University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 30: Lecture Quiz, Project Details, Concept Quiz

University of Colorado Boulder  Exam 2 Friday ◦ Open book, open notes. Like last time, bring a calculator.  Seminar Friday: 2

University of Colorado Boulder  Office Hours this week ◦ Wednesday 4-5pm (different time) ◦ Thursday 11-12noon (usual time)  Guest Lecture Wednesday Nov. 12 ◦ Jason Leonard – Will discuss the processing of real data for NASA spacecraft (Juno and/or Artemis) 3

University of Colorado Boulder 4 Lecture Quiz

University of Colorado Boulder  Correct: 57.14%  As the elements of the Kalman gain matrix decrease towards zero, the filter (select all that apply): ◦ The filter begins to weigh measurements greater than the a priori state in the measurement update. ◦ The filter begins ignoring measurements in favor of the propagated (i.e., a priori) state deviation vector ◦ The measurement update of the state-error covariance matrix (P) begins to yield only slight changes in P ◦ The filter state and covariance matrix experience no changes due to the time or measurement updates. 5

University of Colorado Boulder 6

University of Colorado Boulder  Correct: 57.14%  As the elements of the Kalman gain matrix decrease towards zero, the filter (select all that apply): ◦ The filter begins to weigh measurements greater than the a priori state in the measurement update. ◦ The filter begins ignoring measurements in favor of the propagated (i.e., a priori) state deviation vector ◦ The measurement update of the state-error covariance matrix (P) begins to yield only slight changes in P ◦ The filter state and covariance matrix experience no changes due to the time or measurement updates. 7 0% 90% 81% 33%

University of Colorado Boulder  Correct: 61.9%  Which of the following can cause the EKF to diverge? (For the sake of this problem, assume we are starting with the EKF and will not use a CKF for early observations) ◦ The reference trajectory is a poor approximation of the truth. ◦ Good measurements (trace(R) is small) with a bad a priori state (trace(P) is large) ◦ Bad measurements (trace(R) is large) with a good a priori (trace(P) is small) ◦ Good measurements (trace(R) is small) with a good a priori (trace(P) is small). 8 95% 76% 10%

University of Colorado Boulder  Correct: 76%  We need to estimate a state with an a priori covariance matrix of (using pseudo-MATLAB notation): Pbar = [ 1e16, 0; 0, 1e-1 ] Our computer has 14 significant digits. Can I use the CKF? ◦ No – the condition number of P is too big. ◦ Yes – the condition number of P is small enough ◦ Yes – the condition number does not matter ◦ It depends on the observation-error covariance matrix 9 76% 5% 0% 19%

University of Colorado Boulder  Correct: 81%  We need to estimate a state with an a priori covariance matrix of (using pseudo-MATLAB notation): Pbar = [ 1e16, 0; 0, 1e-1 ] Our computer has 14 significant digits. Can I use the Potter algorithm? ◦ No – the condition number of the matrix square-root of P is not small enough ◦ Yes – the condition number of the matrix square-root of P is small enough ◦ No – we need a triangular form of the updated covariance matrix square root (W) to continue using the Potter algorithm ◦ Yes, but we could also use the CKF if we prefer without any loss of accuracy % 5% 10% 5%

University of Colorado Boulder  Correct: 10%  When solving the linear system: Where N is the normal matrix and M is the information matrix, we can use the Cholesky decomposition of M (M=R^T*R) to solve for xhat. To do this, we set To solve for xhat, we first use a backward substitution to solve for z and then a forward substitution find xhat. We then use the R matrix to solve for S such that P=SS T. ◦ True ◦ False 11

University of Colorado Boulder  Forward Subs:  Backward Subs: 12

University of Colorado Boulder 13 Concept Quiz

University of Colorado Boulder 14 Final Questions?