Understanding Euler's Rotation equation for rigid bodies (Frames Of Reference)

$$\tau_b=I_n\dot\omega_b+\omega_b\times I_b\omega_b$$

Now in the above is Euler's famous rigid body rotation equation, in the body frame of reference ..... this does not make sense to me. How can a body have a rotation in its own frame of reference? For example:

The earth rotates right, we all know it does, and from my understanding of frame of reference, when you pick the earth's frame of reference, you adopt its rotational and translational motion, hence from your point of view, the earth is not rotating or moving (if not for the star's moon and sun, we won't know that we are rotating), so how is it that an object can have a rotation in its own frame of reference?

I think an objects rotation in its own frame of reference is zero.

• the rotation is relatively to inertial frame .
– Eli
Commented Mar 31, 2021 at 7:13

The rotation rate in the equation is the rotational velocity of the body relative to the non-rotating world as measured in a non-rotating frame instantaneously coincident and aligned with the body's frame.

I've recently posted a more detailed description of how this equation works on the Robotics Stack Exchange site,

https://robotics.stackexchange.com/a/21824

• Thanks for the reply, I saw it, but I was trying to understand the given information. "measured in a non-rotating frame instantaneously coincident and aligned with the body's frame." from what I understood in that statement, I will say that, this statement implies that there is a 3rd instantaneous frame of reference that is not rotating, right ? Commented Mar 28, 2021 at 1:00
• You may be thinking of “reference frames” and “coordinate frames” as being the same thing. They are, however different. The world and the body each have a reference frame; you can think of these as two sheets of paper, one stuck to the table, and one stuck to the body. We say the reference frames are rotating with respect to each other if lines in the two reference frames that are parallel to each other at one instant become nonparallel in the next instant. To measure this rotation, we can attach coordinate frames (specific sets of orthogonal lines) to the reference frames.
– RLH
Commented Mar 28, 2021 at 1:14
• For the formula in your question, we start by knowing the coordinate frame attached to the body reference frame, use it to identify the corresponding coordinate frame attached to the world reference frame. In this case we never actually use a “global coordinate frame” attached to the world reference frame (and this idea is tied to physics being invariant wrt translation / having no privileged choice of origin).
– RLH
Commented Mar 28, 2021 at 1:21
• thanks for the answer, i think i just need to understand coordinate systems and frames of references properly, any suggested resource would be helpful. Commented Apr 3, 2021 at 16:02