1. Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION Satellite Tracking Example of SNC and DMC ASEN 5070 LECTURE 32 12/01/08

2. Colorado Center for Astrodynamics Research The University of Colorado 2

3. Colorado Center for Astrodynamics Research The University of Colorado 3

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5. Colorado Center for Astrodynamics Research The University of Colorado 5

6. Colorado Center for Astrodynamics Research The University of Colorado 6

7. Colorado Center for Astrodynamics Research The University of Colorado 7 Again the off diagonal terms are small and the diagonal terms are Aproximately=1. We assume that the velocity is constant over the interval so

8. Colorado Center for Astrodynamics Research The University of Colorado 8

9. Colorado Center for Astrodynamics Research The University of Colorado 9

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11. Colorado Center for Astrodynamics Research The University of Colorado The J 3 Example

12. Colorado Center for Astrodynamics Research The University of Colorado Tracking Configuration

13. Copyright 2005 13 The J 3 Problem Gravitational Potential Function for the J 3 Model where

14. Copyright 2005 14 J 3 Equations of Motion Taking the partials of the Gravitational Potential to get x,y,z accelerations

15. Copyright 2005 15 Conventional Kalman Filter (CKF)

16. Copyright 2005 16 Conventional Kalman Filter (CKF) Position RSS = 50.662 m Velocity RSS = 4.934 x 10 -2 m/s RSS errors are based on X T – (X * + ) at each stage RSS=4.18 m/s

17. Copyright 2005 17 Extended Kalman Filter (EKF) No State Noise Added EKF implemented after 154 minutes Note that pre-fit residuals for the CKF are much larger that the EKF but post-fit residuals are identical indicating that the reference trajectory for the CKF is in the linear range

18. Copyright 2005 18 Extended Kalman Filter (EKF) Position RSS = 50.279 m Velocity RSS = 4.332 x 10 -2 m/s RSS position and velocity errors for CKF and EKF are nearly identical

19. Copyright 2005 19 Band Diagonal State Noise Matrix See web page handouts or previous slides for derivation

20. Copyright 2005 20 Conventional Kalman Filter with SNC Band Diagonal State Noise Matrix results for optimum value:  2 = 10 -13

21. Copyright 2005 21 Conventional Kalman Filter with SNC Position RSS = 8.43 m Velocity RSS = 1.62 x 10 -2 m/s Band Diagonal State Noise Matrix results for optimum value:  2 = 10 -13

22. Copyright 2005 22 Conventional Kalman Filter with SNC Band Diagonal State Noise Matrix results for optimum value:  2 = 10 +13 This is what happens if an input error inject a huge amount of process noise i.e.  2 = 10 + 13 instead of 10 - 13 Kalman gain is so large that data is fit almost perfectly – well below noise levels of 1cm and 1mm/s

23. Copyright 2005 23 Conventional Kalman Filter with input error Position RSS = 12.084 m Velocity RSS = 1.22 x 10 -2 m/s Band Diagonal State Noise Matrix results for optimum value:  2 = 10 +13 Small residuals do not a good orbit Make – RSS errors are larger than Previous results 2  standard deviations of estimation error covariance are red lines

24. Copyright 2005 24 Extended Kalman Filter with State Noise (SNC) Band Diagonal State Noise Matrix

25. Copyright 2005 25 Extended Kalman Filter with State Noise (SNC) Band Diagonal State Noise Matrix Looked for optimum values of  2 by analyzing residuals and errors Optimum value to minimize position errors:  2 = 10 -13

26. Copyright 2005 26 Extended Kalman Filter with State Noise (SNC) Band Diagonal State Noise Matrix results for optimum value:  2 = 10 -13

27. Copyright 2005 27 Extended Kalman Filter with State Noise (SNC) Band Diagonal State Noise Matrix results for optimum value:  2 = 10 -13 Position RSS = 8.520 m Velocity RSS = 1.484 x 10 -2 m/s Position RSS = 8.520 m The CKF and EKF results are Nearly identical indicating that Initial conditions are in the linear range

28. Copyright 2005 28 Extended Kalman Filter with Fading Adds a fading term to the time update – this downweights earlier data by Keeping the Kalman gain elevated where  t = time between measurements = 20 seconds  = age-weighting time constant >1

29. Copyright 2005 29 Extended Kalman Filter with Fading Looked for optimum value of s to minimize the residuals and errors Optimum value of s ~ 1.0339 This corresponds to a  of 10 minutes

30. Copyright 2005 30 Extended Kalman Filter with Fading Optimum values of s to minimize position and velocity errors : s = 1.0339

31. Copyright 2005 31 Extended Kalman Filter with Fading Optimum values of s to minimize position and velocity errors : s = 1.0339 Position RSS = 8.688 m Velocity RSS = 1.545 x 10 -2 m/s Position RSS=8.688 m

32. Copyright 2005 32 Kalman Filter Results Comparison Conclusions: The CKF and EKF produce comparable results indicating that the reference orbit for the CKF is in the linear range for this example. Fading produces comparable results to SNC

