Axes conventions
Encyclopedia
Mobile objects are normally tracked from an external frame considered fixed. Other frames can be defined on those mobile objects to deal with relative positions for other objects. Finally, attitudes or orientation
Orientation
Orientation may refer to:* Orientation , a function of the mind* Orientation , a 1996 short film produced by the Church of Scientology...

s can be described by a relationship between the external frame and the one defined over the mobile object.

The orientation of a vehicle is normally referred to as attitude. It is described normally by the orientation
Orientation (geometry)
In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it is in....

of a frame fixed in the body relative to a fixed reference frame. The attitude is described by attitude coordinates, and consists of at least three coordinates.

While from a geometrical point of view the different methods to describe orientations are defined using only some reference frames, in engineering applications it is important also to describe how these frames are attached to the lab and the body in motion.

Due to the special importance of international conventions in air vehicles, several organizations have published standards to be followed. For example, German DIN has published the DIN
Deutsches Institut für Normung
is the German national organization for standardization and is that country's ISO member body. DIN is a Registered German Association headquartered in Berlin...

9300 norm for aircraft (adopted by ISO as ISO 1151–2:1985).

## Earth bounded axes conventions

### Ground reference frames: ENU and NED

Basically, as lab frame or reference frame, there are two kinds of conventions for the frames (sometimes named LVLH, local vertical, local horizontal):
• East, North, Up, referred as ENU
• North, East, Down, referred as NED, used specially in aerospace

These frames are location dependent. For movements around the globe, like air or sea navigation, the frames are defined as tangent to the lines of coordinates.
• East-West tangent to parallel
Circle of latitude
A circle of latitude, on the Earth, is an imaginary east-west circle connecting all locations that share a given latitude...

s,
• North-South tangent to meridians
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

, and
• Up-Down in the direction to the center of the earth (when using a spherical Earth simplification), or in the direction normal to the local tangent plane (using an oblate spheroidal or geodetic
Geodetic system
Geodetic systems or geodetic data are used in geodesy, navigation, surveying by cartographers and satellite navigation systems to translate positions indicated on their products to their real position on earth....

ellipsoidal model of the earth) which does not generally pass through the center of the Earth.

To establish a standard convention to describe attitudes, it is required to establish at least the axes of the reference system and the axes of the rigid body or vehicle. When an ambiguous notation system is used (such as Euler angles) also the used convention should be stated. Nevertheless most used notations (matrices and quaternions) are unambiguous.
Tait–Bryan angles are often used to describe a vehicle's attitude with respect to a chosen reference frame, though any other notation can be used. The positive x-axis in vehicles points always in the direction of movement. For positive y- and z-axis, we have to face two different conventions:
• In case of land vehicles like cars, tanks etc., which use the ENU-system (East-North-Up) as external reference (world frame), the vehicle's positive y- or pitch axis always points to its left, and the positive z- or yaw axis always points up.

• By contrast, in case of air and sea vehicles like submarines, ships, airplanes etc., which use the NED-system (North-East-Down) as external reference (world frame), the vehicle's positive y- or pitch axis always points to its right, and its positive z- or yaw axis always points down.

• Finally, in case of space vehicles like space shuttles etc., a modification of the latter convention is used, where the vehicle's positive y- or pitch axis again always points to its right, and its positive z- or yaw axis always points down, but “down” now may have two different meanings: If a so-called local frame is used as external reference, its positive z-axis points “down” to the center of the earth as it does in case of the earlier mentioned NED-system, but if the inertial frame is used as reference, its positive z-axis will point now to the North Celestial Pole, and its positive x-axis to the Vernal Equinox or some other reference meridian.

### Frames mounted on vehicles

Specially for aircraft, these frames do not need to agree with the earth-bound frames in the up-down line. It must be agreed what ENU and NED mean in this context.

#### Conventions for land vehicles

For land vehicles is rare to describe their complete orientation, except when speaking about electronic stability control
Electronic stability control
Electronic stability control is a computerized technology that may potentially improve the safety of a vehicle's stability by detecting and minimizing skids. When ESC detects loss of steering control, it automatically applies the brakes to help "steer" the vehicle where the driver intends to go...

or satellite navigation. In this case, the convention is normally the one of the adjacent drawing.

#### Conventions for sea vehicles

As well as aircraft, the same terminology is used for the motion of ships and boats
Ship motions
Ship motions are defined by the six degrees of freedom that a ship, boat or any other craft can experience.- Translation :HeaveSwaySurge-Vertical axis:Vertical axis, or yaw axis — an axis drawn from top to bottom, and perpendicular to the other two axes...

