STATISTICAL
ORBIT DETERMINATION
OD Accuracy Assessment
OD Overlap Example
Project Report
ASEN 5070
LECTURE 34
11/30/09
Colorado Center for Astrodynamics Research
The University of Colorado
Techniques for OD Solution Accuracy Assessment
I. Examine P, the estimation error variance-covariance matrix
1. Plot tracking data residuals
• Are they Gaussian?
• What is the mean and RMS for each data type?
• If these agree with the apriori data statistics, one can believe that P
actually represents the estimation errors – this will probably never happen.
2. In general the correlations in P will be nearly correct but the variances will
be optimistic unless process noise has been added and properly calibrated
(tuned).
3. Do solution overlaps and compute statistics on them
• Any common biases will cancel
4. If the spacecraft has laser tracking compare spacecraft slant range
computed from OD solutions with withheld laser range measurements. Lasers
are generally accurate to 1 or 2 cm.
Colorado Center for Astrodynamics Research
The University of Colorado
Techniques for OD Solution Accuracy Assessment
5. Map P to the high elevation period and compare range standard deviation with
laser range residuals computed in part 4. Use this difference as a scaling factor
for P, i.e., if  r is half as large as RMS of computed range differences, increase
P by a factor of 4.
6. If the spacecraft carries an altimeter, examine cross over differences over the
ocean or any point where the altimeter measures accurately and local surface
elevation is constant.
Orbit #2
Orbit #1
r2
r1
h
s
g
R
x (geocenter)
Colorado Center for Astrodynamics Research
The University of Colorado
r1, r2 = OD solutions, radius
from geocenter
e1, e2=orbit error
h = altimeter measurement
r1-h1 = R+g+s+e1 = C1
r2-h2 = R+g+s+e2= C2
C1-C2 = orbit error less
common biases
surface
geoid
ellipsoid
Term Project overlap study
• In this study, an overlap comparison of solutions for the term project was performed for
tracking data obtained from two different overlapping sets of range and range rate data.
• This data was divided into two arcs as shown in Figure A.
Figure A Two arcs used in the overlap comparison
• The batch filter was used to generate a best estimate for the state and the covariance at
the time corresponding to the two data set epochs.
• The covariance was mapped to the desired time using the equation
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Term Project overlap study
• A comparison was desired during the
overlap time period between 1.5 and 3.5
hours.
• Plots were generated with one standard
deviation (1σ) magnitudes from Pk, and the
differences in the solution overlaps for this
period.
• They are given in the RIC (radial, intrack, cross-track) coordinate system in
Figure 1.
• The actual differences in the radial and
in-track directions generally fell within the
one  values obtained from the early and
the late cases.
Figure 1 Overlap comparison using Range and Range Rate
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Term Project overlap study
• The batch algorithm was also used with each data type separately.
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Figure 2 Overlap comparison using Range
Figure 3 Overlap comparison using Range Rate 6
Term Project overlap study
• Note that in all cases the cross-track standard deviation and overlap differences exhibit a
once per rev sinusoidal behavior.
• In this case the maximum differences occur when the satellite is at its highest latitude,
and the minimum is at the equatorial crossing.
• To first order, only a difference in inclination or right ascension of the ascending node
will cause a cross-track deviation and these will be 90 out of phase.
• A difference in node will cause a maximum deviation at the equatorial crossing while a
difference in inclination will have a maximum effect on cross track deviation at maximum
latitude.
• Therefore, it is concluded that this difference is primarily caused by differences in the
inclination of the two orbits.
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Term Project overlap study
• In general the formal or data noise covariance matrix would not be this indicative of
actual errors in the orbit determination solution because errors would arise from many
sources which are unmodeled or inaccurately modeled.
• In this case the mathematical model contains all quantities affecting the satellite’s motion
and there are no measurement errors except random noise.
• Consequently, the formal covariance matrix should be an accurate measure of estimation
error.
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Term Project overlap study
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Term Project overlap study
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Term Project overlap study
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Term Project overlap study
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Term Project overlap study
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Term Project overlap study
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Term Project overlap study
Summary
• Overlap studies are a consistency but not an absolute accuracy check
• These results demonstrate that in general range is a stronger data type than range rate.
This can be explained by noting that range measurements, which for this study are accurate
to 0.01 m at 20 sec intervals, essentially provides the information equivalent to range rate
with an accuracy of 0.01/20 or 0.5 x 10-3 m/sec.
• Hence, range observations provide not only a measure of the range, but also a measure of
range rate.
• Each range rate measurement is accompanied by an unknown value of range.
• This situation could be mitigated if the range rate were generated by continuous count
Doppler. In this case it could be processing as accumulated or integrated range.
