# Aircraft principal axes

Roll- pitch-yaw angles, English roll -pitch -yaw angle, are a way of describing the orientation of a vehicle in three-dimensional space, which was initially in use only in aircraft, but in the meantime also the location description of land, sea and space vehicles use place.

## Details

Specifically, taken from the flight control angle here describe three successive rotations which convert a fixed reference system (English world- frame ) in an object related right-handed coordinate system (English bodyframe ). Its origin is thinking you are in the center of the vehicle, the angle designations derived directly from the name of the three possible rotations of the vehicle about its body axes ( see right ). At the beginning of the rotations are consistent reference system and body- fixed system.

- Yaw (English yaw): rotation around the z - axis of the reference system ( yaw, high or vertical axis). Also the terms heading or azimuth are doing for the direction angle sometimes needed.
- Nod (English pitch, rarely nick ): rotation around the y- axis of the vehicle ( pitch or lateral axis ).
- Roles (English roll): rotation about the running in the longitudinal direction of the vehicle x axis (roll, roll or longitudinal axis ).

Coincidentally, however, shows only the positive x - axis ( roll axis engl. ) always, ie both in space, air, water and land vehicles, in their flight or movement direction ( engl. heading ). For the positive y -and z- direction of the object system (body frame ), however, there are, depending on which external coordinate system ( engl. world frame ) is used as a reference system, two different conventions:

- For land vehicles that use the ENU - system (East - North -Up ) as a reference system, shows the positive y- axis ( engl. pitch axis ) is always to the left or port side and the perpendicular to the xy plane of the vehicle positive z axis (English yaw axis ) is always upward.
- In space, air and water vehicles including submarines, however, the further (North - East - Down) to use for reasons of compatibility with the traditional compass direction of the NED system as a reference system, shows the positive y- axis ( engl. pitch axis) always in the direction of the right wing and to starboard and the perpendicular to the xy plane of the vehicle positive z - axis ( yaw axis engl. ) always down.

What is "down", is in this case, however, in spacecraft such as space shuttles or satellites defined somewhat differently than in aircraft and vessels, following the objectives of the flight using spacecraft instead of one of the two above-mentioned external coordinate systems either an oriented at the current trajectory absorbed. local reference system ( engl. local frame ) or at the constellation Aries and the Spring turning point (English vernal equinox ) and orienting on the north -south axis of the earth absorbed. Inertial (English inertial frame ):

Is it all about the spacecraft, for example, a space shuttle, to keep in its orbits in a constant position as possible in terms of the firmament, for example, to perform certain astronomical studies, is used as reference system preferably uses the inertial frame in which the RPY angle ( 0 | 0 | 0 ) a layer of the Space Shuttle describe, in this parallel to the earth's equator, pointing the bow toward spring turning point and the belly toward Polaris circles the earth with its wings.

Is it, however, a question of keeping the space shuttle in their orbits in a constant position as possible with respect to the earth's surface to carry out any certain earth-related investigations, instead, is used as reference system preferably uses the local reference system in which the RPY angle ( 0 | 0 | 0) describe a situation of the Space Shuttle, in this like a plane parallel to the earth's surface, the bow to the front and center of the Earth showing the belly orbits the Earth with the wings.

Even with three -axis stabilized satellite, as the earth eg as communication satellites orbit with a fixed orientation to the earth, the language just described has become established for rotation angles and axes. Thus, in this case, the positive x - axis of the satellite - as with the use of a local reference system by means of a shuttle (see above) - in the direction of its flight path to the ground, while the aligned on the center of the earth antennas of the satellite so that in the direction of show its positive z- axis, and finally its solar panels rotate around two axes perpendicular to the y- axis ( see Fig.)

Since rotations of the satellite about its z - axis while having little effect on the orientation of its antennas with respect to the earth, the position control of the z- axis can work with a larger tolerance than the other two axes: Typical values for the yaw angle ( engl. yaw) are ± 0.15 ° for the roll and pitch angle ( engl. roll and pitch) on the other hand only ± 0.05 °.

In robotics roll- pitch - yaw angles are used to describe orientations ( the tool or other objects ), based on a space- fixed base used.

The aerodynamic concept is finally Pitch - notwithstanding the above system - also used for the pitch of a propeller or rotor, see Pitch (aerodynamics ).

## " ZY'X '' Convention"

The rotations are always related to the right-handed orthogonal basis initially fixed in space. The convention of orientation of the angle follows the usual in mathematics right-hand rule, a positive angle is counterclockwise so. All other rotation axes arise dynamically according to the defined sequence of rotations.

If we denote by the yaw angle, the pitch angle and the roll angle, the rotation matrix for this definition is as follows:

Is exactly the same singularities occur, so-called gimbal lock, on which manifest themselves in the fact that there are suddenly infinitely many solutions in these cases and. This results, for example, using the addition theorems for the rotation matrix

And the rotation matrix

## Calculation of rotation matrix

Is a rotation matrix given:

Then, the angle can be calculated as follows ( Craig, pp. 47f ):

In the case, however, the above-mentioned singularities following formulas are useful:

Is it is advantageous to use

If instead, one uses analog expediently

It is the same feature that exists in many mathematical libraries and is compared with the arctan function is often preferred because it makes otherwise needed different cases for the four quadrants superfluous.

Other ways to describe the orientation, are rotation matrix, quaternions or Euler angles.