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

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

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

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

University of Colorado Boulder 11 Bierman Example of Poorly Conditioned System

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University of Colorado Boulder 13  Process the first observation:

University of Colorado Boulder 14  Process the second observation:

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

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?

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 )

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)