1. Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION The Minimum Variance Estimate ASEN 5070 LECTURE 17 10/05/09

2. Colorado Center for Astrodynamics Research The University of Colorado 2 Review problem II. Given the system with the state vector defined by and the deviation vector defined by. Where indicates a small deviation from a reference value. a.Write the linearized equations in state space form, b.How would you determine the state transition matrix for this system? What Additional information is needed to generate the state transition matrix?

3. Colorado Center for Astrodynamics Research The University of Colorado 3 Review problem III. Circle the correct answers or answers. a. Given the observation state equation Where Which of the following state vectors are observable:

4. Colorado Center for Astrodynamics Research The University of Colorado 4 Review problem b. The differential equation is1. 1st order and 1st degree 2. 2nd order and 1st degree 3. linear 4. nonlinear 5. 2nd order and 2nd degree c. Given two uncorrelated observations and the second is twice as accurate as the first, we would use the following weighting matrix

5. Colorado Center for Astrodynamics Research The University of Colorado 5 Review problem d. If the differential equation for the state is given by It would not be necessary to use a state deviation vector. T or F e. The state transition matrix will contain terms such as. The units of this partial derivative are: 1. L/T, 2. L2/T, 3. 1/T, 4. It is dimensionless f. If the state transition matrix is symplectic it can be inverted by inspection. T or F

6. Colorado Center for Astrodynamics Research The University of Colorado Inclusion of Apriori Information in the Batch Processor Computational Algorithm If apriori information and with attendant covariance is given, this Information should be maintained when iterating the batch algorithm i.e., and should be held constant to begin each iteration. Hence, for the first iteration but Hence, Solving for yields,

7. Colorado Center for Astrodynamics Research The University of Colorado Inclusion of Apriori Information in the Batch Processor Computational Algorithm Thus for the n th iteration, the apriori value of is given by (4.6.4) and Finally,

8. Colorado Center for Astrodynamics Research The University of Colorado 8 4.4 THE MINIMUM VARIANCE ESTIMATE The least squares and weighted least squares methods do not include any information on the statistical characteristics of the measurement errors or the a priori errors in the values of the parameters to be estimated. The minimum variance approach is one method for removing this limitation.

9. Colorado Center for Astrodynamics Research The University of Colorado 9 THE MINIMUM VARIANCE ESTIMATE

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13. Colorado Center for Astrodynamics Research The University of Colorado 13 THE MINIMUM VARIANCE ESTIMATE Note that both matrices on the left side of eq (4.4.11) are nxn And symmetric.

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