ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 4: Newton-Raphson and Linear Algebra

2. Announcements • Homework 0 & 1 – Due September 6 • Office Hours Thursday 2-3pm (this week only) • If you planned to come at the normal 3-4pm time and cannot make 2-3pm, please send me an e-mail and we can schedule an appointment

3. Homework 1 Changes • Two changes sent via e-mail • Combined problems 4 and 5 (old assignment numbers) to more clearly define the answer • Ask for code as an appendix to the main file instead of separate software files

4. Numeric Issues • Will get an imaginary number from cos-1(a) if a=1+1e-16 (for example) • The 1e-16 is a result of finite point arithmetic • You may need to use something akin to the pseudocode:

5. Homework Grading • TA Marco Balducci • Office Hours in the Undergrad Lounge 3-4 MWF • Marc.Balducci@Colorado.edu • Grading Will Be Largely Consistent Across Assignments • 33% Presentation • 33% Final Answer • 33% Shown Work • Latex Is Not Required But Highly Encouraged • Automatically covers a lot of presentation points

6. Today’s Lecture • Newton-RaphsonIteration (Chapter 1) • Effects of State Deviations • Linear Algebra (Appendix B)

7. Newton-Raphson Iteration

8. Solving a Nonlinear System • Solving a linear system with the same number of equations as unknowns is easy: • However, what do we do if A is a function of x? For example: • Several tools exist, but we will discuss Newton-Raphson iteration

9. Newton-Raphson (Overview) • Start with the Taylor expansion about x of some (infinitely differentiable) fcn: • To solve for δ, we truncate all but the first two terms and rearrange:

10. You have likely used it before… • Kepler’s Equation: • We want to solve: • Letting f(xn+1)=0, what is δ? Why is this simplification introduced?

11. NR with Vector Inputs • The same method holds for vectors: • HW 1 uses such a method for the flat Earth problem

12. Flat Earth Problem

13. Flat Earth Problem • Assume linear motion:

14. Flat Earth Problem • Given an error-free state at a time t, we can solve for the state at t0 • What about when we have a different observation type?

15. Flat-Earth Problem • Relationship between the estimated state and the observations is no longer linear • For our purposes, let’s assume the station coordinates are known.

16. Homework Problem Soln Outline • Given: • Evaluate the computed observations for ti • Compute cost function:

17. Homework Problem Soln Outline • Compute matrix of partials with current est.: • Update the state estimate: • Repeat until convergence

18. Can we estimate the station location? • No! • There would be an infinite number of possibilities that satisfy:

19. Quantifying Effects of Orbit State Deviations

20. 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)

21. 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)

22. 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)

23. 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)

24. 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)

25. 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)

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

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

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

29. 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.

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

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