Euler312Sequence 2005-07-29 00:06:17 2005-07-29 00:06:17 Definition \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} For more info on Euler Sequences, notation and convention see the generic entry on Euler Angle Sequences. \\ $ R_{312}(\phi, \theta, \psi) = R_2(\psi) R_1(\theta) R_3(\phi) $ \\ The rotation matrices are \begin{equation} R_1(\phi) = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c_{\phi} & s_{\phi} \\ 0 & -s_{\phi} & c_{\phi} \end{array} \right] \end{equation} \begin{equation} R_2(\theta) = \left[ \begin{array}{ccc} c_{\theta} & 0 & -s_{\theta} \\ 0 & 1 & 0 \\ s_{\theta} & 0 & c_{\theta} \end{array} \right] \end{equation} \begin{equation} R_3(\psi) = \left[ \begin{array}{ccc} c_{\psi} & s_{\psi} & 0 \\ -s_{\psi} & c_{\psi} & 0 \\ 0 & 0 & 1 \end{array} \right] \end{equation} Carrying out the multiplication from right to left \\ $ R_1(\theta)R_3(\phi) = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c_{\theta} & s_{\theta} \\ 0 & -s_{\theta} & c_{\theta} \end{array} \right] \left[ \begin{array}{ccc} c_{\phi} & s_{\phi} & 0 \\ -s_{\phi} & c_{\phi} & 0 \\ 0 & 0 & 1 \end{array} \right] = \left[ \begin{array}{ccc} c_{\phi} & s_{\phi} & 0 \\ -s_{\phi} c_{\theta} & c_{\theta} c_{\phi} & s_{\theta} \\ s_{\theta} s_{\phi} & -s_{\theta} c_{\phi} & c_{\theta} \end{array} \right] $ \\ Finaly leaving us with the Euler 123 sequence \\ $ R_2(\psi)R_1(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} - s_{\psi} s_{\theta} s_{\phi} & c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} & -s_{\psi} c_{\theta} \\ -s_{\phi} c_{\theta} & c_{\theta} c_{\phi} & s_{\theta} \\ s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & s_{\psi} s_{\phi} - c_{\psi} s_{\theta} c_{\phi} & c_{\psi} c_{\theta} \end{array} \right] $