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University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 24: Numeric Considerations and Introduction to Square-Root Algorithms

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University of Colorado Boulder Homework 7 Due Friday Lecture Quiz Due by 5pm on Friday 2

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University of Colorado Boulder 3 Filter Saturation

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University of Colorado Boulder 5 Pitfall 2: Beware of collapsing covariance - Prevents new data from influencing solution - More prevalent for longer time-spans

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University of Colorado Boulder 6 Pitfall 2: Beware of collapsing covariance - Prevents new data from influencing solution - More prevalent for longer time-spans

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University of Colorado Boulder To enforce a symmetric result, we may instead use the Potter Formulation: 10 Always yields a symmetric P Assumes that the a priori covariance is positive definite, i.e., not corrupted by previous numeric errors Does not ensure a positive definite covariance matrix You will derive the above in the homework Still applied for unbiased scenarios

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University of Colorado Boulder 11 Bierman Example of Poorly Conditioned System

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University of Colorado Boulder 13 Process the first observation:

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University of Colorado Boulder 14 Process the second observation:

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University of Colorado Boulder Consider the implementation on a computer with a limited precision: 15

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University of Colorado Boulder Exact to order ε 18

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University of Colorado Boulder 20 Potter Algorithm – Motivation and Derivation Chapter 5

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University of Colorado Boulder The condition number of P may be described by 21 With p significant digits, there are estimation difficulties as If we can’t change the condition number, is there something else we can do?

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University of Colorado Boulder For W above, the condition number is 22 Is there something we can do to instead operate on W ?

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University of Colorado Boulder We must process the observations one at a time If we have multiple observations at a single time, this requires that R be diagonal. What can we do if the observations at a single time have a non-zero correlation? 26

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University of Colorado Boulder 31 Process the observations one at a time Repeat if multiple observations available at a single time More computationally expensive than Kalman, but more accurate W after the measurement update is not triangular! (Important for some algorithms) Motivates the derivation of the triangular square-root method (pp. 335-340)

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University of Colorado Boulder If we are given P as a priori information, how do we get W ? If P is diagonal, this is trivial: 32 Great, but what if it isn’t diagonal? Cholesky decomposition (next week)

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