Yaw, pitch, and roll

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http://en.wikipedia.org/wiki/Tait-Bryan_rotations

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. The three angles specified in this formulation are defined as the roll angle, pitch angle, and yaw angle.

## Yaw, pitch and roll in aerospace

 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
 Tait-Bryan angles statically defined. ZXY convention

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. The origin of this intrinsic frame of the vehicle, XYZ system, is located at the vehicle's center of gravity 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 reference frame xyz that shares the same origin as the XYZ system but will be considered fixed in the discussion below. Keeping this plane example in mind, it can be imagined that the fixed xyz frame will always be aligned having x pointing in the direction of true north, y pointing to true east, and the z-axis pointing away from the center of gravity of the earth.

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

1. A right-handed rotation ψ about the z-axis by the yaw angle.
2. A right-handed rotation θ about the new (once-rotated) y-axis by the pitch angle.
3. A right-handed rotation φ about the new (twice-rotated) x-axis by the roll angle.

## Matrix expression

We might be interested to find a transform that expresses any vector with coordinates specified in xyz as a vector with coordinates in XYZ.

To construct such a change of basis matrix $R_{xyz}^{XYZ}$, consider that the target frame XYZ is specified as a change of orientation of the original xyz frame. Imagine the basis xyz and XYZ to coincide first, and consider any vector v as viewed from XYZ: any sequence of rotations to bring XYZ to its final orientation will 'look', from XYZ, as if the opposite sequence of rotations are applied to the vector v.

Thus, by finding the rotation matrix R corresponding to the rotation sequence from the xyz system to the XYZ system, we can find $R_{xyz}^{XYZ}$ as it will just be the inverse of the former transform:

If $v_{XYZ} \,$ represents a column vector expressed in basis XYZ, we have

$Rv_{XYZ}=v_{xyz} \,$

by definition of the frame XYZ, and thus,

$v_{XYZ}=R^{-1}v_{xyz} \,$.

This shows that $R^{-1}=R_{xyz}^{XYZ} \,$, by definition of $R_{xyz}^{XYZ} \,$.

The 3 step rotation sequence above use a new reference frame to specify the axis of each successive rotation, and thus, each rotation associated with each step can't be expressed as a simple rotation matrix multiplication applied to the intermediate result at each step. However, the above sequence is equivalent to doing (in order) a:

1. Right-handed rotation φ, the roll angle, about the x-axis.
2. Right-handed rotation θ, the pitch angle, about the y-axis
3. Right-handed rotation ψ, the yaw angle, about the z-axis.

These previous steps can now be respectively expressed as $R_x(\varphi) \,$, $R_y(\theta) \,$ and $R_z(\psi) \,$, where $R_x(\theta)\,$, $R_y(\theta) \,$, $R_z(\theta) \,$ represent rotation matrices by an angle θ about the axes x, y and z respectively.

These rotations can then be combined successively by matrix multiplication to get a resulting single rotation matrix to apply to vectors to move their coordinates to the new orientation:

$R=R_z(\psi)R_y(\theta)R_x(\varphi)$,

the components of R being

$R= \begin{bmatrix}c_\psi c_\theta & (c_\psi s_\theta s_\varphi - s_\psi c_\varphi) & (c_\psi s_\theta c_\varphi + s_\psi s_\varphi) \\s_\psi c_\theta & (s_\psi s_\theta s_\varphi + c_\psi c_\varphi) & (s_\psi s_\theta c_\varphi - c_\psi s_\varphi) \\-s_\theta & c_\theta s_\varphi & c_\theta c_\varphi \end{bmatrix}$,

where $c_\theta \,$ and $s_\theta \,$ denote $\cos(\theta) \,$ and $\sin(\theta) \,$, respectively.

Using the fact that rotation matrices are orthogonal and that inverses of rotations are just a rotation in the opposite direction, the change of basis matrix $R_{xyz}^{XYZ}$ is thus

$R_{xyz}^{XYZ} =R^{-1} =R^T =(R_z(\psi)R_y(\theta)R_x(\varphi))^T =R_x(\varphi)^T R_y(\theta)^T R_z(\psi)^T=R_x(-\varphi) R_y(-\theta) R_z(-\psi)$.

We obviously only need $R_{xyz}^{XYZ}=R^T$, but the right hand side of the above equalities show our earlier intuition.

The matrix $R_{xyz}^{XYZ}=R^T$ have thus for components

$R_{xyz}^{XYZ} = \begin{bmatrix}c_\psi c_\theta & s_\psi c_\theta & -s_\theta \\(c_\psi s_\theta s_\varphi - s_\psi c_\varphi) & (s_\psi s_\theta s_\varphi + c_\psi c_\varphi) & c_\theta s_\varphi \\(c_\psi s_\theta c_\varphi + s_\psi s_\varphi) & (s_\psi s_\theta c_\varphi - c_\psi s_\varphi) & c_\theta c_\varphi \end{bmatrix}$.

In summary, to transform a column vector $v_{xyz} \,$ expressed in basis xyz into a vector expressed in basis XYZ, we have

$v_{XYZ}=R_{xyz}^{XYZ}v_{xyz}$.

The inverse transform, from XYZ to xyz, is just the transpose of $R_{xyz}^{XYZ}$:

$R_{XYZ}^{xyz}:=(R_{xyz}^{XYZ})^{-1}=(R_{xyz}^{XYZ})^T$

It can be noted that sometimes the names yaw, pitch and roll refer to the three Tait-Bryan angles or rotations of the aircraft local reference frame with regard to a fixed reference frame, while sometimes, they are used to describe rotations (or motion) about the local reference frame of the aircraft, called the aircraft principal axes. Obviously, the second case just considers xyz as the current XYZ, and the specified angles must be taken to move this local reference frame of the aircraft to a new orientation: ie, the angles are never absolutes with regard to an external frame but are used to specify changes in the attitude of the aircraft, or rotations from $XYZ_{\text{current}} \,$ to $XYZ_{\text{new}} \,$, always relative to $XYZ_{\text{current}} \,$.

To understand the distinction in the usage of this terminology, one can picture a space shuttle in the xyz frame above, in its initial position as described, that is, nose pointing towards +x-axis, center of gravity at origin, and wings level with the xy-plane. If we specify a single Tait-Bryan roll angle $\varphi$ with regard to xyz, the shuttle is now described as having tilted wings with regard to the xy-plane. The space shuttle could then be imagined to start a rotating motion about its wings-axis. This later motion could be described as pitching, but it is not a rotation about any of the xyz axes since the term now refers to the local reference frame of the shuttle.

## Yaw, pitch and roll in navigation

 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". Etimologically, it is related with the verb 'to go'. It is typically assigned the shorthand notation ψ.

 The 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 follow 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.

## Yaw, pitch and roll in 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.

The importance of non-singularities in robotics has lead 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 in robot arms that utilize a “triple-roll wrist”. 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 360° 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.

Published - July 2009