Transcript Lecture 10
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 10: Weighted LS and A Priori University of Colorado Boulder Lecture Quiz 5 Due September 18 @ 5pm Lecture Quiz 4 posted over weekend Homework 3 Due September 18 @ 9am Homework 4 Due September 25 Next Week: ◦ Will cover lectures 9 & 10 ◦ Probability and Statistics ◦ Book Appendix A University of Colorado Boulder 2 Weighted Least Squares Estimation University of Colorado Boulder 3 Process all observations over a given time span in a single batch University of Colorado Boulder 4 For each yi, we have some weight wi University of Colorado Boulder 5 Consider the case with two observations (m=2) If w2 > w1, which εi will have a larger influence on J(x) ? Why? University of Colorado Boulder 6 University of Colorado Boulder 7 For the weighted LS estimator: How do we find W ? University of Colorado Boulder 8 University of Colorado Boulder 9 University of Colorado Boulder 10 Weighted Least Squares w/ A Priori University of Colorado Boulder 11 A priori ◦ Relating to or denoting reasoning or knowledge that proceeds from theoretical deduction rather than from observation or experience We have: University of Colorado Boulder 12 You will show in the homework: University of Colorado Boulder 13 University of Colorado Boulder 14 Orbit Determination Algorithm (so far) University of Colorado Boulder 15 We want to get the best estimate of X possible What would we consider when deciding if we should include a solve-for parameter? University of Colorado Boulder 16 Truth Reference Best Estimate (goal) Observations are functions of state parameters, but usually NOT state parameters themselves. Mismodeled dynamics Underdetermined system ◦ l*(n+p) University of Colorado Boulder 17 X* 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. Observed Range Computed Range True Range = ??? University of Colorado Boulder 18 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. University of Colorado Boulder 19 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! University of Colorado Boulder 20 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: University of Colorado Boulder 21 Goal of the Stat OD process: The best fit trajectory is represented by This is what we want University of Colorado Boulder 22 How do we map the state deviation vector from one time to another? X* University of Colorado Boulder 23 How do we map the state deviation vector from one time to another? The state transition matrix. It permits: University of Colorado Boulder 24 X* Now we can relate an observation to the state at an epoch. Observed Range University of Colorado Boulder Computed Range 25 Still need to know how to map measurements from one time to a state at another time! University of Colorado Boulder 26 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 27 Probability and Statistics ◦ Appendix A of the book University of Colorado Boulder 28