Transcript Lecture 4
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 4: Flat-Earth Problem, Newton-Raphson University of Colorado Boulder Homework 1 – Due September 4 Lecture Quiz 1 posted to D2L I am out of town Sept. 15-18 Done with astrodynamics elements required for class ◦ I will not have any office hours ◦ TA will add office hours to compensate ◦ Lecture pre- and post-recorded ◦ If you have any remaining questions, please see us in office hours University of Colorado Boulder 2 Flat-Earth Problem Newton-Raphson Iteration (Chapter 1) State Deviations (Chapter 1 & 4) University of Colorado Boulder 3 Flat-Earth Problem University of Colorado Boulder 4 University of Colorado Boulder 5 Assume linear motion: University of Colorado Boulder 6 University of Colorado Boulder 7 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? University of Colorado Boulder 8 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 NewtonRaphson Iteration University of Colorado Boulder 9 Newton-Raphson Iteration University of Colorado Boulder 10 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 University of Colorado Boulder 11 Start with the Taylor expansion about x of some (infinitely differentiable) function: To solve for δ, we truncate all but the first two terms and rearrange: University of Colorado Boulder 12 University of Colorado Boulder 13 Kepler’s Equation: We want to solve: Letting f(xn+1)=0, what is δ? Why is this simplification introduced? University of Colorado Boulder 14 The same method holds for vectors: HW 1 uses such a method for the flat Earth problem University of Colorado Boulder 15 University of Colorado Boulder 16 Given: Evaluate the computed observations for ti Compute cost function: University of Colorado Boulder 17 Compute matrix of partials with current est.: Update the state estimate: Repeat until converged University of Colorado Boulder 18 University of Colorado Boulder 19 No! There would be an infinite number of possibilities that satisfy: University of Colorado Boulder 20 Quantifying Effects of Orbit State Deviations University of Colorado Boulder 21 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) University of Colorado Boulder 22 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) University of Colorado Boulder 23 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) Initial State: (x0, y0, z0, vx0, vy0, vz0) University of Colorado Boulder ¶x f ¶x0 Final State: (xf, yf, zf, vxf, vyf, vzf) =1 24 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) University of Colorado Boulder é ê ê ê ê ê ê ê ê ë ¶x f ¶x f ¶x0 ¶y0 ¶y f ¶y f ¶x0 ¶y0 ¶z f ¶z f ¶x0 ¶y0 ¶x f ù ú ¶z0 ú ú é 1 0 0 ù ¶y f ú ê ú =ê 0 1 0 ú ¶z0 ú ê ú ë 0 0 1 úû ¶z f ú ¶z0 úû 25 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) University of Colorado Boulder 26 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) Initial State: (x0, y0, z0, vx0, vy0, vz0) University of Colorado Boulder ¶vx f ¶vx0 =1, ¶x f ¶vx0 Final State: (xf, yf, zf, vxf, vyf, vzf) =t 27 What happens if we perturb the position and velocity? Force model: 0 é ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê êë ¶x f ¶x f ¶x f ¶x f ¶x f ¶x f ¶x0 ¶y0 ¶z0 ¶vx0 ¶vy0 ¶vz0 ¶y f ¶y f ¶y f ¶y f ¶y f ¶y f ¶x0 ¶y0 ¶z0 ¶vx0 ¶vy0 ¶vz0 ¶z f ¶z f ¶z f ¶z f ¶z f ¶z f ¶x0 ¶y0 ¶z0 ¶vx0 ¶vy0 ¶vz0 ¶vx f ¶vx f ¶vx f ¶vx f ¶vx f ¶vx f ¶x0 ¶y0 ¶z0 ¶vx0 ¶vy0 ¶vz0 ¶vy f ¶vy f ¶vy f ¶vy f ¶vy f ¶vy f ¶x0 ¶y0 ¶z0 ¶vx0 ¶vy0 ¶vz0 ¶vz f ¶vz f ¶vz f ¶vz f ¶vz f ¶vz f ¶x0 ¶y0 ¶z0 ¶vx0 ¶vy0 ¶vz0 University of Colorado Boulder ù ú ú ú ú ú é ú ê ú ê ú ê ú ê ú=ê ú ê ú ê ú ê ú ë ú ú ú ú úû 1 0 0 0 1 0 0 0 1 t 0 0 0 t 0 0 0 t 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 ù ú ú ú ú ú ú ú úû 28 We could have arrived at this easily enough from the equations of motion. Force model: 0 University of Colorado Boulder é ê ê ê ê ê ê ê ê ê ê ë xf yf zf vx f vy f vz f ù é ú ê ú ê ú ê ú ê ú=ê ú ê ú ê ú ê ú ê ú êë û x0 + vx0 * t y0 + vy0 * t z0 + vz0 * t vx0 vy0 vz0 ù ú ú ú ú ú ú ú ú ú úû 29 This becomes more challenging with nonlinear dynamics Force model: two-body University of Colorado Boulder 30 This becomes more challenging with nonlinear dynamics Force model: two-body Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0) The partial of one Cartesian parameter wrt the partial of another Cartesian parameter is ugly. University of Colorado Boulder 31 This becomes more challenging with nonlinear dynamics Force model: two-body Final State: (xf, yf, zf, vxf, vyf, vzf) -GM r= 3 r r x f = f (x0, y0, z0, x0, y0, z0 ) University of Colorado Boulder 32 Quantification of such effects is fundamental to the OD methods discussed in this course! University of Colorado Boulder 33