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ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor Jeffrey S. Parker Professor George H. Born Lecture 16: Numerical Compensations. Announcements. Homework 5 (?) and the test are graded. Contact me within the week to challenge any grading. - PowerPoint PPT Presentation
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CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder 1
ASEN 5070Statistical Orbit Determination I
Fall 2012
Professor Jeffrey S. ParkerProfessor George H. Born
Lecture 16: Numerical Compensations
CCARColorado Center for
Astrodynamics Research
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Homework 5 (?) and the test are graded.◦ Contact me within the week to challenge any grading.
Homework 6 will be graded shortly Homework 7 due this week.
◦ Points for quality
Guest lecturer this Thursday◦ Jason Leonard will speak about LiAISON Navigation
I’m unavailable Wednesday – Monday◦ Will be around sometimes, so you may catch me.◦ But I will be missing office hours – email me or call me if you have
questions.
Announcements
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I have a few corrections regarding last week’s debrief. Sorry for any confusion – I made a mistake when I announced the answers!
Exam 1 Debrief Again
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Exam 1 ReDebrief
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Exam 1 ReDebrief
True. The transpose of a column of zeros becomes a row of zeros. That row propagates through the whole system and the HTH matrix becomes singular.
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Exam 1 ReDebrief
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Exam 1 ReDebrief
False. Consider H-tilde = [a, b, 0]^T and Phi = 1.
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
Consider estimating c and x1
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Exam 1 Debrief
Consider estimating a and b
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Exam 1 Debrief
Consider estimating b and x0
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Exam 1 Debrief
Consider estimating a and c
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
Can’t use a Laplace Transform because A(t) is time-dependent!
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Exam 1 Debrief
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Exam 1 Debrief
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Quiz Review
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Quiz Review
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Quiz Review
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Quiz Review
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Quiz Review
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Quiz Review
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Quiz Review
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Quiz Review
NOTE: This is a demo for why machines have limits.Computer round-off = bad.
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Conventional Kalman Filter (CKF) Extended Kalman Filter (EKF)
Numerical Issues◦ Machine precision◦ Covariance collapse
Numerical Compensation◦ Joseph, Potter, Cholesky, Square-root free, unscented,
Givens, orthogonal transformation, SVD◦ State Noise Compensation, Dynamical Model Compensation
Topics
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Full, nonlinear system:
Stat OD Conceptualization
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Linearization
Stat OD Conceptualization
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Observations
Stat OD Conceptualization
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Observation Uncertainties
Stat OD Conceptualization
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Least Squares (Batch)
Stat OD Conceptualization
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Least Squares (Batch)
Stat OD Conceptualization
Iterate a few times.• Replace reference trajectory with
best-estimate• Update a priori state• Generate new computed
observations
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Trajectory remains within the linear region longer if you model the dynamics very well.
Always a limit ARTEMIS addressed this by deweighting
older measurements◦ Makes sense because ARTEMIS Nav only cared
about where the spacecraft is and not where it has been.
Side Note
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Conceptualization of the Conventional Kalman Filter (Sequential Filter)
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
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Conventional Kalman
Stat OD Conceptualization
Evolution of covariance
Mapping of final covariance
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Evolution of the covariance matrix as observations are processed.
Evolution of Covariance Matrix
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Conceptualization of the Extended Kalman Filter (EKF)
Major change: the reference trajectory is updated by the best estimate after every measurement.
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
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EKF
Stat OD Conceptualization
Evolution of reference, w/covarianceOriginal Reference
Final mapped Reference
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Stat OD Conceptualization
Pitfall 1: Beware of large a priori covariances with noisy data- Breaks linear approximations- Causes filter to diverge
Linear Regime
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Stat OD Conceptualization
Pitfall 2: Beware of collapsing covariance- Prevents new data from influencing solution- More prevalent for longer time-spans
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Stat OD Conceptualization
Pitfall 2: Beware of collapsing covariance- Prevents new data from influencing solution- More prevalent for longer time-spans
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Stat OD Conceptualization
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Stat OD Conceptualization
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Recall the Kalman Gain, which was formed in the derivation of the Sequential Algorithm.
Kalman Gain’s Influence
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Recall the Kalman Gain, which was formed in the derivation of the Sequential Algorithm.
What happens if or
Kalman Gain’s Influence
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What is the Role of the Kalman Gain?
Let’s examine a simple case where & are both scalars and we observe directly, i.e.
,k k ky x so
Furthermore assume that is a constant so that . , 1i jt t
Then at time k, ˆk jx x
2k j xP P
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What is the Role of the Kalman Gain?
The Kalman gain @ iskt
12 2 2x x
The measurement update becomes
2
2 2x
k k kx
x y x
2 2
2 2 2 2x
k kx x
x y
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What is the Role of the Kalman Gain?
Hence, the best estimate of is a weighted average of the predictedestimate and the measurement.
If the measurement is very noisy(inaccurate) relative to the predicted state ( )
kx
2 2x
Then 2
2 2 1,x
2
2 2 0x
x
and ˆk kx x
Hence, the measurement has little effect.
CCARColorado Center for
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What is the Role of the Kalman Gain?
Conversely if 2 2x ˆk kx y
and the predicted estimate has little effect.
In general the Kalman gain provides a relative weighting between the a priori and the tracking data.
Hence, the best estimate of is a weighted average of the predictedestimate and the measurement.
kx
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Batch◦ Process all observations at once
Sequential / CKF◦ Process one observation at a time
Extended Kalman Filter (EKF)◦ Process one observation at a time and update ref.◦ Can be good for long arcs, if properly implemented
Summary
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Homework 5 (?) and the test are graded.◦ Contact me within the week to challenge any grading.
Homework 6 will be graded shortly Homework 7 due this week.
◦ Points for quality
Guest lecturer this Thursday◦ Jason Leonard will speak about LiAISON Navigation
I’m unavailable Wednesday – Monday◦ Will be around sometimes, so you may catch me.◦ But I will be missing office hours – email me or call me if you have
questions.
The End