PlanetPhysics/Euler 131 Sequence

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

$R_{131}(\varphi ,\theta ,\psi )=R_1(\psi )R_3(\theta )R_1(\varphi )$ \\

The rotation matrices are

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

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

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

Carrying out the matrix multiplication from right to left \\

$R_3(\theta )R_1(\varphi )=\left[\begin{array}{ccc}{c}_{\theta }& s_{\theta }& 0\\ -s_{\theta }& c_{\theta }& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& c_{\varphi }& s_{\varphi }\\ 0& -s_{\varphi }& c_{\varphi }\end{array}\right]=\left[\begin{array}{ccc}{c}_{\theta }& s_{\theta }{c}_{\varphi }& s_{\theta }{s}_{\varphi }\\ -s_{\theta }& c_{\theta }{c}_{\varphi }& c_{\theta }{s}_{\varphi }\\ 0& -s_{\varphi }& c_{\varphi }\end{array}\right]$ \\

Finaly leaving us with the Euler 131 sequence \\

$R_1(\psi )R_3(\theta )R_1(\varphi )=\left[\begin{array}{ccc}{c}_{\theta }& s_{\theta }{c}_{\varphi }& s_{\theta }{s}_{\varphi }\\ -c_{\psi }{s}_{\theta }& c_{\psi }{c}_{\theta }{c}_{\varphi }-s_{\psi }{s}_{\varphi }& c_{\psi }{c}_{\theta }{s}_{\varphi }+s_{\psi }{c}_{\varphi }\\ s_{\psi }{s}_{\theta }& -s_{\psi }{c}_{\theta }{c}_{\varphi }-c_{\psi }{s}_{\varphi }& -s_{\psi }{c}_{\theta }{s}_{\varphi }+c_{\psi }{c}_{\varphi }\end{array}\right]$