# Body angle rate and body angular velocity

Are these terms used interchangeably to mean the same things?

I'm aware of the conversion between body rates (often, $$[p, q, r])$$ and Euler rates (commonly, [$$\dot\phi, \dot\theta, \dot\psi])$$. This answer suggests that Euler rates are inertial angular rates. As written in this answer, it would appear that 'body-frame angle rate' is the same as 'body-frame angular velocity'. I expected angular velocity (as a 3-vector, $$\vec\omega =[\omega_1, \omega_2, \omega_3]$$) to be a different quantity -- referring to wiki for 3D angular velocity.

Is angular velocity (expressed in body frame) the same as angle rate-of-change (expressed in the same body frame)?

In the industry, the elements of the angular velocity vector are called body rates, that is, each element of $$\vec{\omega}=[p,q,r]$$ is called roll rate, pitch rate and yaw rate.

However, they are decidedly not mathematical rates from the perspective that they are derivatives of some quantities. Unless it's a single axis rotation, you can't straight-up integrate the angular velocity in the hopes of getting something meaningful.

I've had an academic professor telling me a mouthful for calling angular velocity "body rates", but alas, that's the industry norm.

• Thanks for this perspective -- this has confused me for a while! Could you clarify if these quantities are numerically the same? That is, if I know the reported angular velocity, $\vec\omega$, of a body, do its three elements really correspond to the rate of change of the three angles? Or is this generally too implementation/sensor specific?
– WesT
Apr 19, 2022 at 7:53
– Jim
Apr 19, 2022 at 11:17
• @WesT No, they do not! That's the whole point! The angular velocity is different than the rates of the attitudes.
– JZYL
Apr 19, 2022 at 13:12
• @JZYL I know; sorry, I should've phrased it differently: I wanted to know if the vector $[p, q, r]$ (body rates reported from a gyro, say) is indeed the angular velocity vector; that is, it numerically obeys the definition of angular velocity (have a direction perpendicular to the plane of rotation and a magnitude corresponding to the speed of rotation). I think I glean from your answer that it does.
– WesT
Apr 19, 2022 at 17:33