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University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 28: Orthogonal Transformations

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University of Colorado Boulder Lecture quiz due at 5pm Exam 2 – Friday, November 7 2

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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

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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

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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

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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

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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 ?

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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

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University of Colorado Boulder 17 We select G to get a zero for the term in red: To achieve this, we use:

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University of Colorado Boulder 18 We select G to get a zero for the term in red: To achieve this, we use:

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University of Colorado Boulder 19 We select G to get a zero for the term in red: To achieve this, we use:

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University of Colorado Boulder 20 We select G to get a zero for the term in red: To achieve this, we use:

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University of Colorado Boulder 21 We select G to get a zero for the term in red: To achieve this, we use:

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University of Colorado Boulder 22 We select G to get a zero for the term in red: To achieve this, we use:

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University of Colorado Boulder 23 We select G to get a zero for the term in red: To achieve this, we use:

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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

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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

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University of Colorado Boulder 32 Batch vs. Givens

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

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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 !

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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

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