. It is interesting to note that some words commonly used were introduced in maritime navigation. For example, the yaw angle or heading, has a nautical
Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

origin, with the meaning of "bending out of the course". Etymologically, it is related with the verb 'to go'. It is related to the concept of bearing
In marine navigation, a bearing is the direction one object is from another object, usually, the direction of an object from one's own vessel. In aircraft navigation, a bearing is the actual compass direction of the forward course of the aircraft...

. It is typically assigned the shorthand notation .

#### Conventions for aircraft local reference frames

Coordinates to describe an aircraft attitude (Heading, Elevation and Bank) are normally given relative to a reference control frame located in a control tower, and therefore ENU, relative to the position of the control tower on the earth surface.

Coordinates to describe observations made from an aircraft are normally given relative to its intrinsic axes, but normally using as positive the coordinate pointing downwards, where the interesting points are located. Therefore they are normally NED.

These axes are normally taken so that X axis is the longitudinal axis pointing ahead, Z axis is the vertical axis pointing downwards, and the Y axis is the lateral one, pointing in such a way that the frame is right handed.

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.

For satellites orbiting the earth it is normal to use the Equatorial coordinate system
Equatorial coordinate system
The equatorial coordinate system is a widely-used method of mapping celestial objects. It functions by projecting the Earth's geographic poles and equator onto the celestial sphere. The projection of the Earth's equator onto the celestial sphere is called the celestial equator...

. The projection of the Earth's equator onto the celestial sphere is called the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

. Similarly, the projections of the Earth's north and south geographic poles become the north and south celestial pole
Celestial pole
The north and south celestial poles are the two imaginary points in the sky where the Earth's axis of rotation, indefinitely extended, intersects the imaginary rotating sphere of stars called the celestial sphere...

s, respectively.

Deep space satellites use other Celestial coordinate system
Celestial coordinate system
In astronomy, a celestial coordinate system is a coordinate system for mapping positions on the celestial sphere.There are different celestial coordinate systems each using a system of spherical coordinates projected on the celestial sphere, in analogy to the geographic coordinate system used on...

, like the Ecliptic coordinate system
Ecliptic coordinate system
The ecliptic coordinate system is a celestial coordinate system that uses the ecliptic for its fundamental plane. The ecliptic is the path that the sun appears to follow across the celestial sphere over the course of a year. It is also the intersection of the Earth's orbital plane and the celestial...

.

### Conventions for space ships as aircraft

If the goal is to keep the shuttle during its orbits in a constant attitude with respect to the sky, e.g. in order to perform certain astronomical observations, the preferred reference is the inertial frame, and the RPY angle vector (0|0|0) describes an attitude then, where the shuttle's wings are kept permanently parallel to the earth's equator, its nose points permanently to the vernal equinox, and its belly towards the Northern polar star (see picture). (Note that rockets and missiles more commonly follow the conventions for aircraft where the RPY angle vector (0|0|0) points north, rather than toward the vernal equinox).

On the other hand, if it's the goal to keep the shuttle during its orbits in an constant attitude with respect to the surface of the earth, the preferred reference will be the local frame, with the RPY angle vector (0|0|0) describing an attitude, where the shuttle's wings are parallel to the earth's surface, its nose points to its heading, and its belly down towards the centre of the earth (see picture).

## Frames used to describe attitudes

Normally the frames used to describe a vehicle's local observations are the same frames used to describe its attitude respect the ground tracking stations.

An important case in which this does not apply are aircraft. Aircraft observations are performed downwards and therefore normally NED axes convention applies. Nevertheless, when attitudes respect ground stations are given, a relationship between the local earth-bound frame and the onboard ENU frame is used.

• Attitude dynamics and control
Attitude dynamics and control
Spacecraft flight dynamics is the science of space vehicle performance, stability, and control. It requires analysis of the six degrees of freedom of the vehicle's flight, which are similar to those of aircraft: translation in three dimensional axes; and its orientation about the vehicle's center...

(spacecraft)
• Euler's rotation theorem
Euler's rotation theorem
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two...

• Gyroscope
Gyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...

The Triad Method, also known as the algebraic method, is a method for calculating the inclination of an object based on two known vectors in relation to that object's axis system. It was developed in 1964 by Harold D. Black and first published in the Journal of the American Institute of Aeronautics...

• Rotation representation (mathematics)
Rotation representation (mathematics)
In geometry a rotation representation expresses a rotation as a mathematical transformation. In physics, this concept extends to classical mechanics where rotational kinematics is the science of describing with numbers the purely rotational motion of an object.According to Euler's rotation theorem...

• Geographic coordinate system
Geographic coordinate system
A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...

• Astronomical coordinate systems
Astronomical coordinate systems
Astronomical coordinate systems are coordinate systems used in astronomy to describe the location of objects in the sky and in the universe. The most commonly occurring such systems are coordinate systems on the celestial sphere, but extragalactic coordinates systems are also important for...