ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 20: Project Discussion and
Introduction to the Kalman Filter
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Homework 6 Due Friday
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No lecture quiz this week
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Exam 1 Returned Monday (10/19)
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Term Project Introduction
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The OD project consists of
◦ Deriving the algorithms for estimating the state of a
spacecraft
◦ Implementing the algorithms
◦ Processing range and range-rate observations from
three ground-stations
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3 Ground-based tracking stations
◦ Equatorial station’s position fixed in filter
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Dynamics Model
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Two-body, unknown μ
Also estimating J2
Drag with unknown CD
Already completed!
◦ Simple spherical satellite with known mass and area
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Estimate satellite position
and velocity in ECI
Constants governing
satellite dynamics
Station positions on
Earth’s surface (ECEF)
18x1 state vector
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Unknown ground tracking station positions
◦ Boat in Pacific at equator
 Equatorial station still estimated, but a priori
covariance is very small
◦ Turkey
◦ Greenland
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Range (m) and range-rate (m/s)
◦ Range: Zero mean, σ = 1 cm
◦ Range-Rate: Zero mean, σ = 1 mm/s
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There is an observation at t=0 !
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Observation intervals do not overlap
◦ Number of observations at a given time always
equals 2
◦ Would add more complexity to the filter
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Derive A and H_tilde and implement the
partials in software
◦ Use the symbolic toolbox with the jacobian()
command
◦ Don’t include calls to symbolic toolbox in ode45()
derivative function!
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Add numeric integration of STM to you HW2
propagator
◦ Check intermediate results with the website
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Implement a batch least squares filter to
estimate the state and get an error covariance
◦ Results from first iteration are provided as your
homework 9
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Implement a Kalman/Sequential filter to
estimate the state and error covariance
◦ Completed with HW 10
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Discussion of OD problem and batch
processor in the report
Discuss the results
◦ Compare results from the Kalman and LS filters
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Plots:
◦ Residuals
◦ Covariance
◦ Error ellipsoids (more in future lecture)
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Process one data type at a time
◦ Range-only processing
◦ Range-rate only
◦ Compare results (which does better?)
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Discussion of results
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The previous tasks get you a maximum score of
90/100
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Additional elements boost your grade from there
◦ Extended Kalman filter (EKF needed for StatOD 2
anyway)
◦ Potter Filter
◦ Givens transformations
◦ Smoothing
◦ Interval processing of data
◦ Others…
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The additional elements is your chance to further
explore a topic of interest covered in class
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Satellite state estimated and propagated in the
inertial frame:
Dynamics solve-for parameters are
(fundamentally) not tied to a coordinate system:
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Ground-station locations are in the Earth-fixed
frame:
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Since the ground stations are in the Earth-fixed
frame, we assume:
Hence, we have:
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The portions of the reference state requiring integration
only includes the spacecraft position and velocity
Strictly speaking, we only need to propagate a 6 × 9 STM
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All of these need to be in
the same reference frame!
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We recommend including this transformation
in the measurement model:
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Although the project description includes an
absolute epoch time (3 Oct. 1999), you do
not need this information
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You can use:
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We give you the rotation rate and t is the
provided measurement time
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In the real world, time and coordinate
systems are not this easy!
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Compare to solution online
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Results available as .txt and .mat
◦ Results generated for the .txt files did not use ode45()!
◦ Results in .mat file appear to have used Rel/Abs
tolerances of 1e-11
◦ Python users can load .mat files using scipy.io.loadmat().
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Note: some elements of the project website will
be updated later in the semester (suggestions
and rubric)
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We ask for relative differences to quickly
identify differences between your result and
the one online:
Example:
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Conventional Kalman Filter (CKF)
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Given from a previous filter:
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We have new a observation and mapping matrix:
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We can update the solution via:
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Two principle phases in any sequential estimator
◦ Time Update
 Map previous state deviation and covariance matrix to the
current time of interest
◦ Measurement Update
 Update the state deviation and covariance matrix given the
new observations at the time of interest
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Jargon can change with communities
◦ Forecast and analysis
◦ Prediction and fusion
◦ others…
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We still have to invert an n × n matrix
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Can be computationally expensive for large n
◦ Gravity field estimation: ~n2+2n-3 coefficients!
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May become sensitive to numeric issues
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Is there a better sequential processing
algorithm?
◦ YES! – This equations above may be manipulated to
yield the Kalman filter
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Today – Outline derivation starting with the
minimum variance estimator
◦ Demonstrates mathematical equivalence of CKF and
Batch
◦ Make sure you understand the derivation presented
in the book!
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Wednesday – Derivation as a solution to Bayes
theorem
◦ Demonstrates strengths of Kalman filter in context
of probability/statistics
◦ Also helps to understand impacts of assumptions
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Schur Identity (Appendix B, Theorem 4):
(Yes, it will simplify things…)
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Kalman Gain
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Instead inverting a p×p matrix
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Mathematically equivalent to the batch least squares, minimum variance
estimator, etc.
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Also provides a solution to the least squares minimization problem
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Yields a new set of problems in filtering (to be covered later)
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Does not map to epoch time!
Note the use of Htilde
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