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?


7 Answers 7


In my experience, "bank angle" and "roll angle" mean exactly the same thing. I've only ever heard of one meaning of either term.

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 (sometimes called "yaw angle") is the horizontal direction that the nose is pointing, measured clockwise from true north.

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.

  • $\begingroup$ How would you express inverted flight in terms of pitch? Shouldn't that be a pitch angle near 180°? $\endgroup$ Dec 5, 2019 at 6:03
  • $\begingroup$ @PeterKämpf Nope, inverted flight is a bank angle greater than 90°. If you're flying completely upside down with the nose pointed at the horizon, then the pitch is 0° and the bank angle is 180°. $\endgroup$ Dec 5, 2019 at 12:37

As an industry standard and in most circumstances, when we say roll or bank, we are referring to body-axis roll. That is, the roll rate, $p$, is the y-axis component of the angular velocity, $\vec{\omega}=\begin{bmatrix}p & q & r\end{bmatrix}^T$, and the roll angle, $\phi$, is the 2nd rotation angle in the 3-2-1 Euler rotation sequence from the ground axis to the body axis. Also, in this parlance, roll angle and bank angle are completely synonymous.

What the OP is referring to is called stability-axis roll. That is, the roll motion is expressed in the stability-axis (or if a sideslip is present, in the wind-axis). We can easily transform the wind-axis angular velocity from the body-axis via:

$$\begin{bmatrix}p_w \\ q_w \\ r_w\end{bmatrix} = \begin{bmatrix} &\cos\alpha \cos\beta & \sin\beta & \sin\alpha \cos\beta \\ &-\cos\alpha\sin\beta & \cos\beta & -\sin\alpha\sin\beta\\ &-\sin\alpha & 0 & \cos\alpha \end{bmatrix} \begin{bmatrix}p \\ q \\ r\end{bmatrix}$$

Or just in the roll axis (quite similar to what's written in the OP, but in roll rate only):

$$p_w = \begin{bmatrix}\cos\alpha \cos\beta & \sin\beta & \sin\alpha \cos\beta\end{bmatrix} \begin{bmatrix}p \\ q \\ r\end{bmatrix}$$

A pure wind-axis roll is enticing because it will not induce any change in sideslip, $\beta$, therefore not creating any adverse yaw or aerodynamic roll-yaw coupling. Of course, pure wind-axis roll is not natural and can only be realized through pilot coordination or active feedback control.

We can also define a wind-axis Euler roll angle (or bank angle), maybe call it $\mu$. But that expression is significantly more complex than the angular velocity expression above. So, no, your expression for $\mu$ is incorrect. If that's your desire, then the expression you should consider is:

$$\textbf{C}_{WE}(\mu, \theta_w, \psi_w) = \textbf{C}_{WB}(\alpha,\beta)\textbf{C}_{BE}(\phi,\theta,\psi)$$


I've just been trying to answer a very similar question, so I'll post what I've found. It's a matter of definition what you describe as bank angle and what as roll angle and they are frequently used interchangeably. However, I have come across instances where they are defined as follows:

Roll Angle: Rotation about the aircraft body x-axis (this is the angle phi in the Euler set psi,theta,phi)

Bank Angle: Rotation about the velocity vector (this is the angle mu in the wind axes set mu, alpha, beta)

If you're working in body axes, you would use phi, theta, psi and if you are working in wind axes (as mentioned in the question), you would use mu, alpha, beta. As an example, the paper AGARD-CP-235 29 defines the two angles as follows: "The term bank angle (mu) is used to indicate rotation about the velocity vector while the term roll angle (phi) is used to indicate rotation about the aircraft x-axis"

mu and phi in these definitions are, in general, not the same.

I don't think you can use the transformation matrix as described in the question to transform angles. This works for vectors, so you need to transform a vector, then work out the angle you want from that.

If phi, theta and psi are the Euler angles taking the aircraft from aligned with the velocity vector to an arbitrary attitude, I derived the following to calculate mu.


I think this may be what the questioner is asking for.

I also have the equations for alpha and beta.

I can supply the equations for alpha and beta and the derivation if anyone is interested.

