ASEN 5070
Statistical Orbit Determination I
Fall 2012
Professor Jeffrey S. Parker
Professor George H. Born
Lecture 19: Numerical Compensations
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
Homework 8 due next week.
◦ Make sure you spend time studying for the exam

Exam 2 in one week (Thursday).
Review on Tuesday.

Exam 2 will cover:

◦ Batch vs. CKF vs. EKF
◦ Probability and statistics (good to keep this up!)
 Haven’t settled on a question yet, but it will probably be a conditional
probability question. I.e., what’s the probability of X given that Y occurs?
◦ Observability
◦ Numerical compensation techniques, such as the Joseph and Potter
formulation.
◦ No calculators should be necessary
◦ Open Book, Open Notes
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Use Matlab and try it out
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This is TRUE for the Batch, but you may run into a problem
with the Kalman filters.
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The best scan of
the book ever:
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
Due in 7 days
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Note that these
plots aren’t 100%
well-labeled!
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
The biggest pitfall

When processing the 2nd observation, set
that is, use the most current covariance you
have as the a priori! Not the original one.
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
Processing an observation vector one element
at a time.
Whitening
Cholesky
Joseph

Today



◦
◦
◦
◦
Positive Definiteness
Conditioning Number
Potter
Householder
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
Definition
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
Properties of PD matrices
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
Properties of PD matrices
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
Quick Break
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Summary of
Exact to order 
1  2
 (1  3 )

 (1  3 ) 
2  4 
Joseph
 (1  3 ) 
1  2
 (1  3 )

2




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P2
Results
Conventional Kalman
1  1
1  2  1
1
 1
Batch
 (1  3 ) 
1  2
 (1  3 )

2

4




Conditioning Number of matrix A: C(A)

where the gammas are the min/max eigenvalue
of the matrix.




Inverting A with p digits of precision becomes
error-prone as C(A) 10p.
If we invert the square root of A:
then C(W)=sqrt(C(A)).
Numerical difficulties will arise as C(W) 102p.
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
Square Root Filter Algorithms
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
Motivation:
◦ Loss of significant digits that occurs in computing
the measurement update of the state error
covariance matrix (P) at the observation epoch
(Kaminski et al., 1971)
◦ If eigenvalues spread a wide range, then the
numerical errors can destroy the symmetry and PDness of the P matrix. Filter divergence can occur.
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


Define W, the square root of P:
Observe that if we have W, then computing P
in this manner will always result in a
symmetric PD matrix.
Note: Square root filters are typically derived
to process one observation at a time. Hence,
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(Sorry for just scanning the text, but it’s a pretty concise description!)
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(Sorry for just scanning the text, but it’s a pretty concise description!)
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This is a key
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(say, using Cholesky)
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Note: if you are given an a priori P
matrix, convert it to W using
Cholesky or equivalent at start.
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And page 339 of Stat OD text.
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Summary of
Exact to order
1  2
 (1  3 )

 (1  3 ) 
2  4 
P2
Results
Potter Algorithm
1  2  (1   ) 
 (1   ) 2  4 


Conventional
Kalman
1  1
1  2  1
Joseph
 (1  3 ) 
1  2
 (1  3 )

2




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1
 1
Batch
 (1  3 ) 
1  2
 (1  3 )

2

4




Homework 8 due next week.
◦ Make sure you spend time studying for the exam

Exam 2 in one week (Thursday).
Review on Tuesday.

Exam 2 will cover:

◦ Batch vs. CKF vs. EKF
◦ Probability and statistics (good to keep this up!)
 Haven’t settled on a question yet, but it will probably be a conditional
probability question. I.e., what’s the probability of X given that Y occurs?
◦ Observability
◦ Numerical compensation techniques, such as the Joseph and Potter
formulation.
◦ No calculators should be necessary
◦ Open Book, Open Notes
University of Colorado
Boulder
51