ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 30: Observability and Introduction to
Process Noise
University of Colorado
Boulder
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Exam 2 – Friday, November 6
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Office hours this week:
◦ Prof. Jones – Th 12:30-1:30pm in Onizuka
Conference Room (ECAE 199)
◦ Eduardo – W 2-4pm, Th 2:30-4:30pm
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Boulder
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Lecture Quizzes
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Correct: 95%
Given infinite precision, the Kalman filter is
more accurate than the batch (after one
iteration).
◦ True
◦ False
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Boulder
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Correct: 93%
We are estimating one-dimensional position and velocity of an
object with linear dynamics moving away from the origin in the
positive X direction. Before receiving any observations, we have
a mean and covariance matrix describing the initial state. We
observe the instantaneous distance from the origin to the object
over time. We also know the variance of the observation error,
which is independent of the state estimate probability density
function.
In the scenario above, using a Kalman filter yields a solution to
Bayes theorem for updating a state estimate probability density
function.
◦ True
◦ False
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Boulder
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Correct: 23%
We want to process observations in the CKF
one at a time. Let R be the observation error
covariance matrix at each point in time.
Which of the following statements are true?
◦ Answers on next slide.
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Boulder
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Which of the following statements are true?
90%
◦ If we don't have a diagonal R matrix, we can still do this by
first using the whitening transformation
65%
◦ If we have a diagonal R at each point in time but there are
non-zero correlations between observations at different
times, then we can still use the CKF with one measurement
at a time.
30%
◦ If we have a diagonal R at each point in time but there are
non-zero correlations between observations at different
times, then we can't use the CKF for this problem.
13%
◦ We can only do this if the R matrix provided by the sensor
operators is diagonal.
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Boulder
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Correct: 22.5%
You are given an a priori estimated state for position and velocity
(mean and positive-definite covariance matrix) and observations
with a full-rank observation error covariance matrix. We can
update the state with a single observation using the minimum
variance estimator because (select all that apply):
70%
◦ The minimum variance estimator is mathematically equivalent to the leastsquares batch estimator.
85%
◦ The information matrix is full rank.
93%
◦ Conceptually, we consider the a priori information as an observation of the
true state, thereby yielding more measurements than unknowns.
3%
◦ The minimum variance estimator actually does not work in the scenario
described.
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Boulder
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Correct: 70%
The conventional Kalman filter updates the
reference trajectory sequentially, e.g., as
observations become available over time.
◦ True
◦ False
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Boulder
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0%

Correct: 69%
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As the elements of the Kalman gain matrix decrease
towards zero, the filter (select all that apply):
◦ The filter begins to weigh measurements greater than the a
priori state in the measurement update.
98%
◦ The filter begins ignoring measurements in favor of the
propagated (i.e., a priori) state deviation vector.
90%
◦ The measurement update of the state-error covariance
matrix (P) begins to yield only slight changes in P.
29%
◦ The filter state and covariance matrix experience no
changes due to the time or measurement updates.
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Correct: 50%
Which of the following can cause the EKF to diverge? (For
the sake of this problem, assume we are starting with the
EKF and will not use a CKF for early observations)
90%
◦ The reference trajectory is a poor approximation of the truth.
76%
◦ Good measurements (trace(R) is small) with a bad a priori state
(trace(P) is large).
74%
◦ Bad measurements (trace(R) is large) with a good a priori (trace(P)
is small).
5%
◦ Good measurements (trace(R) is small) with a good a priori
(trace(P) is small).
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Boulder
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Correct: 76%
When given the same inputs (observations, a
priori information at the initial time, models,
etc.), the estimated state covariance at the
final observation time is the same for the EKF
and CKF.
◦ True
◦ False
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Boulder
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Correct: 79%
When given the same inputs (observations, a priori information
at the initial time, models, etc.), the estimated state covariance
at the final observation time is the same for the EKF and CKF.
95%
◦ An example of bad a priori state (large P) with good measurements (small
R).
7%
◦ An example of bad measurements (large R) with good a priori information
(small P).
2%
◦ An example where the Kalman filter performs better than the Batch filter.
15%
◦ An example where the Kalman filter with the Joseph formulation performs
better than the Batch filter.
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Boulder
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Correct: 76%
(For the sake of this problem, assume the filter converged
sufficiently so that the nonlinear state is close enough to the
truth to begin using the EKF. Also assume that measurements
are provided at a frequency such that propagation does not
increase errors in the state estimate.)
Since the EKF updates the reference trajectory at the end of the
measurement update process, it reduces linearization
errors. Due to this result, the filter can run without fear of
producing a state error such that the measurement update with a
linearized observation model cannot be used. In this respect, we
can run the EKF "forever" without diverging.
◦ True
◦ False
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Boulder
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Observability
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Boulder
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How do I determine what parameters may be
successfully estimated in the filter?
◦ Can I use observations of a spacecraft to estimate
the height of Folsom Field Stadium?
◦ What about observations of a spacecraft to measure
variations in rainfall in the Amazon river basin?
◦ How do I determine if either of these are possible?
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Boulder
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Consider the case of two spacecraft and a
ground station with a fixed inertial position
◦
◦
◦
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Two-body gravity field (no perturbations)
No modeling error
Infinite precision
Little/no error on range observations
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Satellite 1
Satellite 2
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The two plots look similar (this is not a copy/paste error)
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Does anyone think there is a problem?
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Gather more
observations?
◦ Unfortunately, No.
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Gather range-rate to
go with the range
data?
◦ Nope – we run into the
same problem
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Orthogonal data type,
e.g., angles?
◦ Actually that would
work, but how do we
find out?
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We can use the information matrix:
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In other words, when designing our filter, we
should study the information matrix to
determine if we can get a solution
Let’s say you solve for the information matrix
defined by some simulation.
◦ How would you determine if it is positive definite?
◦ Do you need to generate simulated observations?
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Boulder
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What if the condition number of the
information matrix is very large (too large for
any of the more numerically stable methods
to apply)?
◦ Maybe we should reconsider what parameters to
estimate?
◦ This can be the case for gravity field estimation
with spatially sparse measurements
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Boulder
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◦ Can I use observations of a spacecraft to estimate the
height of Folsom Field?
 Only if observations of/from a well-known spacecraft are
gathered with respect to the top of the stadium
◦ What about observations of a spacecraft to measure
variations in rainfall in the Amazon river basin?
 Actually – you can!
 Scientific studies of GRACE data do this type of analysis
regularly
◦ How do I determine if either of these are possible?
 You perform an observability study!
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Boulder
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Can we estimate the absolute position of two
spacecraft in Earth orbit (two-body dynamics)
using relative range and/or range-rate
measurements?
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Boulder
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Can we do it if we put one of the spacecraft
near the Moon and keep one at Earth?
Image Credit: Hill and Born, 2007
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