkins University Press.

    Google Scholar 

  26. BATE, R. R., MUELLER, D. D., and WHITE, J. E. Fundamentals of Astrodynamics, Dover Books on Astronomy, 1971, New York, Dover Publications.

    Google Scholar 

  27. BAR-SHALOM, Y., LI, X. R., and KIRUBARAJAN, T. Estimation with Applications to Tracking and Navigation, 2001, New York, NY, John Wiley and Sons.

    Book  Google Scholar 

  28. TURNER, J. D., MAJJI, M., and JUNKINS, J. L. “High-Order State and Parameter Transition Tensor Calculations,” Proceedings of the 2008 AIAA Guidance Navigation and Control Conference and Exhibit, 2008, American Institute of Aeronautics and Astronautics, Honolulu, HI.

    Google Scholar 

  29. JUNKINS, J. L., MAJJI, M., and TURNER, J.D. “High Order Keplerian State Transition Tensors,” Proceedings of the F. Landis Markley Astronautics Symposium, 2008, Cambridge, MD, Univelt Publishing Company, American Astronautical Society.

    Google Scholar 

  30. JUNKINS, J. L., AKELLA, M. R., and ALFRIEND, K. T. “Non-Gaussian Error Propagation in Orbital Mechanics,” The Journal of the Astronautical Sciences, 1996, Vol. 44, No. 4, pp. 541–563.

    Google Scholar 

  31. JUNKINS, J. L. “Adventures on the Interface of Dynamics and Control,” Journal of Guidance, Control and Dynamics, 1997, Vol. 20, No. 6, pp. 1058–1071.

    Article  MATH  Google Scholar 

  32. LAKSMIKANTHAM, V. and DEO, S.G. Method of Variation of Parameters for Dynamic Systems, Series in Mathematical Analysis and Applications, ed. R. P. Agarwal, and O’Regan, D. Vol. 1. 1998, Boca Raton, FL, Gordon and Breach Science Publishers/CRC Press. 328.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Aerospace Engineering Department, 3141 TAMU, USA

    Manoranjan Majji (Graduate Research Assistant, Student Member, AAS), John L. Junkins (Regents Professor, Distinguished Professor of Aerospace Engineering Holder of the Royce E. Wisenbaker Chair in Engineering, Fellow, AAS) & James D. Turner (Research Professor, Director of Operations, Consortium for Autonomous Space Systems (CASS), Senior Member, AAS)

Authors
  1. Manoranjan Majji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John L. Junkins
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James D. Turner
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Professor Byron D. Tapley, celebrating 50 years of his academic and research contributions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Majji, M., Junkins, J.L. & Turner, J.D. A high order method for estimation of dynamic systems. J of Astronaut Sci 56, 401–440 (2008). https://doi.org/10.1007/BF03256560

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03256560

Keywords

Search

Navigation