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 saying that a rotation matrix can be decomposed as a product of three elemental rotations.
Euler angles are a means of representing the spatial orientation of any frame of the space as a composition of rotations from a reference frame. In the following the fixed system is denoted in lower case (x,y,z) and the rotated system is denoted in upper case letters (X,Y,Z).
The definition is Static. The intersection of the xy and the XY coordinate planes is called the line of nodes (N).
This previous definition is called z-x-z convention and is one of several common conventions; others are x-y-z and z-y-x. Unfortunately the order in which the angles are given and even the axes about which they are applied has never been “agreed” upon. When using Euler angles the order and the axes about which the rotations are applied should be supplied.. This article uses the convention described in the adjacent drawing, unless stated otherwise. It arises from considering Euler Angles as an extension of the spherical coordinate system.
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.
The angles α, β and γ are uniquely determined except for the singular case that the xy and the XY planes are identical, the z axis and the Z axis having 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 = π and only (α-γ) is uniquely defined (not the individual values). These ambiguities are known as gimbal lock in applications.
Relationship with physical motions
Euler rotations are defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never 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 and the third one is an intrinsic rotation around an axis fixed in the body that moves.
of a precession, a nutation and a rotation. Then, the following properties hold true
As consequence of this property, Euler rotations are commutative in the sense that given any frame, it is the same to perform over it any sequence of these rotations regardless of the order. For example, is the same to perform a precession φ followed by a nutation θ , as - instead - to perform the nutation θ followed by the precession φ , as can be shown from the former equations:
This partial commutability can be easily seen for γ =0 using the analogy of the gimbal. The same applies for any combination of all three rotations.
Euler angles as composition of Euler rotations. Gimbal analogy
Intermediate frames: Taking some vectors i, j and k over the axes x, y and z, and vectors I, J, K over X, Y and Z, and a vector N over the line of nodes, some intermediate frames can be defined using the vector cross product, as following:
These intermediate frames are such that they differ from the previous one in just a single elemental rotation. This proves that:
Gimbal analogy: If we suppose a set of frames, able to move each respect the former just one angle, like a gimbal, there will be one initial, one final and two 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.
Euler angles as composition of Euler rotations: When these rotations are performed on a frame whose Euler angles are all zero, the rotating XYZ system starts coincident with the fixed xyz system. The first rotation is performed around z (which is parallel to Z), the second around the line of nodes N (which at this point is over X), and the third around Z.
Euler angles as composition of intrinsic rotations
Assumed some initial conditions, a composition of three intrinsic rotations can be used to reach any target frame from the reference frame. The value of the rotations are the Euler Angles.
Given two coordinate systems xyz and XYZ with common origin, starting with the axis z and Z overlapping, the position of the second can be specified in terms of the first using three rotations with angles α, β, γ in three ways equivalent to the former definition, as follows:
The XYZ system is fixed while the xyz system rotates. Starting with the xyz system coinciding with the XYZ system, the same rotations as before can be performed using only rotations around the moving axis.
Naming conventions.This equivalence is normally used to name the possible conventions of Euler Angles. If we are told that some angles are given using the convention Z-X-Z, this means that they are equivalent to three concatenated intrinsic rotations around the axes Z, X and Z in that order. This composition is non-commutative. It has to be applied in such a way that in the beginning one of the intrinsic axis moves together with the line of nodes. Unfortunately this is also non-standard, and there are different ways to name the same convention, for example, stating explicitly that rotation axis are different for each step, as in Z-X '-Z' '.
Euler angles as composition of extrinsic rotations
Also composition of rotations in the reference frame can be used to reach any target frame. Let xyz system be fixed while the XYZ system rotates. Start with the rotating XYZ system coinciding with the fixed xyz system.
This can be shown to be equivalent to the previous statement:
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:
Composing rotations about fixed axes is to multiply our orientation matrix by the left. Composing rotations about the moving axes is to multiply the orientation matrix by the right. Both methods will lead to the same final decomposition. If M = A.B.C is the orientation matrix (the components of the frame to be described in the reference frame), it can be reached 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 ABC is obtained.
Again, this kind of compositions is non-commutative.
As consequence of the relationship between Euler angles and Euler rotations, we can find a Matrix expression for any frame given its Euler angles, here named as , , and . Using the z-x-z convention, a matrix can be constructed that transforms every vector of the given reference frame in the corresponding vector of the referred frame.
