PlanetPhysics/Euler 123 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences.

$$ R_{123}(\phi, \theta, \psi) = R_3(\psi) R_2(\theta) R_1(\phi) $$ \\

The rotation matrices are

$$ R_3(\psi) = \left[ \begin{matrix} c_{\psi} & s_{\psi} & 0 \\ -s_{\psi} & c_{\psi} & 0 \\ 0 & 0 & 1 \end{matrix} \right] $$

$$ R_2(\theta) = \left[ \begin{matrix} c_{\theta} & 0 & -s_{\theta} \\ 0 & 1 & 0 \\ s_{\theta} & 0 & c_{\theta} \end{matrix} \right] $$

$$ R_1(\phi) = \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & c_{\phi} & s_{\phi} \\ 0 & -s_{\phi} & c_{\phi} \end{matrix} \right] $$

Carrying out the matrix multiplication from right to left

$$ R_2(\theta)R_1(\phi) = \left[ \begin{matrix} c_{\theta} & 0 & -s_{\theta} \\ 0 & 1 & 0 \\ s_{\theta} & 0 & c_{\theta} \end{matrix} \right] \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & c_{\phi} & s_{\phi} \\ 0 & -s_{\phi} & c_{\phi} \end{matrix} \right] = \left[  \begin{matrix} c_{\theta} & s_{\theta}s_{\phi} & -s_{\theta} c_{\phi} \\ 0 & c_{\phi} & s_{\phi} \\ s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{matrix} \right] $$

Finaly leaving us with the Euler 123 sequence

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