Velocity screw description of a robot end-effector question

[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: [tex]\dot{P}=\omega\times a+b[/tex]
Here, the [itex]\omega[/itex] 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:
[tex]\dot{P}=\Omega\times P+T[/tex]
here, P has the coordinates in base frame. T and [itex]\Omega[/itex] is defined as the velocity screw.

Another expression:
[tex]\dot{P}=\omega\times(R\cdot^{1}P)+v[/tex]
here, Here, the R is the rotation matrix of end-effector frame wrt base frame, and [itex]^{1}P[/itex] is the vector wrt end-effector frame. [itex]\omega v[/itex] 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:
https://www.physicsforums.com/showthread.php?t=512697

Hope someone can give me some explanation on why there have two different formulas.

thanks very much!

zyh

 

[eljose79]

Dear zyh,

Thank you for sharing your thoughts and questions on this topic. As a fellow scientist, I understand your confusion and will do my best to explain the different expressions for calculating the velocity of a point on the robot end-effector.

Firstly, let's define what we mean by the end-effector. The end-effector is the part of the robot that interacts with the environment and performs tasks. It is usually located at the tip of the robot arm and can have various tools attached to it, such as a gripper or a sensor.

Now, let's take a closer look at the two different expressions you mentioned. The first one, \dot{P}=\omega\times a+b, is a general formula that describes the velocity of a point P attached to the end-effector. Here, \omega is the angular velocity of the end-effector and b is the translation velocity. This formula is commonly used in robotic kinematics, which is the study of the motion of robots.

The second expression, \dot{P}=\Omega\times P+T, is known as the velocity screw formula. It is derived from the screw theory, which is a mathematical tool used to describe the motion of rigid bodies in space. In this formula, \Omega is the twist or the velocity screw of the end-effector, and T is the translation velocity. The twist is a mathematical representation of the end-effector's motion, and it contains information about both the rotation and translation of the end-effector.

The third expression, \dot{P}=\omega\times(R\cdot^{1}P)+v, is a combination of the two previous expressions. Here, R is the rotation matrix that describes the orientation of the end-effector frame with respect to the base frame, and ^{1}P is the vector representing the coordinates of the point P in the end-effector frame. This expression is also derived from the screw theory and is commonly used in robotic dynamics, which is the study of the forces and torques that affect the motion of robots.

So, why do we have different expressions for calculating the velocity of a point on the end-effector? The reason is that each expression is derived from a different mathematical tool and is used for different purposes. The first formula is simple and intuitive, making it suitable for kinematic analysis. The second formula, on the other hand, provides a more comprehensive description of the end-effector's motion