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 is 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, because Tait-Bryan angles are only a sub-set of the ways Euler angles may be used to define the relative orientation of two coordinate systems.

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 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.

Other conventions

We define Tait-Bryan angles as the rotations needed to get to the target attitude, expressed in a moving frame that moves to its desired orientation. 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. that is: The first rotation is about the -axis (parallel to the -axis), the second angle about the new -axis (denoted as ) and finally the third angle about the new -axis ().

The so-called "x-convention" (also called 3-1-3) is similar to the Euler angle (Z,N,z) convention but with respect to the moving frame. The first rotation is about the -axis, then a second rotation about the -axis and finally a third rotation again about the -axis. is usually restricted to . Notice that if, and only if, we start with Z and z axis coinciding, this is exactly equivalent to Euler angle use.

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.

Relationship with Euler angles

The Tait-Bryan angles are equivalent to the Euler angles (with Z,N,x formalism, (N being 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 undefined, so that the rotation around y (pitch) may be taken as equivalent to rotation about N.

Thus, in a frame co-moving with the rotating system, Euler angles are shown to be equivalent to Tait-Bryan angles.

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 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 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.

See also

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

References

• Wright Air Development Center Technical Report 58-17: On The Use of Quaternions In Simulation of Rigid Body Motion, Dec. 1958 by Alfred C. Robinson (Appendix B)