|
Euler angles are a means of representing the spatial orientation of an object. An object in space is said to have 6 degrees of freedom, meaning it's position can be described with a minimum of 6 numbers, three for position and three for orientation. The three for position are fairly intuitive (see cartesian, spherical, cylindrical, coordinates. Orientation is more complicated. Consider rotations about cartesian coordinates: x, y, z. It is easy enough to imagine rotating an object some amount about each axis and that is the basic concept euler angles. The complicating factor is that the order of rotations is significant. For example assume you were given three angles: 10,20 and 30 (degrees). One conventional euler angle representation (1,2,3 or x,y,z) would dictate you rotate the object 10 degrees about the object's x axis, then 20 degrees about the object's new y axis and then 30 degrees about the objects new z axis. It may be surprising to some that the object would have a different orientation if these rotations were carried out in reverse. Image File history File links Circle-question-red. ...
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
The Euler angles were developed by Leonhard Euler to describe the orientaion of a rigid body in 3-dimensional Euclidean space. A closed book is a reasonable approximation to a rigid body or object; an open book, being leafed through, is not. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles. The motivation for this description is as follows. Euler redirects here. ...
In physics, a rigid body is an idealisation of a solid body of finite size in which deformation is neglected. ...
:For other senses of this word, see dimension (disambiguation). ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
A sphere rotating around its axis. ...
The motion of a rigid body is governed by Newton's Laws of Motion, which apply to inertial frames (i.e. not rotating relative to the "fixed stars"). To describe the motion of the object we apply: (1) Newton's Laws, to obtain its center-of-mass motion (driven by the net force on the object); and (2) Euler's equations -- based on Newton's Laws -- to obtain its rotational motion (driven by the net torque on the object, measured about its center-of-mass). Thus, if you throw an empty cup in the air, it has a complicated motion, but one described by only six numbers: three coordinates in space (for its center-of-mass) and three angles (for its orientation); Newton tells us how to get the first three and Euler tells us how to get the second three.
Let us denote by xyz the inertial frame coordinates (e.g., us), and by XYZ a set of coordinates attached to the rigid body (e.g., the cup). Although all xyz inertial coordinates will work for Newton's Laws, only a special set of coordinates XYZ, called the principal axis coordinates, gives a simple form for Euler's equations. The Euler angles tell us the orientation of XYZ relative to xyz.
Just as Newton's Second Law gives the rate of change of the linear momentum of an object, so Euler's equations give the rate of change of the angular momentum of that object. Euler's equations are expressed in terms of the object's angular velocity, which is the rate of change of its orientation. Although "angular velocity" can mean either the motion of XYZ (e.g., the cup) relative to xyz (e.g., us) or vice-versa, fortunately they are the same when expressed in the same set of coordinates. Thus, at some time in our xyz frame these numbers might be, in units of radians per second, (5,0,0), but in the cup's XYZ frame they might be (3,-4,0); so we must remember to translate the XYZ information to the xyz frame, or vice-versa.
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. In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. ...
Definition Given two coordinate systems xyz and XYZ with common origin, one can specify the position of the second in terms of the first using three angles α, β, γ in three equivalent ways, as follows: Image File history File links Download high resolution version (720x720, 7 KB) This Math image (or all images in this article or category) should be recreated using vector graphics as an SVG file. ...
Image File history File links Download high resolution version (720x720, 7 KB) This Math image (or all images in this article or category) should be recreated using vector graphics as an SVG file. ...
Image File history File links Euler_stereo3. ...
Image File history File links Euler_stereo3. ...
Stereoscopy, stereoscopic imaging or 3-D (three-dimensional) imaging is a technique to create the illusion of depth in a photograph, movie, or other two-dimensional image, by presenting a slightly different image to each eye. ...
Image File history File links Euler_anaglyph. ...
Image File history File links Euler_anaglyph. ...
Stereo image anaglyphed for red (left eye) and cyan (right eye) filters. ...
- Static The intersection of the xy and the XY coordinate planes is called the line of nodes.
- 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. (Note however, that the first figure has left-handed coordinate systems.)
- Fixed axes of rotation Start with the XYZ system equalling the xyz system.