So, to directly answer the questions

  1. Not sure if your understanding is correct but I've given my understanding here. Note I have always distinguished between stability and wind axes and I'm talking about what I call wind axes here. I've supplied the equation I think you need above.

  2. Not answered the second question but I believe it is not valid to use the transformation matrix on angles in the way suggested.


Roll is often used as a verb describing rotation around the longitudinal axis of the aircraft (at least with pilots, especially flight instructors). If the aircraft has zero pitch elevation angle, then bank angle could be determined by the amount of roll that occurred from wings level. When at non-zero pitch elevation angle, the rolling movement is used to establish a bank angle, but roll change and bank change may not be the same. When pitched up at +90 degrees, or down at -90 degrees, then bank is no longer meaningful, but the aircraft can still roll around the longitudinal axis.

When at a pitch elevation angle that is not 90 or -90 degrees, bank is the angle between the aircraft lateral axis and what could be called the "wings level vector", typically positive is considered right wing down. The wings level vector is the cross product of the local longitudinal axis and world Z axis (a Z axis where positive is upwards).

Consider the following image:enter image description here

Near the center is a semi-transparent aircraft. The square pink column represents the longitudinal X axis, the dark green column is the Y axis out the left wing, and the cyan column is the aircraft's vertical or normal axis. (Often in aerospace engineering, the green axis is out the right wing and the vertical axis points downwards. Not in this case.) The thinner and lighter green line out the right wing represents the reverse of the green Y axis, representing the lateral axis. These axes are in a local coordinate system for the aircraft. The orange line is the product of the pink longitudinal axis and black world vertical axis. Bank angle is the angle between the lighter green lateral axis and the orange wings level axis.

At pitch elevation angles of +90 or -90 degrees, the cross product of lateral and wings level vectors is zero, and bank is meaningless. (I had a URL reference for this at one time that was a snippet of source code, but I can't find it at this time).


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.”

  • 1
    $\begingroup$ "To roll" describes a motion. Roll Angle describes a state. Roll speed describes a motion. $\endgroup$
    – Koyovis
    Dec 6, 2019 at 5:35

Plane coordinate systems are typically expressed in three Axes, Longitudinal, lateral, and vertical. Longitudinal is nose to tail, lateral is wingtip to wingtip, Vertical is straight up.

Primary controls effect rotation about the axes. Primary controls are rudder, elevator and aileron.

Gyro instruments such as attitude indicator measure pitch and bank. https://www.faa.gov/regulations_policies/handbooks_manuals/aviation/phak/media/10_phak_ch8.pdf That is rotation about the lateral and longitudinal axis.

The turn and slip indicator shows the rate of turn in degrees per second. The slip indicator shows the roll rate. Thus, it appears to this author that the turn and slip indicator provides an output that is the derivative of Attitude indicator.

Roll is the derivative of bank. Both are about the Longitudinal access. At least that’s how I read the FAA official description of these instruments. This is because the turn coordinator only shows the rate of turn and rate of roll in degrees per second. There is also a non-Gyro instrument Called the inclinometer. It is a curved tube with a ball inside. It is a tristate system in the sense that it is either coordinated, showing slip, or showing skid. This is a measure of rotation about the vertical axis, also known as yaw. Slip occurs when the nose is pointing in the opposite direction of the bank.

Cross controls result in a side slip that may enable you to straighten your aircraft during a crosswind landing. Dip the wing into the wind and use the rudder to align the nose with the centerline. This puts less stress on the landing gear than attempting to land while crabbing into the wind. The forward slip is a maneuver typically used for short field landing’s.

It enables you to lose altitude without increasing speed. This is accomplished by introducing drag. This is because we are showing the side of the airplane to the wind and the vigorous slip serves as a dive break. Over correct the rudder in the other direction and you have a skid. That is, nose to the wind and wing dipped to the wind. Now the lowered wing runs the risk of stalling first. This can lead to a dangerous spin.


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

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

  • $\begingroup$ please use quoteblocks for quotes, not codeblocks $\endgroup$
    – Federico
    Aug 18, 2020 at 11:17

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