University of Colorado Boulder 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 2

University of Colorado Boulder  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 3

University of Colorado Boulder 4 Least Squares via Orthogonal Transformations

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University of Colorado Boulder  Recall the least squares cost function: 6  By property 4 on the previous slide and Q an orthogonal matrix:

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University of Colorado Boulder  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 9

University of Colorado Boulder 10 LS Solution via Givens Transformations

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University of Colorado Boulder  Consider the desired result 13  To achieve this, we select the Givens matrix such that  We then use this transformation in the top equation

University of Colorado Boulder  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: 14

University of Colorado Boulder  After applying the transformation, we get: 15  Repeat for all remaining non-zero elements in the third column  What if the term is already 0 ?

University of Colorado Boulder  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 16

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 select G to get a zero for the term in red: To achieve this, we use:

University of Colorado Boulder  We now have the required Q matrix (for this conceptual example): 24

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University of Colorado Boulder 26 Givens Transformations – An New Example

University of Colorado Boulder  Consider the case where: 27

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University of Colorado Boulder  We then have the matrices needed to solve the system: 31

University of Colorado Boulder 32 Batch vs. Givens

University of Colorado Boulder  Consider the case where: 33  The exact solution is:  After truncation:

University of Colorado Boulder  Well, the Batch can’t handle it. What about Cholesky decomposition? 34  Darn, that’s singular too.  Let’s give Givens a shot!

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University of Colorado Boulder  Hence, Givens transformations give us a solution for the state 38 Home Exercise: Why is this true? Note: R is not equal to H ! Still a problem w/ P !

University of Colorado Boulder  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 39