Euler Angles

Three angles are needed to describe an arbitrary rotation. There are an infinite number of ways to do this but the Euler angles are most often used. This is a slightly complicated problem, no matter how you define the angles. We will go from the Inertial system to the Body system, in three steps from $\vec{r}_I$ to $\vec{r}\;'$ to $\vec{r}\;''$ to $\vec{r}_B$.

The three steps are

1. rotate axes by an angle $\phi$ about the $z_I$ axis.
2. rotate axes by an angle $\theta$ about the $x'$ axis.
3. rotate axes by an angle $\psi$ about the $z''=z_B$ axis.

The three rotations are shown in the figure below.

The line along the $x'$ axis is called the line of nodes'. It is common to the $xy$ plane of both the Inertial and Body coordinates and is key to finding the Euler angles needed for some rotation.

The resulting rotation matrix $R=R_\psi R_\theta R_\phi$ can be straightforwardly calculated.

$$\begin{pmatrix}\cos\psi & \sin\psi & 0\cr -\sin\psi & \cos\psi & ... ...\phi & 0\cr 0&0&1\end{pmatrix} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$$

$$=\begin{pmatrix}\cos\psi\cos\phi-\cos\theta\sin\phi\sin\psi & \co... ...sin\theta\cr \sin\theta\sin\phi & -\sin\theta\cos\phi & \cos\theta\end{pmatrix}$$

$$\begin{pmatrix}x_B\cr y_B\cr z_B\end{pmatrix} = \begin{pmatrix}\c... ...cos\phi & \cos\theta\end{pmatrix} \begin{pmatrix}x_I\cr y_I\cr z_I\end{pmatrix}$$