Transcript Lecture 24
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 24: The EKF and Conceptual Filter Comparisons University of Colorado Boulder Homework 7 Due Friday Lecture Quiz Due by 5pm on Friday University of Colorado Boulder 2 University of Colorado Boulder 3 University of Colorado Boulder 4 University of Colorado Boulder 5 University of Colorado Boulder 6 What are the similarities between the batch and the sequential processor (as discussed up until now)? What are the differences between the batch and the sequential processor (as discussed up until now)? University of Colorado Boulder 7 Extended Kalman Filter (EKF) University of Colorado Boulder 8 No change to the reference trajectory with each observation! University of Colorado Boulder 9 One method of nonlinear filtering Update reference trajectory with each measurement update Reduces linearization errors University of Colorado Boulder 10 University of Colorado Boulder 11 The Batch and the CKF are mathematically equivalent Is the EKF equivalent to the Batch? Is it Bayes optimal? What happens if the updated reference trajectory is a poor approximation? Under what conditions could that happen? University of Colorado Boulder 12 Typically should not update reference trajectory immediately ◦ Initial estimate of state deviation may be unstable until the filter “converges” Must propagate the trajectory between each observation University of Colorado Boulder 13 Filter Conceptual Comparisons University of Colorado Boulder 14 Linearization Observation and Uncertainties University of Colorado Boulder 15 Least Squares (Batch) University of Colorado Boulder 16 Least Squares (Batch) Iterate a few times. • • University of Colorado Boulder Replace reference trajectory with best-estimate Generate new computed observations 17 Conventional Kalman University of Colorado Boulder 18 Conventional Kalman University of Colorado Boulder 19 Conventional Kalman University of Colorado Boulder 20 Conventional Kalman University of Colorado Boulder 21 Conventional Kalman University of Colorado Boulder 22 Conventional Kalman University of Colorado Boulder 23 Conventional Kalman University of Colorado Boulder 24 Conventional Kalman University of Colorado Boulder 25 Conventional Kalman University of Colorado Boulder 26 Conventional Kalman University of Colorado Boulder 27 Conventional Kalman University of Colorado Boulder 28 Conventional Kalman University of Colorado Boulder 29 Conventional Kalman University of Colorado Boulder 30 Conventional Kalman University of Colorado Boulder 31 Conventional Kalman University of Colorado Boulder 32 Conventional Kalman University of Colorado Boulder 33 Conventional Kalman University of Colorado Boulder 34 Conventional Kalman University of Colorado Boulder 35 Conventional Kalman University of Colorado Boulder 36 Conventional Kalman University of Colorado Boulder 37 Conventional Kalman University of Colorado Boulder 38 Conventional Kalman Evolution of covariance Mapping of final covariance University of Colorado Boulder 39 EKF University of Colorado Boulder 40 EKF University of Colorado Boulder 41 EKF University of Colorado Boulder 42 EKF University of Colorado Boulder 43 EKF University of Colorado Boulder 44 EKF University of Colorado Boulder 45 EKF University of Colorado Boulder 46 EKF University of Colorado Boulder 47 EKF University of Colorado Boulder 48 EKF University of Colorado Boulder 49 EKF University of Colorado Boulder 50 EKF University of Colorado Boulder 51 EKF University of Colorado Boulder 52 EKF University of Colorado Boulder 53 EKF University of Colorado Boulder 54 EKF University of Colorado Boulder 55 EKF University of Colorado Boulder 56 EKF University of Colorado Boulder 57 EKF University of Colorado Boulder 58 EKF University of Colorado Boulder 59 EKF University of Colorado Boulder 60 EKF Final mapped Reference Evolution of reference, w/covariance Original Reference University of Colorado Boulder 61 Numeric and Implementation Considerations University of Colorado Boulder 62 Linear Regime Pitfall 1: Beware of large a priori covariances with noisy data - Breaks linear approximations - Causes filter to diverge University of Colorado Boulder 63 As we process more measurements, our filter uncertainty is reduced. This causes the variance to get smaller with each observation epoch What happens to the Kalman filter? University of Colorado Boulder 64 University of Colorado Boulder 65 Pitfall 2: Beware of collapsing covariance - Prevents new data from influencing solution - More prevalent for longer time-spans University of Colorado Boulder 66 Pitfall 2: Beware of collapsing covariance - Prevents new data from influencing solution - More prevalent for longer time-spans University of Colorado Boulder 67 University of Colorado Boulder 68 University of Colorado Boulder 69 University of Colorado Boulder 70 To enforce a symmetric result, we may instead use the Joseph Formulation: Assumes that the a priori covariance is positive definite, e.g., not corrupted by previous numeric errors If assumption is met, always yields a symmetric P Does not ensure a positive definite covariance matrix You will/did derive the above in the homework Still applies for unbiased scenarios University of Colorado Boulder 71