What is the difference between Body Pitch Rate and Inertial Pitch Rate?

Question

ARINC 429 specifies two series of attitude rate labels for ADIRS/AHRS/IRS systems:

Body

• 326 : Body Pitch Rate
• 327 : Body Roll Rate
• 330 : Body Yaw Rate

Inertial

• 336 : Inertial Pitch Rate
• 337 : Inertial Roll Rate
• 340 : Inertial Yaw Rate

What are the differences between these two sets of values? I'm assuming the Body labels are in reference to the airframe, but do the Inertial labels use a different reference?

Answer 1

I'm not 100% sure about ARINC definitions specifically, but in the general aeronautics usage the inertial frame is the ground (flat Earth) frame. The rotation from this inertial ground frame to the body frame is defined by what is known as Euler angles.

Consequently, the inertial angular rates are Euler rates: the rates at which the Euler angles change.

In the body frame, pitch, for example, is what elevator controls, 'up/down' from the point of view of the pilot. Technically, this is pitch motion but not exactly pitch angle control. Euler pitch, the pitch angle per se, is the angle between the horizontal plane and the aircraft body X axis. This angle has a different rate to the body pitch. In an extreme case, on a knife edge (90° roll, zero pitch), body yaw controls pitch angle, and so body yaw rate will match Euler (inertial) pitch rate, and vice versa (minding the sign).

Remember aerospace Euler rotations are in the yaw-pitch-roll order.

Note you cannot rotate angular rates from one frame to another using the same rotation matrix as you use for linear quantities. You need to use the matrix that keeps momentum:

$$ \begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix} = \begin{bmatrix} 1 & sin(\phi)tan(\theta) & cos(\phi)tan(\theta) \\ 0 & cos(\phi) & -sin(\phi) \\ 0 & sin(\phi) / cos(\theta) & cos(\phi) / cos(\theta) \end{bmatrix} \begin{bmatrix} P \\ Q \\ R \end{bmatrix} $$

where $\phi, \theta, \psi$ are Euler roll, pitch, yaw; and P, Q, R are body roll, pitch and yaw rates. The aircraft rate sensors measure the latter, so typically you need conversion this way. But for reverse conversion you need to invert this matrix: transposition is not correct for it!

Note this matrix has singularity at pitch ±90°, and indeed, yaw is undefined at such pitch angles. But there are ways to avoid it, e.g. to integrate the whole matrix at every step, or to use quaternions.