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ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones

ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 15: Statistical Least Squares and Estimation of Nonlinear System. Announcements. Lecture Quiz Due by 5pm Homework 5 Due Friday Exam 1 – Friday, October 11. Today’s Topics.

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ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 15: Statistical Least Squares and Estimation of Nonlinear System

  2. Announcements • Lecture Quiz Due by 5pm • Homework 5 Due Friday • Exam 1 – Friday, October 11

  3. Today’s Topics • Statistical Least Squares w/ a priori • SLS and Estimation of Nonlinear System

  4. Statistical Interpretation of Least Squares

  5. Weighted Least Squares

  6. Observation Errors

  7. State Estimation Error Description

  8. State Estimation Error Description

  9. State Estimation Error Description

  10. Statistical LS w/ a priori

  11. Statistical LS w/ a priori

  12. Statistical LS w/ a priori

  13. Statistical LS w/ a priori

  14. State Estimation Error Description

  15. Measurement Mapping • Still need to know how to map measurements from one time to a state at another time!

  16. State Update • Since we linearized the formulation, we can still improve accuracy through iteration (more on this in a future lecture)

  17. Statistical Least Squares Solution for Nonlinear System

  18. Computation Algorithm of the Batch Processor p. 196-197 of textbook (includes corrections)

  19. Computation Algorithm for the Batch Processor

  20. Why Reuse A Priori Information?

  21. Assumptions with the Iterated Batch • The batch filter depends on the assumptions of linearity • Violations of this assumption may lead to filter divergence • If the reference trajectory is near the truth, this holds just fine • The batch processor must be iterated 2-3 times to get the best estimate • The iteration reduces the linearization error in the approximation • Continue the process until we “converge” • Definition of convergence is an element of filter design

  22. Post-fit Residuals RMS

  23. Convergence via Post-fit Residuals • If we know the observation error, why “fit to the noise”?

  24. Other Convergence Tests • No improvement in observation RMS • No reduction in state deviation vector • Maximum number of iterations

  25. LEO Orbit Determination Example • Instantaneous observation data is taken from three Earth-fixed tracking stations • Why is instantaneous important in this context? • x, y, z – Satellite positionin ECI • xs, ys, zs are tracking station locations in ECEF

  26. Effects of Iteration

  27. Improved Fit to Data

  28. Estimated State Uncertainty

  29. Estimated State Uncertainty

  30. Estimated State Uncertainty

  31. Advantage of Different Data Types • FLIR – Forward-looking infrared (FLIR) imaging sensor Image: Hall and Llinas, “Multisensor Data Fusion”, Handbook of Multisensor Data Fusion: Theory and Practice, 2009.

  32. Batch Processor Issues • Inverting a potentially poorly scaled matrix • Solutions: • Matrix Decomposition (e.g., Singular Value Decomposition) • Orthogonal Transformations • Square-root free Algorithms • Numeric Issues • Resulting covariance matrix not symmetric • Becomes non-positive definite (bad!)

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