Twists

Any rigid body velocity, which consists of a linear and angular component, is equivalent to the instantaneous velocity about some screw axis.

The screw axis $\mathcal{S}$ is defined by a point $q$ on the axis, a unit vector $\hat{s}$ pointing in the direction of the screw axis, and the pitch $h = \frac{\text{linear speed}}{\text{angular speed}}$. For any linear and angular velocity of a rigid body, there is a corresponding screw axis, as if the body's instantaneous motion is twisting about this screw axis.

The screw axis $\mathcal{S}$ defines the direction of the body's movement, and $\dot{\theta}$ indicates how fast the body rotates about the screw axis.

To represent the screw axis, we first choose a reference frame and represent it as a 6-vector in that frame's coordinates, and it is \[ \mathcal{S} = \begin{bmatrix} \mathcal{S}_\omega\\ \mathcal{S}_v \end{bmatrix} = \begin{bmatrix} \text{angular velocity when $\dot{\theta}=1$}\\ \text{linear velocity of origin when $\dot{\theta}=1$} \end{bmatrix}\]

Combining the screw axis with the scalar rate of rotation $\dot{\theta}$ gives the twist $\mathcal{V} = \begin{bmatrix} \omega \\ v \end{bmatrix} = \mathcal{S}\dot{\theta} \in \mathbb{R}^6$. We have 2 cases:

$h = \infty$ : then $\mathcal{S}_\omega= \mathbf{0}$, and $||\mathcal{S}_v|| =1$, and $\dot{\theta}$ is the linear speed.
$h < \infty$ : then $||\mathcal{S}_\omega||=1$, and $\dot{\theta}$ is the rotational speed.

If the screw axis is defined in the $\{b\}$ frame, $\mathcal{V}_b = \mathcal{S}\dot{\theta}$ is called the body twist. When it is defined the $\{s\}$ frame, it is called the spatial twist. The body twist is not affected by the choice of the space frame, and vice versa.

Similar to how angular velocities can be represented via matrices using so(3), twists also have a matrix representation. Namely, we have the $4\times 4$ matrices \[ [\mathcal{V}_b] = T^{-1}\dot{T} = \begin{bmatrix} [\omega_b] & v_b \\ 0 & 0 \end{bmatrix} \] \[ [\mathcal{V}_s] = \dot{T}T^{-1} = \begin{bmatrix} [\omega_s] & v_s \\ 0 & 0 \end{bmatrix} \] Here, the [.] on both the angular velocity vectors $\omega_b$ and the twists $\mathcal{V}_b$ simply refer to their corresponding matrix representations.