ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 15: Statistical Least Squares and
Estimation of Nonlinear System
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Boulder
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Lecture 14
◦ Derivations of Bayes’ Theorem skipped in class
were added to D2L version of slides
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Lecture Quiz Due by 5pm
◦ Posted later this morning

Homework 5 Due Friday

Exam 1 – Friday, October 9
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Statistical Least Squares w/ a priori

SLS and Estimation of Nonlinear System
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Statistical Interpretation of Least Squares
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Still need to know how to map measurements
from one time to a state at another time!
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
Since we linearized the formulation, we can still
improve accuracy through iteration (more on this
in a moment)
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Statistical Least Squares Solution for Nonlinear
System
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p. 196-197 of textbook
(includes corrections)
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
The batch filter depends on the assumptions of
linearity
◦ Violations of this assumption may lead to filter
divergence
◦ If the reference trajectory is near the truth, this holds
just fine
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The batch processor must be iterated 2-3 times
to get the best estimate
◦ The iteration reduces the linearization error in the
approximation

Continue the process until we “converge”
◦ Definition of convergence is an element of filter design
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If we know the observation error, why “fit to
the noise”?
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No improvement in observation RMS
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Magnitude of the state deviation vector
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Maximum number of iterations
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Instantaneous observation data is taken from
three Earth-fixed tracking stations
◦ Why is instantaneous important in this context?
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x, y, z – Satellite position in ECI
xs, ys, zs are tracking station locations in ECEF
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Range Residuals (m)
Range Rate Residuals (m/s)
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0
0
Pass 1
2000
-2000
-20
0
100
200
300
400
0
100
observation number
200
300
400
observation number
RMS Values
(Range σ=0.01 m, Range-Rate σ = 0.001 m/s)
-3
Pass 2
x 10
1
5
0
0
-1
Pass 1
-5
0
100
200
300
400
0
100
observation number
200
300
400
observation number
-3
Pass 3
x 10
0.05
2
0
0
-0.05
Pass 2
Pass 3
Range (m)
732.748
0.319
0.010
Range Rate
(m/s)
2.9002
0.0012
0.0010
-2
0
100
200
300
400
observation number
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100
200
300
400
observation number
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Image: Hall and Llinas, “Multisensor Data Fusion”, Handbook of Multisensor Data Fusion: Theory and Practice, 2009.
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FLIR – Forward-looking infrared (FLIR) imaging sensor
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
Inverting a potentially poorly scaled matrix

Solutions:

Numeric Issues
◦ Matrix Decomposition (e.g., Singular Value
Decomposition)
◦ Orthogonal Transformations
◦ Square-root free Algorithms
◦ Resulting covariance matrix not symmetric
◦ Becomes non-positive definite (bad!)
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