Convert Rotation Velocity Respect from a Certain Coordinate Sytem into a Rotation Velocity Respect from another Coordinate Sytem

Question

So I have a Case:

1. I have two 3D Coordinate Sytems:
   • Real World(Rigid) Coordinate System ~ the Origin of all objects.
   • Certain Object's Local Coordinate System ~ since the object can move or rotate, this 3D coordinate system changes its position and orientations.
2. An Object is found at the Origin(0.0,0.0,0.0).
3. The Variables are:
   • α, β, γ = Pitch(X-Axis), Roll(Y-Axis), Yaw(Z-Axis) ~ Intrinsic Rotations of the Object's Local Coordinate System, from the Real World Coordinate System.
   • u, v, w = rotation velocity of the Object Relative to the Real World Coordinate System.
   • x, y, z = rotation velocity of the Object Relative to the Object's Local Coordinate System.
4. The First Trial was:
   • When the two coordinate systems are in phase(α = 0.0, β = 0.0, γ = 0.0)
   • Then the rotation velocities of the object on both coordinate systems are equal(x = u, y = v, z = w)
5. The Second Trial was:
   • When Object's Local Coordinate System was rotated(α = 131.42, β = -36.7, γ = 317.22)
   • And u = 14.2, v = 5.2, w = -1.32
   • Then x = ?, y = ?, z = ?

I am trying to get the Rotation Velocity of the Object relative to the its Local Coordinate System. I will appreciate the help of the community!

Answer 1

It's worth remembering at this point that the instantaneous rotational velocity of a rigid three-dimensional object is always a rotational speed around a single axis.

If you have two non-rotating frames of reference, the rotational speed is the same in both frames, so you need merely convert the coordinates of the directional vector of the axis from one frame to another.

Since the roll, pitch, and yaw axes are orthogonal, you can combine and decompose the rotational velocities as if they were vectors; that is, the rotation rates $u,$ $v,$ and $w$ around the orthogonal axes combine into a single rotation of magnitude $\sqrt{u^2 + v^2 + w^2}$ around the axis parallel to the vector $(u,v,w).$ In fact, you might as well just use the vector $(u,v,w)$ as a representation of the rotational velocity. Convert this to the other coordinate system and decompose it into its components along that system's orthogonal axes.

If one rotational system is rotating then you have to think about how you're measuring rotation.