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)
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, for a Z-X-Y system,
\[\begin{gather*} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \gamma & -\sin \gamma \\ 0 & \sin \gamma & \cos \gamma \end{bmatrix} * \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix} * \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{gather*}\]
See? The rotation about both the Z axis $\gamma$ and the X axis $\alpha$ will effectively create a combined rotation about the X axis, $\theta$. So, such rotations do not have unique angular values.
Implementations
- OpenCV: OpenCV provides rotation vector -> single rotation matrix. See here
Instantaneous Rotation
According to the Poisson Formula, $R’ = Rw^{\land}$, for a small time period $\Delta t$, the ODE can be solved:
\[\begin{gather*} R(t) = R(t_0)exp(w^{\land}(t - t_0)) = R(t_0) exp(w^{\land} \Delta t) \end{gather*}\]Rotation and Skew Matrices in 2D:
In 2D, skew matrix is simply:
\[\begin{gather*} \begin{aligned} & w^{\land} = \begin{bmatrix} 0 & -a \\ a & 0 \end{bmatrix} \end{aligned} \end{gather*}\]Rotation matrix is:
\[\begin{gather*} \begin{aligned} & R = \begin{bmatrix} cos \theta & -sin \theta \\ sin \theta & cos \theta \end{bmatrix} \end{aligned} \end{gather*}\]- One Key property that doesn’t hold true in 2D is this:
But in 2D, one can easily find that: $\phi^{\land} R = R \phi^{\land}$
