ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 6: Linearization of OD Problem
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Lecture Quiz 1 – Due today by 5pm
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Lecture Quiz 2 – Posted by Monday morning
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Homework 2– Due September 11
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Time of Periapse Passage
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Linearization
◦ Why do we need it?
◦ How do we do it?
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Nonlinear estimated state and observation vectors
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Linear estimated state and observation vectors
◦ By linear, we mean the dynamics AND the observation-state relationship is linear
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Estimated state and observation deviation vectors
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The reasons for the linear and the deviation vectors to use the same
symbol will be more evident shortly
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Time of Periapse Passage – Common HW Issue
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Tp is determined from the following equations:
However, as time t increases, Tp is not constrained to an
orbital period and thus increases as a step function. To
resolve this, MOD Tp with the orbital period.
A situation may arise in which the calculation for the mean
Anomaly, M, and true anomaly, ν, do not agree resulting in
the mean anomaly to be past perigee while the true
anomaly is behind perigee (this is an artifact of numerical
integration).
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To correct this, we will introduce the angle of periapse θp:
From this, one will notice that the artifacts do not occur when
Thus, constraining θp to be between –π to π will remove the
artifacts. The angle of periapse θp can then be converted back to
time of periapse Tp by
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Artifacts
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Linearization – Why do we need it?
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We want to get the best estimate of X possible
◦ Ex. force model parameters: CD, CR, J2, etc.
◦ Ex. measurement params: station coordinates,
observation biases, etc.
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“Solve-for” parameters are usually constant (but
not always…)
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More generally:
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Example measurement types:
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At each epoch ti we have a measurement model G(Xi, ti)
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Range, Range-Rate
Right Ascension/Declination
GPS pseudorange and carrier phase
Star tracker and angular rate gyro
◦ εi represents the model error in G(Xi, ti)
 May result from statistical uncertainty
 Could be a result of modeling error
 What are some examples of modeling error?
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How do we estimate X ?
How do we estimate the errors εi?
How do we account for force and observation model
errors?
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It works for HW 1, why don’t we do it in
practice?
◦ Assumed the same number of observations as
unknowns
 What about when we have more observations than
unknowns?
 What about when we have more unknowns than
observations?
◦ Did not rigorously account for observation errors
 How do we account for statistical uncertainties?
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Known: p×l observations
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Unknowns:
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◦ n×l unknown state variables
◦ p×l unknown observation errors
◦ (n+p)×l total unknown values
We have more unknowns that observations, what
do we do now?
X(t) is a function of X(t0)
◦ Well, now we are down to n+(p×l) unknowns…
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For now, let’s consider a linear problem:
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We introduce a “cost function” that we seek to
minimize
◦ We now select x to minimize J(x)
◦ No longer estimating εi!
◦ This gives us n+p×l equations and only n unknowns
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This is known as: Least Squares Estimation
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What is one way to find the minimum of J ?
◦ Differentiate with respect to x
◦ Find the point where dJ/dx = 0
◦ Make sure the matrix of second-order partials is
positive definite
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Differentiate with respect to x. What is the answer?
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In order for the normal equation to yield a
minimum of the cost function
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In other words, HTH must be what?
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This is the “normal equation” for the least squares estimator
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Other methods of minimizing J(x) exist
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We assumed the state-observation relationship was linear, but
the orbit determination problems is nonlinear
◦ Singular value decomposition
◦ Givens transformations
◦ Etc.
◦ We will linearize the formulation of the problem
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Linearization – How do we do it?
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Vector of Estimated Values
Vector of Observations at ti
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In general, the problems are nonlinear in dynamics and/or observations
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This course primarily discusses methods based on linearization
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What is required to “linearize” the problem?
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Deviations
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Truth
Reference
We will define lower-case vectors as representing a linear system
◦ For a linear system, this is the vector of interest
◦ For a nonlinear system, these are deviation vectors
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In general, we do not know the truth. Hence, we must estimate
deviation relative to the reference
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If we have a nonlinear observation or state dynamics model, we have to
use a fully linearized form!
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We generate linearized models about the reference
trajectory, which are a function of the deviation
vectors
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Which terms are non-zero?
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Which terms equal 1?
What are the partials w.r.t. μ?
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Computed, not measured values!
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