Define three sets of coordinate axes, called intermediate frames with their origin in common in such a way that each one of them differs from the previous frame in an elemental rotation, as if they were mounted on a gimbal. In these conditions, any target can be reached performing three simple rotations, because two first rotations determine the new Z axis and the third rotation will obtain all the orientation possibilities that this Z axis allows. These frames could also be defined statically using the reference frame, the referred frame and the line of nodes.
A matrix representing the end result of all three rotations is formed by successive multiplication of the matrices representing the three simple rotations, as in the following transformation equation
Carrying out the matrix multiplication, and abbreviating the sine and cosine functions as s and c, respectively, results in
There are 12 possible conventions regarding the Euler angles in use. The above description works for the z-x-z form. Similar conventions are obtained by selecting different axes (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.
A second kind of conventions is with the three rotation matrices with a different axis. z-y-x for example. There are also six possibilities of this kind (xyz, xzy, zxy, zyx, yzx, yxz). They behave slightly differently. In the zyx case, the two first rotations determine the line of nodes and the axis x, and the third rotation is around x.
The first convention (zxz) is properly known as Euler angles and the second one is known sometimes as Cardan angles, or yaw, pitch and roll, or Tait-Bryan angles, though sometimes Aviators and aerospace engineers, when referring to rotations about a moving body principal axes, often call these "yaw, pitch and roll" or even "Euler angles".
Table of matrices
The following matrices assume fixed (world) axes and column vectors, with rotations acting on objects rather than on reference frames. A matrix like that for xzy is constructed as a product of three matrices, Rot(y,θ3)Rot(z,θ2)Rot(x,θ1). To obtain a matrix for the same axis order but with referred frame (body) axes, use the matrix for yzx with θ1 and θ3 swapped. In the matrices, c1 represents cos(θ1), s1 represents sin(θ1), and similarly for the other subscripts.
Matrix expression for Euler rotations
As a precession is equivalent to a rotation from the reference frame, both are equivalent to a left-multiplication. We can perform a precession of a given frame 'F' with just the product R.F, where R is the rotation matrix for the angle we want to rotate. In a similar way, we can perform an intrinsic rotation with a right multiplication of the matrix of the frame. Nevertheless for nutation rotations we cannot use directly the matrix of the rotation we want to perform. We have to perform first a change of basis.
As a rotation is a Linear operator and its matrix is known in the rotated frame, it can be put back into the reference frame as B.R.B , being B the matrix of the basis in which R is known. As rotations are orthonormal, B= B and therefore, the matrix of a nutation β in the reference frame is:
The same can be applied to the intrinsic rotation. Being R the intrinsic rotation matrix in the twice-rotated frame, it can be taken back to lab frame as B.R.B , where B=P.N is the basis of the reference frame after precesion and nutation and B= B = N.P:
Derivation of the Euler angles of a given frame
The fastest way to get the Euler Angles of a frame is to write the three given vectors as columns of a matrix and compare it with the expression of the theoretical matrix (see former table of matrices). Hence the three Euler Angles can be calculated.
Nevertheless, the same result can be reached avoiding matrix calculus, which is more geometrical. Given a frame (X, Y, Z) expressed in coordinates of the reference frame (x, y, z), its Euler Angles can be calculated searching for the angles that rotate the unit vectors (x,y,z) to the unit vectors (X,Y,Z)
The inner product between the unit vectors z and Z is
and the cross product vector
has the magnitude
where, in general, arg(u,v) is the polar argument of the vector (u,v), taking values in the range [-π < arg(u,v) < π].
If is parallel or antiparallel to (where β=0 or β=π, respectively), it is a singular case for which α and γ are not individually defined. If this is not the case, is non-zero and has the same direction as the unit vector of the figure above. Therefore,
It can be calculated that
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. The space of rotations is called in general "The Hypersphere of rotations".
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).
Their main advantage over other orientation descriptions is that they are directly measurable from a gimbal mounted in an aircraft. As gyroscopes keep its rotation axis constant, angles meassured in a gyro frame are equivalent to angles meassured in the lab frame. Therefore gyros are used to know the actual position of moving spacecrafts, and Euler angles are directly measurable. Intrinsic rotation angle cannot be read from a single gimbal, so there has to be more than one gimbal in a spacecraft. Normally there are at least three for redundancy. There is also a relation to the well-known gimbal lock problem of Mechanical Engineering .
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:
where IJK are unit vectors for XYZ.
Here the rotation sequence is 3-1-3 (or Z-X-Z using the convention stated above).
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.
Expressing rotations in 3D as unit quaternions instead of matrices has some advantages:
Published - July 2009
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