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ASEN 5070Statistical Orbit Determination I
Fall 2012
Professor Jeffrey S. ParkerProfessor George H. Born
Lecture 20: Exam 2 Review
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Homework 8 due this week.◦ Make sure you spend time studying for the exam
Homework 9 out today. You’re not busy, are you? This one is easy and will push you toward the completion of the final project.
Exam 2 on Thursday.◦ A-H in this classroom◦ I-Z in ECEE 265
Exam 2 will cover:◦ Batch vs. CKF vs. EKF◦ Probability and statistics (good to keep this up!)
Haven’t settled on a question yet, but it will probably be a conditional probability question. I.e., what’s the probability of X given that Y occurs?
◦ Observability◦ Numerical compensation techniques, such as the Joseph and Potter formulation.◦ No calculators should be necessary◦ Open Book, Open Notes
Announcements
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Quiz 16 Review
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Quiz 16 Review
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Quiz 16 Review
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Quiz 16 Review
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Due a week from Thursday
HW#9
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Review
Lots of questions of CKF vs. EKF
Lots of questions on observability
Some questions on clarifications of parameters (bar, hat, P vs R, etc.), n / m / p
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First off, conceptual parameters
If you have n parameters to estimate, you require at least n pieces of information to uniquely estimate those parameters.◦ If you don’t have that you can use the min-norm estimate
Parameters
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First off, conceptual parameters
If you have n parameters to estimate, you require at least n pieces of information to uniquely estimate those parameters.◦ If you don’t have that you can use the min-norm estimate
The sum of all observations = m pieces of information◦ Range = 1 piece◦ Doppler = 1 piece◦ An optical observation may involve 2 pieces (RA and Dec)
Parameters
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First off, conceptual parameters
If you have n parameters to estimate, you require at least n pieces of information to uniquely estimate those parameters.◦ If you don’t have that you can use the min-norm estimate
The sum of all observations = m pieces of information◦ Range = 1 piece◦ Doppler = 1 piece◦ An optical observation may involve 2 pieces (RA and Dec)
Number of observation data types = p
Number of observations = l
l x p = m
Parameters
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So, say you have n parameters and m total observations.
◦ If m < n, min-norm◦ If m = n, deterministic◦ If m > n, least squares
Each observation has an error associated with it, which introduces more unknowns. You end up with n+m unknowns and m pieces of information least squares to minimize the errors.
Parameters
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Least Squares (Batch)
Stat OD Conceptualization
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Least Squares
Weighted Least Squares
Least Squares with a priori
Min Variance
Min Variance with a priori
Least Squares Options
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The Batch processor is just a wrapper around Least Squares.
Accumulate information from all observations and simultaneously process them all (in a batch).
Batch
Note the sizes of each matrix
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Any numerical issues with the Batch?
Batch
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What if the a priori covariance is huge? Tiny?
Batch
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What if we have poorly-modeled dynamics?
Batch
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Stat OD Conceptualization
Batch fits a line to this data. (CONCEPTUAL)
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Stat OD Conceptualization
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Batch◦ Process all observations at once
Sequential◦ Process one observation at a time
Algorithm Options
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Sequential◦ Process one observation at a time
◦ Reformulation
Algorithm Options
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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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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
Evolution of covariance
Mapping of final covariance
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Stat OD Conceptualization
CKF fits a line to this data. (CONCEPTUAL)
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Stat OD Conceptualization
AFTER all observations have been processed.
Imagine what it would look like DURING the process.
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Filter Saturation◦ Causes?◦ Fixes?
Sequential
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Any numerical issues with the Kalman filter?
Sequential
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Any numerical issues with the Kalman filter?
Joseph:
Square Root◦ Potter
Sequential
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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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Every state parameter must be observed somehow◦ Either the observations must be a function of that
parameter, or the observation-state relationship changes over time according to the effects of that parameter.
◦ I.e., it has to be in the A or H matrix!
There have to be enough observations
The state parameters must be distinguishable. That is, they can’t be linearly dependent.
Observability
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Basic
Linearly Dependent
Observability
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Homework 8 due this week.◦ Make sure you spend time studying for the exam
Homework 9 out today. You’re not busy, are you? This one is easy and will push you toward the completion of the final project.
Exam 2 on Thursday.◦ A-H in this classroom◦ I-Z in ECEE 265
Exam 2 will cover:◦ Batch vs. CKF vs. EKF◦ Probability and statistics (good to keep this up!)
Haven’t settled on a question yet, but it will probably be a conditional probability question. I.e., what’s the probability of X given that Y occurs?
◦ Observability◦ Numerical compensation techniques, such as the Joseph and Potter formulation.◦ No calculators should be necessary◦ Open Book, Open Notes
Questions?
