University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 18: Minimum Variance Estimator and the Term Project

University of Colorado Boulder  Exam 1 – Friday, October 10 2

University of Colorado Boulder  Minimum Variance w/ A Priori  Sequential Processing w/ Minimum Variance  Term Project 3

University of Colorado Boulder 4 Minimum Variance w/ A Priori

University of Colorado Boulder  With the least squares solution, we minimized the square of the residuals  Instead, what if we want the estimate that gives us the highest confidence in the solution: ◦ What is the linear, unbiased, minimum variance estimate of the state x? 5

University of Colorado Boulder  What is the linear, unbiased, minimum variance estimate of the state x ? ◦ This encompasses three elements  Linear  Unbiased, and  Minimum Variance  We consider each of these to formulate a solution 6

University of Colorado Boulder 7  Put into the context of scalars:

University of Colorado Boulder  Turns out, we get the weighted, linear least squares!  Hence, the linear least squares gives us the minimum variance solution ◦ Of course, this is predicated on all of our statistical/linearization assumptions 8

University of Colorado Boulder  To add a priori in the least squares, we augment the cost function J(x) to include the minimization of the a priori error.  How do we control the weighting of the a priori solution and the observations in the cost function? 9

University of Colorado Boulder  This is analogous to treating the a priori information as an observation of the estimated state at the epoch time 10

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University of Colorado Boulder  Like the previous case, the statistical least squares w/ a priori is equivalent to the minimum variance estimator  Do I have to use a statistical description of the observation/state errors to estimate the state? 13

University of Colorado Boulder  Least squares does not require a probabilistic definition of the weights/state  The minimum variance estimator demonstrates that, for a Gaussian definition of the observation and state errors, the LS is the best solution  Also know as the Best Linear Unbiased Estimator (BLUE)  Now, we can use the minimum variance estimator as a sequential estimator… 14

University of Colorado Boulder 15 Minimum Variance and Sequential Processing

University of Colorado Boulder 16 X*X*  Batch – process all observations in a single run of the filter  Sequential – process each observations individually (usually as they become available over time)

University of Colorado Boulder  Recall how to map the state deviation and covariance matrix (previous lecture) 17  Can we leverage this information to sequentially process measurements in the minimum variance / least squares algorithm?

University of Colorado Boulder  Given from a previous filter run: 18  We have new a observation and mapping matrix:  We can update the solution via:

University of Colorado Boulder  Two principle phases in any sequential estimator ◦ Time Update  Map previous state deviation and covariance matrix to the current time of interest ◦ Measurement Update  Update the state deviation and covariance matrix given the new observations at the time of interest  Jargon can change with communities ◦ Forecast and analysis ◦ Prediction and fusion ◦ others… 19

University of Colorado Boulder 20  No assumptions on the number of observations at t k.  Wait, but what if we have fewer observations than unknowns at t k ? ◦ Do we have an underdetermined system?

University of Colorado Boulder 21  The a priori may be based on independent analysis or a previous estimate ◦ Independent analysis could be a product of:  Expected launch vehicle performance  Previous analysis of system (a priori gravity field)  Initial orbit determination solution

University of Colorado Boulder 22  We still have to invert a n × n matrix  Can be computationally expensive for large n ◦ Gravity field estimation: ~n 2 +2n-3 coefficients!  May become sensitive to numeric issues

University of Colorado Boulder 23  Is there a better sequential processing algorithm? ◦ YES! – This equations above may be manipulated to yield the Kalman filter (next week)

University of Colorado Boulder 24 Term Project Introduction

University of Colorado Boulder  The OD project consists of ◦ Deriving the algorithms for estimating the state of a spacecraft ◦ Implementing the algorithms ◦ Processing range and range-rate observations from three ground-stations 25

University of Colorado Boulder  3 Ground-based tracking stations ◦ Equatorial station’s position fixed in filter  Dynamics Model ◦ Two-body, unknown μ ◦ Also estimating J 2 ◦ Drag with unknown C D ◦ Already completed! ◦ Simple satellite, spherical with known mass and area 26

University of Colorado Boulder  Estimate satellite position and velocity in ECI  Constants governing satellite dynamics  Station positions on Earth (ECEF)  18x1 state vector 27

University of Colorado Boulder  Unknown ground tracking station positions ◦ Boat in Pacific at equator  Equatorial station still estimated, but a priori covariance is very small ◦ Turkey ◦ Greenland 28

University of Colorado Boulder  Range (m) and range-rate (m/s) ◦ Range: Zero mean, σ = 1 cm ◦ Range-Rate: Zero mean, σ = 1 mm/s  There is an observation at t=0 !  Observation intervals do not overlap ◦ Number of observations at a given time always equals 2 ◦ Would add more complexity to the filter 29

University of Colorado Boulder  Derive A and H_tilde and implement the partials in software ◦ Use the symbolic toolbox with the jacobian() command ◦ Output results to a file to later copy/paste  Add numeric integration of STM ◦ Check intermediate results with the website 30

University of Colorado Boulder  Implement a batch least squares filter to estimate the state and error covariance ◦ First iteration implemented at the end of HW 9  Implement a Kalman/Sequential filter to estimate the state and error covariance ◦ Completed with HW 10 31

University of Colorado Boulder  Discussion of OD problem and batch processor in the report  Discuss the results ◦ Compare results from the Kalman and LS filters  Plots: ◦ Residuals ◦ Covariance ◦ Error ellipsoids (more in future lecture) 32

University of Colorado Boulder  Process one data type at a time ◦ Range-only processing ◦ Range-rate only ◦ Compare results (which does better?)  Discussion of results 33

University of Colorado Boulder  The previous tasks get you in the high-B range  Additional elements boost your grade from there ◦ Extended Kalman filter (EKF needed for StatOD 2 anyway) ◦ Potter Filter ◦ Givens transformations ◦ Smoothing ◦ Interval processing of data ◦ Others…  The additional elements is your chance to further explore a topic of interest covered in class 34