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Named after Peter Guthrie Tait and George Bryan, Tait-Bryan angles are three angles used to describe a general rotation in three-dimensional Euclidean space by three successive elemental rotations around the axis of the moving frame in which they are defined. Usually the order once about the x-axis, once about the y-axis, and once about the z-axis.

They are also called Cardano angles or nautical angles. 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.

In aeronautical and aerospace engineering they are often called Euler angles, but this conflicts with existing usage elsewhere.

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 axis—an 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 axis—an 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.

• Yaw is rotation about the vertical axis—an axis drawn from top to bottom, and perpendicular to the other two axes.

We define Tait-Bryan angles as the rotations needed to get to the described frame, expresed in a moving frame that moves to its position. There are different ways to choose the angles.

The so-called "general convention" (yaw-pitch-roll), illustrated in figure, rotates about the three successive body-fixed axes. For example: The first rotation φ is about the Z₀-axis (parallel to the Z-axis), the second angle θ about the new X-axis (denoted as X₁) and finally the third angle ψ about the new Z-axis (Z₂).

The so-called "x-convention" (also called 3-1-3) is similar to Euler angle (Z,N,z) convention but with respect to the moving frame. The first rotation φ is about the Z-axis, then a second rotation θ about the X-axis and finally a third rotation ψ about the Z-axis (again). θ is usually restricted to . Notice that if we start with Z and z axis overlapping, it is exactly equivalent to Euler angles (Otherwise they are not equivalent)

A rotation represented by nautical angles with (φ,θ,ψ)=(−60◦,30◦,45◦) using the 3-1-3 general (co-moving axes) convention.

The same rotation alternatively expressed by (φ,θ,ψ)=(45◦,30◦,−60◦) using the 3-1-3 (fixed axes) x-convention.

The Tait-Bryan angles are equivalent to the Euler angles (with Z,N,x formalism, being N the line of nodes) when the moving frame initial position is the same as the external reference frame. If xyz are the reference frame and XYZ the moving frame, the first rotation (yaw) around Z leaves the line of nodes N equal to y, so that the rotation around y is equivalent to the pitch.

Therefore, in a frame comoving with the rotating system, Euler angles are equivalent to Tait-Bryan angles.

The main usage is in a part of flight dynamics, called attitude control, due that the three angles can be controlled separately. If we take yaw, pitch and roll independently to a nominal state, then we have reached the nominal attitude of the aircraft. In case of a unmaned spacecraft, this can be performed automatically with a gyroscope and an inertial wheel controller in each axis.

When studying rigid bodies, one calls the xyz system space coordinates, and the XYZ system body coordinates. Calculations are usually easiest in body coordinates, because then the moment of inertia tensor does not change in time. If one also diagonalizes the rigid body's moment of inertia tensor (with nine components, six of which are independent), then one has a set of coordinates (called the principal axes) in which the moment of inertia tensor has only three components. With this considerations, one reaches the Euler's equations.

The angular velocity, in body coordinates, of a rigid body takes a simple form using Tait-Bryan angles:

where IJK are unit vectors for XYZ.

• Yaw angle
• Euler angles
• flight dynamics
• attitude control

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