What is the relation between roll angle and pitch angle?

Question

I was wondering about roll angle and pitch angle values. Will I get same pitch angle at different roll angles and vice-versa?

For example. At its initial position, the aircraft is standing on the runway). I got roll and pitch values from a sensor. Lets say X and Y resp. Now, when aircraft is taking off, it is obvious that it will pitch up so my pitch angle will change to lets say Y1. Now if I look for roll Value, will it be same 'X' or it will be different?

Answer 1

Roll, pitch and yaw are rotations about the principle body axis of the aircraft. Roll angle, pitch angle and yaw angle together describe the attitude of an aircraft.

The principle body axes are:

• X-axis is the longitudinal axis pointing out the nose of the aircraft. Rotation about the x-axis is called roll. Roll rate is denoted $p$, roll angle is denoted $\phi$
• Y-axis is the lateral axis pointing out the right wing. Rotation about the Y-axis is called pitch. Pitch rate is denoted $q$, pitch angle is denoted $\theta$
• Z-axis is the vertical axis pointing down. Rotation about the Z-axis is call yaw. Yaw rate is denoted $r$, yaw angle is denoted $\psi$

Initially when all angles are 0, the aircraft is wings level, nose pointing at the horizon and heading north.

The angles $\phi, \theta, \psi$ describe the attitude with respect to this initial positions. They are called Euler angles

The rotations are applied in backwards order, first yaw ($\psi$), then pitch ($\theta$) and finally roll ($\phi$).

This describes almost every possible attitude uniquely, unless the pitch angle is +/- 90 degrees. Then roll and yaw will become ambiguous.

Now back to your question. Initially on the runway the roll angle of the aircraft is 0 degrees. If the aircraft pitches up ($q$ becomes positive for a while), the roll angle will not change.

However, if initially the roll angle was non-zero, the roll angle will be affected by a pitch-up manoeuvre. You can easily visualise this with your hands.

Open your right hand with the palm facing up, you thumb at a right angle to the fingers. The middle finger is the X-axis, the thumb is the Y axis.

Now roll your hand 45 degrees counter clockwise, keeping the fingers pointed at the horizon. Your thumb will now point 45 degrees into the air. This was a pure roll manoeuvre, the pitch angle is still 0 degrees.

Now flap your hand 90 degrees up using your wrist, this is a pitch up manoeuvre since you rotate about your thumb axis which is the Y axis.

Now you see that your fingers are all in the same vertical plane, pointing 45 degrees up to your left. If you describe this position with respect to the initial position, the pitch angle is 45 degrees, and the roll angle is 90 degrees. So pitching 90 degrees resulted only in 45 degrees of pitch, but also in 45 degrees of extra roll.

Because the roll-angle was non-zero before we started to pitch, the pitch rotation affect the roll-angle.

If your sensor is not aligned with the body axes of the aircraft, you will see a lot of cross-effects between the angles during a take-off.

The mathematical relation is well described on this page, which is also the source of the images above.