PlanetPhysics/Euler 323 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler Angle Sequences. \\

$R_{323}(\varphi ,\theta ,\psi )=R_3(\psi )R_2(\theta )R_3(\varphi )$ \\

The rotation matrices are

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

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

$R_3(\varphi )=\left[\begin{array}{ccc}{c}_{\varphi }& s_{\varphi }& 0\\ -s_{\varphi }& c_{\varphi }& 0\\ 0& 0& 1\end{array}\right]$

Carrying out the matrix multiplication from right to left \\

$R_2(\theta )R_3(\varphi )=\left[\begin{array}{ccc}{c}_{\theta }& 0& -s_{\theta }\\ 0& 1& 0\\ s_{\theta }& 0& c_{\theta }\end{array}\right]\left[\begin{array}{ccc}{c}_{\varphi }& s_{\varphi }& 0\\ -s_{\varphi }& c_{\varphi }& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}{c}_{\theta }{c}_{\varphi }& c_{\theta }{s}_{\varphi }& -s_{\theta }\\ -s_{\varphi }& c_{\varphi }& 0\\ s_{\theta }{c}_{\varphi }& s_{\theta }{s}_{\varphi }& c_{\theta }\end{array}\right]$ \\

Finaly leaving us with the Euler 323 sequence \\

$R_3(\psi )R_2(\theta )R_3(\varphi )=\left[\begin{array}{ccc}{c}_{\psi }{c}_{\theta }{c}_{\varphi }-s_{\psi }{s}_{\varphi }& c_{\psi }{c}_{\theta }{s}_{\varphi }+s_{\psi }{c}_{\varphi }& -c_{\psi }{s}_{\theta }\\ -s_{\psi }{c}_{\theta }{c}_{\varphi }-c_{\psi }{s}_{\varphi }& -s_{\psi }{c}_{\theta }{s}_{\varphi }+c_{\psi }{c}_{\varphi }& s_{\psi }{s}_{\theta }\\ s_{\theta }{c}_{\varphi }& s_{\theta }{s}_{\varphi }& c_{\theta }\end{array}\right]$