University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 10: Weighted LS and A Priori
University of Colorado Boulder Homework 4 – Due September 26 Exam 1 – October 10 ◦ Open book, open notes ◦ Calculator, but no computer 2
University of Colorado Boulder Lecture 10 – WLS w/ a priori and OD Process Lecture 11 – Probability and Statistics I ◦ This 3pm (ECCS 1B14) Lecture 12 – Probability and Statistics II ◦ Monday morning Friday and Next Week: ◦ Probability and Statistics ◦ Book Appendix A 3
University of Colorado Boulder 4 Weighted Least Squares Estimation
University of Colorado Boulder Process all observations over a given time span in a single batch 5
University of Colorado Boulder For each y i, we have some weight w i 6
University of Colorado Boulder 7 Consider the case with two observations (m=2) If w 2 > w 1, which ε i will have a larger influence on J(x) ? Why? 2.0
University of Colorado Boulder For the weighted LS estimator: 8 How do we find W ?
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University of Colorado Boulder 11 Weighted Least Squares w/ A Priori
University of Colorado Boulder A priori ◦ Relating to or denoting reasoning or knowledge that proceeds from theoretical deduction rather than from observation or experience We have: 12
University of Colorado Boulder As you showed in the homework: 13
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University of Colorado Boulder 15 Orbit Determination Algorithm (so far)
University of Colorado Boulder We want to get the best estimate of X possible What would we consider when deciding if we should include a solve-for parameter? 16
University of Colorado Boulder Truth Reference Best Estimate (goal) Observations are functions of state parameters, but usually NOT state parameters themselves. Mismodeled dynamics Underdetermined system ◦ l*(n+p) 17
University of Colorado Boulder We have noisy observations of certain aspects of the system. We need some way to relate each observation to the trajectory that we’re estimating. 18 Observed Range Computed Range True Range = ??? X*X*
University of Colorado Boulder Assumptions: ◦ The reference/nominal trajectory is near the truth trajectory. Why do we introduce this assumption? ◦ Force models are good approximations for the duration of the measurement arc. Why does this matter? ◦ The filter that we are using is unbiased: The filter’s best estimate is consistent with the true trajectory. 19
University of Colorado Boulder Linearization Introduce the state deviation vector If the reference/nominal trajectory is close to the truth trajectory, then a linear approximation is reasonable. If they are not, then higher order terms are no longer negligible! 20
University of Colorado Boulder Goal of the Stat OD process: Find a new state/trajectory that best fits the observations: If the reference is near the truth, then we can assume: 21
University of Colorado Boulder Goal of the Stat OD process: The best fit trajectory is represented by 22 This is what we want
University of Colorado Boulder How do we map the state deviation vector from one time to another? 23 X*X*
University of Colorado Boulder How do we map the state deviation vector from one time to another? The state transition matrix. It permits: 24
University of Colorado Boulder Now we can relate an observation to the state at an epoch. 25 Observed Range Computed Range X*X*
University of Colorado Boulder Still need to know how to map measurements from one time to a state at another time! 26
University of Colorado Boulder 27 Since we linearized the formulation, we can still improve accuracy through iteration (more on this in a future lecture) How do we get the weights? Probability and Statistics
University of Colorado Boulder 28 Concept Exercises
University of Colorado Boulder Work in groups At 11:40am, we will reassemble and discuss The concept quiz will not be turned in for a grade 29