PlanetPhysics/Euler Angles

An Euler angle sequence is a rotation matrix that is completely determined by three parameters, called Euler angles. These Euler angles are represented by the $$ \phi $$, $$ \theta $$ and $$ \psi $$ variables with each corresponding to a rotation about an axis. There are several different conventions. Only one will be shown here, since it is more important to understand the underlying math thoroughly.

A list of the Euler angle rotation matrices for different sequences

Euler 123 sequence \\

$$ R_3(\psi)R_2(\theta)R_1(\phi) = \left[ \begin{matrix} c_{\psi} c_{\theta} & c_{\psi} s_{\theta} s_{\phi} + s_{\psi} c_{\phi} & -c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} \\ -s_{\psi} c_{\theta} & -s_{\psi} s_{\theta} s_{\phi} + c_{\psi} c_{\phi} & s_{\psi} s_{\theta} c_{\phi} + c_{\psi} s_{\phi} \\ s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{matrix} \right] $$

Euler 132 sequence \\

$$ R_2(\psi)R_3(\theta)R_1(\phi) = \left[ \begin{matrix} c_{\psi} c_{\theta} & c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} & c_{\psi} s_{\theta} s_{\phi} - s_{\psi} c_{\phi} \\ -s_{\theta} & c_{\theta} c_{\phi} & c_{\theta} s_{\phi} \\ s_{\psi} c_{\theta} & s_{\psi} s_{\theta} c_{\phi} - c_{\psi} s_{\phi} & s_{\psi} s_{\theta} s_{\phi} + c_{\psi} c_{\phi} \end{matrix} \right] $$

Euler 121 sequence \\

$$ R_1(\psi)R_2(\theta)R_1(\phi) = \left[ \begin{matrix} c_{\theta} & -s_{\theta} s_{\phi}  & s_{\theta} c_{\phi}  \\ -s_{\psi} s_{\theta} & c_{\psi} c_{\phi} - s_{\psi} c_{\theta} s_{\phi} & c_{\psi} s_{\phi} + s_{\psi} c_{\theta} c_{\phi} \\ -s_{\theta} c_{\psi} & -s_{\psi} c_{\phi} - c_{\psi} c_{\theta} s_{\phi} & -s_{\psi} s_{\phi} + c_{\psi} c_{\theta} c_{\phi} \end{matrix} \right] $$

Euler 131 sequence \\

$$ R_1(\psi)R_3(\theta)R_1(\phi) = \left[ \begin{matrix} c_{\theta} & s_{\theta} c_{\phi}   & s_{\theta} s_{\phi}   \\ -c_{\psi} s_{\theta} & c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} c_{\theta} s_{\phi} + s_{\psi} c_{\phi} \\ s_{\psi} s_{\theta} & -s_{\psi} c_{\theta} c_{\phi} - c_{\psi} s_{\phi} &  - s_{\psi} c_{\theta} s_{\phi} +  c_{\psi} c_{\phi}  \end{matrix} \right] $$

Euler 213 sequence \\

$$ R_3(\psi)R_1(\theta)R_2(\phi) = \left[ \begin{matrix} c_{\psi} c_{\phi} + s_{\psi} s_{\theta} s_{\phi} & s_{\psi} c_{\theta} & -c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} \\ -s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & c_{\psi} c_{\theta} & s_{\psi} s_{\phi} + c_{\psi} s_{\theta} c_{\phi} \\ c_{\theta} s_{\phi} & -s_{\theta} & c_{\theta} c_{\phi} \end{matrix} \right] $$

Euler 231 sequence \\

$$ R_1(\psi)R_3(\theta)R_2(\phi) = \left[ \begin{matrix} c_{\theta} c_{\phi} & s_{\theta} & -c_{\theta} s_{\phi}  \\ -c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} & c_{\psi} c_{\theta} & c_{\psi} s_{\theta} s_{\phi} + s_{\psi} s_{\theta} s_{\phi} \\ s_{\psi} s_{\theta} c_{\phi} + c_{\psi} s_{\phi} & -s_{\psi} c_{\theta} & -s_{\psi} s_{\theta} s_{\phi} +c_{\psi} c_{\phi} \end{matrix} \right] $$

