1. Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION Error Ellipsoid B-Plane ASEN 5070 LECTURE 32 11/16/09

2. The Probability Ellipsoid 2 Copyright 2006

3. The Probability Ellipsoid 3 Copyright 2006

4. The Probability Ellipsoid 4 Copyright 2006

5. The Probability Ellipsoid 5 Copyright 2006

6. The Probability Ellipsoid 6 Copyright 2006

7. The Probability Ellipsoid 7 Copyright 2006

8. The Probability Ellipsoid 8 Copyright 2006

9. The Probability Ellipsoid 9 Copyright 2006

10. The Probability Ellipsoid 10 Copyright 2006

11. The Probability Ellipsoid 11 Copyright 2006 Example:

12. The Probability Ellipsoid 12 Copyright 2006 Views of Error Ellipsoid view (0,0)* azimuth =0, elevation =0 view down the negative y-axis *view(azimuth, elevation), azimuth is a clockwise rotation about the positive z-axis

13. The Probability Ellipsoid 13 Copyright 2006 Views of Error Ellipsoid view (90°,0) view down the positive x-axis

14. The Probability Ellipsoid 14 Copyright 2006 Views of Error Ellipsoid view (0,90°) view down the positive z-axis

15. The Probability Ellipsoid 15 Copyright 2006 Views of Error Ellipsoid view (-37.5°,0) standard matlab view

16. The Probability Ellipsoid 16 Copyright 2006 Body Plane (B-Plane)

17. 17 Viking Launch Trajectory

18. 18 Viking Trajectory

19. 19 + OD solutions before L + 12h

20. 20 Viking 1 Departure Control Uncertainty in time of closest approach 25=(21 2 +12 2 ) 1/2 3  Error ellipse for 1 st mid course maneuver

21. 21 Mars Climate Orbiter (MCO) B-Plane CERTAIN DEATH Mars Impact Radius AMD = Angular Momentum Desaturation

22. Colorado Center for Astrodynamics Research The University of Colorado 22 Error Ellipse in the B-plane I. Assume we have done an OD solution at some point,, in the interplanetary phase of the mission and we wish to project our estimation error covariance onto the B-plane (B=Body). 1.Compute the time of B-plane penetration based on conic motion (this is known as the linearized time of filter, LTOF) 2.Map P to this time using: 3.Generally P will be expressed in ECI, J2000 coordinates. Rotate the position portion of into the STR Frame

23. Colorado Center for Astrodynamics Research The University of Colorado 23 Error Ellipse in the B-plane 4. Use the 2x2 portion of corresponding to 5. Compute the Eigenvalues and normalized Eigenvectors 6. Compute the semi-major axis and semi-minor axis of the ellipse (x’,y’) and the rotation angle. Plot the 1,2, or 3 error ellipse on the B-plane. For a 2-D error ellipse the probability of being inside the N error ellipse (N=1,2,3…) is given by

24. Colorado Center for Astrodynamics Research The University of Colorado 24 Error Ellipse in the B-plane For a bivariate normal distribution: 7. The B-plane parameters are chosen because the variation of the and vectors are nearly linear with respect to midcourse orbit parameters. B-plane SMAA