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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 16: Iteration of Least Squares and Propagation of Filter Output. Announcements. No homework or lecture quiz for the next week Exam 1 – Friday, October 11

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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 16: Iteration of Least Squares and Propagation of Filter Output

  2. Announcements • No homework or lecture quiz for the next week • Exam 1 – Friday, October 11 • Exam Review on Monday • E-mail Marco if there is anything you would like him to discuss on Monday • Change in Marco’s office hours • Friday 3-4pm move to: • Thursday, 4-5pm

  3. Orbit Determination via Batch Processor

  4. State Update • Since we linearized the formulation, we can still improve accuracy through iteration • How do we perform this iteration?

  5. Computation Algorithm of the Batch Processor

  6. Computation Algorithm for the Batch Processor

  7. Why Reuse A Priori Information?

  8. 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 • Continue the process until we “converge” • Definition of convergence is an element of filter design

  9. Post-fit Residuals RMS

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

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

  12. 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

  13. Effects of Iteration

  14. Improved Fit to Data

  15. Estimated State Uncertainty

  16. Estimated State Uncertainty

  17. Estimated State Uncertainty

  18. 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.

  19. 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!)

  20. Simple Symmetry Fix • Only to be used when non-symmetric due to floating point issues!

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