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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 15: Numerical Compensations
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Exam 1 Final Project
Homework 5 is not graded…neither is the test. Homework 6 due Today Homework 7 due next week (Tuesday!)
◦ It’s okay to use Matlab to compute partials and to output them. But verify them.
Concept Quizzes to resume Monday!
Guest lecturer next week 10/25
Announcements
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Too easy?
Too short?
Exam 1 Debrief
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Exam 1 Debrief
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Exam 1 Debrief
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 Debrief
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Exam 1 Debrief
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
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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
G(t) = z(t) is the simplest representation of the observation. If you solve the EOM for z, you can also get partials wrt z-dot and c, which is more information (optional).
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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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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 coming up
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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
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Least Squares (Batch)
Stat OD Conceptualization
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Least Squares (Batch)
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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Least Squares (Batch)
Stat OD Conceptualization
Note: the linearization assumption will gradually break down.Toward the end, the truth will begin to deviate further from the nominal.Select a measurement interval that balances the number of observations with the length of time being used.
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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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Final Project
Homework 5 is not graded…neither is the test. Homework 6 due Today Homework 7 due next week (Tuesday!)
◦ It’s okay to use Matlab to compute partials and to output them. But verify them.
Concept Quizzes to resume Monday!
Guest lecturer next week 10/25
The End