How to Calculate the Rotation Axis Incorporating Translation?

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Discussion Overview

The discussion revolves around calculating the rotation axis of a body in 3D space as it moves from one position to another, incorporating both translation and rotation. Participants explore the mathematical and conceptual frameworks for understanding this motion, including the use of Rodriguez' rotation formula and the concept of the Instantaneous Centre of Revolution (ICR).

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant, Daniel, seeks to calculate the rotation axis during a small displacement from point A to point B, while incorporating translation components.
  • Another participant suggests that every displacement can be decomposed into translation and rotation, using the center of mass as a reference point.
  • Daniel acknowledges the decomposition but questions how to compute a single rotation axis that encompasses both translation and rotation.
  • Some participants argue against the idea that a movement could be described solely by a single rotation, suggesting that translation could imply rotation around an axis at an infinite distance.
  • There is a discussion about the implications of interpreting translation as a rotation at an infinite distance, with some participants expressing skepticism about its practical relevance.
  • One participant introduces the concept of the Instantaneous Centre of Revolution (ICR) and notes that finding the ICR for complex motions can lead to challenges in practical applications.
  • It is mentioned that the addition of constraints may simplify the problem of determining the rotation axis.

Areas of Agreement / Disagreement

Participants express differing views on whether a movement can be described solely by a single rotation axis. There is no consensus on the best approach to calculate the rotation axis when both translation and rotation are involved, and the discussion remains unresolved.

Contextual Notes

Participants highlight the complexity of determining the rotation axis in non-simple motions and the potential for significant changes in the location of the ICR with small movements. The discussion also touches on the implications of constraints in the motion.

Daniel Thuresson
I have a body in 3D-space and I would like to calculate the rotation axis when the body moves from A to B. I know the location (x, y and z) and the orientation (rx, ry and rz (axis angles)) at both A and B. The difference between A and B is small. The time instant during a dynamics simulation. However, if the small difference is a problem I can make it bigger (within the linear range of this specific simulation).

I have read about Rodriguez' rotation formula and know how to find the axis of rotation, but how can I incorporate the translation components as well? Any help is appreciated!

/Daniel
 
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Every displacement can be decomposed into a translation plus a rotation. For specificity, consider the center of mass of the body for a reference point. In the initial location, the CM has a particular position. In the second location, the CM has a second particular location. The translation from the first to the second is the required translation. Then the change in orientation is simply a rotation about the CM.
 
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Dr.D said:
Every displacement can be decomposed into a translation plus a rotation. For specificity, consider the center of mass of the body for a reference point. In the initial location, the CM has a particular position. In the second location, the CM has a second particular location. The translation from the first to the second is the required translation. Then the change in orientation is simply a rotation about the CM.

Hi,
I understand this, but the movement could also be described by a single rotation around some axis in space. I am wondering how I can compute this axis?

Rgds /Daniel
 
Daniel Thuresson said:
the movement could also be described by a single rotation around some axis in space

um,... I don't think so. Suppose the movement is nothing more than a simple translation with no rotation at all?
 
Dr.D said:
um,... I don't think so. Suppose the movement is nothing more than a simple translation with no rotation at all?

Would'nt translation correspond to a rotation around an axis at infinite distance? Anyway, I know that is not the case for me. The movement will be a combination of translation and rotation.
 
If rotation about an axis at an infinite distance means anything, then I suppose you could interpret it this way. In reality, I don't think it means anything at all, because we cannot realize it.
 
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Daniel Thuresson said:
Would'nt translation correspond to a rotation around an axis at infinite distance? Anyway, I know that is not the case for me. The movement will be a combination of translation and rotation.
Conceptually a rotation around an axis at an infinite distance may mean something, but I don't think you want to go in that direction -- especially if you have to do real calculations.
 
I think it may be interpreted: as r goes to ∞ the curvature approaches a straight line.
Anyways, this is not my question: Howto find the principal axis of rotation from a combination of rotation and translation?

/Daniel
 
(1) You are looking at the Instantaneous Centre of Revolution concept but in 3D instead of the 2D where it is more commonly used ?

Finding the ICR for any particular motion is just maths but for non simple motions and continuous motions it usually leads to a dead end as far as practical use and getting accurate answers is concerned .

For complex motions even a small movement of the body can change the location of the ICR by a large amount . In a general motion it can even flip from one side of a body to another or go to infinity and back again .

(2) The above is for an unconstrained motion .

Your actual problem may involve a partially constrained motion . Addition of constraints generally makes this type of problem easier to deal with .
 
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