Euler angles

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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.

Contents

• 1 Definition
  • 1.1 Angles signs and ranges
• 2 Relationship with physical motions
  • 2.1 Euler rotations
  • 2.2 Euler angles as composition of Euler rotations. Gimbal analogy
  • 2.3 Euler angles as composition of intrinsic rotations
  • 2.4 Euler angles as composition of extrinsic rotations
• 3 Other conventions
  • 3.1 Naming conventions
• 4 Matrix notation
  • 4.1 Table of matrices
  • 4.2 Matrix expression for Euler rotations
• 5 Derivation of the Euler angles of a given frame
• 6 Properties of Euler angles
• 7 Higher dimensions
• 8 Applications
• 9 See also
• 10 Notes
• 11 References
• 12 External links

[edit] Definition

Euler angles are a means of representing the spatial orientation of any frame of reference (coordinate system) as a composition of rotations from a reference frame of reference(coordinate system). In the following the fixed system is denoted inlower case (x,y,z) and the rotated system is denoted in upper caseletters (X,Y,Z).

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.

Euler angles are one of several waysof specifying the relative orientation of two such coordinate systems.Moreover, different authors may use different sets of angles todescribe these orientations, or different names for the same angles,leading to different conventions. Therefore any discussion employing Euler angles should always be preceded by their definition^[1].

Unless otherwise stated, this article will use the convention described in the adjacent drawing, usually named Z-X-Z.

[edit] Angles signs and ranges

Normally, angles are defined in such a way that they are positivewhen they rotate counter-clock-wise (how they rotate depends on whichside of the rotation plane we observe them from. The positive side willbe the one of the positive axis of rotation)

About the ranges:

• α and γ range are defined modulo 2π radians. A valid range could be (-π, π].
• β range covers π radians (but can't be said to be modulo π). For example could be [0, π] or [-π/2, π/2].

The angles α, β and γ are uniquely determined except for the singular case that the xy and the XYplanes are identical, the z axis and the Z axis having the same oropposite 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 andonly (α-γ) is uniquely defined (not the individual values). Theseambiguities are known as gimbal lock in applications.

[edit] Relationship with physical motions

[edit] Euler rotations

Euler rotations are defined as the movement obtained by changing oneof the Euler angles while leaving the other two constant. Eulerrotations are never expressed in terms of the external frame, or interms of the co-moving rotated body frame, but in a mixture. Theyconstitute a mixed axes of rotation system, where the firstangle moves the line of nodes around the external axis z, the secondrotates around the line of nodes and the third one is an intrinsicrotation around an axis fixed in the body that moves.

These rotations are called Precession, Nutation, and intrinsic rotation. They satisfy the following remarkable property: Write the rotation about the given angles (φ,θ,ψ) as a composition

A(φ,θ,ψ) = R(φ,θ,ψ)N(φ,θ)P(φ)

of a precession, a nutation and a rotation. Then, the following properties hold true

A(δφ + φ,θ,ψ) = P(δφ)A(φ,θ,ψ)

A(φ,δθ + θ,ψ) = N(φ,δθ)A(φ,θ,ψ)

A(φ,θ,δψ + ψ) = R(φ,θ,δψ)A(φ,θ,ψ)

As consequence of this property, Euler rotations are commutativeamong them; that is, given any frame, after performing over it anysequence of these rotations yields the same result, regardless of theirorder. For example, it 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 . This can be shown from the former equations:

P(δφ)N(φ,δθ)A(φ,θ,ψ) = A(δφ + φ,δθ + θ,ψ) = N(φ,δθ)P(δφ)A(φ,θ,ψ)

This restricted commutativity can be easily seen using the analogyof the gimbal. The same applies for any combination of all threerotations. Nevertheless these three rotations cannot be said to be acommutative subgroup into the set of rotations, because normally only fixed-axis rotations are considered in this group.

[edit] Euler angles as composition of Euler rotations. Gimbal analogy

Intermediate frames: Taking some vectors i, j and k over theaxes x, y and z, and vectors I, J, K over X, Y and Z, and a vector Nover the line of nodes, some intermediate frames can be defined usingthe vector cross product, as following:

• origin: [i,j,k] (where k = i x j)
• first: [N,kxN,k]
• second: [N,KxN,K]
• final: [I,J,K]

These intermediate frames are such that they differ from the previous one in just a single elemental rotation. This proves that:

• Any target frame can be reached from the reference frame just composing three rotations.
• The values of these three rotations are exactly the Euler angles of the target frame.

Gimbal analogy: If we suppose a set of frames, able to moveeach with respect to the former according to just one angle, like agimbal, there will be one initial, one final and two in the middle,which are called intermediate frames. The two in the middle work as twogimbal rings that allow the last frame to reach any orientation inspace.

