asen 5070: statistical orbit determination i fall 2014 professor brandon a. jones
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ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 5: State Deviations and Fundamentals of Linear Algebra. Announcements. Homework 2– Due September 12 Make-up Lecture Today @ 3pm, here. Today’s Lecture. Effects of State Deviations - PowerPoint PPT PresentationTRANSCRIPT
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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