Tait-Bryan rotations
Named after Peter Guthrie Tait and George Bryan, they are three elemental rotations about each one of the principal axis of a body. For a craft moving in the positive x direction, with the right side corresponding to the positive y direction, and the vertical underside corresponding to the positive z direction, these three angles are individually called roll, pitch and yaw.
They can be used to describe a general rotation in three-dimensional Euclidean space using usually the order "once about the x-axis, once about the y-axis, and once about the z-axis". They are also called nautical rotations.
In aeronautical and aerospace engineering they are often called Euler angles, but this conflicts with existing usage elsewhere, because Tait-Bryan rotations have differences with Euler angles described below.
They are intrinsic rotations and the calculus behind them is similar to the Frenet-Serret formulas.

# Definition
The three critical flight dynamics parameters are rotations in three dimensions around the vehicle's coordinate system origin, the center of mass. These angles are pitch, roll and yaw:
- Pitch is rotation around the lateral or transverse axisan axis running from the pilot's left to right in  piloted aircraft, and parallel to the wings of a winged aircraft; thus the nose pitches up and the tail down, or vice-versa.
- Roll is rotation around the longitudinal axisan axis drawn through the body of the vehicle from tail to nose in the normal direction of flight, or the direction the pilot faces.
The roll angle is also known as bank angle on a fixed wing aircraft, which "banks" to change the horizontal direction of flight.
- The roll angle is also known as bank angle on a fixed wing aircraft, which "banks" to change the horizontal direction of flight.
- Yaw  is rotation about the vertical axisan axis drawn from top to bottom, and perpendicular to the other two axes.

# Composition of intrinsical rotations
To perform a rotation in an intrinsical reference frame is equivalent to right-multiply its characteristic matrix (the matrix that has the vector of the reference frame as columns) by the matrix of the rotation

## Proof
The composition of rotations in the fixed axes is a left-multiplication, because the usage of matrices as operators (left-multiply rotates all that is at the left of the operator). Suppose that you write the characteristic matrix of the frame as product of fixed axes rotations. Then, let x() and z() denote the rotations of angle  about the x-axis and z-axis, respectively. In the moving axes description, let Z()=z(), X() be the rotation of angle  about the once-rotated X-axis, and let Z() be the rotation of angle  about the twice-rotated Z-axis. Then:
Therefore rotations in the intrinsical frame can be performed right-multiplying the matrix of the frame, as we wanted to prove.
A simpler way to see this is changing the rotation operator in the intrinsical basis to the external basis. To change an operator to a basis given by a matrix P, we have the expresion R'=P1.R.P and we need its inverse R=P.R'.P1, and applying this to the rotation matrix we have P'=P.R.P1.P=P.R

# Differences and similarities with Euler angles
The main difference is that as rotation axis are not fixed, their position depends of the first rotation. This complicates calculus, but allow us to reach any final position with only two of the three elemental rotations. For example, a satellite could be stabilized with only two reaction wheels. The existing MAP Safehold/CSS (CSS:coarse Sun sensor) controller can work with only two wheels if the system momentum bias is small.
You can visualize why this can be achieved with an example. An aeroplane doesn't need to perform a yaw to turn. It is enough to make a roll. Then the lift on the wings will force a pitch upwards. At the end, the plane will perform a roll in the opposite direction of the previous one to get the wings horizontal. The whole maneouvre is equivalent to a yaw, but only pitch and roll were performed.
Other good example that any final position can be achieved with only two inertial wheels could be the following picture. With a wheel on the z axis and other in the y axis, we can prove that any position can be reached because starting with z over Z we can perform with a single inertial wheel the first and third Euler angles.
Tait-Bryan rotations, as any other intrinsic rotation, can be composed with each other with no limit. Nevertheless, some compositions of three of them are equivalent to the Euler angles, and share with them their properties. Euler angles therefore can be considered a particular application of Tait-Bryan rotations when the moving frame initial position is the same as the external reference frame and the order in which the rotations are applied is the proper one (if xyz are the reference frame and XYZ the moving frame, the first rotation (yaw) around Z leaves the line of nodes over y, so that the rotation around y (pitch) may be taken as equivalent to rotation about N)
As in a moving frame all this things are true only for an instant, we can only assert them in the limit when time goes to zero. Thus, in a frame co-moving with the rotating system, Euler angles are equivalent to a special combination of Tait-Bryan angles in the limit when delta time goes to zero,

# Applications
The main usage is in a part of flight dynamics, called attitude control, because the three angles can be controlled separately. If we correct small errors in yaw, roll and pitch individually, then we have achieved the nominal attitude of the aircraft. In case of a unmanned spacecraft, this can be performed automatically with a gyroscope and a reaction wheel controller in each axis.