# Velocity screw description of a robot end-effector question

• zyh
In summary: The third formula combines the advantages of both and is often used for more complex robotic systems.In conclusion, both expressions are valid and serve different purposes. It is important to understand the underlying principles and tools behind each formula to fully grasp their meaning and applications. I hope this clears up your confusion and helps you in your research. Best of luck!
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:

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

thanks very much!

zyh

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

## 1. What is a velocity screw description of a robot end-effector?

A velocity screw description of a robot end-effector is a mathematical representation of the movement of the end-effector, which is the tool or device attached to the end of a robot arm. It describes the direction and magnitude of the end-effector's motion at any given point in time.

## 2. How is a velocity screw calculated?

A velocity screw is calculated by taking the cross product of a twist and a joint velocity vector. The twist represents the direction and magnitude of the end-effector's motion, while the joint velocity vector represents the speed at which each individual joint of the robot is moving.

## 3. What is the significance of using a velocity screw description?

A velocity screw description allows for a more efficient and accurate representation of the movement of a robot end-effector. It takes into account both the rotational and translational components of the end-effector's motion, making it a more comprehensive description.

## 4. How does a velocity screw description affect the robot's precision and accuracy?

Using a velocity screw description can improve a robot's precision and accuracy by providing a more detailed and precise representation of the end-effector's motion. This allows for better control and manipulation of the end-effector, resulting in more precise and accurate movements.

## 5. Are there any limitations to using a velocity screw description?

While a velocity screw description is a powerful tool for describing the movement of a robot end-effector, it does have some limitations. It assumes that the robot's joints are rigid and do not experience any friction or backlash, which may not always be the case in real-world scenarios.

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