PlanetPhysics/Euler Angles

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 $\varphi$ , $\theta$ 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 \\

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

Euler 132 sequence \\

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

Euler 121 sequence \\

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

Euler 131 sequence \\

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

Euler 213 sequence \\

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

Euler 231 sequence \\

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

Euler 212 sequence \\

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

Euler 232 sequence \\

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

Euler 312 Sequence \\

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

Euler 321 sequence \\

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

Euler 313 sequence \\

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

Euler 323 sequence \\

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