Consider Covariance Analysis Example 6.9, Spring-Mass

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Consider Covariance AnalysisExample 6.9,
Spring-Mass

• George H. Born
• ASEN5080 Statistical Orbit Determination II
• 3/14/07, Lecture 6,

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Example 6.9, page 420
Solution Steps
1. Iterate batch processor to convergence
solving for
with IC (for 20 perturbation)
using 1.8kg for mass and 3.0 for , as
indicated in the text, reduces ? to the point
where the initial conditions are out of the
linear range and the Batch Processor fails to
converge.
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Example 6.9, page 420
,
(6.5.23)
Iterate to convergence using
(4.6.4)
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Example 6.9, page 420
2. Compute
where
(6.9.9)
Use true values, those in Eq. 6.9.13, for all
parameters to compute
is computed using
(Eq. A)
and Eqns (6.9.9)
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Example 6.9, page 420
3. Compute the sensitivity and consider
covariance matrix at t0 using initial conditions
from Eq. A, and
Where is the covariance matrix of the
consider parameters and remains constant since we
are not estimating the consider parameters
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Example 6.9, page 420
The sensitivity matrix is given by
(6.4.4)
where
(6.5.24)
The consider covariance for the estimated
parameters is given by
(6.4.6)
(6.4.7)
note
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Example 6.9, page 420
4. The complete consider covariance is given by
(6.6.16)
where
5. Map through the observation and
prediction span
(6.6.15)
where
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Example 6.9, page 420
For this example we compute
where
and
Note typo in Eq (6.9.11), i.e.,
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Example 6.9, page 420
6. Plot results as in Fig 6.9.1
and are computed from
where
Plot the bounds on and
from the data noise covariance, , and
and from the consider covariance,
. Also place the position and velocity errors on
this plot.
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Example 6.9, page 420, Results
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Example 6.9, page 420, Results
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Example 6.9, page 420, Results
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Example 6.9, page 420, Results
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Example 6.9, page 420, Results