Which Coordinate System Should Be Used for Force in Torque Calculations?

AI Thread Summary
When calculating torque on a rigid body, it is essential to use a consistent coordinate system for both the position vector (r) and the force (f). The torque is defined as t = r X f, and using different systems can lead to confusion. The discussion highlights the importance of expressing all components in the same coordinate system, whether body-fixed or space-fixed. Additionally, a formula for torque in the body-fixed frame is presented, involving potential energy and quaternions, prompting a request for clarification on its derivation. Understanding these concepts is crucial for accurate torque calculations in rigid body dynamics.
alextex
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Hi!
I have been working on a rigid body subject quite a long. But till now there is an unresolved question for me. When we calculating the torque acting on the rigid body we use the following definition of the torque: t = r X f, X - is a cross product. So if I calculate the torque in a body-fixed system I use r in the body-fixed system too, but for force f - I am not sure. In which coordinate system should it be expressed - in the body-fixed or in space-fixed. Are this forces different in both systems? Sorry, for probably stupid questions. Thanks
 
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Sorry, I've found - we really need to do all in the same coordinate system.

But now, I have some other question - probably, more interesting. The torque acting on the rigid body (expressed in the body-fixed frame) is given by: t = -0.5*S(q)*(dU/dq), gde U - potential, q - is a quaternion, ans S(q) - is a matrix such that: dq/dt = 0.5*S(q)w, where w - is a 4-dimensional angular velocity. Can anybody explain how to get that formula for the torque?
 

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