33. Copyright 2005 33 Dynamic Model Compensation (DMC) DMC accounts for unmodeled or inaccurately modeled accelerations acting on the spacecraft - J 3 in this problem. The state vector was augmented to the following: where  x,  y, and  z are the accelerations A Gauss-Markov process is used to account for these accelerations: where u(t) is white Gaussian noise with and Where  is a time constant

34. Copyright 2005 34 Dynamic Model Compensation  and  were optimized to give the lowest position and velocity errors:  2 = 10 -16

35. Copyright 2005 35 Dynamic Model Compensation Optimized Results Position RSS = 8.195 m Velocity RSS = 1.429 x 10 -2 m/s

36. Copyright 2005 36 Dynamic Model Compensation Optimized Results Errors in acceleration estimates in x,y, And z directions Note: the DMC did a poor job of recovering accelerations. More work is needed on optimizing  and  We should do a better job of recovering accelerations and the state while reducing tracking residuals. Actual vs estimated accelerations

37. Copyright 2005 37 Filter Comparisons All filters achieved comparable results; however we should do better with the DMC

38. Copyright 2005 38 Added deviations to the initial conditions so that they are outside linear range to show Convergence for EKF with SNC Divergence for CKF with SNC Original Initial Conditions [757700.301, 5222606.566, 4851499.77, 2213.250611, 4678.372741, -5371.314404] m and m/s Perturbed Initial Conditions Deviation [8, 5, 5, 8, 5, 5] m and m/s [757708.301, 5222611.566, 4851504.77, 2221.250611, 4683.372741, -5366.314404] m and m/s

39. Copyright 2005 39 Conventional Kalman Filter Band Diagonal State Noise Matrix results for optimum value:  2 = 10 -13 Deviated Initial Conditions

40. Copyright 2005 40 Conventional Kalman Filter Position RSS = 2353.49 m Velocity RSS = 2.88 m/s Band Diagonal State Noise Matrix results for optimum value:  2 = 10 -13 Deviated Initial Conditions

41. Copyright 2005 41 Extended Kalman Filter Band Diagonal State Noise Matrix results for optimum value:  2 = 10 -13 Deviated Initial Conditions

42. Copyright 2005 42 Extended Kalman Filter Position RSS = 17.79 m Velocity RSS = 2.53 x 10 -2 m/s Band Diagonal State Noise Matrix results for optimum value:  2 = 10 -13 Deviated Initial Conditions

43. Copyright 2005 43 Filter Comparisons While the CKF does not diverge, its solution is significantly In error

44. Copyright 2005 44 Added deviations to the initial conditions to show Convergence for EKF with SNC Divergence for the Batch Processor Original Initial Conditions [757700.301, 5222606.566, 4851499.77, 2213.250611, 4678.372741, -5371.314404] m, m/s Perturbed Initial Conditions Deviation [1000, 1000, 1000, 500, 500, 500] m,m/s [758700.301, 5223606.566, 4852499.77, 2713.250611, 5178.372741, -4871.314404] m, m/s

45. Copyright 2005 45 Batch Processor Pass 1 Range RMS = 3672.36 km Range rate RMS = 4.304 km/s Pass 3 Range RMS = 4457.33 km Range rate RMS = 4.710 km/s Note that the batch processor is not converging and successive Iterations show divergence

46. Copyright 2005 46 Extended Kalman Filter Band Diagonal State Noise Matrix results for various values of  2 Deviated Initial Conditions Trajectory updated after 30 minutes Note: Values of  2 from 10 -5 to 10 -20 were tested. However, the residuals and errors were orders of magnitude higher for value of  2 between 10 -5 to 10 -9. Therefore, those values are not shown on the plots. Note that the optimal value for  2 is the same as for small initial condition errors.

47. Copyright 2005 47 Extended Kalman Filter Band Diagonal State Noise Matrix results for  2 = 10 -13 Deviated Initial Conditions Trajectory updated after 30 minutes

48. Copyright 2005 48 Extended Kalman Filter Band Diagonal State Noise Matrix results for  2 = 10 -13 Deviated Initial Conditions Position Errors Position Errors after 250 minutes Position RSS = 8.643 m

49. Copyright 2005 49 Extended Kalman Filter Band Diagonal State Noise Matrix results for  2 = 10 -13 Deviated Initial Conditions Velocity Errors Velocity Errors after 250 minutes Velocity RSS = 1.258 x 10 -2 m/s Note that position and velocity RSS are comparable to those on slide 40. If IC errors are large it may be Advantageous to use the EKF to Obtain ICs for the batch

50. Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 50

51. Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 51

52. Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 52

53. Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 53

54. Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 54

55. Copyright 2005 Assume we wish to transform Φ(t i, t j ) from the A frame to the B frame Let Recall that and Transformation of State Transition Matrix to Alternate Frames 55

56. Copyright 2005 But Hence, where Transformation of State Transition Matrix to Alternate Frames 56