Transcript Lecture 5
ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 5: State Deviations and Fundamentals of
Linear Algebra
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Lecture Quiz – Due Friday by 5pm
◦ Covers lectures 1-4
Homework 1– Due Friday by 9am
Make-up Lectures in ECCS 1B12
◦ Monday 9/14 @ 4pm
◦ Monday 9/12 @ 4pm
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Effects of State Deviations
Linear Algebra (Appendix B)
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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!
Time
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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)
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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)
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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)
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¶x f
¶x0
Final State:
(xf, yf, zf, vxf, vyf, vzf)
=1
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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)
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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
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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)
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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)
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¶vx f
¶vx0
=1,
¶x f
¶vx0
Final State:
(xf, yf, zf, vxf, vyf, vzf)
=t
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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
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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
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We could have arrived at this easily enough
from the equations of motion.
Force model: 0
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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
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This becomes more challenging with
nonlinear dynamics
Force model: two-body
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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 the final Cartesian state
w.r.t. the initial Cartesian state is not
pretty.
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This becomes more challenging with
nonlinear dynamics
Force model: two-body
Final State:
(xf, yf, zf, vxf, vyf, vzf)
x f = f (x0, y0, z0, x0, y0, z0 )
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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
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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
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If we have a 2x2, nonsingular matrix:
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:
It also describes the change in area/volume/etc. due to a
linear operation:
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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,
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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)
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What is the rank of the following matrices:
é
ê
A =ê
ê
ê
ë
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
ù
ú
ú
ú
ú
û
é
ê
B =ê
ê
ê
ë
1
0
0
0
0
1
2
0
0
1
2
0
0
0
0
1
ù
ú
ú
ú
ú
û
é
ê
C =ê
ê
ê
ë
0
0
0
1
0
1
2
0
0
1
2
0
1
0
1
0
ù
ú
ú
ú
ú
û
é
ê
D =ê
ê
ê
ë
1
1
1
1
2
2
2
2
0
0
0
0
0
0
0
0
ù
ú
ú
ú
ú
û
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When differentiating a scalar function w.r.t. a
vector:
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When differentiating a function with vector
output w.r.t. a vector:
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If A and B are n×1 vectors that are functions
of X:
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The n×n matrix A is positive definite if and only if:
The n×n matrix A is positive semi-definite if and only
if:
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The point x is a minimum if
and
is positive definite.
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Given the n×n matrix A, there are n eigenvalues λ
and vectors X≠0 where
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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
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