# Tait-Bryan angles

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.

## 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)
Peter Guthrie Tait (April 28, 1831 - July 4, 1901) was a Scottish mathematical physicist, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with Kelvin.

## Early years

He was born at Dalkeith.
A rotation is a movement of an object in a circular motion.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
Gerolamo Cardano or Girolamo Cardano (English Jerome Cardan, Latin Hieronymus Cardanus; September 24, 1501 - September 21 1576) was a celebrated Italian Renaissance mathematician, physician, astrologer, and gambler.
Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body (a body in which the relative position of all its points is constant) in 3-dimensional Euclidean space.
dimension (Latin, "measured out") is a parameter or measurement required to define the characteristics of an object—i.e., length, width, and height or size and shape.
coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring.
origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.
center of mass of a system of particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated. The center of mass is a function only of the positions and masses of the particles that comprise the system.
Transverse may refer to:
• Transversality, a concept related to the intersection of manifolds in topology
• Transverse City, an album by Warren Zevon

A rotation is a movement of an object in a circular motion.
Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate.
A wing is an appendage used for flight by an animal or an apparatus used to create lift in aeronautics.

Wing or wings may also refer to:
• Insect wing, outgrowths of the insect exoskeleton that enable insects to fly

The term nose cone is used to refer to the forwardmost section of a rocket, guided missile or aircraft. The cone is shaped to offer minimum aerodynamic resistance. Nose cones are also designed for travel in and under water and in high speed land vehicles.
Empennage is an aviation term used to describe the tail portion of an aircraft. ("Empennage", "tail", and "tail assembly" may be interchangeably used.) The empennage gives stability to the aircraft and controls the flight dynamics: pitch and yaw.
The term, longitudinal means "along the major (or long) axis" as opposed to latitudinal which means "along the width", transverse, or across.
• In automotive engineering, a longitudinal engine

yaw angle is the angle between a vehicle's heading and a reference heading (normally true or magnetic North). One of the Tait-Bryan angles. In aeronautics, robotics and marine control, it is typically assigned the shorthand notation .
Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body (a body in which the relative position of all its points is constant) in 3-dimensional Euclidean space.
For the dynamics of flight, see Flight dynamics.

Rockwell Collins Flight Dynamics is a subdivision of the aerospace giant Rockwell Collins. They manufacture and develop heads-up displays for civilian applications.
Attitude control is control of the orientation of a spacecraft, or other flight vehicle, either relative to the celestial sphere or to a gravitating body influencing its flight path.
A gyroscope is a device for measuring or maintaining orientation, based on the principle of conservation of angular momentum. The device is a spinning wheel whose axle is free to take any orientation.
Euler's equations describe the rotation of a rigid body in a frame of reference fixed in the rotating body

where are the applied torques, are the principal moments of inertia and are the components of the angular velocity vector along the principal
angular velocity is a vector quantity (more precisely, a pseudovector) which specifies the angular speed at which an object is rotating along with the direction in which it is rotating.
yaw angle is the angle between a vehicle's heading and a reference heading (normally true or magnetic North). One of the Tait-Bryan angles. In aeronautics, robotics and marine control, it is typically assigned the shorthand notation .
Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body (a body in which the relative position of all its points is constant) in 3-dimensional Euclidean space.
For the dynamics of flight, see Flight dynamics.

Rockwell Collins Flight Dynamics is a subdivision of the aerospace giant Rockwell Collins. They manufacture and develop heads-up displays for civilian applications.
Attitude control is control of the orientation of a spacecraft, or other flight vehicle, either relative to the celestial sphere or to a gravitating body influencing its flight path.