EulerAngleSequences 2005-03-03 18:54:43 2005-08-14 16:20:08 Definition A list of the Euler angle rotation matrics for different sequences 1-2-3 $ R_3(\psi)R_2(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} & c_{\psi} s_{\theta} s_{\phi} + s_{\psi} c_{\phi} & -c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} \\ -s_{\psi} c_{\theta} & -s_{\psi} s_{\theta} s_{\phi} + c_{\psi} c_{\phi} & s_{\psi} s_{\theta} c_{\phi} + c_{\psi} s_{\phi} \\ s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{array} \right] $ 2-1-3 $ R_3(\psi)R_1(\theta)R_2(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} + s_{\psi} s_{\theta} s_{\phi} & s_{\psi} c_{\theta} & -c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} \\ -s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & c_{\psi} c_{\theta} & s_{\psi} s_{\phi} + c_{\psi} s_{\theta} c_{\phi} \\ c_{\theta} s_{\phi} & -s_{\theta} & c_{\theta} c_{\phi} \end{array} \right] $ 3-2-1 $ R_1(\psi)R_2(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\theta} c_{\phi} & c_{\theta} s_{\phi} & s_{\theta} \\ - c_{\psi} s_{\phi} - s_{\psi} s_{\theta} c_{\phi} & c_{\psi} c_{\phi} - s_{\psi} s_{\theta} s_{\phi} & s_{\psi} c_{\theta} \\ s_{\psi} c_{\phi} - c_{\psi} s_{\theta} s_{\phi} & -s_{\psi} c_{\phi} - c_{\psi} s_{\theta} s_{\phi} & c_{\psi} c_{\theta} \end{array} \right] $ 3-1-3 3-2-3 2-3-2 $$ \left[ \begin{array}{ c c c } -s_3 s_1 + c_2 c_3 c_1 & s_2 c_3 & -s_3 c_1 - c_2 c_3 s_1 \\ -s_2 c_1 & c_2 & s_2 s_1 \\ c_3 s_1 + c_2 s_3 c_1 & s_2 s_3 & c_3 c_1 - c_2 s_3 s_1 \end{array} \right] $$ 1-3-1 $$ \left[ \begin{array}{ c c c } c_2 & s_2 c_1 s_2 & s_2 s_1 \\ -s_2 c_3 & s_3 s_1 + c_2 c_3 c_1 & c_2 c_3 s_1 + s_3 c_1 \\ s_2 s_3 & -c_2 s_3 c_1 - c_3 s_1 & c_3 c_1 - c_2 s_3 s_1 \end{array} \right] $$ 2-1-2 $$ \left[ \begin{array}{ c c c } c_3 c_1 - c_2 s_3 s_1 & s_2 s_3 & -c_2 s_3 c_1 - c_3 s_1 \\ s_2 s_1 & c_2 & s_2 c_1 \\ c_2 c_3 s_1 + s_3 c_1 & -s_2 c_3 & -s_3 s_1 + c_2 c_3 c_1 \end{array} \right] $$ 2-3-1 $$ \left[ \begin{array}{ c c c } c_2 c_1 & s_2 & -c_2 s_1 \\ s_3 s_1 - s_2 c_3 c_1 & c_2 c_3 & s_3 c_1 + s_2 c_3 s_1 \\ c_3 s_1 + s_2 s_3 c_1 & -c_2 s_3 & c_3 c_1 - s_2 s_3 s_1 \end{array} \right] $$ 3-1-2 $$ \left[ \begin{array}{ c c c } c_3 c_1 - s_2 s_3 s_1 & c_3 s_1 + s_2 s_3 c_1 & -c_2 s_3 \\ -c_2 s_1 & c_2 c_1 & s_2 \\ s_3 c_1 + s_2 c_3 s_1 & s_3 s_1 - s_2 c_3 c_1 & c_2 c_3 \end{array} \right] $$ 1-2-1 $$ \left[ \begin{array}{ c c c } c_2 & s_2 s_1 & -s_2 c_1 \\ s_2 s_3 & c_3 c_1 - c_2 s_3 s_1 & c_3 s_1 + c_2 s_3 c_1 \\ s_2 c_3 & -s_3 c_1 - c_2 c_3 s_1 & -s_3 s_1 + c_2 c_3 c_1 \end{array} \right] $$ 1-3-2 $$ \left[ \begin{array}{ c c c } c_2 c_3 & s_2 c_3 c_1 + s_3 s_1 & s_2 c_3 s_1 - s_3 c_1 \\ -s_2 & c_2 c_1 & c_2 s_1 \\ c_2 s_3 & s_2 s_3 c_1 - c_3 s_1 & s_2 s_3 s_1 + c_3 c_1 \end{array} \right] $$