1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 41: Initial Orbit Determination

2. University of Colorado Boulder  Exam 3 ◦ Available under “Assignments” on the public website ◦ 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 Project Q&A

4. University of Colorado Boulder 4 Lecture Quiz 9

5. University of Colorado Boulder  Correct: 74%  Consider the case of a bi-variate Gaussian distribution. The two random variables are x and y. If there is no correlation and the variance of each random variable is 1, then (in MATLAB notation): ◦ P = [ 1, 0; 0, 1 ] When we plot the probability ellipsoid, we get a circle. If the correlation is instead non-zero, then the semimajor axis of the ellipsoid is rotated w.r.t. the original x-y axes. Since a rotated circle is still a circle, then the ellipsoid for the correlated case is also a circle. Is the last paragraph true or false? 5

6. University of Colorado Boulder  Correct: 57%  When plotting the error ellipsoid, we use the principal axis frame because it is only defined in that frame. ◦ True ◦ False 6

7. University of Colorado Boulder  Correct: 35%  A fixed-interval smoother has the greatest effect on a solution that was generated while using a relatively large process noise term (e.g., trace(Γ*Q*Γ T ) >> trace(Φ*P*Φ T )). ◦ True ◦ False 7

8. University of Colorado Boulder  Correct: 17%  Which of these quantities/filter outputs are required as part of the fixed-interval smoother? ◦ The filter state estimate (for a nonlinear filter, the state deviation vector) at each measurement time. ◦ The state-error covariance matrix computed via the measurement update at each measurement time. ◦ The observation error covariance matrix at each measurement time. ◦ The state transition matrix from the initial time to the time of the last measurement, i.e., Phi(t_l,t_0) where t_l is the final measurement time. 8 70% 83% 57% 52%

9. University of Colorado Boulder 9 Initial Orbit Determination (IOD)

10. University of Colorado Boulder  Up to now, we’ve assume a priori knowledge of the spacecraft state ◦ Required to establish a reference trajectory and a priori information ◦ Need a probabilistic description of the trajectory  Mean  Covariance  How do we get this in the real world? 10

11. University of Colorado Boulder  Without sufficient observations, we have an underdetermined problem ◦ Angle measurements provide no range data 11

12. University of Colorado Boulder  Without sufficient observations, we have an underdetermined problem ◦ Range measurements provide one degree of freedom per measurement ◦ Need a sufficiently large number of observations to resolve orbit in the presence of noise 12

13. University of Colorado Boulder  Without sufficient observations, we have an underdetermined problem ◦ Range measurements provide one degree of freedom per measurement ◦ Need a sufficiently large number of observations to resolve orbit in the presence of noise 13

14. University of Colorado Boulder  Without sufficient observations, we have an underdetermined problem ◦ Range and range-rate help, but there is still too much ambiguity 14

15. University of Colorado Boulder  Different methods for different measurement types ◦ Angles-only IOD  Gauss’s Method  Double r-iteration  Admissible Region ◦ Range-only  Homotopy Continuation ◦ Range and Range-Rate  Trilateration 15

16. University of Colorado Boulder  Different methods for different measurement types ◦ Position Vectors  Gibbs Method  Herrick-Gibbs  Lambert’s Problem ◦ Angles, range, and range-rate  Admissible Region 16

17. University of Colorado Boulder  Angles-only IOD required for surveys of GEO debris ◦ Radar requires too much power ◦ Optical observations typically used for objects at higher altitudes  If we see a new debris object, how do we determine an initial orbit for follow-up tracking? 17

18. University of Colorado Boulder  Use three non-zero, coplanar observations of position to estimate the velocity for one observation ◦ Is there a situation where we could have such measurements? 18

19. University of Colorado Boulder  By assuming the position vectors are coplanar, one may be represented as a linear combination of the other two  Method breaks down for small separation due to cross products in algorithm 19

20. University of Colorado Boulder  More accurate with small angular separation in the measurements  Based on a truncated Taylor series expansion ◦ Represent first and third vectors as deviations from the second ◦ Allows for estimating the velocity for the reference ◦ What happens with large separations?  Close small angule separations yield solutions more sensitive to measurement errors ◦ Leads to the “too-short arc” (TSA) problem 20

21. University of Colorado Boulder  Assumptions ◦ Three observations ◦ Less than 60 deg separation (aids accuracy) ◦ All three observations lie in a single plane: 21 Why is this equation valid?

