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University of Colorado Boulder 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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University of Colorado Boulder Homework 2– Due September 12 Make-up Lecture ◦ 3pm, here 2

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University of Colorado Boulder Effects of State Deviations Linear Algebra (Appendix B) 3

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University of Colorado Boulder 4 Quantifying Effects of Orbit State Deviations

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University of Colorado Boulder Quantification of such effects is fundamental to the OD methods discussed in this course! 5 Time

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University of Colorado Boulder Let’s think about the effects of small variations in coordinates, and how these impact future states. 6 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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University of Colorado Boulder What happens if we perturb the value of x0? 7 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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University of Colorado Boulder What happens if we perturb the value of x0? 8 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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University of Colorado Boulder What happens if we perturb the position? 9 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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University of Colorado Boulder What happens if we perturb the value of vx0? 10 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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University of Colorado Boulder What happens if we perturb the value of vx0? 11 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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University of Colorado Boulder What happens if we perturb the position and velocity? 12 Force model: 0

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University of Colorado Boulder We could have arrived at this easily enough from the equations of motion. 13 Force model: 0

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University of Colorado Boulder This becomes more challenging with nonlinear dynamics 14 Force model: two-body

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University of Colorado Boulder This becomes more challenging with nonlinear dynamics 15 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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University of Colorado Boulder This becomes more challenging with nonlinear dynamics 16 Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: two-body

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University of Colorado Boulder 17 Select Topics in Linear Algebra

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University of Colorado Boulder Matrix A is comprised of elements a i,j The matrix transpose swaps the indices 18

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University of Colorado Boulder 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 19

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University of Colorado Boulder If we have a 2x2, nonsingular matrix: 22 Asking you to invert a full 2x2 matrix on an exam is fair game!

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University of Colorado Boulder The square matrix determinant, |A|, describes if a solution to a linear system exists: 23 It also describes the change in area/volume/etc. due to a linear operation:

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University of Colorado Boulder 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 α i exist such that for some vector v j in the set {v i }, i=1,…,n, 25

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University of Colorado Boulder 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) 26

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University of Colorado Boulder When differentiating a scalar function w.r.t. a vector: 29

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University of Colorado Boulder When differentiating a function with vector output w.r.t. a vector: 30

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University of Colorado Boulder If A and B are n×1 vectors that are functions of X: 31

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University of Colorado Boulder The n×n matrix A is positive definite if and only if: 32 The n×n matrix A is positive semi-definite if and only if:

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University of Colorado Boulder The point x is a minimum if 33 and is positive definite.

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University of Colorado Boulder Given the n×n matrix A, there are n eigenvalues λ and vectors X≠0 where 34

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University of Colorado Boulder 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 35

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