1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 34: Probability Ellipsoids

2. University of Colorado Boulder  Homework 10 due on Friday  Lecture quiz due by 5pm on Friday ◦ Posted later today  Guest Lecture on Wednesday  No lecture on Friday ◦ Make-up nominally posted over the break 2

3. University of Colorado Boulder 3 Probability Ellipsoids

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5. University of Colorado Boulder  An “ellipsoid” is an n-dimensional ellipse or, more generally, a hyperellipsoid.  P, the variance-covariance matrix, represents the uncertainty in the state estimate.  The truth is that there is a relationship between the variances in each component, and visualization of such relationships provides information on quality of the state 5

6. University of Colorado Boulder  Generally the best to represent the probability ellipsoid using the covariance matrix’s principal axes. 6

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8. University of Colorado Boulder  For a filter estimated Cartesian state, it is easier to use the principal axes of P to construct the probability ellipsoid  For this we use the eigenvector/value decomposition: 8

9. University of Colorado Boulder  The matrix U may be used to diagonalize P ◦ Called a principal axis transformation  Any realization of the random vector x may be rotated into the principal axes via 9

10. University of Colorado Boulder  This is really useful, because if P is oriented in Cartesian coordinates, we don’t really know what the size of a probability ellipsoid is. 10 ???

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13. University of Colorado Boulder  The axis sizes of the l  σ ellipsoid are then 13  The orientation of the ellipsoids is determined by the eigenvector matrix U.  We may interpret this matrix as an Euler angle sequence of rotations:

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16. University of Colorado Boulder 16  The Euler angles are defined by:

17. University of Colorado Boulder  Although the Euler-rotation interpretation provides some understanding of the probability ellipsoid, we do not have to compute the angles ◦ The eigenvector matrix U defines the transformation from the frame used to define P and the principal axis frame 17

18. University of Colorado Boulder  Consider the case: 18  We will plot this case in MATLAB using:

19. University of Colorado Boulder 19 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

20. University of Colorado Boulder 20 Views of Error Ellipsoid view (90°,0) view down the positive x-axis

21. University of Colorado Boulder 21 Views of Error Ellipsoid view (0,90°) view down the positive z-axis

22. University of Colorado Boulder 22 Views of Error Ellipsoid view (-37.5°,0) standard matlab view

23. University of Colorado Boulder  Probability of being within 1σ, 2σ, and 3σ differs based on dimension ◦ Univariate:  1σ – 0.683  2σ – 0.954  3σ – 0.997 ◦ Trivariate  1σ – 0.200  2σ – 0.739  3σ – 0.971 23

24. University of Colorado Boulder  Output of eig() (in almost any language) is not always consistent with the presented formulation ◦ Always inspect the covariance matrix and compare it to the plotted ellipsoid ◦ For example, does the standard deviation in the x- component in P match the plot 24

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27. University of Colorado Boulder  Problem first identified in 1996: ◦ Junkins, et al., “Non-Gaussian Error Propagation in Orbital Mechanics”, Journal of Astronautical Sciences, V. 44, N. 4, 1996 pp. 541-563  Multiple methods exists for nonlinear propagation: ◦ Monte Carlo ◦ State transition tensors (STT) ◦ Gaussian Mixtures ◦ Polynomial Chaos 27

28. University of Colorado Boulder  The STM represents a 2 nd -order tensor ◦ Generated via the first derivative of the force model  Accuracy improved with the inclusion of higher-order effects  A STT maintains higher order derivatives for mapping of the a priori p.d.f. 28

29. University of Colorado Boulder 29 Fujimoto, et al., 2011

30. University of Colorado Boulder 30 Horwood, et al., JGCD, Nov.-Dec., 2011

31. University of Colorado Boulder  Based on Weiner’s Homogeneous Chaos (1938)  Generates an approximate solution to a stochastic ODE: 31  More commonly used is structures, CFD, applied physics, and other fields  We are applying it to orbital mechanics

32. University of Colorado Boulder 32  Use polynomial surrogate to approximate the p.d.f.  PC requires ~100-200 ODE evaluations  Monte Carlo requires more than 100,000 evaluations Image: Jones, et al., 2013

33. University of Colorado Boulder 33  Part of NASA/GSFC-based navigation team for the Magnetospheric Multi-Scale (MMS) mission ◦ Leveraging CU-developed methods and applications of uncertainty quantification ◦ Applying polynomial chaos (PC) to the estimation of collision probabilities ◦ Includes post-maneuver uncertainty quantification Collision Risk Relative Accuracy