1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 15: Statistical Least Squares and Estimation of Nonlinear System

2. University of Colorado Boulder  Lecture 14 ◦ Derivations of Bayes’ Theorem skipped in class were added to D2L version of slides  Lecture Quiz Due by 5pm ◦ Posted later this morning  Homework 5 Due Friday  Exam 1 – Friday, October 9 2

3. University of Colorado Boulder  Statistical Least Squares w/ a priori  SLS and Estimation of Nonlinear System 3

4. University of Colorado Boulder 4 Statistical Interpretation of Least Squares

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15. University of Colorado Boulder  Still need to know how to map measurements from one time to a state at another time! 15

16. University of Colorado Boulder 16  Since we linearized the formulation, we can still improve accuracy through iteration (more on this in a moment)

17. University of Colorado Boulder 17 Statistical Least Squares Solution for Nonlinear System

18. University of Colorado Boulder 18 p. 196-197 of textbook (includes corrections)

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21. University of Colorado Boulder  The batch filter depends on the assumptions of linearity ◦ Violations of this assumption may lead to filter divergence ◦ If the reference trajectory is near the truth, this holds just fine  The batch processor must be iterated 2-3 times to get the best estimate ◦ The iteration reduces the linearization error in the approximation  Continue the process until we “converge” ◦ Definition of convergence is an element of filter design 21

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23. University of Colorado Boulder 23  If we know the observation error, why “fit to the noise”?

24. University of Colorado Boulder  No improvement in observation RMS 24  Magnitude of the state deviation vector  Maximum number of iterations

25. University of Colorado Boulder  Instantaneous observation data is taken from three Earth-fixed tracking stations ◦ Why is instantaneous important in this context? 25  x, y, z – Satellite position in ECI  x s, y s, z s are tracking station locations in ECEF

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27. University of Colorado Boulder 27 RMS Values (Range σ=0.01 m, Range-Rate σ = 0.001 m/s) Pass 1Pass 2Pass 3 Range (m)732.7480.3190.010 Range Rate (m/s) 2.90020.00120.0010

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29. University of Colorado Boulder 29 Image: Hall and Llinas, “Multisensor Data Fusion”, Handbook of Multisensor Data Fusion: Theory and Practice, 2009.  FLIR – Forward-looking infrared (FLIR) imaging sensor

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32. University of Colorado Boulder  Inverting a potentially poorly scaled matrix  Solutions: ◦ Matrix Decomposition (e.g., Singular Value Decomposition) ◦ Orthogonal Transformations ◦ Square-root free Algorithms  Numeric Issues ◦ Resulting covariance matrix not symmetric ◦ Becomes non-positive definite (bad!) 32