1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 29: Observability and Introduction to Process Noise

2. University of Colorado Boulder  Exam 2 Friday  Seminar Friday: 2

3. University of Colorado Boulder 3 Givens Transformations – Lingering Question

4. University of Colorado Boulder  We do not want to add non-zero terms to the previously altered rows, so we use the identity matrix except in the rows of interest: 4

5. University of Colorado Boulder  Apply a series of rotations such that: 5

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7. University of Colorado Boulder  A commonly used tool in linear algebra is the QR factorization of a matrix:  Using the Given rotations, we have:  Givens is one way to get the QR solutions where: 7

8. University of Colorado Boulder 8 Observability

9. University of Colorado Boulder  How do I determine what parameters may be successfully estimated in the filter? ◦ Can I use observations of a spacecraft to estimate the height of Folsom Field Stadium? ◦ What about observations of a spacecraft to measure variations in rainfall in the Amazon river basin? ◦ How do I determine if either of these are possible? 9

10. University of Colorado Boulder  Consider the case of two spacecraft and a ground station with a fixed inertial position ◦ Two-body gravity field (no perturbations) ◦ No modeling error ◦ Infinite precision ◦ Little/no error on range observations 10

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12. University of Colorado Boulder 12  The two plots look similar (this is not a copy/paste error)  Does anyone think there is a problem? Satellite 1Satellite 2

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14. University of Colorado Boulder  Gather more observations? ◦ Unfortunately, No.  Gather range-rate to go with the range data? ◦ Nope – we run into the same problem  Orthogonal data type, e.g., angles? ◦ Actually that would work, but how do we find out? 14

15. University of Colorado Boulder  We can use the information matrix: 15

16. University of Colorado Boulder  In other words, when designing our filter, we should study the information matrix to determine if we can get a solution  Let’s say you solve for the information matrix defined by some simulation. ◦ How would you determine if it is positive definite? ◦ Do you need to generate simulated observations? 16

17. University of Colorado Boulder  What if the condition number of the information matrix is very large (too large for any of the more numerically stable methods to apply)? ◦ Maybe we should reconsider what parameters to estimate? ◦ This can be the case for gravity field estimation with spatially sparse measurements 17

18. University of Colorado Boulder ◦ Can I use observations of a spacecraft to estimate the height of Folsom Field?  Only if observations of/from a well-known spacecraft are gathered with respect to the top of the stadium ◦ What about observations of a spacecraft to measure variations in rainfall in the Amazon river basin?  Actually – you can!  Scientific studies of GRACE data do this type of analysis regularly ◦ How do I determine if either of these are possible?  You perform an observability study! 18

19. University of Colorado Boulder  Can we estimate the absolute position of two spacecraft in Earth orbit (two-body dynamics) using relative range and/or range-rate measurements? 19

20. University of Colorado Boulder  Can we do it if we put one of the spacecraft near the Moon and keep one at Earth? 20 Image Credit: Hill and Born, 2007

21. University of Colorado Boulder 21 Process Noise

22. University of Colorado Boulder 22  What happened to u (modeling error) ? ◦ This is true process noise…  Can we ignore it?  How do we account for it?

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24. University of Colorado Boulder  Random process u maps to the state through the matrix B ◦ Consider it a random process for our purposes  Usually (for OD), we consider random accelerations: 24

25. University of Colorado Boulder  For the sake of our discussion, assume: 25

26. University of Colorado Boulder  This is a non-homogenous differential equation  The derivation of the general (continuous) solution to this equation is derived in the book (Section 4.9), and is: 26

27. University of Colorado Boulder 27  If we want to instead map between two discrete times:

28. University of Colorado Boulder 28  For the case of a noise process with zero mean:  The zero-mean noise process does not change the mapping of the mean state

29. University of Colorado Boulder  What about the covariance matrix?  The derivation of the general (continuous) solution to this equation is derived in the book (Section 4.9), and is: 29

30. University of Colorado Boulder  The previous discussion considered the case where the noise process is continuous, i.e, 30  Things may be simplified if we instead consider a discrete process:

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32. University of Colorado Boulder  Using the discrete noise process, we instead get (for zero mean process): 32

33. University of Colorado Boulder  This defines, mathematically, how we can select the minimum covariance to prevent saturation ◦ Saturation is typically dominated by dynamic model error ◦ With a stochastic (probabilistic) description of the modeling error, we have our minimum 33

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35. University of Colorado Boulder  The addition of a noise process is better suited for a sequential filter ◦ Must include the process noise transition matrix in the Batch formulation ◦ Changes the mapping of the state (deviation) back to the epoch time, which requires alterations to the H matrix definition ◦ Tapley, Schutz, and Born (p. 229) argue that this is cumbersome and impractical for real-world application  Advantage: Kalman, EKF, Potter, and others 35