# Help on velocity screw description and velocity propagation

• zyh
In summary: Both equations have the same form, but the variables have different meanings. In summary, there are two different ways of calculating a point's velocity in robotics: one described in paper A and the other in paper B. Both methods use the concept of rotation, but the point of rotation is different in each case. Additionally, the velocity screw in paper A is equivalent to the 6-D Spatial Vector defined by Roy Featherstone.
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

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.

## 1. What is a velocity screw in robotics?

A velocity screw is a mathematical representation of the instantaneous motion of a rigid body. It is a six-dimensional vector that describes both the translational and rotational components of the body's velocity.

## 2. How is velocity screw described?

A velocity screw is typically described using the screw axis, which is a line in space that indicates the direction and magnitude of the velocity. It also includes a pitch, which is the amount of rotational motion around the screw axis per unit of translation.

## 3. What is the purpose of velocity propagation?

Velocity propagation is used in robotics to calculate the velocity of a robot's end-effector based on the velocity of its joints. This allows for precise control and coordination of movements.

## 4. How is velocity screw related to kinematics?

Velocity screw is a fundamental concept in kinematics, which is the study of motion without considering the forces that cause it. It is used to describe the motion of rigid bodies in space and is essential for analyzing the movements of robots.

## 5. Can velocity screws be used for non-rigid bodies?

No, velocity screws are specific to rigid bodies, which are objects that do not deform under external forces. They cannot be used for non-rigid bodies, such as fluids or soft materials.

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