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.
Mathematical definition
Tait-Bryan angles statically
defined.
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).
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 this construction 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 \pi/2.
Local attitude description
Aircraft attitude
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 as follows:
- right-handed rotation \psi \in (-180, 180] about the
z-axis by the yaw angle
- right-handed rotation \theta \in [-90, 90] about the new
(once-rotated) y-axis by the pitch angle
- 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 R_x(\phi) R_y(-\theta) R_z(-\psi). In order to
convert a point in
xyz to
XYZ coordinates, one
applies the matrix R_z(\psi) R_y(\theta) R_x(-\phi) to the
point.
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 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.
Air and maritime 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". Etymologically, it is related with the verb 'to
go'. It is typically assigned the shorthand notation \psi.
[[Image:Tangentialvektor.svg|thumb|left|200px|
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.
See also
References
- Orientation, Rotation, Velocity, and Acceleration and the SRM,
Paul Verner, [1]
- NASA slides on Yaw motion
http://www.grc.nasa.gov/WWW/K-12/airplane/yaw.html
- NASA slides on Pitch motion
http://www.grc.nasa.gov/WWW/K-12/airplane/pitch.html
- NASA slides on Roll motion
http://www.grc.nasa.gov/WWW/K-12/airplane/roll.html
- Etimology online dictionary
http://www.20kweb.com/etymology_dictionary_Y/origin_of_the_word_yaw.htm