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Created page with "== Introduction == Obtaining quaternion information from q-method involves solving the eigenvalue problem for a 4X4 Matrix. This can be done on MATLAB or other computer tools...."
== Introduction ==
Obtaining quaternion information from q-method involves solving the eigenvalue problem for a 4X4 Matrix. This can be done on MATLAB or other computer tools. But the solution is numerically intensive. On-board computing is a serious concern for any satellite designer and hence a better algorithm to solve this problem using less computing is required. <br \>
The QuEST algorithm provides a less efficient but a ‘faster’ way to solve the eigenvalue problem.
== The Algorithm ==
Recall from [[Q-Method and Wahba's Problem]] that the optimal attitude minimizes the loss function,
[[File:Equation40.JPG|frame|center]]
And maximizes the gain function,
[[File:Equation41.png|frame|center]]
Thus,
[[File:Equation42.png|frame|center]]
[[File:Equation43.png]] where [[File:Equation44.png]] is the maximum eigenvalue of the K matrix defined in [[Q-Method and Whab's Problem]].
With this [[File:Equation44.png]] can be written as
[[File:Equation45.png|frame|center]]
If we approximate J to be very small (this approximation is valid as we want to minimize J and now if after completing this method, we calculate J, it would be very small),
[[File:Equation46.png|frame|center]]
Defining and writing the equation in terms of Rodrigue’s parameter '''p''' simplifies the equation
[[File:Equation47.png|frame|center]]
where a is the vector along axis of rotation and Φ is the angle of rotation. Eigenvalue equation, which was to be solved is now written as
[[File:Equation48.png|frame|center]]
Which can be solved by gauss elimination. <br \>
Once the '''p''' has been obtained, the quaternion can be found out by noting
[[File:Equation49.png|frame|center]]
Hence the estimated quaternion is obtained without solving eigenvalue equation. <br \>
If you are done reading this page, you can go back to [[Attitude Determination and Control Subsystem]]
Obtaining quaternion information from q-method involves solving the eigenvalue problem for a 4X4 Matrix. This can be done on MATLAB or other computer tools. But the solution is numerically intensive. On-board computing is a serious concern for any satellite designer and hence a better algorithm to solve this problem using less computing is required. <br \>
The QuEST algorithm provides a less efficient but a ‘faster’ way to solve the eigenvalue problem.
== The Algorithm ==
Recall from [[Q-Method and Wahba's Problem]] that the optimal attitude minimizes the loss function,
[[File:Equation40.JPG|frame|center]]
And maximizes the gain function,
[[File:Equation41.png|frame|center]]
Thus,
[[File:Equation42.png|frame|center]]
[[File:Equation43.png]] where [[File:Equation44.png]] is the maximum eigenvalue of the K matrix defined in [[Q-Method and Whab's Problem]].
With this [[File:Equation44.png]] can be written as
[[File:Equation45.png|frame|center]]
If we approximate J to be very small (this approximation is valid as we want to minimize J and now if after completing this method, we calculate J, it would be very small),
[[File:Equation46.png|frame|center]]
Defining and writing the equation in terms of Rodrigue’s parameter '''p''' simplifies the equation
[[File:Equation47.png|frame|center]]
where a is the vector along axis of rotation and Φ is the angle of rotation. Eigenvalue equation, which was to be solved is now written as
[[File:Equation48.png|frame|center]]
Which can be solved by gauss elimination. <br \>
Once the '''p''' has been obtained, the quaternion can be found out by noting
[[File:Equation49.png|frame|center]]
Hence the estimated quaternion is obtained without solving eigenvalue equation. <br \>
If you are done reading this page, you can go back to [[Attitude Determination and Control Subsystem]]