Velocity description in Screw Motion Theory

Question

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.

Answer 1

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.

For an explanation of why the spatial velocity is useful, see my answer to this question.