paparazzi/doc/pprz_algebra/transformations/axisangle2matrix.tex

With a known axis of rotation $\vect u$ and an angle $\alpha$ it is possible to compute a rotational matrix with
\begin{equation}
\mat R_m = \begin{pmatrix}
0 & -u_z & u_y \\
u_z & 0 & -u_x \\
-u_y & u_x & 0
\end{pmatrix} \sin \alpha + \left( \eye - \vect u \transp{\vect u} \right)\cos \alpha +  \vect u \transp{\vect u}.
\end{equation}
Please note again, that all angles are from the perspective of the aircraft (see section \ref{Important definition}). Therefore the angle is defined ngeative, leading to
\begin{equation}
\mat R_m = \begin{pmatrix}
0 & u_z & -u_y \\
-u_z & 0 & u_x \\
u_y & -u_x & 0
\end{pmatrix} \sin \alpha + \left( \eye - \vect u \transp{\vect u} \right)\cos \alpha +  \vect u \transp{\vect u}.
\end{equation}
Rearranging this equation leads to
\begin{equation}
\mat R_m = \begin{pmatrix}
u_x^2+(1-u_x^2)\cos \alpha				& u_xu_y(1-\cos \alpha) + u_z \sin \alpha	& u_xu_z(1-\cos \alpha) - u_y \sin \alpha \\
u_xu_y(1-\cos \alpha) - u_z \sin \alpha	& u_y^2+(1-u_y^2)\cos \alpha				& u_yu_z(1-\cos \alpha) + u_x \sin \alpha \\
u_xu_z(1-\cos \alpha) + u_y \sin \alpha & u_yu_z(1-\cos \alpha) - u_x \sin \alpha	& u_z^2+(1-u_z^2)\cos \alpha
\end{pmatrix}.
\end{equation}
\inHfile{FLOAT\_RMAT\_OF\_AXIS\_ANGLE(rm, uv, an)}{pprz\_algebra\_float}