ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor Jeffrey S. Parker

1. ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor Jeffrey S. Parker Professor George H. Born Lecture 27: Final Lecture!

2. Announcements • Today’s the final lecture! • Check your grades (for those graded anyway), especially quizzes. • Exam 3 out today – due Monday at midnight unless you get permission otherwise. • Thursday work day in this room. • Everything else due Dec 20. Let me know if this is a problem! Take-Home Exam Due Final Project Due All HW Due

3. Review Topics • The Art of Stat OD • Probability and Statistics • The Process • Then we’ll conclude with a final visit of spaceflight operations in practice.

4. The Art of Stat OD • Set up your scenario • What are you estimating? • Set up • What observations do you have? • Set up • Do you have any a priori information? • Any consider parameters?

5. The Art of Stat OD • Set up your scenario • What are the dynamics? • What are the observation state equations?

6. The Art of Stat OD • Select a Filter • Least Squares • Weighted Least Squares • Minimum-Variance with a priori • P-Norm Filters • Select an algorithm • Batch • Conventional Kalman • Extended Kalman • Others

7. The Art of Stat OD • Make adjustments to account for numerical issues • Joseph formulation to help the P matrix • Square root processes to keep P positive definite • Potter, Cholesky, Givens, Householder • Also improves the conditioning of any matrices • Square root free algorithms to speed up numerical computations • Process noise compensation to avoid filter saturation • State Noise Compensation • Dynamical Model Compensation

8. Analysis • After processing your observations, observe the results. • Is everything observable? • Did the covariance collapse? Diverge? • Do the post-fit residuals appear Gaussian? • If not, perhaps you need to adjust the tuning parameters or the dynamical model. • How do the results compare with other solutions? • Try out all sorts of set-ups, arcs, etc, to see if everything is consistent. • Can you safely reduce the amount of tracking in the future? • Are you meeting your scientific/engineering objectives?

9. Review Topics • The Art of Stat OD • Probability and Statistics • The Process • Then we’ll conclude with a final visit of spaceflight operations in practice.

10. Probability and Statistics • Why do we care? • We need to know something about the expected value and distribution of an error! • Given: • An uncertain state (hopefully with a corresponding covariance) • Noisy observations • Errors in dynamical model • We’d like to know what the “expected value” and “variance-covariance” of our state estimation errors are!

11. Probability and Statistics • What is a probability density function? • What is a cumulative distribution function?

12. Probability and Statistics • What is a conditional probability function? • Why do we care? • Bayesian statistics • We want to be able to compute the best estimate of our state in the presence of our observations!

13. Expected Value and Variance • Given a probability density function, we’d like to be able to compute E[x] and E[(x-xb)(x-xb)T]

14. The Gaussian or Normal Density Function ~68.3% within 1 sigma of mean ~95.4% within 2 sigmas of mean ~99.7% within 3 sigmas of mean ~99.994% within 4 ~99.99994% within 5 ~99.9999998% within 6

15. Variance-Covariance Matrix

16. Review Topics • The Art of Stat OD • Probability and Statistics • The Process • Then we’ll conclude with a final visit of spaceflight operations in practice.

17. Fitting the data • How do we best fit the data? • A good solution, and one easy to code up, is the least-squares solution

18. Fitting the data • How do we best fit the data? Residuals = ε = O-C No No Not bad

19. State Deviation and Linearization • Linearization • Introduce the state deviation vector • If the reference/nominal trajectory is close to the truth trajectory, then a linear approximation is reasonable.

20. State Transition Matrix • The state transition matrix maps a deviation in the state from one epoch to another. • It is constructed via numerical integration, in parallel with the trajectory itself.

21. Measurement Mapping Matrix • The Mapping Matrix

22. Mapping an observation • After these mapping matrices are defined, we can relate an observation to the satellite’s state at any epoch! X* Observed Range Computed Range ε = O-C = “Residual”

23. Different ways to get a best estimate • Least Squares • Weighted Least Squares • Least Squares with a priori • Min Variance • Min Variance with a priori

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

25. 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 =

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

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

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

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

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

31. 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

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

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

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

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

36. 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

37. 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

38. 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

39. 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

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

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

42. 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.

43. 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

44. Filter Options • Batch • Using any of the Least-Squares derivations • Sequential • CKF • EKF • UKF (others)

45. Stat OD Conceptualization • Least Squares (Batch)

46. Stat OD Conceptualization • Conventional Kalman

47. Stat OD Conceptualization • EKF

48. Numerical Issues • As we’ve seen, computers aren’t perfect and Stat OD is a very sensitive subject!

49. Joseph Formulation • Replacewith • This formulation will always retain a symmetric matrix, but it may still lose positive definiteness.

50. Square Root Filter Algorithms • Define W, the square root of P: • Observe that if we have W, then computing P in this manner will always result in a symmetric PD matrix. • Perform measurement updates and time updates on W rather than P. • Often process one observation at a time (rather than a vector of observations). • W may be constructed via Givens, Householder, Cholesky, or other methods.