HCR - Rotation Matrix Descriptor 'Angle and Axis'

Formula:$R(\theta ,\overline{v})=R_z(\alpha )R_y(\beta )R_z(\theta )R_y(-\beta )R_z(-\alpha )$Where:

• $R_z$: rotation matrix along the $z$ axis.
• $\theta$ is the angle of rotation along the $\overline{v}$ axis (in the figure this axis/vector it’s represented by the vector $r$ but i wanted to avoid confusion between this axis of rotation and the elements of the rotation matrix $R$ : $r_{i,j}$, so i called it $\vec{v}$).
  The angle of rotation $\theta$ can be found as:$\theta ={\cos }^{-1}\left(\frac{r_{1,1}+r_{2,2}+r_{3,3}-1}{2}\right)$And the axis of rotation $\vec{v}$ as:$\overline{v}=\frac{1}{2\sin (\theta )}\left[\begin{array}{c}{r}_{3,2}-r_{2,3}\\ r_{1,3}-r_{3,1}\\ r_{2,1}-r_{1,2}\end{array}\right]$Where:
  • $r_{i,j}$ is the $i,j$ element of the rotation matrix $R$

So to find the $R$ rotation matrix we can impose $\theta$ and $\overline{v}$ as constraints and find:

$$R=\left[\begin{array}{lcc}{r}_{1,1}& r_{1,2}& r_{1,3}\\ r_{2,1}& r_{2,2}& r_{2,3}\\ r_{3,1}& r_{3,2}& r_{3,3}\end{array}\right]$$

NOTE:IMPORTANTE For $r_{1,1}+r_{2,2}+r_{3,3}=1$ → $\theta =0$ (or $\theta =\pi$) → $\overline{v}=\infty$ (Indeterminate) The Quaternion descriptor is more complicated but can avoid this problem.