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ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 28: Orthogonal Transformations University of Colorado Boulder Lecture quiz due at 5pm Exam 2 – Friday, November 7 University of Colorado Boulder 2 Minimum Variance Conventional Kalman Filter Extended Kalman Filter Prediction Residuals Handling Observation Biases Numeric Considerations in the Kalman Batch vs. CKF vs. EKF Effects of Uncertainties on Estimation Potter Square-Root Filter Cholesky Decomposition w/ Forward and Backward substitution Singular Value Decomposition Methods University of Colorado Boulder 3 Least Squares via Orthogonal Transformations University of Colorado Boulder 4 University of Colorado Boulder 5 Recall the least squares cost function: By property 4 on the previous slide and Q an orthogonal matrix: University of Colorado Boulder 6 University of Colorado Boulder 7 University of Colorado Boulder 8 The method for selecting R defines a particular algorithm ◦ Givens Transformations (Section 5.4) ◦ Householder Transformation (Section 5.5) ◦ Gram-Schmidt Orthogonalization Not in the book and we won’t cover it University of Colorado Boulder 9 LS Solution via Givens Transformations University of Colorado Boulder 10 University of Colorado Boulder 11 University of Colorado Boulder 12 Consider the desired result To achieve this, we select the Givens matrix such that We then use this transformation in the top equation University of Colorado Boulder 13 We do not want to add non-zero terms to the previously altered rows, so we use the identity matrix except in the rows of interest: University of Colorado Boulder 14 After applying the transformation, we get: Repeat for all remaining non-zero elements in the third column What if the term is already 0 ? University of Colorado Boulder 15 Need to find the orthogonal matrix Q to yield a matrix of the form of the RHS Q is generated using a series of Givens transformations G University of Colorado Boulder 16 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 17 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 18 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 19 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 20 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 21 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 22 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 23 We now have the required Q matrix (for this conceptual example): University of Colorado Boulder 24 University of Colorado Boulder 25 Givens Transformations – An New Example University of Colorado Boulder 26 Consider the case where: University of Colorado Boulder 27 University of Colorado Boulder 28 University of Colorado Boulder 29 University of Colorado Boulder 30 We then have the matrices needed to solve the system: University of Colorado Boulder 31 Batch vs. Givens University of Colorado Boulder 32 Consider the case where: The exact solution is: After truncation: University of Colorado Boulder 33 Well, the Batch can’t handle it. What about Cholesky decomposition? Darn, that’s singular too. Let’s give Givens a shot! University of Colorado Boulder 34 University of Colorado Boulder 35 University of Colorado Boulder 36 University of Colorado Boulder 37 Hence, Givens transformations give us a solution for the state Home Exercise: Why is this true? Note: R is not equal to H ! University of Colorado Boulder Still a problem w/ P ! 38 Givens uses a sequence of rotations to generate the R matrix Instead, Householder transformations use a sequence of reflections to generate R ◦ See the book for details University of Colorado Boulder 39