Transcript Lecture 25
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 25: Potter Algorithm and Decomposition Methods University of Colorado Boulder Homework 8 Due Friday (10/30) ◦ The assignment has been update, so be sure to grab the new one! Lecture Quizzes ◦ Due by 5pm Today ◦ Next one due by 5pm 10/30 Exam 2 – Friday, November 6 University of Colorado Boulder 2 Numeric and Implementation Considerations University of Colorado Boulder 3 Least Squares (Batch) Iterate a few times. • • University of Colorado Boulder Replace reference trajectory with bestestimate Generate new computed observations, STM, and H matrices 4 Conventional Kalman Evolution of covariance Mapping of final covariance University of Colorado Boulder 5 EKF Final mapped Reference Evolution of reference, w/covariance Original Reference University of Colorado Boulder 6 The KF has some problems that are unique or exacerbated by its nature as a sequential processor For example: ◦ Sensitivity to Input P and R ◦ Numeric Considerations ◦ Saturation University of Colorado Boulder 7 Linear Regime Pitfall 1: Beware of large a priori covariances with noisy data - Breaks linear approximations - Causes filter to diverge University of Colorado Boulder 8 As we process more measurements in CKF, 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 9 University of Colorado Boulder 10 Pitfall 2: Beware of collapsing covariance - Prevents new data from influencing solution - More prevalent for longer time-spans University of Colorado Boulder 11 Pitfall 2: Beware of collapsing covariance - Prevents new data from influencing solution - More prevalent for longer time-spans University of Colorado Boulder 12 University of Colorado Boulder 13 University of Colorado Boulder 14 University of Colorado Boulder 15 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 16 Bierman Example of Poorly Conditioned System University of Colorado Boulder 17 University of Colorado Boulder 18 Process the first observation: University of Colorado Boulder 19 Process the second observation: University of Colorado Boulder 20 Consider the implementation on a computer with a limited precision: University of Colorado Boulder 21 University of Colorado Boulder 22 University of Colorado Boulder 23 Exact to order ε University of Colorado Boulder 24 University of Colorado Boulder 25 The assignment asks that you observe the numeric problems created by the Bierman example ◦ ◦ ◦ ◦ Batch Processor CKF CKF w/ Joseph Formulation Potter’s Algorithm We will cover this one on Monday University of Colorado Boulder 26 Bias Estimation University of Colorado Boulder 27 As shown in the homework, i.e., biased observations, yields a biased estimator. To compensate, we can estimate the bias: University of Colorado Boulder 28 What are some example sources of bias in an observation? University of Colorado Boulder 29 GPS receiver solutions for Jason-2 Antenna is offset ~1.4 meters from COM What could be causing the bias change after 80 hours? University of Colorado Boulder 30