Lecture_20_ASEN_5070_2014F_Post - CCAR

advertisement
ASEN 5070: Statistical Orbit Determination I
Fall 2014
Professor Brandon A. Jones
Lecture 20: Project Discussion and
the Kalman Filter
University of Colorado
Boulder

Homework 6 Due Friday
University of Colorado
Boulder
2
Project/Homework Discussion
University of Colorado
Boulder
3


Satellite state estimated and propagated in the
inertial frame:
Dynamics solve-for parameters are
(fundamentally) not tied to a coordinate system:

Ground-station locations are in the Earth-fixed
frame:
University of Colorado
Boulder
4


Since the ground stations are in the Earth-fixed
frame, we assume:
Hence, we have:
University of Colorado
Boulder
5


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!
University of Colorado
Boulder
6
All of these need to be in
the same reference frame!

We recommend including this transformation
in the measurement model:
University of Colorado
Boulder
7


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)
University of Colorado
Boulder
The STM is a
function of
these values
8

Compare to solution online

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

Note: some elements of the project website
need to be updated (suggestions and rubric)
University of Colorado
Boulder
9


We ask for relative differences to quickly
identify differences between your result and
the one online:
Example:
University of Colorado
Boulder
10
Conventional Kalman Filter (CKF)
University of Colorado
Boulder
11

Given from a previous filter:

We have new a observation and mapping matrix:

We can update the solution via:
University of Colorado
Boulder
12

Is there a better sequential processing
algorithm?
◦ YES! – The equations above may be manipulated to
yield the Kalman filter
University of Colorado
Boulder
13

Today – Outline derivation from minimum
variance estimator
◦ Demonstrates mathematical equivalence of CKF and
Batch

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
University of Colorado
Boulder
14

Schur Identity (Appendix B, Theorem 4):
(Yes, it will simplify things…)
University of Colorado
Boulder
15
Kalman Gain
University of Colorado
Boulder
16
University of Colorado
Boulder
17

Instead inverting a p×p matrix

Mathematically equivalent to the batch least squares

Also provides a solution to the least squares minimization problem

Yields a new set of problems in filtering (to be covered later)
University of Colorado
Boulder
18
Does not map to epoch time!
Note the use of Htilde
University of Colorado
Boulder
19


Reinitialize integrator after each observation:
Alternatively, we can use already generated
output:
University of Colorado
Boulder
20



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…
University of Colorado
Boulder
21
University of Colorado
Boulder
22
University of Colorado
Boulder
23

Whitening Transformation
Use new values in Kalman filter
University of Colorado
Boulder
24

Whitening Transformation
University of Colorado
Boulder
25
The Kalman Filter – Prediction Residuals
University of Colorado
Boulder
26


Previously, we have discussed the pre-fit and
post-fit residuals:
How can this change in the context of the
CKF?
University of Colorado
Boulder
27


At each measurement time in the CKF, we can
take a look at the prediction residual:
Covariance of the prediction residual:
University of Colorado
Boulder
28

How might we use the prediction residual
PDF?
University of Colorado
Boulder
29
Download