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ASEN 5070Statistical Orbit Determination I
Fall 2012
Professor George H. BornProfessor Jeffrey S. Parker
Lecture 12: The Kalman Filter
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Homework 5 due Today
Exam on 10/11. (Anyone going to miss it?)◦ Eduardo and/or Paul will be reviewing subjects on Tuesday – send
them emails with questions/subjects that you’d like them to cover.
◦ 1 hour, open book, open notes.
◦ Topics: Definitions of variables,Probability/StatisticsObservability,LinearizationLeast squares, Batch processor
Announcements
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Quiz Results
If you took the quiz, you scored 100%
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Some feedback:
Biggest issues:◦ Moving pretty fast◦ Lectures and HW don’t correlate well
Review next week:◦ Review of variables◦ Stat OD example from start to finish (something
plain and easy to follow)
Quiz Results
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Setup.◦ Given: an initial state◦ Optional: an initial covariance
Review of the Stat OD Process
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Setup.◦ Given: an initial state◦ Optional: an initial covariance
◦ The satellite will not be there, but will (hopefully) be nearby True state =
Review of the Stat OD Process
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What really happens◦ Satellite travels according to the real forces in the universe
Review of the Stat OD Process
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What really happens◦ Of course, we don’t know this!
Review of the Stat OD Process
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Model reality as best as possible Propagate our initial guess of the state
Review of the Stat OD Process
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Goal: Determine how to modify to match
Review of the Stat OD Process
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Goal: Determine how to modify to match
Review of the Stat OD Process
Define
Want
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations
Review of the Stat OD Process
Define
Want
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Process:1. Track satellite
Review of the Stat OD Process
Perfect Observations
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Process:1. Track satellite
Review of the Stat OD Process
Perfect Observations
Computed Observations
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Process:1. Track satellite
Review of the Stat OD Process
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Process:1. Track satellite2. Map observations to state deviation
Review of the Stat OD Process
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CCARColorado Center for
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University of ColoradoBoulder 17
Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations
Review of the Stat OD Process
Least Squares
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
Least Squares
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
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CCARColorado Center for
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
Small errors due to mismodeled dynamics
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Process:1. Track satellite
Review of the Stat OD Process
Perfect Observations
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Process:1. Track satellite
Review of the Stat OD Process
Imperfect Observations
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
Same process, but the best estimate trajectory will never quite match the truth, since the observations have noise.
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
Least Squares
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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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Batch◦ Process all observations at once
Sequential◦ Process one observation at a time
Algorithm Options
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Collect mapped information
Batch Processor
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Collect mapped information
Batch Processor
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Collect mapped information
Batch Processor
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Collect mapped information
Batch Processor
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Collect mapped information
Batch Processor
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Collect mapped information
Batch Processor
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(Break)
TAs will go through an end-to-end Batch run to demo equations, etc.
Next up: Kalman Filter
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Consider
Rather than mapping all observations to one epoch and processing them simultaneously, what if we processed each separately and mapped the best estimate through each?
Sequential Processor
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Consider
Rather than mapping all observations to one epoch and processing them simultaneously, what if we processed each separately and mapped the best estimate through each?
Sequential Processor
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Given an a priori state and covariance, we know how to generate a new estimate of the state:
Need a way to generate the a posteriori covariance matrix as well.
Recall
The trouble is inverting the n x n matrix.
Sequential Processor
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We can use the Schur Identity and a bunch of math (see Section 4.7) and obtain:
Sequential Processor
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We can use the Schur Identity and a bunch of math (see Section 4.7) and obtain:
Sequential Processor
Kalman Gain
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After some more math, we can simplify to obtain:
Sequential Processor
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1. Initialize the first run
2. Start at the reference epoch
3. Time Update◦ Integrate from the current time to the next time of interest◦ Map the state estimate and the covariance to the new time
4. Measurement Update◦ If there is a new measurement, process it.◦ Update the state estimate and covariance with this new information
Repeat 3-4 until all measurements have been processed and all times of interest have been recorded.
Optional: Map the estimate and covariance back to the reference epoch and iterate the whole process.
Sequential Algorithm
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Initialization
Sequential Algorithm
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Time Update
◦ Integration
◦ Mapping
Sequential Algorithm
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Measurement Update
◦ Collect measurement
◦ Compute update
Sequential Algorithm
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Repeat just like the Batch
◦ Replace the reference trajectory with the new best estimate.
◦ Make sure to update the a priori state deviation vector to retain any information.
◦ Recompute all observation residuals
Sequential Processor
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Sequential Flow-Chart
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Collect mapped information
Batch Processor
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Collect mapped information
Batch Processor
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Collect mapped information
Batch Processor
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Collect mapped information
Batch Processor
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Collect mapped information
Kalman Filter
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Collect mapped information
Kalman Filter
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Collect mapped information
Kalman Filter
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Collect mapped information
Kalman Filter
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Collect mapped information
Kalman Filter
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Evolution of the covariance matrix as observations are processed.
◦ Q: How do you imagine it would change?
◦ Q: What would cause it to shrink? To grow?
Kalman Filter
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Evolution of the covariance matrix as observations are processed.
Kalman Filter
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Least squares estimation began with Gauss
1963: Kalman’s sequential approach◦ Introduced minimum variance◦ Introduced process noise◦ Permitted covariance analyses without data
Schmidt proposed a linearization method that would work for OD problems◦ Supposed that linearizing around the best estimate trajectory is better than linearizing
around the nominal trajectory
1970: Extended Kalman Filter
Gradually, researchers identified problems.◦ (a) Divergence due to the use of incorrect a priori statistics and unmodeled parameters.◦ (b) Divergence due to the presence of nonlinearities.◦ (c) Divergence due to the effects of computer round-off.
Kalman Filter History
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Numerical issues cause the covariance matrix to lose their symmetry and nonnegativity
Possible corrections:◦ (a) Compute only the upper (or lower) triangular entries and force symmetry◦ (b) Compute the entire matrix and then average the upper and lower fields◦ (c) Periodically test and reset the matrix◦ (d) Replace the optimal Kalman measurement update by other expressions
(Joseph, Potter, etc)◦ (e) Use larger process noise and measurement noise covariances.
Kalman Filter History
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Potter is credited with introducing square root factorization.◦ Worked for the Apollo missions!
1968: Andrews extended Potter’s algorithms to include process noise and correlated measurements.
1965 – 1969: Development of the Householder transformation◦ Worked for Mariner 9 in 1971!
1969: Dyer-McReynolds filter added additional process noise effects.◦ Worked for Mariner 10 in 1973 for Venus and Mercury!
Kalman Filter History
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Homework 5 due Today
Exam on 10/11.◦ Eduardo and/or Paul will be reviewing subjects on Tuesday – send
them emails with questions/subjects that you’d like them to cover.
◦ 1 hour, open book, open notes.
◦ Topics: Definitions of variables,Probability/StatisticsObservability,LinearizationLeast squares, Batch processor
Final Statements