- Eigenvector slew
In
aerospace engineering , especially those areas dealing withspacecraft , the eigenvector slew is a method to calculate a steering correction (called a slew) by rotating the spacecraft around one fixed axis. This correponds in general to the fastest and most efficient way to reach the desired target orientation as there is only one acceleration phase and one braking phase for the angular rate. If this fixed axis is not aprincipal axis a time varying torque must be applied to force the spacecraft to rotate as desired, though. Also the gyroscopic effect ofmomentum wheels must be compensated for.That such a rotation exists corresponds precisely to a main result of the mathematical theory of rotation operators, the (only)
eigenvector of the rotation operator corresponding to the desired re-orientation is this axis.Given the current orientation of the craft, and the desired orientation of the craft in
cartesian coordinates , the requiredaxis of rotation and corresponding rotation angle to achieve the new orientation is determined by computing the eigenvector of the rotation operator.Problem
Let
:
be a body fixed reference system for a
3 axis stabilized spacecraft . The initial attitudeis given by:::
One wants to find an axis relative the spacecraft body
:
and a rotation angle such that after the rotation with the angle one has that
:::
where
:
are the new target directions.
In vector form this means that
:::
Solution
In terms of
linear algebra this means that one wants to find aneigenvector with theeigenvalue = 1 for thelinear map ping defined by:::
which relative to the
:
coordinate system has the matrix
:
Because this is the matrix of the rotation operator relative the base vector system the eigenvalue can be determined with the algorithm described in "
Rotation operator (vector space) ".With the notations used here this is:
::::::
The rotation angle is
:
where "" is the polar argument of the vector corresponding to the function function ATAN2(y,x) (or in
double precision DATAN2(y,x)) available in for example the programming language FORTRAN.The resulting will be in the interval .
If then and the uniquely defined rotation (unit) vector is:
:
Note that :
is the trace of the matrix defined by the orthogonal linear mapping and that the components of the "
eigenvector " are fixed and constant during the rotation, i.e. :where are moving with time during the slew.See also
*
Rotation operator (vector space)
*Slew (spacecraft)
Wikimedia Foundation. 2010.