ASEN 5070: Statistical Orbit Determination I
Fall 2014
Professor Brandon A. Jones
Lecture 31: State Noise Compensation
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Homework 9 due Friday
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Lecture Quiz due Friday at 5pm
◦ It will be posted tonight
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Last homework due 12/5 (in-class students)
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I need to turn grades in on Thursday December 18
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Exam 3 will be take home
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Will be posted on Friday Dec. 5
In-class: Due by 5pm, Friday Dec. 12
CAETE students: Due by 11:59pm (Mountain), Sunday Dec. 14
-10 pts for each 24 hours late (includes weekend days!)
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Final Projects
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“Freebies” not applicable for exam or project
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We are happy to accept your exam and project earlier!
◦ In-class & CAETE students: Due by noon, Monday Dec. 15
◦ -10 pts for each 24 hours late
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Further Project Details
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Grading rubric generated for the projects
◦ Will be posted to the Project Report Suggestions
page
◦ Reserve the right to edit/clarify, but the core
content will not change
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Process Noise
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What happened to u (modeling error) ?
◦ This is true process noise…
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Can we ignore it?
How do we account for it?
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Random process u maps to the state through the
matrix B
◦ Consider it a random process for our purposes
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Usually (for OD), we consider random accelerations:
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For the sake of our discussion, assume:
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For the sake of our discussion, assume:
In other words, Gaussian with zero mean and
uncorrelated in time
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This is a non-homogenous differential equation
The derivation of the general (continuous) solution
to this equation is derived in the book (Section
4.9), and is:
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If we want to instead map between two
discrete times:
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For the case of a noise process with zero
mean:
The zero-mean noise process does not
change the mapping of the mean state
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What about the covariance matrix?
The derivation of the general (continuous)
solution to this equation is derived in the book
(Section 4.9), and is:
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The previous discussion considered the case
where the noise process is continuous, i.e,
Things may be simplified if we instead
consider a discrete process:
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Using the discrete noise process, we instead
get (for zero mean process):
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This defines, mathematically, how we can select
the minimum covariance to prevent saturation
◦ Saturation is typically dominated by dynamic model error
◦ With a stochastic (probabilistic) description of the
modeling error, we have our minimum
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The addition of a noise process is better suited
for a sequential filter
◦ Must include the process noise transition matrix in the
Batch formulation
◦ Changes the mapping of the state (deviation) back to the
epoch time, which requires alterations to the H matrix
definition
◦ Tapley, Schutz, and Born (p. 229) argue that this is
cumbersome and impractical for real-world application
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Advantage: Kalman, EKF, Potter, and others
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Let’s derive the process noise model for a simple
case
◦ Noise process defined in the acceleration
◦ Time between measurement small enough to treat
velocity as constant
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What is the process noise transition matrix
(PNTM)?
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Recall that:
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Recall that we are assuming a small time
between observations, i.e., velocity is
constant
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If velocity is constant, change in position is linear
in time:
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Derived under the assumptions that:
◦ Noise process defined only in the acceleration
◦ Time between measurement small enough to treat
velocity as constant
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This definition of Q is in the inertial Cartesian
frame
◦ Is that always a good idea?
◦ What are some of the things you should consider?
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We can define Q in any frame, and rotate the
matrix:
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The previous derivation assumed Q was known
◦ How do we select Q ?
◦ Ideally, Q describes the magnitude of the uncertainty of
the acceleration acting on the satellite
◦ What if we don’t know the magnitude? (after all, we are
trying to account for an unknown acceleration)
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Often determined by trial and error
◦ You will do this in Homework 11
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Can we estimate the Q matrix or other
parameters of the process noise?
◦ Gauss Markov Process (Dynamic Model
Compensation)
◦ Multiple Model Adaptive Estimation (MMAE) and
Heirarchical Mixture of Experts (HME)
◦ Others in the literature
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