ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones

1. ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 5: State Deviations and Fundamentals of Linear Algebra

2. Announcements • Homework 2– Due September 12 • Make-up Lecture • Today @ 3pm, here

3. Today’s Lecture • Effects of State Deviations • Linear Algebra (Appendix B)

4. Quantifying Effects of Orbit State Deviations

5. Effects of Small Variations • Quantification of such effects is fundamental to the OD methods discussed in this course! Time

6. Effects of Small Variations • Let’s think about the effects of small variations in coordinates, and how these impact future states. Example: Propagating a state in the presence of NO forces Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

7. Effects of Small Variations • What happens if we perturb the value of x0? Force model: 0 Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

8. Effects of Small Variations • What happens if we perturb the value of x0? Force model: 0 Final State: (xf+Δx, yf, zf, vxf, vyf, vzf) Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

9. Effects of Small Variations • What happens if we perturb the position? Force model: 0 Final State: (xf+Δx, yf+Δy, zf+Δz, vxf, vyf, vzf) Initial State: (x0+Δx, y0+Δy, z0+Δz, vx0, vy0, vz0) Initial State: (x0, y0, z0, vx0, vy0, vz0)

10. Effects of Small Variations • What happens if we perturb the value of vx0? Force model: 0 Initial State: (x0, y0, z0, vx0-Δvx, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

11. Effects of Small Variations • What happens if we perturb the value of vx0? Force model: 0 Final State: (xf+tΔvx, yf, zf, vxf+Δvx, vyf, vzf) Initial State: (x0, y0, z0, vx0+Δvx, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

12. Effects of Small Variations • What happens if we perturb the position and velocity? Force model: 0

13. Effects of Small Variations • We could have arrived at this easily enough from the equations of motion. Force model: 0

14. Effects of Small Variations • This becomes more challenging with nonlinear dynamics Force model: two-body

15. Effects of Small Variations • This becomes more challenging with nonlinear dynamics Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: two-body Initial State: (x0, y0, z0, vx0, vy0, vz0) The partial of one Cartesian parameter wrt the partial of another Cartesian parameter is ugly.

16. Effects of Small Variations • This becomes more challenging with nonlinear dynamics Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: two-body

17. Select Topics in Linear Algebra

18. Matrix Basics • Matrix A is comprised of elements ai,j • The matrix transpose swaps the indices

19. Matrix Basics • Matrix inverse A-1 is the matrix such that • For the inverse to exist, A must be square • We will treat vectors as n×1 matrices

20. Matrix Basics

21. Transpose/Inverse Identities

22. 2x2 Matrix Inverse Trick • If we have a 2x2, nonsingular matrix: Asking you to invert a full 2x2 matrix on an exam is fair game!

23. Matrix Determinant • The square matrix determinant, |A|, describes if a solution to a linear system exists: • It also describes the change in area/volume/etc. due to a linear operation:

24. Matrix Determinant Identities

25. Linear Independence • A set of vectors are linearly independent if none of them can be expressed as a linear combination of other vectors in the set • In other words, no scalars αi exist such that for some vector vj in the set {vi}, i=1,…,n,

26. Matrix Rank • The matrix column rank is the number of linearly independent columns of a matrix • The matrix row rank is the number of linearly independent rows of a matrix • rank(A) = min( col. rank of A, row rank of A)

27. Examples

28. Rank Identities

29. Vector Differentiation • When differentiating a scalar function w.r.t. a vector:

30. Vector Differentiation • When differentiating a function with vector output w.r.t. a vector:

31. Matrix Derivative Identities • If A and B are n×1 vectors that are functions of X:

32. Positive Definite Matrices • The n×n matrix A is positive definite if and only if: • The n×n matrix A is positive semi-definite if and only if:

33. Minimum of a function • The point x is a minimum if and is positive definite.

34. Eigenvalues/vectors • Given the n×n matrix A, there are n eigenvalues λ and vectors X≠0 where

35. Book Appendix B • Other identities/definitions in Appendix B of the book • Matrix Trace • Maximum/Minimum Properties • Matrix Inversion Theorems • Review the appendix and make sure you understand the material