Euler angles

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This article deals with the mathematics usage of the word. For aerospace use of the word see Tait-Bryan angles

The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body (a body in which the relative position of all its points is constant) in 3-dimensional Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles. This is equivalent to say that a rotation matrix can be decomposed as a product of three elemental rotations.

1 Definition
- 1.1 Angle ranges
2 Relation to physical motions
- 2.1 Geometric meaning of the motions
3 Equivalence of the definitions
4 Matricial notation
5 Other conventions
6 Properties of Euler angles
7 Applications
8 See also
9 References
10 External link

[edit] Definition

Euler angles - The xyz (fixed) system is shown in blue, the XYZ (rotated) system is shown in red. The line of nodes, labelled N, is shown in green.

Stereoscopic image of Euler angles

Anaglyph image of Euler angles

Euler angles (pronounced "Oiler") are a means of representing the spatial orientation of any frame of the space as a composition of rotations from a reference frame.

The definition is Static. The intersection of the xy and the XY coordinate planes is called the line of nodes (N).

α is the angle between the x-axis and the line of nodes.
β is the angle between the z-axis and the Z-axis.
γ is the angle between the line of nodes and the X-axis.

Unfortunately the order in which the rotations are applied and even the axis about which they are applied has never been “agreed” upon. When using Euler angles the order and axes the rotations are applied should be supplied. Mathworld does a good job describing this issue: http://mathworld.wolfram.com/EulerAngles.html

Euler angles are one of several ways of specifying the relative orientation of two such coordinate systems. Moreover, different authors may use different sets of angles to describe these orientations, or different names for the same angles. Therefore a discussion employing Euler angles should always be preceded by their definition.

[edit] Angle ranges

α and γ range from 0 to 2π radians.
β ranges from 0 to π radians.

These angles are uniquely determined, with certain exceptions.

With α and γ, 0 and 2π radians give the same 3D rotation.
With β, 0 and π give the same 3D rotation.

This corresponds to the xy and the XY planes being identical, so the rotation is just a rotation of α+γ about the z-axis. (This last ambiguity is known as gimbal lock in applications.)

[edit] Relation to physical motions

The former definition can be seen as composition of three rotations around the intrinsecal (moving) or extrinsecal (fixed) axis in some initial conditions.

Given two coordinate systems xyz and XYZ with common origin, starting with the axis z and Z overlaping, one can specify the position of the second in terms of the first using three rotations with angles α, β, γ in three ways equivalent to the former definition, as follows:

Mixed axes of rotation - The xyz system is fixed while the XYZ system rotates. Start with the rotating XYZ system coinciding with the fixed xyz system. Perform the first rotation around z, the second around the line of nodes N and the third around Z. In this way we will reach the final frame starting from the initial one.

Moving axes of rotation (See Tait-Bryan angles) The xyz system is fixed while the XYZ system rotates. If we start with the XYZ system coinciding with the xyz system we can perform the same rotations than before using only rotations around the moving axis.
- Rotate the XYZ-system about the Z-axis by α. The X-axis now lies on the line of nodes.
- Rotate the XYZ-system again about the now rotated X-axis by β. The Z-axis is now in its final orientation, and the X-axis remains on the line of nodes.
- Rotate the XYZ-system a third time about the new Z-axis by γ.
  
  (Note that the angles are in reverse order.)

Fixed axes of rotation - The xyz system is fixed while the XYZ system rotates. Start with the rotating XYZ system coinciding with the fixed xyz system.
- Rotate the XYZ-system about the z-axis by γ. The X-axis is now at angle γ with respect to the x-axis.
- Rotate the XYZ-system again about the x-axis by β. The Z-axis is now at angle β with respect to the z-axis.
- Rotate the XYZ-system a third time about the z-axis by α.
  
  (Note that the first and third axes are identical - the z-axis. This is why Z and z have to start overlapping)

These three angles α, β, γ are the Euler angles. The equivalence of these three definitions is verified below.

[edit] Geometric meaning of the motions

Having emphasized that "physical motions" have been abstracted away, their reappearance in two of the above definitions might seem inconsistent. In fact, these three motions are simply "nominal". The actual motion of an object may or may not follow the three Euler angles literally.

If, for example, a satellite has spin control in two orthogonal directions, then reorienting the satellite can be accomplished by using the Euler angles directly, in the moving axes definition. But if engineering reasons dictated a different control mechanism, Euler angles will still describe the before and after relative positions.

This is like using ordinary rectangular coordinates. A given x,y specifies x to the right, y forward, which may be used directly, as on street grids, or not.

[edit] Equivalence of the definitions

The static description is usually used in conjunction with spherical trigonometry. It is the only form in older sources. The two rotating axes descriptions are usually used in conjunction with matrices, since 2D coordinate rotations have a simple form. These last two are easily seen to be equivalent, since rotation about a moved axis is the conjugation of the original rotation by the move in question.

To be explicit, in the fixed axes description, let x(φ) and z(φ) denote the rotations of angle φ about the x-axis and z-axis, respectively. In the moving axes description, let Z(φ)=z(φ), X′(φ) be the rotation of angle φ about the once-rotated X-axis, and let Z″(φ) be the rotation of angle φ about the twice-rotated Z-axis. Then:

Z″(α)oX′(β)oZ(γ) = [ (X′(β)z(γ)) o z(α) o (X′(β)z(γ))⁻¹ ] o X′(β) o z(γ)

= [ {z(γ)x(β)z(−γ) z(γ)} o z(α) o {z(−γ) z(γ)x(−β)z(−γ)} ] o [ z(γ)x(β)z(−γ) ] o z(γ)

= z(γ)x(β)z(α)x(−β)x(β) = z(γ)x(β)z(α) .

