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University of ColoradoBoulder
ASEN 5070: Statistical Orbit Determination I
Fall 2014
Professor Brandon A. Jones
Lecture 15: Statistical Least Squares and
Estimation of Nonlinear System
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Lecture Quiz Due by 5pm
Homework 5 Due Friday
Exam 1 – Friday, October 11
Announcements
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Statistical Least Squares w/ a priori
SLS and Estimation of Nonlinear System
Today’s Topics
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Statistical Interpretation of Least Squares
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Weighted Least Squares
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Observation Errors
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State Estimation Error Description
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State Estimation Error Description
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State Estimation Error Description
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Statistical LS w/ a priori
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Statistical LS w/ a priori
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Statistical LS w/ a priori
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Statistical LS w/ a priori
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State Estimation Error Description
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Still need to know how to map measurements from one time to a state at another time!
Measurement Mapping
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State Update
Since we linearized the formulation, we can still improve accuracy through iteration (more on this in a future lecture)
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Statistical Least Squares Solution for NonlinearSystem
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Computation Algorithm of the Batch Processor
p. 196-197 of textbook (includes corrections)
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Computation Algorithm for the Batch Processor
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Why Reuse A Priori Information?
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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
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
Assumptions with the Iterated Batch
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Post-fit Residuals RMS
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Convergence via Post-fit Residuals
If we know the observation error, why “fit to the noise”?
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No improvement in observation RMS
Other Convergence Tests
No reduction in state deviation vector
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?
LEO Orbit Determination Example
x, y, z – Satellite position in ECI xs, ys, zs are tracking station locations in ECEF
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Effects of Iteration
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Improved Fit to Data
0 100 200 300 400-2000
0
2000Range Residuals (m)
Pas
s 1
observation number0 100 200 300 400
-20
0
20Range Rate Residuals (m/s)
observation number
0 100 200 300 400-1
0
1
Pas
s 2
observation number0 100 200 300 400
-5
0
5x 10
-3
observation number
0 100 200 300 400-0.05
0
0.05
Pas
s 3
observation number0 100 200 300 400
-2
0
2x 10
-3
observation number
RMS Values (Range σ=0.01 m, Range-Rate σ = 0.001 m/s)
Pass 1 Pass 2 Pass 3
Range (m) 732.748
0.319 0.010
Range Rate (m/s)
2.9002 0.0012 0.0010
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Estimated State Uncertainty
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Estimated State Uncertainty
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Estimated State Uncertainty
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Advantage of Different Data Types
Image: Hall and Llinas, “Multisensor Data Fusion”, Handbook of Multisensor Data Fusion: Theory and Practice, 2009.
FLIR – Forward-looking infrared (FLIR) imaging sensor
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Inverting a potentially poorly scaled matrix
Solutions:◦ Matrix Decomposition (e.g., Singular Value Decomposition)◦ Orthogonal Transformations◦ Square-root free Algorithms
Numeric Issues◦ Resulting covariance matrix not symmetric◦ Becomes non-positive definite (bad!)
Batch Processor Issues