1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 39: Measurement Modeling and Combining State Estimates

2. University of Colorado Boulder  No lecture quiz next week.  Exam 3 Posted Today ◦ 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  Your solutions must be uploaded to D2L as a searchable PDF ◦ Same rules as homework apply in regards to format, code appendices, etc.  Open-book, open notes  You may use a computer, MATLAB, etc.  Honor code rules apply ◦ Do not give or ask for help from your peers  The TA has been instructed to redirect all questions to the instructor  I can answer questions to clarify what is being asked, but cannot provide guidance on solutions 3

4. University of Colorado Boulder 4 Project Q&A

5. University of Colorado Boulder 5 Kalman Filter Discussion

6. University of Colorado Boulder  In the project, we are estimating C D. However, the partial of the measurements w.r.t. C D is zero. How are we able to estimate this parameter in the Kalman filter? 6

7. University of Colorado Boulder  We use a process noise model where What happens in our filter if Q k = σ 2 I and the second term above is much bigger than the first? 7

8. University of Colorado Boulder 8 Modeling Measurements Tapley, Schutz, and Born, Chapter 3 Montenbruck and Gill, Satellite Orbits, Chapter 6

9. University of Colorado Boulder  One-way Range ◦ Example: GNSS ◦ Signal travels to/from reference from/to satellite 9

10. University of Colorado Boulder  Two-way Range ◦ Examples: SLR, DSN ◦ Satellite is a relay for signal 10

11. University of Colorado Boulder  Multi-way Range ◦ Examples: DSN, TDRSS ◦ Multiple satellite and/or ground stations used 11

12. University of Colorado Boulder  We have been using range and range-rate:  In the real world, what is wrong with these equations? 12

13. University of Colorado Boulder  At best, a signal travels at the speed of light  We must approximate the signal propagation time δt  Approximately 0.06 seconds for GPS signal to reach Earth  A LEO spacecraft will have moved approximately 500 meters in that time 13

14. University of Colorado Boulder  Assume we have estimates of our satellite trajectory and the reference station/satellite  We need to solve for δt  No analytic solution so we solve for the correction using iteration 14

15. University of Colorado Boulder  Start with δt=0  Compute the distance with the satellite state at time t and the reference state at t-δt  Given that distance, compute the light propagation time Δδt  Set δt=δt+Δδt  Continue until Δδt is sufficiently small 15

16. University of Colorado Boulder  We have taken care of light-time correction assuming the speed of light in a vacuum.  Any other things we should account for? ◦ Signal does not always propagate through a vacuum  Ionosphere  Troposphere  Charged particle interactions  Solar corona  etc. ◦ Coordinate and time systems  This requires a very careful treatment in the filter 16

17. University of Colorado Boulder  Not including accurate coordinate system information creates systematic errors. ◦ Violates our assumption of random errors! ◦ Creates a time-varying bias in the measurement 17 Table courtesy of Bradley, et al., 2011

18. University of Colorado Boulder  Is it possible to measure range-rate instantaneously? ◦ No! (at least not that I am aware of) ◦ We have to observe this indirectly  Instead, we look at the change in a signal over time to approximate the change in velocity. 18

19. University of Colorado Boulder  Satellite sends pulse at fixed interval 19

20. University of Colorado Boulder 20

21. University of Colorado Boulder  The velocity of the spacecraft affects the frequency of any radar signal ◦ Some of the effects illustrated in Jason Leonard’s talk on using real data  Requires us to observe the change in frequency over some period of time ◦ Known as integrated Doppler shift 21 Image Courtesy of WikiCommons

22. University of Colorado Boulder  The velocity of the spacecraft affects the frequency of any radar signal ◦ Some of the effects illustrated in Jason Leonard’s talk on using real data  Requires us to observe the change in frequency over some period of time ◦ Known as integrated Doppler shift 22 Image Courtesy of WikiCommons

23. University of Colorado Boulder  Assumes: ◦ Linear change in range over integration time ◦ Constant transmission frequency over integration time 23

24. University of Colorado Boulder  Do I need to perform any light time correction?  Is there anything different about this case when compared to range? 24

25. University of Colorado Boulder 25 Combining State Estimates

26. University of Colorado Boulder  A ground station in Maui observed our satellite several times over the past week ◦ Generated a filtered solution using their observations  A ground station in Florida also observed the satellite several times over the past week ◦ Generated a filter solution using their observations  What is the best approach to fusing this information? 26

27. University of Colorado Boulder  Treat one solution as the a priori and the other as the observation ◦ Does it matter which one is which?  For this case, H=I 27

28. University of Colorado Boulder  Does not require the additional processing of observations 28

29. University of Colorado Boulder 29