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. In fact, the order in this case does not matter, the former rotations being commutative.

[edit] Euler angles as composition of intrinsic rotations

Starting with an initial set of reference axes, say XYZ, a composition of three intrinsic rotations can be used to reach any target frame with an origin coincident with that of XYZ from the reference frame. The value of the rotations are the Euler Angles.

Given two coordinate systems xyz and XYZ with commonorigin, starting with the axis z and Z overlapping, the position of thesecond can be specified in terms of the first using three rotationswith 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.

• 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.

This equivalence is normally used to name conventions.

[edit] 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.

• • 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. The first and third axes are identical.

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:

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) .

Composing rotations about fixed axes is to multiply our orientationmatrix by the left. Composing rotations about the moving axes is tomultiply the orientation matrix by the right. Both methods will lead tothe 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.

[edit] Other conventions

There are 12 possible conventions regarding the Euler angles in use. The above description works for the z-x-zform. Similar conventions are obtained by selecting different axes(zyz, xyx, xzx, yzy, yxy). There are six possible combinations of thiskind, and all of them behave in an identical way to the one describedbefore.

A second kind of conventions is with the three rotation matrices with a different axis. z-y-xfor 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, andthe third rotation is around x.

The first convention (zxz) is properly known as Euler angles and the second one is known sometimes as Nautical angles, Cardan angles, Tait-Bryan angles or yaw, pitch and roll.They are defined using as line of nodes the intersection of twonon-homologous planes (for example XZ and xz are homologous while XZand xy are not), unlike proper Euler angles^[2].

Aviators and aerospace engineers, when referring to rotations about a moving body principal axes, often call these "yaw, pitch and roll".

[edit] Naming conventions

The previous intrinsic rotations equivalenceis normally used to name the possible conventions of Euler Angles. Ifwe are told that some angles are given using the convention Z-X-Z, thismeans that they are equivalent to three concatenated intrinsicrotations around the axes Z, X and Z in that order. This composition isnon-commutative. It has to be applied in such a way that in thebeginning one of the intrinsic axis moves together with the line ofnodes.

Nevertheless, sometimes the extrinsic rotations equivalencecould be used. If this is the case, the given angles are backwards,meaning that the first angle is the intrinsic rotation and the last onethe precession. The name of the convention would be indistinguishablefrom the previous one, even if the angles' order is the opposite, beingsomething like Z-X-Z.

To specify that the given order means intrinsic composition,sometimes a similar notation is used, but stating explicitly whichrotation axis are different for each step, as in Z-X’-Z’’. Using thisnotation, Z-X-Z would mean extrinsic composition.

For Tait-Bryan angles, also intrinsic and extrinsic conventions canbe used, giving therefore two meanings for every convention name. Forexample, X-Y-Z, using intrinsic convention, means that a X-rotation isperformed, composing intrinsic rotations Y and Z later, but usingextrinsic convention means that after the X rotation, extrinsicrotations Y and Z are performed. The meaning is different in both cases.

The static parameters of the convention can be calculated from thename. For example, given the convention X-Y’-Z’’, the first rotation isperpendicular to "x" and the last one to "Z". Therefore the planes are the yz and the XY, and the line of nodes is the intersection of these two.

[edit] Matrix notation

As consequence of the relationship between Euler angles and Eulerrotations, we can find a Matrix expression for any frame given itsEuler angles, here named as , , and . Using the z-x-zconvention, a matrix can be constructed that transforms every vector ofthe given reference frame in the corresponding vector of the referredframe.

Define three sets of coordinate axes, called intermediate frameswith their origin in common in such a way that each one of them differsfrom the previous frame in an elemental rotation, as if they weremounted on a gimbal.In these conditions, any target can be reached performing three simplerotations, because two first rotations determine the new Z axis and thethird rotation will obtain all the orientation possibilities that thisZ axis allows. These frames could also be defined statically using thereference frame, the referred frame and the line of nodes.

A matrix representing the end result of all three rotations isformed by successive multiplication of the matrices representing thethree simple rotations, as in the following transformation equation

where

where

• The rightmost matrix represents the α rotation around the axis ' z ' of the original reference frame
• The middle matrix represents the β rotation around an intermediate ' x ' axis which is the "line of nodes".
• The leftmost matrix represents the γ rotation around the axis ' Z ' of the final reference frame

Carrying out the matrix multiplication, and abbreviating the sine and cosine functions as s and c, respectively, results in the following:

[edit] Table of matrices

The following matrices assume fixed (world) axes and column vectors,with rotations acting on objects rather than on reference frames. Amatrix 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.

