The Euler angles are three angles introduced by
Leonhard Euler to describe the
orientation of a
rigid body with respect to a fixed
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
.
[Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478]
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They can also represent the orientation of a mobile frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...
in physics or the orientation of a general basis in 3-dimensional linear algebra. Alternative forms were later introduced by Peter Guthrie Tait and George H. Bryan
George Hartley Bryan FRS (1 March 1864 – 13 October 1928) was an English applied mathematician who was an authority on thermodynamics and aeronautics. He was born in Cambridge, and was educated at Peterhouse, Cambridge, obtaining his BA in 188 ...
intended for use in aeronautics and engineering.
Chained rotations equivalence
Euler angles can be defined by elemental geometry or by composition of rotations. The geometrical definition demonstrates that three composed '' elemental rotations'' (rotations about the axes of a coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
) are always sufficient to reach any target frame.
The three elemental rotations may be extrinsic (rotations about the axes ''xyz'' of the original coordinate system, which is assumed to remain motionless), or intrinsic (rotations about the axes of the rotating coordinate system ''XYZ'', solidary with the moving body, which changes its orientation with respect to the extrinsic frame after each elemental rotation).
Euler angles are typically denoted as ''α'', ''β'', ''γ'', or ''ψ'', ''θ'', ''φ''. Different authors may use different sets of rotation axes to define Euler angles, or different names for the same angles. Therefore, any discussion employing Euler angles should always be preceded by their definition.
Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups:
* Proper Euler angles
* Tait–Bryan angles .
Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called "Euler angles". In that case, the sequences of the first group are called ''proper'' or ''classic'' Euler angles.
Proper Euler angles
Geometrical definition
The axes of the original frame are denoted as ''x'', ''y'', ''z'' and the axes of the rotated frame as ''X'', ''Y'', ''Z''. The geometrical definition (sometimes referred to as static) begins by defining the line of nodes
An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes.
Planes of reference
Common planes of reference ...
(N) as the intersection of the planes ''xy'' and ''XY'' (it can also be defined as the common perpendicular to the axes ''z'' and ''Z'' and then written as the vector product ''N'' = ''z'' ''Z''). Using it, the three Euler angles can be defined as follows:
* (or ) is the signed angle between the ''x'' axis and the ''N'' axis (''x''-convention – it could also be defined between ''y'' and ''N'', called ''y''-convention).
* (or ) is the angle between the ''z'' axis and the ''Z'' axis.
* (or ) is the signed angle between the ''N'' axis and the ''X'' axis (''x''-convention).
Euler angles between two reference frames are defined only if both frames have the same handedness
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more Fine motor skill, dextrous. The other hand, comparatively often the weaker, less dextrous or sim ...
.
Conventions by intrinsic rotations
Intrinsic rotations are elemental rotations that occur about the axes of a coordinate system ''XYZ'' attached to a moving body. Therefore, they change their orientation after each elemental rotation. The ''XYZ'' system rotates, while ''xyz'' is fixed. Starting with ''XYZ'' overlapping ''xyz'', a composition of three intrinsic rotations can be used to reach any target orientation for ''XYZ''.
Euler angles can be defined by intrinsic rotations. The rotated frame ''XYZ'' may be imagined to be initially aligned with ''xyz'', before undergoing the three elemental rotations represented by Euler angles. Its successive orientations may be denoted as follows:
* ''x''-''y''-''z'', or ''x''0-''y''0-''z''0 (initial)
* ''x''′-''y''′-''z''′, or ''x''1-''y''1-''z''1 (after first rotation)
* ''x''″-''y''″-''z''″, or ''x''2-''y''2-''z''2 (after second rotation)
* ''X''-''Y''-''Z'', or ''x''3-''y''3-''z''3 (final)
For the above-listed sequence of rotations, the line of nodes
An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes.
Planes of reference
Common planes of reference ...
''N'' can be simply defined as the orientation of ''X'' after the first elemental rotation. Hence, ''N'' can be simply denoted ''x''′. Moreover, since the third elemental rotation occurs about ''Z'', it does not change the orientation of ''Z''. Hence ''Z'' coincides with ''z''″. This allows us to simplify the definition of the Euler angles as follows:
* ''α'' (or ) represents a rotation around the ''z'' axis,
* ''β'' (or ) represents a rotation around the ''x''′ axis,
* ''γ'' (or ) represents a rotation around the ''z''″ axis.
