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ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor George H. Born

ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 12: The Kalman Filter. Announcements. Homework 5 due Today Exam on 10/11. (Anyone going to miss it?)

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ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor George H. Born

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  1. ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 12: The Kalman Filter

  2. Announcements • Homework 5 due Today • Exam on 10/11. (Anyone going to miss it?) • Eduardo and/or Paul will be reviewing subjects on Tuesday – send them emails with questions/subjects that you’d like them to cover. • 1 hour, open book, open notes. • Topics: Definitions of variables, Probability/Statistics Observability, Linearization Least squares, Batch processor

  3. Quiz Results If you took the quiz, you scored 100%

  4. Quiz Results • Some feedback: • Biggest issues: • Moving pretty fast • Lectures and HW don’t correlate well • Review next week: • Review of variables • Stat OD example from start to finish (something plain and easy to follow)

  5. Review of the Stat OD Process • Setup. • Given: an initial state • Optional: an initial covariance

  6. Review of the Stat OD Process • Setup. • Given: an initial state • Optional: an initial covariance • The satellite will not be there, but will (hopefully) be nearby • True state =

  7. Review of the Stat OD Process • What really happens • Satellite travels according to the real forces in the universe

  8. Review of the Stat OD Process • What really happens • Of course, we don’t know this!

  9. Review of the Stat OD Process • Model reality as best as possible • Propagate our initial guess of the state

  10. Review of the Stat OD Process • Goal: Determine how to modify to match

  11. Review of the Stat OD Process • Goal: Determine how to modify to match • Define • Want

  12. Review of the Stat OD Process • Process: • Track satellite • Map observations to state deviation • Determine how to adjust the state to bestfit the observations • Define • Want

  13. Review of the Stat OD Process • Process: • Track satellite Perfect Observations

  14. Review of the Stat OD Process • Process: • Track satellite Perfect Observations Computed Observations

  15. Review of the Stat OD Process • Process: • Track satellite

  16. Review of the Stat OD Process • Process: • Track satellite • Map observations to state deviation

  17. Review of the Stat OD Process • Process: • Track satellite • Map observations to state deviation • Determine how to adjust the state to bestfit the observations Least Squares

  18. Review of the Stat OD Process • Process: • Track satellite • Map observations to state deviation • Determine how to adjust the state to bestfit the observations • Apply and repeat Least Squares

  19. Review of the Stat OD Process • Process: • Track satellite • Map observations to state deviation • Determine how to adjust the state to bestfit the observations • Apply and repeat

  20. Review of the Stat OD Process • Process: • Track satellite • Map observations to state deviation • Determine how to adjust the state to bestfit the observations • Apply and repeat

  21. Review of the Stat OD Process • Process: • Track satellite • Map observations to state deviation • Determine how to adjust the state to bestfit the observations • Apply and repeat Small errors due to mismodeled dynamics

  22. Review of the Stat OD Process • Process: • Track satellite Perfect Observations

  23. Review of the Stat OD Process • Process: • Track satellite Imperfect Observations

  24. Review of the Stat OD Process • Process: • Track satellite • Map observations to state deviation • Determine how to adjust the state to bestfit the observations • Apply and repeat Same process, but the best estimate trajectory will never quite match the truth, since the observations have noise.

  25. Review of the Stat OD Process • Process: • Track satellite • Map observations to state deviation • Determine how to adjust the state to bestfit the observations • Apply and repeat Least Squares

  26. Least Squares Options • Least Squares • Weighted Least Squares • Least Squares with a priori • Min Variance • Min Variance with a priori

  27. Algorithm Options • Batch • Process all observations at once • Sequential • Process one observation at a time

  28. Batch Processor • Collect mapped information

  29. Batch Processor • Collect mapped information

  30. Batch Processor • Collect mapped information

  31. Batch Processor • Collect mapped information

  32. Batch Processor • Collect mapped information

  33. Batch Processor • Collect mapped information

  34. (Break) • TAs will go through an end-to-end Batch run to demo equations, etc. • Next up: Kalman Filter

  35. Sequential Processor • Consider • Rather than mapping all observations to one epoch and processing them simultaneously, what if we processed each separately and mapped the best estimate through each?

  36. Sequential Processor • Consider • Rather than mapping all observations to one epoch and processing them simultaneously, what if we processed each separately and mapped the best estimate through each?

  37. Sequential Processor • Given an a priori state and covariance, we know how to generate a new estimate of the state: • Need a way to generate the a posteriori covariance matrix as well. • Recall • The trouble is inverting the n x n matrix.

  38. Sequential Processor • We can use the Schur Identity and a bunch of math (see Section 4.7) and obtain:

  39. Sequential Processor • We can use the Schur Identity and a bunch of math (see Section 4.7) and obtain: Kalman Gain

  40. Sequential Processor • After some more math, we can simplify to obtain:

  41. Sequential Algorithm • 1. Initialize the first run • 2. Start at the reference epoch • 3. Time Update • Integrate from the current time to the next time of interest • Map the state estimate and the covariance to the new time • 4. Measurement Update • If there is a new measurement, process it. • Update the state estimate and covariance with this new information • Repeat 3-4 until all measurements have been processed and all times of interest have been recorded. • Optional: Map the estimate and covariance back to the reference epoch and iterate the whole process.

  42. Sequential Algorithm • Initialization

  43. Sequential Algorithm • Time Update • Integration • Mapping

  44. Sequential Algorithm • Measurement Update • Collect measurement • Compute update

  45. Sequential Processor • Repeat just like the Batch • Replace the reference trajectory with the new best estimate. • Make sure to update the a priori state deviation vector to retain any information. • Recompute all observation residuals

  46. Sequential Flow-Chart

  47. Batch Processor • Collect mapped information

  48. Batch Processor • Collect mapped information

  49. Batch Processor • Collect mapped information

  50. Batch Processor • Collect mapped information

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