Euler 212 sequence \\

$$ R_2(\psi)R_1(\theta)R_2(\phi) = \left[ \begin{matrix} c_{\psi} c_{\phi} - s_{\psi} c_{\theta} s_{\phi} & s_{\psi} s_{\theta} & -c_{\psi} s_{\phi} - s_{\psi} c_{\theta} c_{\phi} \\ s_{\theta} s_{\phi} & c_{\theta} & s_{\theta} c_{\phi} \\ s_{\psi} c_{\phi} + c_{\psi} c_{\theta} s_{\phi} & -c_{\psi} s_{\theta} & -s_{\psi} s_{\phi} + c_{\psi} c_{\theta} c_{\phi} \end{matrix} \right] $$

Euler 232 sequence \\

$$ R_2(\psi)R_3(\theta)R_2(\phi) = \left[ \begin{matrix} c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} s_{\theta}  & -c_{\psi} c_{\theta} s_{\phi} - s_{\psi} c_{\phi}   \\ -s_{\theta} c_{\phi} & c_{\theta} & s_{\theta} s_{\phi} \\ s_{\psi} c_{\theta} c_{\phi} + c_{\psi} s_{\phi} & s_{\psi} s_{\theta} &  - s_{\psi} c_{\theta} s_{\phi} +  c_{\psi} c_{\phi}  \end{matrix} \right] $$

Euler 312 Sequence \\

$$ R_2(\psi)R_1(\theta)R_3(\phi) = \left[ \begin{matrix} c_{\psi} c_{\phi} - s_{\psi} s_{\theta} s_{\phi} & c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} & -s_{\psi} c_{\theta} \\ -s_{\phi} c_{\theta} & c_{\theta} c_{\phi} & s_{\theta} \\ s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & s_{\psi} s_{\phi} - c_{\psi} s_{\theta} c_{\phi} & c_{\psi} c_{\theta} \end{matrix} \right] $$

Euler 321 sequence \\

$$ R_1(\psi)R_2(\theta)R_3(\phi) = \left[ \begin{matrix} c_{\theta} c_{\phi} & c_{\theta} s_{\phi} & -s_{\theta}  \\ - c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} & c_{\psi} c_{\phi} + s_{\psi} s_{\theta} s_{\phi} & s_{\psi} c_{\theta} \\ s_{\psi} s_{\phi} - c_{\psi} s_{\theta} c_{\phi} & -s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & c_{\psi} c_{\theta}  \end{matrix} \right]  $$

Euler 313 sequence \\

$$ R_3(\psi)R_1(\theta)R_3(\phi) = \left[ \begin{matrix} c_{\psi} c_{\phi} - s_{\psi} s_{\phi} c_{\theta} & c_{\psi} s_{\phi} + s_{\psi} c_{\theta} c_{\phi} & s_{\psi} s_{\theta} \\ -s_{\psi} c_{\phi} - c_{\psi} s_{\phi} c_{\theta} & -s_{\psi} s_{\phi} + c_{\psi} c_{\theta} c_{\phi} & c_{\psi} s_{\theta} \\ s_{\theta} s_{\phi} & -s_{\theta} c_{\phi} & c_{\theta} \end{matrix} \right] $$

Euler 323 sequence \\

$$ R_3(\psi)R_2(\theta)R_3(\phi) = \left[ \begin{matrix} c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} c_{\theta} s_{\phi} + s_{\psi} c_{\phi} & -c_{\psi} s_{\theta} \\ - s_{\psi} c_{\theta} c_{\phi} - c_{\psi} s_{\phi} & -s_{\psi} c_{\theta} s_{\phi} + c_{\psi} c_{\phi} & s_{\psi} s_{\theta} \\ s_{\theta} c_{\phi} & s_{\theta} s_{\phi} & c_{\theta} \end{matrix} \right] $$