University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 19: Examples with the Batch Processor

University of Colorado Boulder  Exam 1 – Friday, October 9 ◦ Any exam related questions?  My office hours today in CCAR Meeting room instead of ECNT 420 2

University of Colorado Boulder  Everyone did well on these two quizzes  All answers are included in the slides as an appendix, but we will only go over two questions from Quiz 4 3

University of Colorado Boulder  Percent Correct: D2L Error  Consider the observation-state equation: In the case of a nonlinear estimation problem, which of the following are true: ◦ The observation-state relationship is linear with respect to deviation vectors ◦ The observation-state relationship is linear with respect to the nonlinear estimated state X ◦ We require an a priori x (deviation vector) to estimate the state using least squares ◦ We require an a priori X (nonlinear state) to estimate the state using least squares 4

University of Colorado Boulder  Percent Correct: 30%  In the case of nonlinear estimation using the linear batch filter, we attempt to estimate a state deviation vector x by solving for the vector that minimizes the sum of the observation residuals. By adding the state deviation vector to our best guess for the initial trajectory (X*), we get an updated state. To get this estimated state deviation vector, we require the observation deviation vector y. To solve for this, we use the predicted measurement G(X*,t). ◦ True ◦ False 5

University of Colorado Boulder 6 Illustration – Object in Ballistic Trajectory

University of Colorado Boulder  A cannonball has been launched with some uncertainty on the initial trajectory. We wish to: ◦ Estimate the initial state of the cannonball for future calibrations ◦ Determine where the cannonball went  We have some observations near the peak of the trajectory. 7

University of Colorado Boulder 8 Start of measurements  Object in ballistic trajectory under the influence of gravity Start of filter

University of Colorado Boulder  Object in ballistic trajectory under the influence of gravity  Equations of motion: 9  EOMs: Linear or Nonlinear?

University of Colorado Boulder  Object in ballistic trajectory under the influence of gravity  Observation Equations: 10  Obs. Eqns: Linear or Nonlinear?  Filter: Linear or Nonlinear?

University of Colorado Boulder  Filter: Linear or Nonlinear?  What do we need to solve via least squares? 11

University of Colorado Boulder  How do we get the STM for this problem? 12

University of Colorado Boulder  How do we get H_tilde for this problem?  What is H ? 13

University of Colorado Boulder  We have initial uncertainties on the a priori and the observations: 14

University of Colorado Boulder 15 Pierson Correlation Coeffs

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University of Colorado Boulder 17  Filter error smaller than measurement errors  Why does the uncertainty decrease and then increase?

University of Colorado Boulder  Illustrates error and 3σ bounds for data fit and prediction 18 Filter Span Predicted

University of Colorado Boulder  Should examine both the pre- and post-fit residuals: 19

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University of Colorado Boulder  Observation Equations: 22 Station 1 Station 2

University of Colorado Boulder  How do we get H_tilde for this problem? 23

University of Colorado Boulder 24  Filter error smaller than measurement errors  Uncertainty decreases and then increases

University of Colorado Boulder  Illustrates error 3σ bounds for data fit and prediction 25

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University of Colorado Boulder 32 Appendix: Lecture Quizzes

University of Colorado Boulder 33 Lecture Quiz 4

University of Colorado Boulder  Percent Correct: D2L Error  Consider the observation-state equation: In the case of a nonlinear estimation problem, which of the following are true: ◦ The observation-state relationship is linear with respect to deviation vectors ◦ The observation-state relationship is linear with respect to the nonlinear estimated state X ◦ We require an a priori x (deviation vector) to estimate the state using least squares ◦ We require an a priori X (nonlinear state) to estimate the state using least squares 34 89% 25% 20%

University of Colorado Boulder  Percent Correct: 30%  In the case of nonlinear estimation using the linear batch filter, we attempt to estimate a state deviation vector x by solving for the vector that minimizes the sum of the observation residuals. By adding the state deviation vector to our best guess for the initial trajectory (X*), we get an updated state. To get this estimated state deviation vector, we require the observation deviation vector y. To solve for this, we use the predicted measurement G(X*,t). ◦ True ◦ False 35

University of Colorado Boulder  Percent Correct: 93%  Consider the weighted least-squares cost function J(x). We have two observation errors e 1 and e 2. The weights for those observations are w 1 =3 and w 2 =2. Which of the following provides the best solution? ◦ e 1 =1, e 2 =1 ◦ e 1 =2, e 2 =1 ◦ e 1 =1, e 2 =2 ◦ e 1 =1, e 2 =1/2 36

University of Colorado Boulder  Percent Correct: 84%  In the weighted least-squares estimator, the H matrix no longer needs to be full rank. ◦ True ◦ False 37

University of Colorado Boulder  Percent Correct: 91%  For a linear estimation problem solved via the batch filter, we require a priori information to obtain a solution. ◦ True ◦ False 38

University of Colorado Boulder 39 Lecture Quiz 5

University of Colorado Boulder  Percent Correct: 95%  The inverse of the variance-covariance matrix is symmetric ◦ True ◦ False 40

University of Colorado Boulder  Percent Correct: 76%  Which of the following lists of numbers has the largest variance? A: [ 0.0, 0.5, 1.0 ] B: [ 0.0, 0.25, 0.5, 0.75, 1.0 ] C: [ 0.0, 0.1, 0.2, 0.3, …, 0.8, 0.9, 1.0 ] ◦ A ◦ B ◦ C ◦ They all have the same variance 41

University of Colorado Boulder  Percent Correct: 86%  If X and Y are independent random variables drawn from the standard normal distribution and Z = X+Y, which of the following best describes the probability density of Z? ◦ U(0,1) (uniform distribution with range [0,1]) ◦ U(0,2) (uniform distribution with range [0,2]) ◦ A normal distribution with mean 0.0 ◦ A normal distribution with mean

University of Colorado Boulder  Percent Correct: 90%  If X is the number of people who fall asleep during an average ASEN 5070 lecture and “X” is drawn from U(0,2), then what is the expected value for the total number of instances of people falling asleep after 42 independent lectures? ◦ 0 ◦ 16 ◦ 30 ◦ Incorrect, but brownie points awarded!

University of Colorado Boulder  Percent Correct: 88% ◦ Let f(x) be a probability density function. Which of the following are true? 44 90% 98% 100% 2%