Eigenvector slew

Let

${\displaystyle {\hat {x}}\ ,\ {\hat {y}}\ ,\ {\hat {z}}}$

be a body fixed reference system for a 3 axis stabilized spacecraft. The initial attitude is given by

${\displaystyle {\hat {x}}={\hat {a}}}$

${\displaystyle {\hat {y}}={\hat {b}}}$

${\displaystyle {\hat {z}}={\hat {c}}.}$

One wants to find an axis relative the spacecraft body

${\displaystyle {\hat {r}}=r_{x}\cdot {\hat {x}}+r_{y}\cdot {\hat {y}}+r_{z}\cdot {\hat {z}}}$

and a rotation angle ${\displaystyle \alpha }$ such that after the rotation with the angle ${\displaystyle \alpha }$ one has that

${\displaystyle {\hat {x}}={\hat {d}}}$

${\displaystyle {\hat {y}}={\hat {e}}}$

${\displaystyle {\hat {z}}={\hat {f}}}$

where

${\displaystyle {\hat {d}}\ ,\ {\hat {e}}\ ,\ {\hat {f}}}$

are the new target directions.

In vector form this means that

${\displaystyle {\hat {d}}=r_{a}\cdot {\hat {r}}+\cos \alpha \cdot ({\hat {a}}-r_{a}\cdot {\hat {r}})+\sin \alpha \cdot {\hat {r}}\times {\hat {a}}}$

${\displaystyle {\hat {e}}=r_{b}\cdot {\hat {r}}+\cos \alpha \cdot ({\hat {b}}-r_{b}\cdot {\hat {r}})+\sin \alpha \cdot {\hat {r}}\times {\hat {b}}}$

${\displaystyle {\hat {f}}=r_{c}\cdot {\hat {r}}+\cos \alpha \cdot ({\hat {c}}-r_{c}\cdot {\hat {r}})+\sin \alpha \cdot {\hat {r}}\times {\hat {c}}.}$

In terms of linear algebra this means that one wants to find an eigenvector with the eigenvalue = 1 for the linear mapping defined by

${\displaystyle {\hat {a}}\longrightarrow {\hat {d}}}$

${\displaystyle {\hat {b}}\longrightarrow {\hat {e}}}$

${\displaystyle {\hat {c}}\longrightarrow {\hat {f}}}$

which relative to the

${\displaystyle {\hat {a}}\ ,\ {\hat {b}}\ ,\ {\hat {c}}}$

coordinate system has the matrix

${\displaystyle {\begin{bmatrix}\langle {\hat {d}}|{\hat {a}}\rangle &\langle {\hat {e}}|{\hat {a}}\rangle &\langle {\hat {f}}|{\hat {a}}\rangle \\\langle {\hat {d}}|{\hat {b}}\rangle &\langle {\hat {e}}|{\hat {b}}\rangle &\langle {\hat {f}}|{\hat {b}}\rangle \\\langle {\hat {d}}|{\hat {c}}\rangle &\langle {\hat {e}}|{\hat {c}}\rangle &\langle {\hat {f}}|{\hat {c}}\rangle \end{bmatrix}}}$

Because this is the matrix of the rotation operator relative the base vector system ${\displaystyle {\hat {a}}\ ,\ {\hat {b}}\ ,\ {\hat {c}}}$ the eigenvalue can be determined with the algorithm described in "Rotation operator (vector space)".

With the notations used here this is:

${\displaystyle \cos \alpha ={\frac {\langle {\hat {d}}|{\hat {a}}\rangle +\langle {\hat {e}}|{\hat {b}}\rangle +\langle {\hat {f}}|{\hat {c}}\rangle -1}{2}}}$

${\displaystyle r_{a}=\langle {\hat {f}}|{\hat {b}}\rangle -\langle {\hat {e}}|{\hat {c}}\rangle }$

${\displaystyle r_{b}=\langle {\hat {d}}|{\hat {c}}\rangle -\langle {\hat {f}}|{\hat {a}}\rangle }$

${\displaystyle r_{c}=\langle {\hat {e}}|{\hat {a}}\rangle -\langle {\hat {d}}|{\hat {b}}\rangle }$

${\displaystyle |{\bar {r}}|={\sqrt {{r_{a}}^{2}+{r_{b}}^{2}+{r_{c}}^{2}}}}$

${\displaystyle \sin \alpha ={\frac {|{\bar {r}}|}{2}}}$

The rotation angle ${\displaystyle \alpha }$ is

${\displaystyle \alpha =\operatorname {arg} (\cos \alpha ,\sin \alpha )}$

where "${\displaystyle \operatorname {arg} (x\ ,\ y)}$" is the polar argument of the vector ${\displaystyle (\ x\ ,\ y\ )}$ corresponding to the function ATAN2(y,x) (or in double precision DATAN2(y,x)) available in for example the programming language FORTRAN.

The resulting ${\displaystyle \alpha }$ will be in the interval ${\displaystyle 0\leq \alpha \leq \pi }$.

If ${\displaystyle 0<\alpha <\pi }$ then ${\displaystyle |{\bar {r}}|>0}$ and the uniquely defined rotation (unit) vector is:

${\displaystyle {\hat {r}}={\frac {\bar {r}}{|{\bar {r}}|}}}$

Note that

${\displaystyle \langle {\hat {d}}|{\hat {a}}\rangle +\langle {\hat {e}}|{\hat {b}}\rangle +\langle {\hat {f}}|{\hat {c}}\rangle }$

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.

${\displaystyle {\hat {r}}=r_{x}\cdot {\hat {x}}(t)+r_{y}\cdot {\hat {y}}(t)+r_{z}\cdot {\hat {z}}(t)=r_{x}\cdot {\hat {a}}+r_{y}\cdot {\hat {b}}+r_{z}\cdot {\hat {c}}=r_{x}\cdot {\hat {d}}+r_{y}\cdot {\hat {e}}+r_{z}\cdot {\hat {f}}}$

where ${\displaystyle {\hat {x}}\ ,\ {\hat {y}}\ ,\ {\hat {z}}}$ are moving with time ${\displaystyle t}$ during the slew.

• Rotation operator (vector space)
• Slew (spacecraft)