Transcript Lecture 40
ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 40: Elements of Attitude Estimation
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Exam 3
◦ In-class Students: Due December 11 by 5pm
◦ CAETE Students: Due 11:59pm (Mountain) on 12/13
Final Project Due December 14 by noon
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Project Q&A
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Elements of Attitude Estimation
Follows (with slight changes in notation to match class):
•
Crassidis and Junkins, Optimal Estimation of Dynamic Systems,
Chapman and Hall, Section 4.2, 2004.
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Like the orbit determination problem, we
must estimate the attitude of a spacecraft to
meet requirements
Why would we require an accurate attitude
estimate?
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Attitude estimation can be difficult
◦ Highly non-linear dynamics
◦ State ambiguities (3, 4, or 9 estimated states?)
◦ More design considerations based on the dynamics of
your problem (spacecraft/mission dependent!)
Different sensor types providing different
information
◦ Direct attitude information
Star trackers
Sun sensors
◦ Direct attitude rate information
Rate gyros
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Image Credit: Crassidis and Junkins, Fig. 4.1
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For the current discussion, assume we are
estimating the attitude matrix:
This matrix is orthogonal (AAT=I) with
differential equation
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Dynamics of the angle rates
is described by the Euler equations
The Euler equations are the primary source of
nonlinearity in the dynamics
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What are some example torques acting on a
spacecraft that we could model?
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A star’s location may be described by its right
ascension and declination
Image Credit: Wiki Commons
These angles are a projection
of the position onto the
unit sphere.
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Assume A is the transformation from the inertial
frame to the camera frame (z along the
boresight)
◦ If we estimate A, then we know our attitude
Given star j’s location xj and yj in the camera
frame
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It may then be shown that
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Least-squares cost function
Subject to the constraint
Hence, we instead have a constrained least
squares problem
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We may instead write the cost function as
Due to constants, we can instead maximize
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Quaternions leverage Euler’s theorem to yield
a non-singular attitude state representation
Constraint on magnitude:
Matrix/quaternion relationship:
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Using the relationship between the attitude
matrix and a quaternion
Combined with the previous identities
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Okay, we have:
Along with our constraint:
When we have an equality constraint, what
can we do to find the maximum of J()?
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We incorporate our constraint in the cost
function:
Now, we solve for the multiplier
Do we recognize the equation on the right?
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We know that the quaternion needs to be an
eigenvector of K and the Lagrange multiplier
is an eigenvalue of K
Substituting the previous equation into the
original cost function:
We maximize J() when we pick the eigenvector
with the maximum eigenvalue!
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It can be shown that, if there are at least two
non-collinear observations then we have at
least two unique eigenvalues
◦ In other words we need two stars with adequate
separation in the field of view to fully estimate the
attitude
Each observation is 2-D, so we need two independent
observations to estimate the 4-D quaternion
◦ Improved accuracy with larger separation of the
stars (similar to measurement diversity) and more
measurements (measurement volume)
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Here, we discussed a relatively easy attitude
problem
◦ No propagation, i.e., more observations than estimated
states at a single time
◦ Relatively simple solution to constrained estimation
problem
◦ Did not incorporate angle rates, modeling of torque, etc.
◦ More complex methods typically need an EKF or other
nonlinear filter to be tractable
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FCQs
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