Conventions by extrinsic rotations
Extrinsic rotations are elemental rotations that occur about the axes of the fixed coordinate system ''xyz''. The ''XYZ'' system rotates, while ''xyz'' is fixed. Starting with ''XYZ'' overlapping ''xyz'', a composition of three extrinsic rotations can be used to reach any target orientation for ''XYZ''. The Euler or Tait–Bryan angles (''α'', ''β'', ''γ'') are the amplitudes of these elemental rotations. For instance, the target orientation can be reached as follows (note the reversed order of Euler angle application):
* The ''XYZ'' system rotates about the ''z'' axis by ''γ''. The ''X'' axis is now at angle ''γ'' with respect to the ''x'' axis.
* The ''XYZ'' system rotates again, but this time about the ''x'' axis by ''β''. The ''Z'' axis is now at angle ''β'' with respect to the ''z'' axis.
* The ''XYZ'' system rotates a third time, about the ''z'' axis again, by angle ''α''.
In sum, the three elemental rotations occur about ''z'', ''x'' and ''z''. Indeed, this sequence is often denoted ''z''-''x''-''z'' (or 3-1-3). Sets of rotation axes associated with both proper Euler angles and Tait–Bryan angles are commonly named using this notation (see above for details).
Signs, ranges and conventions
Angles are commonly defined according to the right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of th ...
. Namely, they have positive values when they represent a rotation that appears clockwise when looking in the positive direction of the axis, and negative values when the rotation appears counter-clockwise. The opposite convention (left hand rule) is less frequently adopted.
About the ranges (using interval notation):
* for ''α'' and ''γ'', the range is defined modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is t ...
2 radians. For instance, a valid range could be .
* for ''β'', the range covers radians (but can't be said to be modulo ). For example, it could be or .
The angles ''α'', ''β'' and ''γ'' are uniquely determined except for the singular case that the ''xy'' and the ''XY'' planes are identical, i.e. when the ''z'' axis and the ''Z'' axis have the same or opposite directions. Indeed, if the ''z'' axis and the ''Z'' axis are the same, ''β'' = 0 and only (''α'' + ''γ'') is uniquely defined (not the individual values), and, similarly, if the ''z'' axis and the ''Z'' axis are opposite, ''β'' = and only (''α'' − ''γ'') is uniquely defined (not the individual values). These ambiguities are known as gimbal lock in applications.
There are six possibilities of choosing the rotation axes for proper Euler angles. In all of them, the first and third rotation axes are the same. The six possible sequences are:
# ''z''1-''x''′-''z''2″ (intrinsic rotations) or ''z''2-''x''-''z''1 (extrinsic rotations)
# ''x''1-''y''′-''x''2″ (intrinsic rotations) or ''x''2-''y''-''x''1 (extrinsic rotations)
# ''y''1-''z''′-''y''2″ (intrinsic rotations) or ''y''2-''z''-''y''1 (extrinsic rotations)
# ''z''1-''y''′-''z''2″ (intrinsic rotations) or ''z''2-''y''-''z''1 (extrinsic rotations)
# ''x''1-''z''′-''x''2″ (intrinsic rotations) or ''x''2-''z''-''x''1 (extrinsic rotations)
# ''y''1-''x''′-''y''2″ (intrinsic rotations) or ''y''2-''x''-''y''1 (extrinsic rotations)
Precession, nutation and intrinsic rotation
Precession, nutation, and intrinsic rotation (spin) are defined as the movements obtained by changing one of the Euler angles while leaving the other two constant. These motions are not expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis ''z'', the second rotates around the line of nodes ''N'' and the third one is an intrinsic rotation around ''Z'', an axis fixed in the body that moves.
The static definition implies that:
* ''α'' (precession) represents a rotation around the ''z'' axis,
* ''β'' (nutation) represents a rotation around the ''N'' or x′ axis,
* ''γ'' (intrinsic rotation) represents a rotation around the ''Z'' or z″ axis.
If ''β'' is zero, there is no rotation about ''N''. As a consequence, ''Z'' coincides with ''z'', ''α'' and ''γ'' represent rotations about the same axis (''z''), and the final orientation can be obtained with a single rotation about ''z'', by an angle equal to .
As an example, consider a top. The top spins around its own axis of symmetry; this corresponds to its intrinsic rotation. It also rotates around its pivotal axis, with its center of mass orbiting the pivotal axis; this rotation is a precession. Finally, the top can wobble up and down; the inclination angle is the nutation angle. The same example can be seen with the movements of the earth.
Though all three movements can be represented by a rotation operator with constant coefficients in some frame, they cannot be represented by these operators all at the same time. Given a reference frame, at most one of them will be coefficient-free. Only precession can be expressed in general as a matrix in the basis of the space without dependencies of the other angles.
These movements also behave as a gimbal set. If we suppose a set of frames, able to move each with respect to the former according to just one angle, like a gimbal, there will exist an external fixed frame, one final frame and two frames in the middle, which are called "intermediate frames". The two in the middle work as two gimbal rings that allow the last frame to reach any orientation in space.