xzx xzy xyx xyz yxy yxz yzy yzx zyz zyx zxz zxy

[edit] Matrix expression for Euler rotations

As a precession is equivalent to a rotation from the referenceframe, both are equivalent to a left-multiplication. We can perform aprecession of a given frame 'F' with just the product R.F, where R isthe rotation matrix for the angle we want to rotate. In a similar way,we can perform an intrinsic rotation with a right multiplication of thematrix of the frame. Nevertheless for nutation rotations we cannot usedirectly the matrix of the rotation we want to perform. We first haveto perform 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 ^{− 1}, being B the matrix of the basis in which R is known. As rotations are orthonormal, B ^{− 1} = B^t and therefore, the matrix of a nutation β in the reference frame is:

The same can be applied to the intrinsic rotation. Being R theintrinsic rotation matrix in the twice-rotated frame, it can be takenback to lab frame as B.R.B ^{− 1}, where B=P.N is the basis of the reference frame after precession and nutation and B ^{− 1} = B^t = N^t.P^t:

[edit] Derivation of the Euler angles of a given frame

The fastest way to get the Euler Angles of a frame is to write thethree given vectors as columns of a matrix and compare it with theexpression 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 matrixcalculus, which is more geometrical. Given a frame (X, Y, Z) expressedin coordinates of the reference frame (x, y, z), its Euler Angles canbe 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

Therefore,

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,

For the numerical computation of arg(u,v), the standard function ATAN2(v,u) (or in double precision DATAN2(v,u)), available in the programming language FORTRAN for example, can be used. In case

and

It can be calculated that

and that

and that

and that

In summary,

[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. The space of rotations is called in general "The Hypersphere of rotations".

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.

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

[edit] Higher dimensions

It is possible to define parameters analogous to the Euler angles in dimensions higher than three.

The number of degrees of freedom of a rotation matrix is always lessthan the dimension of the matrix squared. That is, the elements of arotation matrix are not all completely independent. For example, therotation matrix in dimension 2 has only one degree of freedom, sinceall four of its elements depend on a single angle of rotation. Arotation matrix in dimension 3 (which has nine elements) has threedegrees of freedom, corresponding to each independent rotation, forexample by its three Euler angles or a magnitude one (unit) quaternion.

In SO(4) the rotation matrix is defined by two quaternions,and is therefore 6-parametric (three degrees of freedom for everyquaternion). The 4x4 rotation matrices have therefore 6 out of 16independent components.

Any set of 6 parameters that define the rotation matrix could be considered an extension of Euler angles to dimension 4.

In general, the number of euler angles in dimension D is quadraticin D; since any one rotation consists of choosing two dimensions torotate between, the total number of rotations available in dimension Dis , which for D=2,3,4 yields N_rot = 1,3,6.

[edit] Applications

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 measured in agyro frame are equivalent to angles measured in the lab frame.Therefore gyros are used to know the actual orientation of movingspacecrafts, and Euler angles are directly measurable. Intrinsicrotation angle cannot be read from a single gimbal, so there has to bemore than one gimbal in a spacecraft. Normally there are at least threefor redundancy. There is also a relation to the well-known gimbal lock problem of Mechanical Engineering ^[3] .

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 bodycoordinates are considered embedded in the moving body. Calculationsinvolving kinetic energyare usually easiest in body coordinates, because then the moment ofinertia tensor does not change in time. If one also diagonalizes therigid body's moment of inertia tensor (with nine components, six ofwhich are independent), then one has a set of coordinates (called theprincipal axes) in which the moment of inertia tensor has only threecomponents.

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 beendone using Euler angles. In the early history of quantum mechanics,when physicists and chemists had a sharply negative reaction towardsabstract group theoretic methods (called the Gruppenpest), reliance on Euler angles was also essential for basic theoretical work.

In materials science, crystallographic texture(or preferred orientation) can be described using Euler angles. Intexture analysis, the Euler angles provide the necessary mathematicaldepiction of the orientation of individual crystallites within apolycrystalline material, allowing for the quantitative description ofthe macroscopic material.^[4]

Naval gun fire control systems require corrections to gun-orderangles (bearing and elevation) to compensate for deck tilt (pitch androll). In traditional systems, a stabilizing gyroscope with a verticalspin axis corrects for deck tilt, and stabilizes the optical sights andradar antenna. However, gun barrels point in a direction different fromthe line of sight to the target, to anticipate target movement and fallof the projectile due to gravity, among other factors. Gun mounts rolland pitch with the deck plane, but also require stabilization. Gunorders include angles computed from the vertical gyro data, and thosecomputations involve Euler angles.

Unit quaternions, also known as Euler-Rodrigues parameters, provide another mechanism for representing 3D rotations. This is equivalent to the special unitary group description.

Expressing rotations in 3D as unit quaternions instead of matrices has some advantages:

• Concatenating rotations is faster and more stable.
• Extracting the angle and axis of rotation is simpler.
• Interpolation is more straightforward. See for example slerp.