The equivalence of the static description with the rotating axes descriptions can be verified by direct geometric construction, or by showing that the nine direction cosines (between the three xyz axes and the three XYZ axes) form the correct rotation matrix.

The equivalence of the static description with the rotating axes descriptions can be understood as external or internal composition of matrices. Composing rotations about fixed axes is multiply our orientation matrix by the left.Composing rotations about the moving axes is to multiply the orientation matrix by the left.

Both methods will lead to the same final decomposition. If M = A.B.C is our orientation matrix (the components of the frame to be described in the reference frame), we can reach it from composing C, B and A at the left of I (identity, reference frame on itself), or composing A, B and C at the right of I. Both ways we obtain ABC.

[edit] Matricial notation

We call $[\mathbf{R}]$ to the matrix that represents the coordinates of the referred frame respect the reference frame. We have seen before that its matrix is equivalent to the composition of three rotations. Therefore:

$[\mathbf{R}] = \begin{bmatrix} \cos \psi & \sin \psi & 0 \\ -\sin \psi & \cos \psi & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta\\ 0 & -\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} \cos \phi & \sin \phi & 0 \\ -\sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Notice that independently each matrix refers to a rotation around an axis, but when they are multiplied,

The inner (rightmost) matrix refers to a rotation around Z
The outer (leftmost) matrix refers a to rotation around the "fixed" axis z
The one in the middle represents a rotation around the line of nodes.

[edit] Other conventions

There are numerous conventions regarding the Euler angles in use. The above description works for the z-x-z form. Similar conventions are obtained just renaming the axis (zyz, xyx, xzx, yzy, yxy). There are six possible combinations of this kind, and all of them behave in an identical way to the one described before. Just imagine the three axis with other names.

A second kind of convention is with the three rotation matrices with a different axis. z-y-x for example. There are six possibilities of this kind. They behave slightly different. In the zyx case, the two first rotations determine the line of nodes and the axis x, and the third rotation is around x.

Therefore there are 12 possible combinations. The second kind of converntion sometimes are refered to as "yaw, pitch and roll", though usually this name is reserved for intrinsecal angles in a moving frame.

To add to the confusion, flight and aerospace engineers, when using yaw, pitch and roll to refer to rotations about moving frames, often call these the Euler angles. These angles in moving frames are properly known as the Tait-Bryan angles, also called Cardan angles or nautical angles.

[edit] Properties of Euler angles

The Euler angles form a chart on all of SO(3), the special orthogonal group of rotations in 3D space. The chart is smooth except for a polar coordinate style singularity along β=0. See charts on SO(3) for a more complete treatment.

A similar three angle decomposition applies to SU(2), the special unitary group of rotations in complex 2D space, with the difference that β ranges from 0 to 2π. These are also called Euler angles.

[edit] Applications

Euler angles are used extensively in the classical mechanics of rigid bodies, and in the quantum mechanics of angular momentum.

When studying rigid bodies, one calls the xyz system space coordinates, and the XYZ system body coordinates. The space coordinates are treated as unmoving, while the body coordinates are considered embedded in the moving body. Calculations involving kinetic energy are usually easiest in body coordinates, because then the moment of inertia tensor does not change in time. If one also diagonalizes the rigid body's moment of inertia tensor (with nine components, six of which are independent), then one has a set of coordinates (called the principal axes) in which the moment of inertia tensor has only three components.

The angular velocity, in body coordinates, of a rigid body takes a simple form using Euler angles:

$(\dot\alpha\sin\beta\sin\gamma+\dot\beta\cos\gamma){\bold I} +(\dot\alpha\sin\beta\cos\gamma-\dot\beta\sin\gamma){\bold J} +(\dot\alpha\cos\beta+\dot\gamma){\bold K},$

where IJK are unit vectors for XYZ.

In quantum mechanics, explicit descriptions of the representations of SO(3) are very important for calculations, and almost all the work has been done using Euler angles. In the early history of quantum mechanics, when physicists and chemists had a sharply negative reaction towards abstract group theoretic methods (called the Gruppenpest), reliance on Euler angles was also essential for basic theoretical work.

Haar measure for Euler angles has the simple form sin(β)dαdβdγ, usually normalized by a factor of 1/8π². For example, to generate uniformly randomized orientations, let α and γ be uniform from 0 to 2π, let z be uniform from −1 to 1, and let β = arccos(z).

Unit quaternions, also known as Euler-Rodrigues parameters, provide another mechanism for representing 3D rotations. This is equivalent to the special unitary group description. Quaternions are generally quicker for most calculations, conceptually simpler to interpolate, and are not subject to gimbal lock. Much high speed 3D graphics programming (gaming, for example) uses quaternions because of this.

[edit] See also

[edit] References

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Addison-Wesley, Reading, MA, 1981.
Herbert Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1980.
Andrew Gray, A Treatise on Gyrostatics and Rotational Motion, MacMillan, London, 1918.
M. E. Rose, Elementary Theory of Angular Momentum, John Wiley, New York, NY, 1957.
Symon, Keith (1971). Mechanics. Addison-Wesley, Reading, MA. ISBN 0-201-07392-7.
Landau, L.D.; Lifshitz, E.M. (1997). Mechanics. Butterworth-Heinemann. ISBN 0-750-62896-0.