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

ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 4: Flat-Earth Problem, Newton - Raphson. Announcements. Homework 0 & 1 – Due September 5 I am out of town Sept. 9-12 No office hours Lecture pre- recorded

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

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 4: Flat-Earth Problem, Newton-Raphson

  2. Announcements • Homework 0 & 1 – Due September 5 • I am out of town Sept. 9-12 • No office hours • Lecture pre-recorded • Done with astrodynamics elements required for class • If you have any remaining questions, please see us in office hours

  3. Today’s Lecture • Flat-Earth Problem • Newton-RaphsonIteration (Chapter 1) • State Deviations (Chapter 1 & 4)

  4. Flat-Earth Problem

  5. Flat-Earth Problem

  6. Flat Earth Problem • Assume linear motion:

  7. Equations of Motion – Linear System

  8. Flat-Earth Problem – Solution with Measured State • 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, e.g., range?

  9. 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. • You will solve one case of this problem for HW 1, Prob. 6 via Newton-Raphson Iteration

  10. Newton-Raphson Iteration

  11. 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

  12. 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:

  13. 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?

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

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

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

  17. Estimation Problem - Observability

  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!

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