University of ColoradoBoulder

ASEN 5070: Statistical Orbit Determination I

Fall 2014

Professor Brandon A. Jones

Lecture 5: State Deviations and Fundamentals of Linear Algebra

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Homework 2– Due September 12

Make-up Lecture◦ Today @ 3pm, here

Announcements

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Effects of State Deviations

Linear Algebra (Appendix B)

Today’s Lecture

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Quantifying Effects of Orbit State Deviations

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

Effects of Small Variations

Time

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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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Select Topics in Linear Algebra

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Matrix A is comprised of elements ai,j

The matrix transpose swaps the indices

Matrix Basics

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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

Matrix Basics

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Matrix Basics

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Transpose/Inverse Identities

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If we have a 2x2, nonsingular matrix:

2x2 Matrix Inverse Trick

Asking you to invert a full 2x2 matrix on an exam is fair game!

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The square matrix determinant, |A|, describes if a solution to a linear system exists:

Matrix Determinant

It also describes the change in area/volume/etc. due to a linear operation:

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Matrix Determinant Identities

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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,

Linear Independence

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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)

Matrix Rank

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Examples

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Rank Identities

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When differentiating a scalar function w.r.t. a vector:

Vector Differentiation

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When differentiating a function with vector output w.r.t. a vector:

Vector Differentiation

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If A and B are n×1 vectors that are functions of X:

Matrix Derivative Identities

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The n×n matrix A is positive definite if and only if:

Positive Definite Matrices

The n×n matrix A is positive semi-definite if and only if:

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The point x is a minimum if

Minimum of a function

and

is positive definite.

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Given the n×n matrix A, there are n eigenvalues λ and vectors X≠0 where

Eigenvalues/vectors

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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

Book Appendix B