University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 41: Information Filter

University of Colorado Boulder  Exam 3 ◦ In-class Students: Due Friday by 5pm ◦ CAETE Students: Due 11:59pm (Mountain) on 12/13  Final Project Due December 14 by noon 2

University of Colorado Boulder 3 Project Q&A

University of Colorado Boulder 4 Information Filter

University of Colorado Boulder 5  Well, we know that the CKF has problems… Negative Values

University of Colorado Boulder 6  How about the Joseph formulation of the measurement update? Negative Values

University of Colorado Boulder  How about the EKF?  How about the Potter square-root filter? 7

University of Colorado Boulder  Time Update 8  Measurement Update:

University of Colorado Boulder 9  What if we go back to the minimum variance?

University of Colorado Boulder 10  If I don’t want to invert the information matrix, do I have another option?

University of Colorado Boulder  Well, that was easy.  What about the time update? 11

University of Colorado Boulder  What can we do to simplify this? 12 (Assume Q k-1 non-singular)

University of Colorado Boulder  Require that Q k-1 be non-singular  Do not need to invert the n×n information matrix 13 Still need to maintain information matrix separate from D !

University of Colorado Boulder  From the time update of the information matrix: 14

University of Colorado Boulder 15  Can I initialize the filter with an infinite a priori state covariance matrix?  What happens if we have very accurate measurements?

University of Colorado Boulder  Once the information matrix has a sufficiently small condition number: 16

University of Colorado Boulder 17  Provides a more numerically stable solution  Stability equals that of the Batch, but in a sequential implementation  Don’t need to generate state/covariance until needed  Square-root information filter (SRIF) ◦ Refined through extensive use in POD ◦ Includes smoothing capabilities

University of Colorado Boulder 18 Information Filter with Bierman’s Problem

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University of Colorado Boulder 22 Monte Carlo Analysis

University of Colorado Boulder  There are many unknowns in orbit determination. What are some? 23

University of Colorado Boulder  There are many unknowns in orbit determination ◦ Dynamics Model ◦ Dynamics Errors (systematic and stochastic) ◦ Measurement Model ◦ Measurement Noise  Many of these may be characterized using covariance analysis (CH. 6, StatOD 2)  Given the large number of random inputs, how would we characterize the possible OD performance when covariance analysis is limited? 24

University of Colorado Boulder  Consider many different types of models and model errors  What about the accuracy of input models? ◦ Example: Gravity Field ◦ Our best estimate of the gravity field still has a variance.  How do we consider the filter performance with such errors? 25

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