# Velocity screw description of a robot end-effector question

1. Jul 8, 2011

### zyh

look at the above image, the red coordinates is attached to the robot end-effector, and the other one is the base.coordinates.

Now, As we know, the end-effector can do any rigid motion, so if a point P is attached on the robot end-effector. we can calculate the velocity of the point P.

The general formula is just like: $$\dot{P}=\omega\times a+b$$
Here, the $\omega$ is just the angular velocity, and b is the translation velocity.

But when reading some papers, I found the general formula has different expression.

One is:
$$\dot{P}=\Omega\times P+T$$
here, P has the coordinates in base frame. T and $\Omega$ is defined as the velocity screw.

Another expression:
$$\dot{P}=\omega\times(R\cdot^{1}P)+v$$
here, Here, the R is the rotation matrix of end-effector frame wrt base frame, and $^{1}P$ is the vector wrt end-effector frame. $\omega v$ is also some kind of velocity description about the end-effector. In-fact, this is the most convenient way I can understand.

I'm quite confusing that it seems both expression is valid, but what exactly does the velocity screw means?

For more details about the paper, I have also wrote another post, see here: