Euler angle sequences

An Euler angle sequence is a rotation matrix that is completely determined by three parameters, called Euler angles. These Euler angles are represented by the $ \phi $ , $ \theta $ and $ \psi $ variables with each corresponding to a rotation about an axis. There are several different conventions. Only one will be shown here, since it is more important to understand the underlying math thoroughly.

A list of the Euler angle rotation matrices for different sequences

Euler 123 sequence

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

Euler 132 sequence

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

Euler 121 sequence

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

Euler 131 sequence

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

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] $

Euler 231 sequence

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

Euler 212 sequence

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

Euler 232 sequence

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

Euler 312 Sequence

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

Euler 321 sequence

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

Euler 313 sequence

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

Euler 323 sequence

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



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