1. Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION ASEN 5070 LECTURE 11 9/16,18/09

2. Colorado Center for Astrodynamics Research The University of Colorado 2 Concept Test True or False 1) If the relationship between the observations and the state is linear we do not have to iterate the Newton-Raphson equation. ____________ 2) Given observations of range rate, j=1----5, all elements of the following state vectors can be solved for (indicate T or F for case A and B) Case A: T or F __.Case B: T or F __.

3. Colorado Center for Astrodynamics Research The University of Colorado 3 Concept Test 3) The state vector in case 2.B could be solved for uniquely with one observation each of at one instant in time. _______ 4) For problem 2.B we may use any initial guess for the state but it may take many iterations to converge. _______ Since the observation-state and state propagation equations are linear we do not have to use a state deviation vector. ________ 5) Given:

4. Colorado Center for Astrodynamics Research The University of Colorado 4 Concept Test 6) The differential equation in each column of is independent of the equations in other columns. _________ 7) The least squares solution minimizes the sum of the residuals. ______ 8)For the equation we always have more unknowns than equations. ________ 9) If the determinant of a symmetric matrix is negative (answer T or F) a) It is not positive definite _______ b) It’s inverse does not exist. _______ c) Some eigenvalues are imaginary. ______ 10) The derivative of a scalar WRT a vector is a scalar. _____

5. Colorado Center for Astrodynamics Research The University of Colorado 5 Concept Test 11) The rank of is 2. _______ 12) What is the one topic or concept you are having the most trouble with in this class?

6. Colorado Center for Astrodynamics Research The University of Colorado 6 Constant of the motion when potential function contains tesseral and sectoral harmonics

7. Colorado Center for Astrodynamics Research The University of Colorado Cubed Sphere vs. Spherical Harmonics In some cases, integration constant fluctuations decreased by as much as 2 orders of magnitude. Spherical harmonic model is 70x70

8. Colorado Center for Astrodynamics Research The University of Colorado 8 Least Squares with apriori Information If an apriori value is available for (call it ) and an associated symmetric weighting matrix, the weighted least squares estimate of can be obtained.

9. Colorado Center for Astrodynamics Research The University of Colorado 9 Least Squares with apriori Information Given Where is the error in and its accuracy is reflected in the weighting matrix and Choose to minimize the performance index

10. Colorado Center for Astrodynamics Research The University of Colorado 10 Least Squares with apriori Information Writing explicitly in terms of (4.3.24) Results in (See Eq B.7.4)

11. Colorado Center for Astrodynamics Research The University of Colorado 11 Least Squares with apriori Information Solving for yields (4.3.25)

12. Colorado Center for Astrodynamics Research The University of Colorado 12 Least Squares with apriori Information Note that is symmetric also which will be positive definite if a solution exists

13. Colorado Center for Astrodynamics Research The University of Colorado 13 Computational Algorithm for the Batch Processor Look at, Likewise, and

14. Colorado Center for Astrodynamics Research The University of Colorado 14 Computational Algorithm for the Batch Processor Hence, the computational algorithm involves forming the indicated summations, performing the Matrix inversion and solving for. We will see later that there are more computationally efficient ways to solve for without inverting the normal matrix.

15. 15 Copyright 2006 LEO Orbit Determination Example The Batch (Least Squares) Algorithm

16. 16 Copyright 2006 LEO Orbit Determination Example The Batch (Least Squares) Algorithm We want to maintain apriori Information; hence,

17. 17 Copyright 2006 LEO Orbit Determination Example The batch (or least squares) processor processes all observation data at once and, then, determines the best estimate of the state deviation, thereby estimating. The best estimate of x is chosen such that it minimizes the sum of the squares of the calculated observation errors. Estimating x k begins with the following state propagation and observation-state relationships: The H matrix relates the state deviation vector at an epoch time to the observation deviation vector at another time.

