University of Colorado Boulder ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 8: Stat OD Processes 1

University of Colorado Boulder  Homework 1 Graded ◦ Comments included on D2L ◦ Any questions, talk with us this week  Homework 2 CAETE due Today ◦ Graded soon after  Homework 3 due Today  Homework 4 due next week 2

University of Colorado Boulder  A few questions on Problem 4’s Laplace Transform.  There’s one Transform missing from the table:  Note: This has been posted in the HW3 discussions on D2L for a few days now. Check there if you have a commonly-asked question! 3

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University of Colorado Boulder  Truth  Reference  Best Estimate  Observations are functions of state parameters, but usually NOT state parameters.  Mismodeled dynamics  Underdetermined system. 13

University of Colorado Boulder  We have noisy observations of certain aspects of the system.  We need some way to relate each observation to the trajectory that we’re estimating. 14 Observed Range Computed Range True Range = ??? X*X*

University of Colorado Boulder  We have noisy observations of certain aspects of the system.  We need some way to relate each observation to the trajectory that we’re estimating. 15 Observed Range Computed Range True Range = ??? ε = O-C = “Residual” X*X*

University of Colorado Boulder  Assumptions: ◦ The reference/nominal trajectory is near the truth trajectory.  Linear approximations are decent ◦ Force models are good approximations for the duration of the measurement arc. ◦ The filter that we’re using is unbiased:  The filter’s best estimate is consistent with the true trajectory. 16

University of Colorado Boulder  How do we best fit the data? 17 Residuals = ε = O-C

University of Colorado Boulder  How do we best fit the data? 18 Residuals = ε = O-C

University of Colorado Boulder  How do we best fit the data? 19 Residuals = ε = O-C No

University of Colorado Boulder  How do we best fit the data? 20 Residuals = ε = O-C No

University of Colorado Boulder  How do we best fit the data? 21 Residuals = ε = O-C No Not bad

University of Colorado Boulder  How do we best fit the data? 22 Residuals = ε = O-C No Not bad

University of Colorado Boulder  How do we best fit the data?  A good solution, and one easy to code up, is the least-squares solution 23

University of Colorado Boulder  How do we map an observation to the trajectory? 24 Observed Range Computed Range ε = O-C = “Residual” X*X*

University of Colorado Boulder  How do we map an observation to the trajectory? 25

University of Colorado Boulder  How do we map an observation to the trajectory?  Very non-linear relationships!  Need to linearize to make a practical algorithm. 26

University of Colorado Boulder  Linearization  Introduce the state deviation vector  If the reference/nominal trajectory is close to the truth trajectory, then a linear approximation is reasonable. 27

University of Colorado Boulder  Goal of the Stat OD process:  Find a new state/trajectory that best fits the observations:  If the reference is near the truth, then we can assume: 28

University of Colorado Boulder  Goal of the Stat OD process:  The best fit trajectory is represented by 29 This is what we want

University of Colorado Boulder  How do we map the state deviation vector from one time to another? 30 X*X*

University of Colorado Boulder  How do we map the state deviation vector from one time to another?  The state transition matrix. 31

University of Colorado Boulder  How do we map the state deviation vector from one time to another?  The state transition matrix.  It permits: 32

University of Colorado Boulder  The state transition matrix maps a deviation in the state from one epoch to another.  It is constructed via numerical integration, in parallel with the trajectory itself. 33

University of Colorado Boulder  The “A” Matrix: 34

University of Colorado Boulder  Still need to know how to map measurements from one time to a state at another time! 35

University of Colorado Boulder  Still need to know how to map measurements from one time to a state at another time!  Would like this: 36

University of Colorado Boulder  Still need to know how to map measurements from one time to a state at another time! 37

University of Colorado Boulder  Still need to know how to map measurements from one time to a state at another time!  Define: 38

University of Colorado Boulder  The Mapping Matrix 39

University of Colorado Boulder  Now we can map an observation to the state at an epoch. 40 Observed Range Computed Range ε = O-C = “Residual” X*X*

University of Colorado Boulder  We have the method of least squares

University of Colorado Boulder  We have the method of least squares

University of Colorado Boulder  We have the method of least squares

University of Colorado Boulder  We have the method of least squares

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University of Colorado Boulder Or in component form: Expressed in first order form:

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University of Colorado Boulder  Homework 1 Graded ◦ Comments included on D2L ◦ Any questions, talk with us this week  Homework 2 CAETE due Today ◦ Graded soon after  Homework 3 due Today  Homework 4 due next week 50