3.2 Rotation Vectors and the Euler Angles

3.2.1 Rotation Vectors

The matrix representation has at least the following disadvantages:

1. SO(3) has a rotation matrix of 9 quantities, but a 3D rotation only has 3 degrees of freedom. Therefore the matrix expression is redundant. Similarity, the transformation matrix expresses a 6-degree-of-freedom transformation with 16 quantities.
2. The rotation matrix itself has constraints: it must be an orthogonal matrix with a determination of 1. The same is true for the transformation matrix.

Rotation

Can be described by a rotation axis ($\textbf{n}$) and a rotation angle ($\theta$).

The conversion from the rotation vector to the rotation matrix: $$\pmb{R} = \cos \theta \pmb{I} + \left({ 1 - \cos \theta } \right) \pmb{n} { \pmb {n} ^T } + \sin \theta { \pmb{n}^ \wedge }$$ The conversion from the rotation matrix to rotation vector: $$\begin{aligned} \mathrm{tr} \left( \pmb{R} \right) &= \cos \theta \mathop{}\!\mathrm{tr}\left( \pmb{I} \right) + \left( {1 - \cos \theta } \right) \mathop{}\!\mathrm{tr} \left( { \pmb{n} {\pmb{n}^T}} \right) + \sin \theta \mathop{}\!\mathrm{tr} ({\pmb{n}^ \wedge })\\ &= 3\cos \theta + (1 - \cos \theta )\\ &= 1 + 2\cos \theta . \end{aligned}$$ Therefore: $$\theta = \arccos ( \frac{\mathrm{tr}(\pmb{R}) - 1}{2} ) .$$

$$\pmb{R} \pmb{n} = \pmb{n}.$$

The axis $\pmb{n}$ is the eigenvector corresponding to the matrix $\pmb{R}$'s eigenvalue 1. Solving this equation and normalizing it gives the axis of rotation.

3.2.2 Euler Angles

One of the most commonly used Euler angles is the yaw-pitch-roll angles. Since it is equivalent to the rotation of the ZYX axis, we take the ZY X Euler angle as an example.

1. Rotate around the $Z$ axis of the object to get the yaw angle $\theta_{\mathrm{yaw}}=y$;
2. Rotate around the $Y$ axis of after rotation to get the pitch angle $\theta_{\mathrm{pitch}}=p$;
3. Rotate around the $X$ axis of after rotation to get the roll angle $\theta_{\mathrm{roll}}=r$

A major drawback of Euler Angle is that it encounters the famous Gimbal lock