1. University of Colorado Boulder ASEN 6008 Interplanetary Mission Design Statistical Orbit Determination A brief overview 1

2. University of Colorado Boulder  Before we start Stat OD, let’s talk about homework.  Eduardo is angry. 2

3. University of Colorado Boulder  Watch your significant figures!  We don’t need to know velocities to the pm/s.  For most situations (especially HW in this class) ◦ Positions: meter level accuracy ◦ Velocities: m/s or cm/s accuracy 3

4. University of Colorado Boulder  Dawn spacecraft is nearing dwarf planet Ceres  Ceres orbit is between Mars and Jupiter in the main asteroid belt  Ceres was discovered by Father Giuseppe Piazzi 1801. ◦ Ceres was initially classified as a planet and later demoted an asteroid ◦ Ceres was upgraded a dwarf planet in 2006, along with Pluto (downgraded) and Eris. 4

5. University of Colorado Boulder  Ceres will enter into orbit on March 6, 2015  First mission to visit a dwarf planet ◦ Barely beats New Horizons. 5

6. University of Colorado Boulder  Which missions require spacecraft navigation? 6

7. University of Colorado Boulder  Which missions require spacecraft navigation?  ALL OF THEM. 7

8. University of Colorado Boulder  Orbit determination is an essential part of any mission  There are (multiple) courses here at CU devoted to the art of OD ◦ ASEN 5050, ASEN 6080  If you have any future interest in trajectory design, orbit determination is a highly useful (i.e., essential) skill to have 8

9. University of Colorado Boulder  Do we actually know exactly where a spacecraft is? ◦ No, there are many sources of error ◦ Modeling errors ◦ Launch errors ◦ Spacecraft performance ◦ Observation errors 9

10. University of Colorado Boulder  Tracking data may include many types of data – and often should include many types of data: ◦ Ground observations:  Doppler  Range  1-Way, 2-Way, 3-Way  Angles when very near the Earth  Delta-DOR when further from Earth ◦ Relative to other spacecraft, vehicles, bodies  GPS  Autonav  LiAISON  Formation Flying ◦ Spacecraft measurements:  Accelerations, including drag-free corrections, thrust, etc.  Measured mass-flow  Attitude measurements 10

11. University of Colorado Boulder  We have observations of a spacecraft at different points in time. How can we estimate its state? 11 X*X* Measurements

12. University of Colorado Boulder  Estimate the state using a filter 12 Observed Range Computed Range ε = O-C = “Residual” X*X*

13. University of Colorado Boulder  What really happens ◦ Satellite travels according to the real forces in the universe ◦ We model the motion to the best of our ability, but our force models contain errors 13

14. University of Colorado Boulder  Setup. ◦ Given: an initial state ◦ Optional: an initial covariance 14

15. University of Colorado Boulder  Setup. ◦ Given: an initial state ◦ Optional: an initial covariance ◦ The satellite will not be there, but will (hopefully) be nearby  True state = 15

16. University of Colorado Boulder  What really happens ◦ Of course, we don’t know this! 16

17. University of Colorado Boulder  Model reality as best as possible  Propagate our initial guess of the state 17

18. University of Colorado Boulder  Goal: Determine how to modify to match 18

19. University of Colorado Boulder  Goal: Determine how to modify to match 19  Define  Want

20. University of Colorado Boulder  Process: 1.Track satellite 2.Map observations to state deviation 3.Determine how to adjust the state to best fit the observations 20  Define  Want

21. University of Colorado Boulder  Process: 1.Track satellite 21 Perfect Observations

22. University of Colorado Boulder  Process: 1.Track satellite 22 Perfect Observations Computed Observations

23. University of Colorado Boulder  Process: 1.Track satellite 23

24. University of Colorado Boulder  Process: 1.Track satellite 2.Map observations to state deviation 24

25. University of Colorado Boulder  Process: 1.Track satellite 2.Map observations to state deviation 3.Determine how to adjust the state to best fit the observations 25 Least Squares

26. University of Colorado Boulder  Process: 1.Track satellite 2.Map observations to state deviation 3.Determine how to adjust the state to best fit the observations 4.Apply and repeat 26 Least Squares

27. University of Colorado Boulder  Process: 1.Track satellite 2.Map observations to state deviation 3.Determine how to adjust the state to best fit the observations 4.Apply and repeat 27

28. University of Colorado Boulder  Process: 1.Track satellite 2.Map observations to state deviation 3.Determine how to adjust the state to best fit the observations 4.Apply and repeat 28

29. University of Colorado Boulder  Process: 1.Track satellite 29 Perfect Observations

30. University of Colorado Boulder  Process: 1.Track satellite 30 Imperfect Observations

31. University of Colorado Boulder  Process: 1.Track satellite 2.Map observations to state deviation 3.Determine how to adjust the state to best fit the observations 4.Apply and repeat 31 Same process, but the best estimate trajectory will never quite match the truth, since the observations have noise.