18. 18 Copyright 2006 LEO Orbit Determination Example The normal equations are formed R is the observation error covariance matrix. and are the a priori state estimate and state error covariance, respectively. Eqn. (13) may now be solved for The RMS of the observation residuals is given by

19. 19 Copyright 2006 LEO Orbit Determination Example Instantaneous observation data is taken from three Earth fixed tracking stations over an approximate 5 hour time span (light time is ignored). where x, y, and z represent the spacecraft Earth Centered Inertial (ECI) coordinates and are the tracking station Earth Centered, Earth Fixed (ECEF) coordinates.

20. 20 Copyright 2006 LEO Orbit Determination Example

21. 21 Copyright 2006 LEO Orbit Determination Example Batch (Least Squares) Residuals RMS Values Pass 1Pass 2Pass 3 Range (m)732.7483502252640.3195707667262650.00974562719122707 Range Rate (m/s)2.900165318977110.001199729785847210.000997930398398708

22. Colorado Center for Astrodynamics Research The University of Colorado 22 Given the following system and a)Could we obtain a solution for using the least squares equation? Why or why not? Will minimum norm yield a solution? b)Assume we are given a priori information and. Can we now solve for ? Least Squares Example

23. Colorado Center for Astrodynamics Research The University of Colorado 23 Least Squares Example

24. Colorado Center for Astrodynamics Research The University of Colorado 24 CH 4 Problem (19) Given the observation-state relation and the observation sequence at,, and at,. Find the “best” estimate of.

25. Colorado Center for Astrodynamics Research The University of Colorado 25 CH 4 Problem (19) Find: We have 2 observations and 3 unknowns; hence, we use the minimum norm solution given by Eq (4.3.13) **Note the rank of H is 2

26. Colorado Center for Astrodynamics Research The University of Colorado CH 4 Problem (19) 26 We have 2 observations and 3 unknowns; hence, we use the minimum norm solution given by Eq (4.3.13). Note that the rank of H is 2 (4.3.13)

27. Colorado Center for Astrodynamics Research The University of Colorado 27 CH 4 Problem (19) Note that because, i.e., the state vector is not a function of time, and. Remember that we derived with the constraint that i.e.

28. Colorado Center for Astrodynamics Research The University of Colorado 28 CH 4 Problem (19) What if we had apriori information, e.g., Can we now have a least squares solution?

29. Colorado Center for Astrodynamics Research The University of Colorado 29 CH 4 Problem (19) Note that the rank of A+B is less than or equal to the rank of A plus the rank of B. Hence, in this example the rank of need not be rank 3 but their sum must be rank 3.

30. Colorado Center for Astrodynamics Research The University of Colorado 30 Given with and a, b, c, and e are constants. a)Write as a first order system and derive the A matrix

31. Colorado Center for Astrodynamics Research The University of Colorado 31 Given with and a, b, c, and e are constants. a)If what is ? Assume initial conditions, are given at. Write as a 1 st order system

32. Colorado Center for Astrodynamics Research The University of Colorado 32 If the matrix is not of full rank, i.e. does not exist, can we make exist by the proper choice of a weighting matrix, ? 1.T or F 2.Justify your answer The rank of the product AB of two matrices is less than or equal to The rank of A and is less than or equal to the rank of B. Hence the answer to (1) is false.

33. Colorado Center for Astrodynamics Research The University of Colorado 33 Given range observations in the 2-D flat earth problem, i.e. 1.Assume all parameters except and are known. We can solve for both and from range measurements taken simultaneously from two well separated tracking stations. T or F? 2.Justify your answer in terms of the rank of.

34. Colorado Center for Astrodynamics Research The University of Colorado 34 y x Station 1 Station 2 x s1, y s1 x s2, y s2 y o = y o * Note: may lie anywhere on this line and modified to accommodate it. Spacecraft at t = t i Incorrect y o

35. Colorado Center for Astrodynamics Research The University of Colorado 35

36. Colorado Center for Astrodynamics Research The University of Colorado 36 Assume we had both range and range rate; can we now solve for ?

37. Colorado Center for Astrodynamics Research The University of Colorado 37 The differential equation is (choose all correct answers) 1.2 nd order and 2 nd degree 2.2 nd order and 1 st degree 3.linear 4.nonlinear