Tait–Bryan angles
The second type of formalism is called Tait–Bryan angles, after Peter Guthrie Tait and George H. Bryan
George Hartley Bryan FRS (1 March 1864 – 13 October 1928) was an English applied mathematician who was an authority on thermodynamics and aeronautics. He was born in Cambridge, and was educated at Peterhouse, Cambridge, obtaining his BA in 188 ...
. It is the convention normally used for aerospace applications, so that zero degrees elevation represents the horizontal attitude. Tait–Bryan angles represent the orientation of the aircraft with respect to the world frame. When dealing with other vehicles, different axes conventions are possible.
Definitions
The definitions and notations used for Tait–Bryan angles are similar to those described above for proper Euler angles ( geometrical definition, intrinsic rotation definition, extrinsic rotation definition). The only difference is that Tait–Bryan angles represent rotations about three distinct axes (e.g. ''x''-''y''-''z'', or ''x''-''y''′-''z''″), while proper Euler angles use the same axis for both the first and third elemental rotations (e.g., ''z''-''x''-''z'', or ''z''-''x''′-''z''″).
This implies a different definition for the line of nodes
An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes.
Planes of reference
Common planes of reference ...
in the geometrical construction. In the proper Euler angles case it was defined as the intersection between two homologous Cartesian planes (parallel when Euler angles are zero; e.g. ''xy'' and ''XY''). In the Tait–Bryan angles case, it is defined as the intersection of two non-homologous planes (perpendicular when Euler angles are zero; e.g. ''xy'' and ''YZ'').
Conventions
The three elemental rotations may occur either about the axes of the original coordinate system, which remains motionless (extrinsic rotations
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref>
They ...
), or about the axes of the rotating coordinate system, which changes its orientation after each elemental rotation ( intrinsic rotations).
There are six possibilities of choosing the rotation axes for Tait–Bryan angles. The six possible sequences are:
* ''x''-''y''′-''z''″ (intrinsic rotations) or ''z''-''y''-''x'' (extrinsic rotations)
* ''y''-''z''′-''x''″ (intrinsic rotations) or ''x''-''z''-''y'' (extrinsic rotations)
* ''z''-''x''′-''y''″ (intrinsic rotations) or ''y''-''x''-''z'' (extrinsic rotations)
* ''x''-''z''′-''y''″ (intrinsic rotations) or ''y''-''z''-''x'' (extrinsic rotations)
* ''z''-''y''′-''x''″ (intrinsic rotations) or ''x''-''y''-''z'' (extrinsic rotations): the intrinsic rotations are known as: yaw, pitch and roll
* ''y''-''x''′-''z''″ (intrinsic rotations) or ''z''-''x''-''y'' (extrinsic rotations)
Signs and ranges
Tait–Bryan convention is widely used in engineering with different purposes. There are several axes conventions in practice for choosing the mobile and fixed axes, and these conventions determine the signs of the angles. Therefore, signs must be studied in each case carefully.
The range for the angles ''ψ'' and ''φ'' covers 2 radians. For ''θ'' the range covers radians.
Alternative names
These angles are normally taken as one in the external reference frame ( heading, bearing), one in the intrinsic moving frame ( bank) and one in a middle frame, representing an elevation or inclination with respect to the horizontal plane, which is equivalent to the line of nodes for this purpose.
For an aircraft, they can be obtained with three rotations around its principal axes if done in the proper order. A yaw will obtain the bearing, a pitch will yield the elevation and a roll gives the bank angle. Therefore, in aerospace they are sometimes called yaw, pitch and roll. Notice that this will not work if the rotations are applied in any other order or if the airplane axes start in any position non-equivalent to the reference frame.
Tait–Bryan angles, following ''z''-''y''′-''x''″ (intrinsic rotations) convention, are also known as nautical angles, because they can be used to describe the orientation of a ship or aircraft, or Cardan angles, after the Italian mathematician and physicist Gerolamo Cardano, who first described in detail the Cardan suspension
A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of ...
and the Cardan joint.
Angles of a given frame
A common problem is to find the Euler angles of a given frame. The fastest way to get them is to write the three given vectors as columns of a matrix and compare it with the expression of the theoretical matrix (see later table of matrices). Hence the three Euler Angles can be calculated. Nevertheless, the same result can be reached avoiding matrix algebra and using only elemental geometry. Here we present the results for the two most commonly used conventions: ''ZXZ'' for proper Euler angles and ''ZYX'' for Tait–Bryan. Notice that any other convention can be obtained just changing the name of the axes.
Proper Euler angles
Assuming a frame with unit vectors (''X'', ''Y'', ''Z'') given by their coordinates as in the main diagram, it can be seen that:
:
And, since
:
for