PlanetPhysics/Euler 321 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on [[../EulerAngleSequence/|Euler angle sequences]]. \\

$R_{321}(\varphi ,\theta ,\psi )=R_1(\varphi )R_2(\theta )R_3(\psi )$ \\

The rotation [[../Matrix/|matrices]] are

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

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

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

Carrying out the [[../Matrix/|matrix multiplication]] from right to left \\

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

Finaly leaving us with the Euler 321 sequence \\

$R_1(\varphi )R_2(\theta )R_3(\psi )=\left[\begin{array}{ccc}{c}_{\theta }{c}_{\psi }& c_{\theta }{s}_{\psi }& -s_{\theta }\\ -c_{\varphi }{s}_{\psi }+s_{\varphi }{s}_{\theta }{c}_{\psi }& c_{\varphi }{c}_{\psi }+s_{\varphi }{s}_{\theta }{s}_{\psi }& s_{\varphi }{c}_{\theta }\\ s_{\varphi }{s}_{\psi }+c_{\varphi }{s}_{\theta }{c}_{\psi }& -s_{\varphi }{c}_{\psi }+c_{\varphi }{s}_{\theta }{s}_{\psi }& c_{\varphi }{c}_{\theta }\end{array}\right]$