Euler 213 sequence

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

$ R_{213}(\phi, \theta, \psi) = R_3(\psi) R_1(\theta) R_2(\phi) $

The rotation matrices are

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

$\displaystyle R_1(\theta) = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c_{\theta} & s_{\theta} \\ 0 & -s_{\theta} & c_{\theta} \end{array} \right]$ (2)

$\displaystyle R_2(\phi) = \left[ \begin{array}{ccc} c_{\phi} & 0 & -s_{\phi} \\ 0 & 1 & 0 \\ s_{\phi} & 0 & c_{\phi} \end{array} \right]$ (3)

Carrying out the matrix multiplication from right to left

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

Finaly leaving us with the Euler 213 sequence

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



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