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

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SUMMARY

The discussion centers on the calculation of torque in rigid body dynamics, specifically addressing the coordinate systems used for force representation. The torque is defined as t = r X f, where r is the position vector and f is the force vector. It is established that both vectors must be expressed in the same coordinate system, either body-fixed or space-fixed, to ensure accurate calculations. Additionally, a formula for torque in the body-fixed frame is presented as t = -0.5*S(q)*(dU/dq), where S(q) is a matrix related to angular velocity and U is the potential energy.

PREREQUISITES
  • Understanding of rigid body dynamics
  • Familiarity with cross products in vector mathematics
  • Knowledge of quaternions and their applications in 3D rotations
  • Basic principles of potential energy in physics
NEXT STEPS
  • Study the application of quaternions in rigid body motion
  • Learn about the derivation of torque formulas in different coordinate systems
  • Explore the properties and applications of the S(q) matrix in angular velocity calculations
  • Investigate the relationship between potential energy and torque in mechanical systems
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Physics students, mechanical engineers, and researchers in robotics or aerospace engineering who are involved in torque calculations and 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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