- Rotate the XYZ-system about the z-axis by α; the xyz-system does not move, now or later.
- Rotate it again about the x-axis by β.
- Rotate it a third time about the z-axis by γ.
- (Note that the first and third axes are identical.)
- Moving axes of rotation Start with the XYZ system equalling the xyz system.
- Rotate the XYZ-system about the Z-axis by γ; the xyz-system does not move, now or later.
- Rotate it again about the now rotated X-axis by β.
- Rotate it a third time about the now doubly rotated Z-axis by α.
- (Note that the angles are in reverse order.)
These three angles α, β, γ are the Euler angles. The equivalence of these three definitions is verified below.
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. (This is most easily verified using the static description.) The radian is a unit of plane angle. ...
- 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.) In gyroscopic devices controlled by Euler mechanics or Euler angles, gimbal lock is caused by the alignment of two of the three gimbals together so that one of the rotation references (pitch/yaw/roll, often yaw) is cancelled. ...
Relation to physical 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.
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. Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...
In linear algebra and geometry, a coordinate rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. ...
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.
Conventions There are numerous conventions regarding the Euler angles in use. The above three descriptions are for the z-x-z form ("x-convention"). z-y-z ("y-convention") is also common, especially in quantum mechanics. One also finds variation over the use of left- versus right-handed coordinate systems, clockwise versus counterclockwise angles, and active versus passive coordinate transformations.
Even within a given mathematical choice of convention, one frequently finds different and conflicting choice of notation. Authors have been known to use conventions incorrectly.
To add to the confusion, flight and aerospace engineers, when using yaw, pitch, and roll (also called heading, attitude, bank) to refer to rotations about the z, y, x axes, respectively, often call these the Euler angles. These x-y-z angles are properly known as the Tait-Bryan angles, also called Cardan angles or nautical angles. Aerospace engineering is the branch of engineering concerning aircraft, spacecraft and related topics. ...
Flight dynamics is the study of orientation of air and space vehicles and how to control the critical flight parameters, typically named pitch, roll and yaw. ...
In geometry, Tait-Bryan angles are three angles used to describe a general rotation in three-dimensional Euclidean space by three successive rotations, once about the x-axis, once about the y-axis, and once about the z-axis. ...
Gerolamo Cardano or Jerome Cardan or Girolamo Cardan (September 24, 1501 - September 21, 1576) was a celebrated Italian Renaissance mathematician, physician, astrologer, and gambler. ...
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. In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...
In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ...
In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. ...
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. In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ...
In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ...
Applications Euler angles are used extensively in the classical mechanics of rigid bodies, and in the quantum mechanics of angular momentum. In physics, a rigid body is an idealisation of a solid body of finite size in which deformation is neglected. ...
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. Kinetic energy is the energy by virtue of the motion of an object. ...
The angular velocity, in body coordinates, of a rigid body takes a simple form using Euler angles: Angular velocity describes the speed of rotation. ...
 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). In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ...
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. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
In mathematics, Euler-Rodrigues parameters, also called just Euler parameters, are four numbers a, b, c, d such that a2+b2+c2+d2=1. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
In gyroscopic devices controlled by Euler mechanics or Euler angles, gimbal lock is caused by the alignment of two of the three gimbals together so that one of the rotation references (pitch/yaw/roll, often yaw) is cancelled. ...
See also In geometry a rotation representation expresses the orientation of an object (or coordinate frame) relative to a coordinate reference frame. ...
In mathematics, Eulers rotation theorem states that any rotation has an axis. ...
A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. ...
In geometry, Tait-Bryan angles are three angles used to describe a general rotation in three-dimensional Euclidean space by three successive rotations, once about the x-axis, once about the y-axis, and once about the z-axis. ...
A point plotted using the spherical coordinate system In Mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates, (Ï, φ, θ), where Ï represents the radial distance of a point from a fixed origin, φ represents the zenith angle from the positive z-axis and...
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.
- Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, W. H. Freeman, San Francisco, CA, 1973.
- M. E. Rose, Elementary Theory of Angular Momentum, John Wiley, New York, NY, 1957.
External link |
Feeds |
Contact