STATISTICAL
ORBIT DETERMINATION
Project Report
Unscented kalman Filter
Information Filter
Combining Estimates
Concept Exam
ASEN 5070
LECTURE 38, 39
12/07, 09/09
Colorado Center for Astrodynamics Research
The University of Colorado
Requirements/Suggestions for Term Project Report
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**1. General description of the OD problem and the batch and sequential algorithms
**2. Discussion of the results - contrasting the batch processor, and sequential
filter. Discuss the relative advantages, shortcomings, applications, etc. of the algorithms.
**3. Show plots of residuals for all algorithms. Plot the trace of the covariance for
position and velocity for the sequential filter for the first iteration. You may want to use
a log scale.
**4. When plotting the trace of P for the position and velocity, do any numerical
problems show up? If so discuss briefly how they may be avoided.
**5. Contrast the relative strengths of the range and range rate data. Generate
solutions with both data types alone for the batch and discuss the solutions. How do the
final covariances differ? You could plot the two error ellipsoids for position. What does
this tell you about the solutions and the relative data strength?
**6. Why did you fix one of the stations? Would the same result be obtained by not
solving for one of the stations i.e., leaving it out of the solution list? Does it matter which
station is fixed?
**7. A discussion of what you learned from the term project and suggestions for
improving it.
**Required items for the final report
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Requirements/Suggestions for Term Project Report
• Extra Credit Items
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8. Code the Extended Kalman Filter
9. How does varying the a priori covariance and data noise covariance affect
the solution? What would happen if we used an a priori more compatible with
the actual errors in the initial conditions, i. e. a few meters in position etc.
10. Do an overlap study (see lecture 34).
11. Code the Potter algorithm and compare results to the conventional Kalman
filter.
12. Solve for the state deviation vector using the Givens square root free
algorithm. Compare solution and RMS residuals for range and range rate
from Givens solution with results from conventional Kalman and Potter filters.
13. Plot pre and post fit residuals for the Kalman filter. Include the 1 sigma
pre-fit standard deviations on the plot (See Eqn. 4.7.34 of text).
14. Convert the estimation error covariance matrix into classical orbit element
and nonsingular orbit element space. What elements are most in error? Can
you see a reason for this. The coordinate transformation code is being mailed
to the class.
15. Examine the effects of fixing various parameters such as J2 and seeing the
results on the solution and residuals. They will no longer be Gaussian.
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Unscented Kalman Filter
• Introduced by Julier & Uhlmann circa 1997
• Minimal set of samples (sigma points) are
deterministically drawn from the square root
decomposition of prior covariance, using a
specific algorithm
– Monte Carlo draws samples randomly (based upon a priori
statistics)
• Sigma points are propagated through the
modeled non-linear function
– No Jacobians need be computed
• State estimate and covariance are accurate to 2nd
order for all non-linearities
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– 3rd order accuracy is expected for Gaussian distributions
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Unscented Kalman Filter
(1) Compute weights:
W0m   / L   

W0c   / L     1   2  
Wi m  Wi c  1 /2L   

i  1,........,2 L
where:
   2 L     L
 =3-L
(2) Initialize the UKF given:
1e  4   1

 2
L   
Use  =1
(3) Compute Sigma Points (matrix) for t-1

Pt 1 , Xˆ t 1 , Rt
 t 1  Xˆ t 1
Xˆ t 1   Pt 1
Xˆ t 1   Pt 1

L x ( 2 L1)
Use matlab fn.
Sqrtm to get square
Root of P
(4) Propagate Sigma Points through non-linear system
note: this implies integrating 2L+1 state vectors at each time step (i.e. each column)
t / t 1  F t 1 , ut 1 
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Unscented Kalman Filter
2L
X t   Wi m  i ,t / t 1
(5) Perform a state and covariance time
update
i 0
Pt  Q   Wi c  i ,t / t 1  X t  i ,t / t 1  X t 
2L
T
i 0
where: Q is the process noise covariance
(6) Re-compute Sigma Points to incorporate effects of process noise

 t / t 1  X t
Xt  
Pt
Xt  
Pt

L x ( 2 L 1)
(7) Compute the measurements associated with each Sigma Point vector and their weighted average
Yt / t 1  G(  t / t 1 )
2L
yt   Wi m Yi ,t / t 1
i 0
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Unscented Kalman Filter
(8) Compute the innovation and cross-correlation covariances
Pyy  R   Wi c Yi ,t / t 1  yt Yi ,t / t 1  yt 
2L
T
i 0
Pxy   Wi c  i ,t / t 1  X t Yi ,t / t 1  yt 
2L
T
i 0
(9) Compute the Kalman Gain, the best estimate of the state and the a posteriori state covariance
K  Pxy Pyy
t
1
Xˆ t  X t  Kt  yt  yt 
Pt  P t  K t Pyy K tT
where:
yt
are the observations
(10) Replace t with t-1 and return to (3) until all the observations have been processed.
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Information Filter
Statistical Orbit Determination
University of Colorado at Boulder
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Recall that the equations for the standard sequential algorithm
(also called the covariance filter) are:
Statistical Orbit Determination
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The equations for the information filter are as follows.
Statistical Orbit Determination
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Statistical Orbit Determination
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Statistical Orbit Determination
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Statistical Orbit Determination
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Statistical Orbit Determination
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Statistical Orbit Determination
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Combining Estimates
Statistical Orbit Determination
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• Treat estimates as observations.
• Write observation equations.
• Then use Linear Unbiased Minimum
Variance Estimate Equation, as follows.
Statistical Orbit Determination
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 x^    
1 
1
 ^    x   
 x 2    
2 
y  Hx  
 x^ 
1 

1
y  ^  , H    ,    
 x 2 

2 
 P1 0 
R

0
P
2

^
x  ( H T R 1H ) 1 H T R 1 y
1
1
1 1
2
1
1
^
1
2
^
 [ P  P ] [ P x1  P x 2 ]
P  [ P11  P21 ]1
Statistical Orbit Determination
University of Colorado at Boulder
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:
Statistical Orbit Determination
University of Colorado at Boulder
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Statistical Orbit Determination
University of Colorado at Boulder
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