# How does bank angle differ from roll angle with respect to stability in fixed-wing aircraft?

I would like a clarification on bank angle and how its different from roll angle with respect to to fixed wing aircraft. It is my understanding that the bank angle is a result of rotating the aircraft body to the stability frame, implying that if the angle of attack $\alpha$ and the side slip angle $\beta$ are zero then, and only then the bank and angle and roll angle are the same. Based on Stevens and Lewis Aircraft Control and Simulation (which doesn't define bank angle) the rotation from body to stability(wind) frame is given by $$C_{w-b} = \begin{bmatrix} &\cos\alpha \cos\beta & \sin\beta & \sin\alpha cos\beta \\ &-\cos\alpha\sin\beta & \cos\beta & -\sin\alpha\sin\beta\\ &-\sin\alpha & 0 & \cos\alpha \end{bmatrix}$$

Therefore, the bank angle $\mu$ I think is given by: $$\mu \triangleq \begin{bmatrix} &\cos\alpha \cos\beta & \sin\beta & \sin\alpha \cos\beta \end{bmatrix} \begin{bmatrix} \phi \\ \theta \\ \psi\end{bmatrix}$$

where $\phi,\, \theta,\, \psi$ are standard aerospace Euler angles defined based on a 3-2-1 rotation. My specific questions are:

1) is my understanding and calculation of bank angle correct? If not can someone point me to a good resource for this. I was surprised to be able to find a clear definition and formula quickly.

2) by multiplying the second and third row of $C_{w-b}$ with Euler angles we obtain a set of other two angles relative to stability frame. Do these angles have any names or specific role in aerospace dynamics/control?

I've only ever heard of one meaning of the term "bank angle", and I've never heard "roll angle" used to mean anything different from "bank angle".

For completeness, here are some "plain English" definitions of all three Euler angles:

Pitch is the angle that the nose (the longitudinal axis) forms with the horizon. If the nose is pointed at the horizon, the pitch is 0°; if it's pointed straight up, the pitch is 90°; and if it's pointed straight down, the pitch is -90°.

Heading is the horizontal direction that the nose is pointing, measured clockwise from true north. (Video game developers sometimes refer to the heading as the "yaw angle".)

Bank angle (sometimes called "roll angle") is the amount that the aircraft would have to roll (rotate around its longitudinal axis) in order to bring the wings level (which is to say, bring the lateral axis parallel with the horizon), with the top side of the aircraft facing above rather than below the horizon.

Pitch ranges from -90 to 90. The other two angles have a range consisting of a full circle; I think bank is usually expressed as a number from -180 to 180, and heading as a number from 0 to 360.

You may find more in the Wikipedia Flight Dynamcs page, though I have some heartburn with it. The names for the earth to wind transform are bank angle, flight path angle and heading angle. I don't have the Cw-b matrix handy now but have the Cb-w from an old reference: Dynamics of Atmospheric flight by Bernard Etkin 1972, pg 117  Cbw cos a cos b -cos a sin b -sin a sin b cos b 0 sin a cos b -sin a sin b cos a  I don't have Matlab installed or I would invert it for you.

If you are simulating an aircraft you should be aware that the Euler angle rotations have a problem and for certain sets of rotations a singularity will arise causing an instantaneous pi/2 change in angles. This typically only is a problem for tumbling spacecraft or wild aerobatics. If interested, you should study Euler symmetric or Euler–Rodrigues parameters and Hamilton quaternions.

Most of my references are too old but may be of interest. For basic flight dynamics: Dynamics of Atmospheric flight by Bernard Etkin 1972 Airplane Performance Stability and Control, Perkins and Hage 1949

These are of less interest for the full nonlinear dynamic problem:  Aerodyamics for Naval Aviators 1965 Theory of Flight R. Von Mises 1959 Automatic Control of Aircraft and Missiles, J Blakelock 1965 Aerodynamics for Engineering Students by Houghton & Brock 1972 Advanced Pilot's Flight Manual by W. Kershner 1977 Aircraft Dynamics and Automatic Control, McRuer Ashkenas & Graham, 1973 

Happy Trails

This is one of semantics. Bank is a state, whereas roll is an action; e.g., “The pilot rolled into a steep turn and held a 50° bank angle.”