# Tait-Bryan rotations: Wikis

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# Encyclopedia

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Yaw, pitch, and roll, also known as Tait–Bryan angles, named after Peter Guthrie Tait and George Bryan, are a specific kind of Euler angles very often used in aerospace applications to define the relative orientation of a vehicle respect a reference frame. The three angles specified in this formulation are defined as the roll angle, pitch angle, and yaw angle.

## Mathematical definition

Tait-Bryan angles statically defined. ZXY convention

These angles are called Tait-Bryan angles in mathematics. They can be statically defined using a line of nodes constructed by the intersection of two non-homologous planes (for example XZ and xy are not homologous planes), unlike proper Euler angles which use homologous planes (for example XZ and xz)[1].

This second kind of Euler angles is such as it is equivalent to three rotations composed with a different axis. z-y-x for example. There are therefore six possibilities of this kind (xyz, xzy, zxy, zyx, yzx, yxz). They behave slightly differently than Euler angles. In the zyx case, the two first rotations determine the line of nodes and the axis x, and the third rotation is around the axis x.

Because the line of nodes is the intersection of two non-homologous planes the pitch angle is meassured from the horizontal plane instead of the vertical axis. Therefore this kind of constructions would give a pitch equal to zero for an airplane flying horizontal while the first kind of Euler angles would assign it an angle of π/2.

## Local attitude description

### Aircraft attitude

Yaw, pitch and roll angles for an aircraft. Fixed frame xyz has been moved backwards from center of gravity (preserving angles) for clarity. Axes Y and Z are not shown. The convention used here for axis definition would give the name z-y-x to the convention of angles shown

Yaw, pitch and roll are used in aerospace to define a rotation between a reference axis system and a vehicle-fixed axis system.

Consider an aircraft-body coordinate system with axes XYZ (sometimes named roll, pitch and yaw axes, though these names will not be used in this article) which is fixed to the vehicle, rotating and translating with it. This intrinsic frame of the vehicle, XYZ system, is oriented such that the X-axis points forward along some convenient reference line along the body, the Y-axis points to the right of the vehicle along the wing, and the Z-axis points downward to form an orthogonal right-handed system.

Consider a coordinate system xyz, aligned having x pointing in the direction of true north, y pointing to true east, and the z-axis pointing down, normal to the local horizontal direction.

Given this setting, the rotation sequence from xyz to XYZ is specified by and defines the angles yaw, pitch and roll[2][3][4] as follows:

1. right-handed rotation $\psi \in (-180, 180]$ about the z-axis by the yaw angle
2. right-handed rotation $\theta \in [-90, 90]$ about the new (once-rotated) y-axis by the pitch angle
3. right-handed rotation $\phi \in (-180, 180]$ about the new (twice-rotated) x-axis by the roll angle

Readers wishing to see a matrix representation of the conversion from the frame xyz to XYZ are advised to read the article on Rotation matrices and to form the matrix Rx(φ)Ry( − θ)Rz( − ψ). In order to convert a point in xyz to XYZ coordinates, one applies the matrix Rz(ψ)Ry(θ)Rx( − φ) to the point.

The motion of an aircraft is often described in terms of rotation about these axes, so rotation about the X-axis is called rolling, rotation about the Y-axis is called pitching, and rotation about the Z-axis is called yawing. As well as aircraft, the same terminology is used for the motion of ships and boats. For locomotives the motions are called nosing, pitch and roll.

### Robotics

Industrial robot operating in a foundry.

These three angles are also used in robotics for speaking about the degrees of freedom of a wrist. It is also used in Electronic stability control in a similar way.

As with proper Euler angles, gimbal lock can appear. The importance of non-singularities in robotics has led the American National Standard for Industrial Robots and Robot Systems — Safety Requirements to define it as “a condition caused by the collinear alignment of two or more robot axes resulting in unpredictable robot motion and velocities”. (ANSI/RIA R15.06-1999)

It is common to use a "triple-roll wrist" in robot arms. This is a wrist about which the three axes of the wrist, controlling yaw, pitch, and roll, all pass through a common point.

An example of a wrist singularity is when the path through which the robot is traveling causes the first and third axes of the robot's wrist to line up. The second wrist axis then attempts to spin 180° in zero time to maintain the orientation of the end effector. Another common term for this singularity is a “wrist flip”. The result of a singularity can be quite dramatic and can have adverse effects on the robot arm, the end effector, and the process.

Representation of the earth with parallels and meridians

In maritime navigation only the yaw angle is important. In fact, the word has a nautical origin, with the meaning of "bending out of the course". Etymologically, it is related with the verb 'to go'[5]. It is typically assigned the shorthand notation ψ.[6]

Tangent space $\scriptstyle T_xM$ and a tangent vector $\scriptstyle v\in T_xM$, along a curve traveling through $\scriptstyle x\in M$

It is defined as the angle between a vehicle's heading and a reference heading (normally true or magnetic North).

When used over the earth surface in long distances, the orientation of reference frame used depends on the latitude and longitude, and it is usually defined on the tangent space of the earth at that point, using as tangent vectors the derivatives of the lines of coordinates.

Given the difficult problem of following a geodesic course, sailors used to follow lines of constant yaw at sea, called Rhumb lines or Loxodromes. On a Mercator projection map, a loxodrome is a straight line; beyond the right edge of the map it continues on the left with the same slope.

Given the spherical geometry of the surface of earth, some unexpected effects as parallel translation can happen.

## References

1. ^ Unpublished Draft: Orientation, Rotation, Velocity, and Acceleration and the SRM, Paul Verner
2. ^ NASA slides on Yaw motion
3. ^ NASA slides on Pitch motion
4. ^ NASA slides on Roll motion
5. ^ Etymology online dictionary
6. ^ Hurt, H. H., Jr. (January 1965) [1960]. Aerodynamics for Naval Aviators. U.S. Government Printing Office, Washington D.C.: U.S. Navy, Aviation Training Division. p. 284. NAVWEPS 00-80T-80.