ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 19: Examples with the Batch Processor
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Exam 1 – Friday, October 9
◦ Any exam related questions?

My office hours today in CCAR Meeting room
instead of ECNT 420
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Everyone did well on these two quizzes
All answers are included in the slides as an
appendix, but we will only go over two
questions from Quiz 4
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Percent Correct: D2L Error

Consider the observation-state equation:
In the case of a nonlinear estimation problem, which of the
following are true:
◦ The observation-state relationship is linear with respect to deviation
vectors
◦ The observation-state relationship is linear with respect to the nonlinear
estimated state X
◦ We require an a priori x (deviation vector) to estimate the state using least
squares
◦ We require an a priori X (nonlinear state) to estimate the state using least
squares
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Percent Correct: 30%
In the case of nonlinear estimation using the linear
batch filter, we attempt to estimate a state deviation
vector x by solving for the vector that minimizes the
sum of the observation residuals. By adding the state
deviation vector to our best guess for the initial
trajectory (X*), we get an updated state. To get this
estimated state deviation vector, we require the
observation deviation vector y. To solve for this, we
use the predicted measurement G(X*,t).
◦ True
◦ False
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Illustration – Object in Ballistic Trajectory
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A cannonball has been launched with some
uncertainty on the initial trajectory. We wish
to:
◦ Estimate the initial state of the cannonball for
future calibrations
◦ Determine where the cannonball went

We have some observations near the peak of
the trajectory.
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Object in ballistic trajectory under the
influence of gravity
Start of measurements
Start of filter
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Object in ballistic trajectory under the
influence of gravity
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Equations of motion:

EOMs: Linear or Nonlinear?
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Object in ballistic trajectory under the
influence of gravity
Observation Equations:
Obs. Eqns: Linear or Nonlinear?
Filter: Linear or Nonlinear?
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Filter: Linear or Nonlinear?
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What do we need to solve via least squares?
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How do we get the STM for this problem?
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How do we get H_tilde for this problem?
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What is H ?
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We have initial uncertainties on the a priori
and the observations:
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Pierson Correlation Coeffs
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Filter error
smaller than
measurement
errors
Why does the
uncertainty
decrease and
then increase?
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Predicted
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Illustrates error
and 3σ bounds
for data fit and
prediction
Filter Span
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Should examine both the pre- and post-fit
residuals:
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Observation Equations:
Station 2
Station 1
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How do we get H_tilde for this problem?
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
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Filter error
smaller than
measurement
errors
Uncertainty
decreases and
then increases
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Illustrates error
3σ bounds for
data fit and
prediction
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Appendix: Lecture Quizzes
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Lecture Quiz 4
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
Percent Correct: D2L Error

Consider the observation-state equation:
In the case of a nonlinear estimation problem, which of the
following are true:
89%
◦ The observation-state relationship is linear with respect to deviation
vectors
25%
◦ The observation-state relationship is linear with respect to the nonlinear
estimated state X
20%
◦ We require an a priori x (deviation vector) to estimate the state using least
squares
20%
◦ We require an a priori X (nonlinear state) to estimate the state using least
squares
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

Percent Correct: 30%
In the case of nonlinear estimation using the linear
batch filter, we attempt to estimate a state deviation
vector x by solving for the vector that minimizes the
sum of the observation residuals. By adding the state
deviation vector to our best guess for the initial
trajectory (X*), we get an updated state. To get this
estimated state deviation vector, we require the
observation deviation vector y. To solve for this, we
use the predicted measurement G(X*,t).
◦ True
◦ False
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

Percent Correct: 93%
Consider the weighted least-squares cost
function J(x). We have two observation errors e1
and e2. The weights for those observations are
w1=3 and w2=2. Which of the following provides
the best solution?
◦
◦
◦
◦
e1=1,
e1=2,
e1=1,
e1=1,
e2=1
e2=1
e2=2
e2=1/2
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Percent Correct: 84%
In the weighted least-squares estimator, the
H matrix no longer needs to be full rank.
◦ True
◦ False
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Percent Correct: 91%
For a linear estimation problem solved via the
batch filter, we require a priori information to
obtain a solution.
◦ True
◦ False
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Lecture Quiz 5
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Percent Correct: 95%
The inverse of the variance-covariance matrix
is symmetric
◦ True
◦ False
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Percent Correct: 76%
Which of the following lists of numbers has the
largest variance?
A: [ 0.0, 0.5, 1.0 ]
B: [ 0.0, 0.25, 0.5, 0.75, 1.0 ]
C: [ 0.0, 0.1, 0.2, 0.3, … , 0.8, 0.9, 1.0 ]
◦
◦
◦
◦
A
B
C
They all have the same variance
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Percent Correct: 86%
If X and Y are independent random variables
drawn from the standard normal distribution
and Z = X+Y, which of the following best
describes the probability density of Z?
◦
◦
◦
◦
U(0,1) (uniform distribution with
U(0,2) (uniform distribution with
A normal distribution with mean
A normal distribution with mean
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range [0,1])
range [0,2])
0.0
1.0
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Percent Correct: 90%
If X is the number of people who fall asleep
during an average ASEN 5070 lecture and “X” is
drawn from U(0,2), then what is the expected
value for the total number of instances of people
falling asleep after 42 independent lectures?
◦
◦
◦
◦
0
16
30
42
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Incorrect, but brownie points awarded!
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Percent Correct: 88%
◦ Let f(x) be a probability density function. Which of
the following are true?
90%
98%
2%
100%
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