• This would result in only one unknown epoch value of range and would increase the
strength of the range rate data.
• It is useful to examine covariances in alternate coordinate systems
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Effects of eliminating parameters from the
solution list of the term project
What will be the effect of fixing J2 at ½ its value and not solving for it?
How will this affect
1. The residuals?
2. The estimation error covariance matrix variations for position and velocity?
3. Variances for the remaining parameters in the solution list?
4. Correlation coefficients between J2 and the position and velocity?
5. Correlation coefficients between J2 and the remaining solved for parameters?
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Post-fit residuals when estimating J2
as in the term Project
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Post-fit residuals when fixing J2
at ½ its actual value
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Effects of eliminating parameters from the
solution list of the term project, cont.
Trace of Position variances (P(1:3,1:3)):
J2 estimated: 0.000416001592394197 m2
J2 fixed: 0.000340165200752961 m2
Trace of Velocity variances (P(4:6,4:6)):
J2 estimated: 3.87987125981823e-10 m2/s2
J2 fixed: 3.33395326676567e-10 m2/s2
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Effects of eliminating parameters from the solution
list of the term project, J2 estimated
Columns 1 through 3
5.66248951105043e-05
-0.715470305515882
0.522626641263219
-0.807236306276535
0.680645690588889
0.392935147605768
0.716300177506644
Columns 4 through 6
-5.24764107312374e-08
9.10346423198167e-08
-9.62669934551078e-08
7.46310345033749e-11
-0.860091223256874
-0.684436116140288
-0.329099080725397
Column 7
1.3181716058567e-12
-1.21979716308379e-12
1.27398213805882e-12
-6.95279762260685e-16
1.63180955939867e-15
5.983788442396e-16
5.98062231305506e-20
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-6.34995290877777e-05
5.83676474867691e-05
0.000139107444083783 -0.000147700652606973
-0.843782063029082
0.000220269253199909
0.893453128315322
-0.750828480428999
-0.96548285467716
0.936698223473284
-0.819102417018464
0.971368676263619
-0.422901518948097
0.351004925603219
7.39743457168762e-08
3.0263270986426e-08
-1.64465980897838e-07 -9.88791391360134e-08
2.0078578570408e-07
1.47554452172925e-07
-1.07315035843011e-10 -6.05179116137273e-11
2.08599208551714e-10
1.32038618166525e-10
0.893209551987995
1.04756882926734e-10
0.461997852034025
0.239062666054139
Position, velocity, and J2
Variance-covariance, and correlation matrix.
Upper triangular portion – covariances
Diagonals – variances
Lower triangular portion – correlation coefficients
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Effects of eliminating parameters from the solution
list of the term project, J2 fixed
Columns 1 through 3
2.82921847411582e-05
-0.658339808283788
0.423275868659951
-0.869076488582465
0.571507757196623
0.331168664496907
4.17715956478505e-06
Columns 4 through 6
-3.79783381401357e-08
7.80015166618908e-08
-8.34297511663449e-08
6.74977702869742e-11
-0.848524459471428
-0.663024667331318
-1.37879730334042e-06
Column 7
2.22184781121686e-23
-2.0054769127344e-23
2.11662230204251e-23
-1.13277886800604e-26
2.7084958657571e-26
9.55384133773851e-27
9.99999999982403e-31
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-3.76090896815745e-05
3.15618952055929e-05
0.000115350352103804 -0.000123860986228135
-0.822656910466205
0.000196522663907999
0.883993092809911
-0.7243857756462
-0.958652500072026
0.934340316962549
-0.815921027632663
0.975734846624612
-1.86727466352177e-06
1.50986127545734e-06
3.93284681315099e-08
1.74839479237644e-08
-1.33205553046635e-07 -8.69790405985169e-08
1.69458387259265e-07
1.35767293813083e-07
-9.01904881666218e-11 -5.40668944167407e-11
1.67379905007772e-10
1.16672317146435e-10
0.90857156259011
9.85176513818207e-11
2.0935171284686e-06
9.62544905311382e-07
Position, velocity, and J2
Variance-covariance, and correlation matrix.
Upper triangular portion – covariances
Diagonals – variances
Lower triangular portion – correlation coefficients
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Effects of eliminating parameters from the
solution list of the term project
Conclusions
1. Fixing J2 at an incorrect value increases the tracking residuals and
introduces systemic signatures
2. Fixing J2 eliminates the correlations between J2 and all other
estimated parameters
3. Fixing J2 at any value reduces the variances on all estimated
parameters that are correlated with J2. The filter has fewer
parameters to solve for so it thinks it can solve for these
parameters to greater accuracy
4. These statements are true for any parameter in the solution list
By fixing J2 we mean removing it from the estimation list or
setting it’s a priori variance nearly to zero
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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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