1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 37: SNC Example and Solution Characterization

2. University of Colorado Boulder  Homework 11 due on Friday ◦ Sample solutions will be posted online  Lecture quiz due by 5pm on Friday  Exam 3 Posted On Friday ◦ In-class Students: Due December 12 by 5pm ◦ CAETE Students: Due 11:59pm (Mountain) on 12/14  Final Project Due December 15 by noon 2

3. University of Colorado Boulder 3 Homework 11

4. University of Colorado Boulder  Leverage code from HW10 ◦ New data set generated with a different force model ◦ Otherwise, same format, data noise, etc.  Process observations in existing filter ◦ Do not add J 3 to your filter model! ◦ Observe the effects of such errors on OD ◦ Add process noise to improve state estimation accuracy 4

5. University of Colorado Boulder 5

6. University of Colorado Boulder 6 3σ

7. University of Colorado Boulder 7 Application of SNC to Ballistic Trajectory

8. University of Colorado Boulder 8  Ballistic trajectory with unknown start/stop  Red band indicates time with available observations Start of filter Obs. Stations

9. University of Colorado Boulder  Object in ballistic trajectory under the influence of drag and gravity 9  Nonlinear observation model ◦ Two observations stations

10. University of Colorado Boulder 10

11. University of Colorado Boulder  Now use an EKF  We will vary the truth model to study the benefits of SNC ◦ Look at two cases:  Run each with and without a process noise model  Error in gravity (g = 9.8 m/s vs. 9.9 m/s)  Error in drag (b = 1e-4 vs. 1.1e-4) 11

12. University of Colorado Boulder 12  Blue – Range  Green – Range-Rate Station 1 Station 2

13. University of Colorado Boulder 13

14. University of Colorado Boulder  Added SNC to the filter: 14  Why is the term for x-acceleration smaller?

15. University of Colorado Boulder 15  Blue – Range  Green – Range-Rate Station 1 Station 2

16. University of Colorado Boulder 16  178.3 vs. 0.8 meters RMS

17. University of Colorado Boulder 17  Blue – Range  Green – Range-Rate Station 1 Station 2

18. University of Colorado Boulder 18

19. University of Colorado Boulder  Added SNC to the filter: 19

20. University of Colorado Boulder 20  Blue – Range  Green – Range-Rate Station 1 Station 2

21. University of Colorado Boulder 21  27.6 vs. 1.26 meters RMS

22. University of Colorado Boulder  Mitigation of the gravity acceleration error yielded better results than the drag error case. Why could that be? 22

23. University of Colorado Boulder 23 Solution Characterization

24. University of Colorado Boulder  Truncation error (linearization)  Round-off error (fixed precision arithmetic)  Mathematical model simplifications (dynamics and measurement model)  Errors in input parameters (e.g., J 2 )  Amount, type, and accuracy of tracking data 24

25. University of Colorado Boulder  For the Jason-2 / OSTM mission, the OD fits are quoted to have errors less than centimeter (in radial) ◦ How do they get an approximation accuracy? ◦ Residuals?  Depends on how much we trust the data  Provides information on fit to data, but solution accuracy? ◦ Covariance Matrix?  How realistic is the output covariance matrix?  (Actually, I can make the output matrix whatever I want through process noise or other means.) 25

26. University of Colorado Boulder  Characterization requires a comparison to an independent solution ◦ Different solution methods, models, etc. ◦ Different observations data sets:  Global Navigation Satellite Systems (GNSS) (e.g., GPS)  Doppler Orbitography and Radio-positioning Integrated by Satellite (DORIS)  Satellite Laser Ranging (SLR)  Deep Space Network (DSN)  Delta-DOR  Others…  Provides a measure based on solution precision 26

27. University of Colorado Boulder  Jason-2 / OSTM positions solutions generated by/at: ◦ JPL – GPS only ◦ GSFC – SLR, DORIS, and GPS ◦ CNES – SLR, DORIS, and GPS  Algorithms/tools differ by team: ◦ Different filters ◦ Different dynamic/stochastic models ◦ Different measurement models 27

28. University of Colorado Boulder  1 Cycle = approximately 10 days  Differences on the order of millimeters 28 Image: Bertiger, et al., 2010

29. University of Colorado Boulder  Compare different fit intervals: 29

30. University of Colorado Boulder  Consider the “abutment test”: 30

31. University of Colorado Boulder  Each data fit at JPL uses 30 hrs of data, centered at noon 31  This means that each data fit overlaps with the previous/next fit by six hours  Compare the solutions over the middle four hours ◦ Why?

32. University of Colorado Boulder  Histogram of daily overlaps for almost one year  Imply solution consistency of ~1.7 mm  This an example of why it is called “precise orbit determination” instead of “accurate orbit determination” 32 Image: Bertiger, et al., 2010

33. University of Colorado Boulder  In some case, we can leverage observations (ideally not included in the data fit) to estimate accuracy  How might we use SLR to characterize radial accuracy of a GNSS-based solution? 33

34. University of Colorado Boulder  Results imply that the GPS-based radial error is on the order of millimeters  Why is the DORIS/SLR/GPS solution better here? 34 Image: Bertiger, et al., 2010

35. University of Colorado Boulder  Must consider independent state estimates and/or observations  Not an easy problem, and the method of characterization is often problem dependent ◦ How do you think they do it for interplanetary missions? 35