Representations of Rotation
A rotation, can be respresented as $so(3)$ (Lie Algebra of Special Orthogonal Group), or $SO(3)$, (Special Orthogonal Group) and rotation vector.
Representation 1 A rotation vector is $s = \theta [s_x, s_y, s_z] = [\omega_x, \omega_y, \omega_z]$, where:
\[\begin{gather*} \theta = \sqrt{\omega_x^2 + \omega_y^2 + \omega_z^2} \end{gather*}\]$[s_x, s_y, s_z]$ here is the axis of rotation, which is a unit vector.
Representation 2 Then we can write this rotation vector in the form of $so(3)$. It’s also called “skew symmetric matrix” of a rotation axis (notice how the matrix diagonal serve as the axis of symmetry and sign?)
\[\begin{gather*} \hat{\omega} = \begin{pmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{pmatrix} \end{gather*}\]Representation 3 SO(3)!! This is the most common representation of rotation I’ve seen so far.
\[\begin{gather*} \exp(\hat{\omega}) = I + \frac{sin(\theta)}{\theta} \hat{\omega} + \frac{1 - cos(\theta)}{\theta} \hat{\omega}^2 \end{gather*}\]Common rotation matrices are:
\[\begin{gather*} R_x(\gamma) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \gamma & -\sin \gamma \\ 0 & \sin \gamma & \cos \gamma \end{bmatrix} R_y(\beta) = \begin{bmatrix} \cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ -\sin \beta & 0 & \cos \beta \end{bmatrix} R_z(\alpha) = \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{gather*}\]Representation 4 Quaternion. TODO
Order is important
Different orders of Rotation about x, y, z axis could yield different final orientations. Try it yourself: first, rotate a random object you see now by z axis by +90 degrees followed by rotating about x axis by +90 degrees. Then, try rotating about x axis followed by about the z axis. A side note is that in robotics, people follow the right hand convention for frames. That is: x forward, y left, z up
So, it’s important to specify the order or rotation, if a single rotation can be decomposed into rotations about x,y,z axes.. Also, there are rotations about fixed axes, and Euler angles. Fixed axes refer to the axes of a fixed world frame, and euler angles refer to the X,Y,Z in the body frame. In either case, x,y,z are also known as “roll-pitch-yaw”.
In the robotics community:
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Use world frame fixed axes. ROS uses the X-Y-Z order, so there is no ambiguity on order. There is no gimbal lock, either.
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Euler angles can represent any orientation. There are common orders such as Z-X-Y, X-Y-Z, etc. There are 24 valid combinations. Below (from Wikipedia) is an illustration of Z-X-Z’
Multiple Rotations Leads To One Rotation
The final rotation of 3 rotations about fixed axes is \(\begin{gather*} R = R_z(\theta_z) R_y(\theta_y) R_x(\theta_x) \end{gather*}\)
Then we can get $\theta$ and rotation axis $u=[u_x, u_y, u_z]$
\[\begin{gather*} \theta = \cos^{-1} \left( \frac{\text{trace}(R) - 1}{2} \right) \\ \mathbf{u} = \frac{1}{2 \sin \theta} \begin{pmatrix} R_{32} - R_{23} \\ R_{13} - R_{31} \\ R_{21} - R_{12} \end{pmatrix} \end{gather*}\]Gimbal Lock, Singularities, and Quaternion
When using euler angles, certain axes in the body frame could align to each other. E.g., when a plane has a pitch of 90 degrees (as below), its z and x axes are aligned. Then, rotation about z and rotation about x are the same. From this configuration, the plane cannot rotate about the axis that are perpendicular to x,y,z axes, hence it loses 1 degree of freedom.
Mathematically,
\[\begin{gather*} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \gamma & -\sin \gamma \\ 0 & \sin \gamma & \cos \gamma \end{bmatrix} * \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix} * \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{gather*}\]
See? If in the plane’s control system orientation is represented by Z-X-Y euler angles, when the plane is in this initial position, rotating about X will affect the reading of yaw (We would want the to be independent at all times).
Quaternion
Implementations
- OpenCV: OpenCV provides rotation vector -> single rotation matrix. See here
