Euler angles

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.

They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in three dimensional linear algebra.

Classic Euler angles usually take the inclination angle in such a way that zero degrees represent the vertical orientation. Alternative forms were later introduced by Peter Guthrie Tait and George H. Bryan intended for use in aeronautics and engineering in which zero degrees represent the horizontal position.

Chained rotations equivalence

Euler angles can be defined by elemental geometry or by composition of rotations (i.e. chained rotations). The geometrical definition demonstrates that three consecutive elemental rotations (rotations about the axes of a coordinate system) are always sufficient to reach any target frame.

The three elemental rotations may be extrinsic (rotations about the axes of the original coordinate system, which is assumed to remain motionless), or intrinsic (rotations about the axes of the rotating coordinate system , solidary with the moving body, which changes its orientation with respect to the extrinsic frame after each elemental rotation).

In the sections below, an axis designation with a prime mark superscript (e.g., ) denotes the new axis after an elemental rotation.

Euler angles are typically denoted as , or , respectively. 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.

Classic Euler angles The Euler angles are three angles introduced by Swiss mathematician Leonhard Euler (1707–1783) to describe the orientation of a rigid body with respect to a fixed coordinate system.

Geometrical definition The axes of the original frame are denoted as and the axes of the rotated frame as . The geometrical definition (sometimes referred to as static) begins by defining the line of nodes () as the intersection of the planes and (it can also be defined as the common perpendicular to the axes and and then written as the vector product ). Using it, the three Euler angles can be defined as follows:

(or ) is the signed angle between the axis and the axis (-convention — it could also be defined between and , called -convention). (or ) is the angle between the axis and the axis. (or ) is the signed angle between the axis and the axis (-convention).

Euler angles between two reference frames are defined only if both frames have the same handedness.

Definition by intrinsic rotations

Intrinsic rotations are elemental rotations that occur about the axes of a coordinate system attached to a moving body. Therefore, they change their orientation after each elemental rotation. The system rotates, while is fixed. Starting with overlapping , a composition of three intrinsic rotations can be used to reach any target orientation for .

Euler angles can be defined by intrinsic rotations. The rotated frame may be imagined to be initially aligned with , before undergoing the three elemental rotations represented by Euler angles. Its successive orientations may be denoted as follows:

or (initial) or (after first rotation) or (after second rotation) or (final)

For the above-listed sequence of rotations, the line of nodes can be simply defined as the orientation of after the first elemental rotation. Hence, can be simply denoted . Moreover, since the third elemental rotation occurs about , it does not change the orientation of . Hence coincides with . This allows us to simplify the definition of the Euler angles as follows:

(or ) represents a rotation around the axis, (or ) represents a rotation around the axis, (or ) represents a rotation around the axis.

Examples: Wigner uses a order to define the Euler angles. Hirschfelder et al. use a order to define the Euler angles.

Definition by extrinsic rotations

Extrinsic rotations are elemental rotations that occur about the axes of the fixed coordinate system . The system rotates, while is fixed. Starting with overlapping , a composition of three extrinsic rotations can be used to reach any target orientation for . The Euler or Tait–Bryan angles () 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 system rotates about the axis by . The axis is now at angle with respect to the axis. The system rotates again, but this time about the axis by . The axis is now at angle with respect to the axis. The system rotates a third time, about the axis again, by angle .

In sum, the three elemental rotations occur about , and . This sequence is often denoted (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 the six possibilities for each).

Examples: Bradley and Cracknell and Evarestov use rotations around fixed (non-primed) axes.

If each step of the rotation acts on the rotating coordinate system XYZ, the rotation is intrinsic (). Intrinsic rotation can also be denoted 3-1-3.

Signs, ranges and conventions Angles are commonly defined according to the right-hand rule. 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 and , the range is defined modulo radians. For instance, a valid range could be . for , the range covers radians (but can not be said to be modulo ). For example, it could be or .

The angles , and are uniquely determined except for the singular case that the and the planes are identical, i.e. when the axis and the axis have the same or opposite directions. Indeed, if the axis and the axis are the same, and only is uniquely defined (not the individual values), and, similarly, if the axis and the axis are opposite, and only 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:

(intrinsic rotations) or (extrinsic rotations) (intrinsic rotations) or (extrinsic rotations) (intrinsic rotations) or (extrinsic rotations) (intrinsic rotations) or (extrinsic rotations) (intrinsic rotations) or (extrinsic rotations) (intrinsic rotations) or (extrinsic rotations)

Precession, nutation and intrinsic rotation

Precession, nutation, and intrinsic rotation are defined as the movements obtained by changing one of the Euler angles while leaving the other two constant. These motions are not all expressed in terms of the external frame, or all in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation systemprecession moves the line of nodes around the external axis , nutation rotates around the line of nodes , and intrinsic rotation is around , an axis fixed in the body that moves.

Note that if an object undergoes a certain change of orientation this can be described as a combination of precession, nutation, and internal rotation, but how much of each depends on what coordinate system one has chosen for the object.

As an example, consider a top. If we define the axis to be the symmetry axis of the top, then the top spinning around its own axis of symmetry corresponds to 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 may wobble up and down (if it is not what is called a symmetric top); the change of inclination angle is nutation. The same example can be seen with the movements of the earth.

Though all three movements can be represented by rotation matrices, only precession can be expressed in general as a matrix in the basis of the space without dependencies on the other angles.

These movements also behave as a gimbal set. Given 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 Scottish mathematical physicist Peter Guthrie Tait (1831–1901) and English applied mathematician George H. Bryan (1864–1928). 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. , or ), while proper Euler angles use the same axis for both the first and third elemental rotations (e.g., , or ).

This implies a different definition for the line of nodes 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. and ). 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. and ).

Conventions

The three elemental rotations may occur either about the axes of the original coordinate system, which remains motionless (extrinsic rotations), 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:

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 and covers radians. For the range covers radians.

Alternative names

Euler angles

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