# Velocity description in Screw Motion Theory

As described in the book Modern Robotics by Frank C. Park the velocity of a reference frame can be described as :

$$V_s = J_s(\theta)\dot{\theta}$$ $$V_b = J_b(\theta)\dot{\theta}$$

However in the definition of the spatial velocities $$V_s$$ and $$V_b$$ we have that:

$$V_s = [\omega_s, v_s]^T \in \mathcal{R}^6$$ $$V_b = [\omega_b, v_b]^T \in \mathcal{R}^6$$

I get the idea that $$\omega_s$$ and $$\omega_b$$ are the angular velocities as seen from each reference frame, however I quite dont get what does $$v_s$$ and $$v_b$$ are making reference to.

The body velocity $$V_{b}$$ is the velocity of the frame with respect to the world, as seen from the frame's perspective. Its rotational component $$\omega_{b}$$ contains the rotation rates around the world-fixed axes instantaneously pointing in the frame's forward, lateral, and dorsal directions (local $$x$$, $$y$$, and $$z$$), and its translational component $$v_{b}$$ contains the frame's translational velocities along these axes.
The spatial velocity $$V_{s}$$ is the velocity of the frame with respect to the world, as seen from the perspective of a second frame rigidly attached to the first frame, but currently located at and aligned with the frame at the origin. Its rotational component $$\omega_{s}$$ contains the rotation rates of this second frame around the world axes, and its translational component $$v_{s}$$ contains the translational velocity of this second frame along the world axes.