# Screw theory and Chasles' theorem

• Liferider
In summary, Chasles' theorem or Mozzi's theorem states that any rigid body motion can be produced by a translation along a line and rotation about the same line, and vice versa. However, I find this hard to believe. Consider a body falling towards the ground while also rotating about an axis parallel to the ground and passing through the objects center of mass. How is it possible to represent this motion in screw-terms?I believe I am thinking of the translational displacement as along, say z-axis towards the ground and then expect the rotation to be around this axis (for Chasles' theorem to be corrrect), and of course that is not always the case... But, is there some other line that can represent
Liferider
I am currently working on my masters and happened to stumble upon screw theory, of which I have no previous experience with. Fundemental to screw theory is Chasles' theorem or Mozzi's theorem that states that any rigid body motion can be produced by a translation along a line and rotation about the same line, and vice versa. However, I find this hard to believe. Consider a body falling towards the ground while also rotating about an axis parallel to the ground and passing through the objects center of mass. How is it possible to represent this motion in screw-terms?

I believe I am thinking of the translational displacement as along, say z-axis towards the ground and then expect the rotation to be around this axis (for Chasles' theorem to be corrrect), and of course that is not always the case... But, is there some other line that can represent the translation, where the rotation axis can also be placed?

Question 1:
Does a line through the object have to lie on the axis of translation and rotation? Or can the trans./rot. axis lie outside the object (relative to an inertial frame)?

Question 2:
Which coordinate systems do I have to consider in order to understand the notions of twists, wrenches and screws?

Try this. I found it fascinating.

Philip Wood said:

Try this. I found it fascinating.
Thanks, I think I learned something just there. Still not quite there though, will try to explain when I truly understand and feel confident enough.

Your example can be described with just a rotation around an axis parallel to the ground (way out). as the object accelerates (free fall), the location of the rotation axis changes.

dauto said:
Your example can be described with just a rotation around an axis parallel to the ground (way out). as the object accelerates (free fall), the location of the rotation axis changes.
That was my conclusion as well, in the end.. Now that I realize that the theorem actually have two versions, the infintesimal one and the configuration-to-configuration version, the theorem seems pretty obvious to me. Following the different proofs out there comes naturally. I thank you both for explaining this to me.

## 1. What is screw theory?

Screw theory is a mathematical framework used to describe and analyze the motion of rigid bodies in space. It uses the concept of a "screw" which is a combination of a rotational and translational motion along a common axis, to represent the movement of a rigid body.

## 2. What is the significance of screw theory?

Screw theory is significant because it provides a unified approach to describe the motion of both single and multiple rigid bodies. It also allows for the analysis of complex mechanical systems and the prediction of their behavior.

## 3. What is Chasles' theorem in screw theory?

Chasles' theorem states that any rigid body motion can be represented as a single screw displacement along a unique screw axis. This means that the motion of a rigid body can be described by a single "screw" instead of multiple individual motions.

## 4. How is screw theory used in robotics?

Screw theory is used in robotics to model the motion of robot manipulators, which are complex mechanical systems made up of multiple rigid bodies. It allows for the analysis and optimization of robot movements, making them more efficient and accurate.

## 5. What are some real-world applications of screw theory?

Screw theory has a wide range of applications in fields such as robotics, biomechanics, aerospace engineering, and computer graphics. It is used to design and analyze mechanisms, study the movement of human joints, and simulate the motion of objects in computer-generated animations.

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