22. University of Colorado Boulder  Provides an estimate of the state at the time of the second observation  No state covariance information 22

23. University of Colorado Boulder  Key elements of derivation: ◦ Use the angles and the ground stations to generate approximate position vectors ◦ Use the three position vectors with Gibbs or Herrick-Gibbs to get velocity ◦ Iterate to account for errors in position vector approximation ◦ Assume that the observations are perfect, i.e., no statistical error 23

24. University of Colorado Boulder  Each method has different strengths and weaknesses ◦ Method selected is typically based on the observations available  None fundamentally provide a PDF ◦ All assume deterministic trajectories  Vallado provides a good summary of the classic methods in his book 24

25. University of Colorado Boulder 25 IOD via the Admissible Region

26. University of Colorado Boulder  Introduced in early-2000s to address the TSA problem  Leverages an observation as a constraint on the solution “parallel” to the measurement  Use constraints to restrict the space of possible solutions in the directions “orthogonal” to the measurement  Refine knowledge of the orbit with follow-up tracking 26

27. University of Colorado Boulder  For optical observations:  Given a time series of right ascension and declination measurements, how do we get the angle rates? 27

28. University of Colorado Boulder  For optical observations:  What do we do about the range and range- rate directions? 28

29. University of Colorado Boulder  What are some reasonable constraints on an orbit? 29

30. University of Colorado Boulder  We can include a constraint based on upper/lower limits of the semimajor axis 30

31. University of Colorado Boulder  We can include a constraint based on an upper limits in the eccentricity 31

32. University of Colorado Boulder  We can combine them to further constrain the space of solutions 32

33. University of Colorado Boulder  “Direct Bayesian” Method ◦ Fujimoto and Scheeres ◦ Following slides from Kohei Fujimoto as part of ASEN 6519: Orbital Debris (Fall 2012)  (Currently plan to teach it again in Fall 2015) 33

34. Adm. Region ASEN 6519 - Orbital Debris34 Direct Bayesian approach

35. Adm. Region ASEN 6519 - Orbital Debris35 Direct Bayesian approach

36. University of Colorado Boulder  “Direct Bayesian” Method ◦ Fujimoto and Scheeres  Allows for hypothesis-free correlation of two tracks ◦ From there, IOD may be performed ◦ Allow for correlation of tracks over relatively large time spans 36

37. University of Colorado Boulder  “Virtual Asteroids” ◦ Milani, et al. 37 Image Credit: Milani and Knežević, 2005

38. University of Colorado Boulder  “Probabilistic Description” ◦ DeMars and Jah 38 Image Credit: Jones, et al., 2014

39. University of Colorado Boulder  Admissible-region based methods in the literature for radar-based measurements 39 Image Credit: Tommei, et al., 2007

40. University of Colorado Boulder  Relatively new way to approach the IOD problem  Has its own set of advantages and disadvantages ◦ Advantages:  Only one observation required to create track hypotheses  May be use to generate an initial-state PDF ◦ Disadvantages:  Still need follow-up observation to refine orbit (not always easy)  Make assumptions on the class of orbit based on constraints  Needs some pre-processing to get 4-D observation vector 40

41. University of Colorado Boulder 41 Summary

42. University of Colorado Boulder  In the case of an operational spacecraft with adequate communication coverage, IOD is relatively easy ◦ Of course, this assumes that you can communicate with the spacecraft  For the case of passive tracking, IOD is not trivial ◦ This is a very active area of research right now in the debris and asteroid communities 42