University of ColoradoBoulder

ASEN 5070: Statistical Orbit Determination I

Fall 2014

Professor Brandon A. Jones

Lecture 4: Flat-Earth Problem, Newton-Raphson

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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

Announcements

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Flat-Earth Problem

Newton-Raphson Iteration (Chapter 1)

State Deviations (Chapter 1 & 4)

Today’s Lecture

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Flat-Earth Problem

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Flat-Earth Problem

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Assume linear motion:

Flat Earth Problem

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Equations of Motion – Linear System

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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?

Flat-Earth Problem – Solution with Measured State

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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

Flat-Earth Problem

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Newton-Raphson Iteration

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Solving a linear system with the same number of equations as unknowns is easy:

Solving a Nonlinear System

However, what do we do if A is a function of x? For example:

Several tools exist, but we will discuss Newton-Raphson iteration

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Start with the Taylor expansion about x of some (infinitely differentiable) fcn:

Newton-Raphson (Overview)

To solve for δ, we truncate all but the first two terms and rearrange:

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Kepler’s Equation:

You have likely used it before…

We want to solve:

Letting f(xn+1)=0, what is δ? Why is this simplification

introduced?

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The same method holds for vectors:

NR with Vector Inputs

HW 1 uses such a method for the flat Earth problem

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Given:

Evaluate the computed observations for ti Compute cost function:

Homework Problem Soln Outline

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Compute matrix of partials with current est.:

Homework Problem Soln Outline

Update the state estimate:

Repeat until converged

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Estimation Problem - Observability

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No! There would be an infinite number of

possibilities that satisfy:

Can we estimate the station location?

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Quantifying Effects of Orbit State Deviations

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Let’s think about the effects of small variations in coordinates, and how these impact future states.

Effects of Small Variations

Initial State:(x0, y0, z0, vx0, vy0, vz0)

Final State:(xf, yf, zf, vxf, vyf, vzf)

Example: Propagating a state in the presence of NO forces

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What happens if we perturb the value of x0?

Effects of Small Variations

Initial State:(x0, y0, z0, vx0, vy0, vz0)

Final State:(xf, yf, zf, vxf, vyf, vzf)

Force model: 0

Initial State:(x0+Δx, y0, z0, vx0, vy0, vz0)

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What happens if we perturb the value of x0?

Effects of Small Variations

Initial State:(x0, y0, z0, vx0, vy0, vz0)

Final State:(xf, yf, zf, vxf, vyf, vzf)

Force model: 0

Initial State:(x0+Δx, y0, z0, vx0, vy0, vz0)

Final State:(xf+Δx, yf, zf, vxf, vyf, vzf)

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What happens if we perturb the position?

Effects of Small Variations

Initial State:(x0, y0, z0, vx0, vy0, vz0)

Force model: 0

Initial State:(x0+Δx, y0+Δy, z0+Δz,

vx0, vy0, vz0)

Final State:(xf+Δx, yf+Δy, zf+Δz,

vxf, vyf, vzf)

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What happens if we perturb the value of vx0?

Effects of Small Variations

Initial State:(x0, y0, z0, vx0, vy0, vz0)

Final State:(xf, yf, zf, vxf, vyf, vzf)

Force model: 0

Initial State:(x0, y0, z0, vx0-Δvx, vy0, vz0)

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What happens if we perturb the value of vx0?

Effects of Small Variations

Initial State:(x0, y0, z0, vx0, vy0, vz0)

Final State:(xf, yf, zf, vxf, vyf, vzf)

Force model: 0

Final State:(xf+tΔvx, yf, zf, vxf+Δvx, vyf, vzf)

Initial State:(x0, y0, z0, vx0+Δvx, vy0, vz0)

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What happens if we perturb the position and velocity?

Effects of Small Variations

Force model: 0

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We could have arrived at this easily enough from the equations of motion.

Effects of Small Variations

Force model: 0

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This becomes more challenging with nonlinear dynamics

Effects of Small Variations

Force model: two-body

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This becomes more challenging with nonlinear dynamics

Effects of Small Variations

Force model: two-body

Initial State:(x0, y0, z0, vx0, vy0, vz0)

Final State:(xf, yf, zf, vxf, vyf, vzf)

The partial of one Cartesian parameter wrt the partial of another Cartesian parameter is ugly.

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This becomes more challenging with nonlinear dynamics

Effects of Small Variations

Final State:(xf, yf, zf, vxf, vyf, vzf)

Force model: two-body

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Quantification of such effects is fundamental to the OD methods discussed in this course!

Effects of Small Variations