32. University of Colorado Boulder  Process: 1.Track satellite 2.Map observations to state deviation 3.Determine how to adjust the state to best fit the observations 4.Apply and repeat 32 Least Squares

33. University of Colorado Boulder  Batch ◦ Using Least-Squares or a variant  Sequential ◦ CKF ◦ EKF ◦ UKF (others) 33

34. University of Colorado Boulder  How do we best fit the data?  A good solution, and one easy to code up, is the least-squares solution 34

35. University of Colorado Boulder 35  Least Squares  Weighted Least Squares  Least Squares with a priori  Min Variance  Min Variance with a priori

36. University of Colorado Boulder  How can we map state deviation errors? 36 Final State: (xf, yf, zf, vxf, vyf, vzf) Example: Propagating a state in the presence of NO forces Initial State: (x0, y0, z0, vx0, vy0, vz0)

37. University of Colorado Boulder  Perturb the initial state in the x direction ◦ x0 = x0 +  x 37 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Force model: 0

38. University of Colorado Boulder  Propagate the deviated state 38 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Final State: (xf+Δx, yf, zf, vxf, vyf, vzf)

39. University of Colorado Boulder  Propagate the deviated state  How is the final state altered? 39 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Final State: (xf+Δx, yf, zf, vxf, vyf, vzf)

40. University of Colorado Boulder  Propagate the deviated state  How is the final state altered? 40 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Final State: (xf+Δx, yf, zf, vxf, vyf, vzf)

41. University of Colorado Boulder  Perturb the position in all 3 directions  Now we have a matrix of partials relating the initial state to the final state 41 Initial State: (x0, y0, z0, vx0, vy0, vz0) Initial State: (x0+Δx, y0+Δy, z0+Δz, vx0, vy0, vz0) Final State: (xf+Δx, yf+Δy, zf+Δz, vxf, vyf, vzf) Force model: 0

42. University of Colorado Boulder  Perturb the x velocity 42 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0-Δvx, vy0, vz0) Force model: 0

43. University of Colorado Boulder  Perturb the x velocity 43 Force model: 0 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Final State: (xf+tΔvx, yf, zf, vxf+Δvx, vyf, vzf) Initial State: (x0, y0, z0, vx0+Δvx, vy0, vz0)

44. University of Colorado Boulder  The relationships describing the change to the final state based on deviations to the initial state are simple because we assumed no forces  The relationships are more complicated with non-linear dynamics 44

45. University of Colorado Boulder  How can we map state deviation errors? 45

46. University of Colorado Boulder  Linearization  Introduce the state deviation vector  If the reference/nominal trajectory is close to the truth trajectory, then a linear approximation is reasonable.  Taylor Series Expansion 46

47. University of Colorado Boulder  The state transition matrix maps a deviation in the state from one epoch to another. 47

48. University of Colorado Boulder  The state transition matrix is constructed via numerical integration, in parallel with the trajectory itself. 48

49. University of Colorado Boulder  Computing the individual partials of the A matrix  You can do it by hand if you enjoy that sort of thing  Alternatively, use MATLAB’s symbolic toolbox ◦ A = jacobian(F,X) ◦ See MATLAB help file for the jacobian function 49

50. University of Colorado Boulder 50 syms mu x y r xdot ydot xddot yddot r = sqrt(x^2 + y^2); xddot = -mu*x/r^3; yddot = -mu*y/r^3; X = [x, y, xdot, ydot]; Xdot = [xdot, ydot, xddot yddot]; A = jacobian(Xdot, X); A = subs(A,'(x^2+y^2)^(5/2)','r^5'); A = subs(A,'(x^2+y^2)^(3/2)','r^3'); A = subs(A,'(x^2+y^2)^(1/2)','r'); A = subs(A,'(x^2+y^2)','r^2'); Use the symbolic toolbox to compute the partials for you

51. University of Colorado Boulder 51 A = [ 0, 0, 1, 0] [ 0, 0, 0, 1] [ (3*mu*x^2)/r^5 - mu/r^3, (3*mu*x*y)/r^5, 0, 0] [ (3*mu*x*y)/r^5, (3*mu*y^2)/r^5 - mu/r^3, 0, 0]

52. University of Colorado Boulder  If you haven’t taken ASEN 5070, it’s a good idea to do so.  For more information on Stat OD visit:  http://ccar.colorado.edu/asen5070/ http://ccar.colorado.edu/asen5070/ 52