ASEN 5070: Statistical Orbit Determination I
Fall 2014
Professor Brandon A. Jones
Lecture 20: Project Discussion and
the Kalman Filter
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Homework 6 Due Friday
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Project/Homework Discussion
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Satellite state estimated and propagated in the
inertial frame:
Dynamics solve-for parameters are
(fundamentally) not tied to a coordinate system:
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Ground-station locations are in the Earth-fixed
frame:
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Since the ground stations are in the Earth-fixed
frame, we assume:
Hence, we have:
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The portions of the reference state requiring integration
only includes the spacecraft position and velocity
Strictly speaking, we only need to propagate a 6 × 9 matrix!
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All of these need to be in
the same reference frame!
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We recommend including this transformation
in the measurement model:
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How can we estimate the filter solve-for
parameters since the observations do not seem
to depend on them?
How/why can we
estimate these values?
(conceptual and
mathematical answers)
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The STM is a
function of
these values
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Compare to solution online
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Results available as .txt and .mat
◦ Results generated for the .txt files did not use
ode45()!
◦ Results in .mat file appear to have used Rel/Abs
tolerances of 1e-11
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Note: some elements of the project website
need to be updated (suggestions and rubric)
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We ask for relative differences to quickly
identify differences between your result and
the one online:
Example:
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Conventional Kalman Filter (CKF)
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Given from a previous filter:
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We have new a observation and mapping matrix:
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We can update the solution via:
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Is there a better sequential processing
algorithm?
◦ YES! – The equations above may be manipulated to
yield the Kalman filter
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Today – Outline derivation from minimum
variance estimator
◦ Demonstrates mathematical equivalence of CKF and
Batch
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Wednesday – Derivation as a solution to Bayes
theorem
◦ Demonstrates strengths of Kalman filter in context
of probability/statistics
◦ Also helps to understand impacts of assumptions
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Schur Identity (Appendix B, Theorem 4):
(Yes, it will simplify things…)
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Kalman Gain
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Instead inverting a p×p matrix
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Mathematically equivalent to the batch least squares
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Also provides a solution to the least squares minimization problem
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Yields a new set of problems in filtering (to be covered later)
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Does not map to epoch time!
Note the use of Htilde
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Reinitialize integrator after each observation:
Alternatively, we can use already generated
output:
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We have to invert a p×p matrix, which is
likely more efficient and stable than a n×n
matrix inversion
Can we further reduce the computation
overhead?
Yes – under certain conditions…
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Whitening Transformation
Use new values in Kalman filter
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Whitening Transformation
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The Kalman Filter – Prediction Residuals
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Previously, we have discussed the pre-fit and
post-fit residuals:
How can this change in the context of the
CKF?
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At each measurement time in the CKF, we can
take a look at the prediction residual:
Covariance of the prediction residual:
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How might we use the prediction residual
PDF?
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