1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 6: Linearization of OD Problem

2. University of Colorado Boulder  Lecture Quiz 1 – Due today by 5pm  Lecture Quiz 2 – Posted by Monday morning  Homework 2– Due September 11 2

3. University of Colorado Boulder  Time of Periapse Passage  Linearization ◦ Why do we need it? ◦ How do we do it? 3

4. University of Colorado Boulder  Nonlinear estimated state and observation vectors  Linear estimated state and observation vectors ◦ By linear, we mean the dynamics AND the observation-state relationship is linear  Estimated state and observation deviation vectors  The reasons for the linear and the deviation vectors to use the same symbol will be more evident shortly 4

5. University of Colorado Boulder 5 Time of Periapse Passage – Common HW Issue

6. University of Colorado Boulder T p is determined from the following equations: However, as time t increases, T p is not constrained to an orbital period and thus increases as a step function. To resolve this, MOD T p with the orbital period. A situation may arise in which the calculation for the mean Anomaly, M, and true anomaly, ν, do not agree resulting in the mean anomaly to be past perigee while the true anomaly is behind perigee (this is an artifact of numerical integration).

7. University of Colorado Boulder To correct this, we will introduce the angle of periapse θ p : From this, one will notice that the artifacts do not occur when Thus, constraining θ p to be between –π to π will remove the artifacts. The angle of periapse θ p can then be converted back to time of periapse T p by

8. University of Colorado Boulder Artifacts

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11. University of Colorado Boulder 11 Linearization – Why do we need it?

12. University of Colorado Boulder  We want to get the best estimate of X possible ◦ Ex. force model parameters: C D, C R, J 2, etc. ◦ Ex. measurement params: station coordinates, observation biases, etc. 12

13. University of Colorado Boulder  “Solve-for” parameters are usually constant (but not always…)  More generally: 13

14. University of Colorado Boulder  Example measurement types: ◦ Range, Range-Rate ◦ Right Ascension/Declination ◦ GPS pseudorange and carrier phase ◦ Star tracker and angular rate gyro  At each epoch t i we have a measurement model G(X i, t i ) ◦ ε i represents the model error in G(X i, t i )  May result from statistical uncertainty  Could be a result of modeling error  What are some examples of modeling error? 14

15. University of Colorado Boulder 15  How do we estimate X ?  How do we estimate the errors ε i ?  How do we account for force and observation model errors?

16. University of Colorado Boulder  It works for HW 1, why don’t we do it in practice? ◦ Assumed the same number of observations as unknowns  What about when we have more observations than unknowns?  What about when we have more unknowns than observations? ◦ Did not rigorously account for observation errors  How do we account for statistical uncertainties? 16

17. University of Colorado Boulder  Known: p × l observations  Unknowns: ◦ n×l unknown state variables ◦ p×l unknown observation errors ◦ (n+p)×l total unknown values  We have more unknowns that observations, what do we do now?  X(t) is a function of X(t 0 ) ◦ Well, now we are down to n+(p×l) unknowns… 17

18. University of Colorado Boulder  For now, let’s consider a linear problem: 18

19. University of Colorado Boulder  We introduce a “cost function” that we seek to minimize ◦ We now select x to minimize J(x) ◦ No longer estimating ε i ! ◦ This gives us n+p×l equations and only n unknowns  This is known as: Least Squares Estimation 19

20. University of Colorado Boulder  What is one way to find the minimum of J ? ◦ Differentiate with respect to x ◦ Find the point where dJ/dx = 0 ◦ Make sure the matrix of second-order partials is positive definite 20

21. University of Colorado Boulder 21  Differentiate with respect to x. What is the answer?

22. University of Colorado Boulder  In order for the normal equation to yield a minimum of the cost function 22  In other words, H T H must be what?

23. University of Colorado Boulder  This is the “normal equation” for the least squares estimator  Other methods of minimizing J(x) exist ◦ Singular value decomposition ◦ Givens transformations ◦ Etc.  We assumed the state-observation relationship was linear, but the orbit determination problems is nonlinear ◦ We will linearize the formulation of the problem 23

24. University of Colorado Boulder 24 Linearization – How do we do it?

25. University of Colorado Boulder 25  In general, the problems are nonlinear in dynamics and/or observations  This course primarily discusses methods based on linearization  What is required to “linearize” the problem? Vector of Estimated Values Vector of Observations at t i

26. University of Colorado Boulder  We will define lower-case vectors as representing a linear system ◦ For a linear system, this is the vector of interest ◦ For a nonlinear system, these are deviation vectors  In general, we do not know the truth. Hence, we must estimate deviation relative to the reference  If we have a nonlinear observation or state dynamics model, we have to use a fully linearized form! 26 TruthReferenceDeviations

27. University of Colorado Boulder  We generate linearized models about the reference trajectory, which are a function of the deviation vectors 27

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34. University of Colorado Boulder 34  Which terms are non-zero?

35. University of Colorado Boulder 35  Which terms equal 1?  What are the partials w.r.t. μ?

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37. University of Colorado Boulder 37 Computed, not measured values!