Help on velocity screw description and velocity propagation

[zyh]

When reading some papers in robotics, I found two different ways of the velocity screw and velocity propagation.

This is one description of calculation a point's velocity in the paper "A tutorial on Visual Servo Control", we say paper A.

The author said: The rotation acts about a point which, unless otherwise indicated, we take to be the origin of the base coordinate system. The author use the distance vector P directly, the P is coordinates of the target point relative to base frame.

Why does the rotation should concern a point? As I have read some other robotics books, a rotation is just a matrix R, in which each column of the R is the moving frame's unit x,y,z axis expressed in the base frame.

Now, here is another description of calculation a point's velocity in the book "Robot modeling and control", we say paper B.

In the above definition, look at the equation (4.39), it has the same express form as the equation (12) in paper A.

$$\dot{P}=\omega\times a+b$$

but the meaning of the variables are quite different. Here, the r in (4.39) is the vector from o1 to p expressed in the orientations of the frame o0x0y0z0. So, this distance vector is in-fact relative to the moving frame. I think here the rotation is acts on the original of the frame o1x1y1z1.

PS: Does this definition in paper A of velocity screw is the same as Roy Featherstone's definition of 6-D Spatial Vector in http://users.cecs.anu.edu.au/~roy/spatial/index.html ?
I look at the definition of P, it is the point attach to the moving frame (end effector), but it has the coordinates in base frame.

Thanks for your time and help.

zyh

 

[Navin A S]

Yes, the velocity screw described in paper A is the same as the 6-D Spatial Vector defined by Roy Featherstone in the link you provided. In paper A, the rotation acts about a point which is the origin of the base coordinate system. This means that the distance vector P is expressed in terms of coordinates relative to the base frame. In paper B, however, the rotation acts about the origin of the moving frame (end effector). The vector r in equation (4.39) is the vector from o1 to p expressed in the orientations of the frame o0x0y0z0. This means that the distance vector is relative to the moving frame.