described as a counterclockwise **rotation** by an **angle** θ about the z-axis. The **matrix** representation of this three-dimensional **rotation** is given by the real 3 × 3 special orthogonal **matrix**, R(zˆ,θ) ≡ cosθ −sinθ 0 sinθ cosθ 0 0 0 1 , (1) where the axis of **rotation** and the **angle** of **rotation** are speciﬁed as arguments of R. The most. eul = rotm2eul (rotm) converts a rotation matrix, rotm, to the corresponding Euler angles, eul. The input rotation matrix must be in the premultiply form for rotations. The default order for Euler angle rotations is "ZYX". example eul = rotm2eul (rotm,sequence) converts. The **Rotation Matrices** of elemental **rotations** along x-axis, y-axis and z-axis by **angle** α, β and ɣ respectively are given as below, Now to represent the orientation of the target frame w.r.t reference frame in **Rotation matrix** form, we need to multiply the three **Euler angles** (α β, ɣ) **rotation matrices** in order. The order of **matrix** multiplication of **rotational matrices** is of extreme importance. **Euler angles** are used to specify the orientation of one reference frame relative to another. I got stucked with the same problem and Zwähnia's solution is great and works perfectly in the case you works with **Euler** **angle** but couldn't help me with Tait-Bryan **angle**. There is kind of a bug and I couldn't track it down yet, so there is a workaround. This remarkable theorem works for any number of rotations and for other axis sequences than the Euler angles. There is a reference on https://www.researchgate.net/profile/Edward_Barile called Rotation Dyads and Coordinate Transformations for Moving Radar Platforms. . However, when things get more complex, **Euler** **angle** will be hard to work with. For instance : Interpolating smoothly between 2 orientations is hard. Naively interpolating the X,Y and Z **angles** will be ugly. Applying several **rotations** is complicated and unprecise: you have to compute the final **rotation** **matrix**, and guess the **Euler** **angles** from this. The **Euler angles** are used to define a sequence of three **rotations** , by the **angles** about the , , or , and axes, respectively. If the second **rotation** is about the axis, this is called the. tableau download. 1 day ago · Rotate Object by Joystick **Rotation**/Position on Gamepad | Unity C# Подробнее unity rotate towards analog stick, Tilting the stick forward and backward pivots the Y-axis shaft from side to side Discover the best assets for game making The transform tool, new to Unity 2018, combines the move, rotate, and scale tool into one swiss army knife Add the free.. Sep 18, 2021 · import math import numpy as np # RPY/**Euler** **angles** **to Rotation** Vector def **euler**_to_rotVec(yaw, pitch, roll): # compute the **rotation** matr... Level up your programming skills with exercises across 52 languages, and insightful discussion with our dedicated team of welcoming mentors.. Quaternions don't suffer from gimbal lock, unlike **Euler angles**. They can be represented as 4 numbers, in contrast to the 9 numbers of a **rotations matrix**. The conversion to and from axis/**angle** representation is trivial. Smooth interpolation between two quaternions is easy (in contrast to axis/**angle** or **rotation matrices**). For **euler** **angles**, if all three of them have the domain 0<=angle<=2*pi, then for every physical **rotation** there are two different sets of **angles** that give the same 3d **rotation**. This is not a good situation. Restricting one of the **angles** **to** 0<=angle<=pi gets rid of that problem, and the restriced set of **angles** still cover every possible 3d **rotation**. The **Rotations** Conversion Library (RCL) is a collection of C++ functions that address common computations and numerical handling of **rotations** in 3D Euclidean space, including support for.

# Euler angle to rotation matrix

Given a **rotation** **matrix** R, we can compute the **Euler** **angles**, ψ, θ, and φ by equating each element in Rwith the corresponding element in the **matrix** product R z(φ)R y(θ)R x(ψ). This results in nine equations that can be used to ﬁnd the **Euler** **angles**. Finding two possible **angles** for θ Starting with R 31, we ﬁnd R 31 = −sinθ.. Hmm, looks like we don’t have any results for this search term. Try searching for a related term below. But this post is a quick reference for **rotation** using z-y-x **Euler** **angles**. For further details, you can refer to this. **Euler** **Angle** Transformation. The most important thing you must remember before reading further about transformations using **Euler** **angles** is: The order of **matrix** multiplication of rotational matrices is of extreme importance. **Euler**. **Euler's** **angles** Therefore a generic **rotation** is described in turn by a **rotation** **matrix** R. Any **matrix** of this type can be described as the product of successive **rotations** around the principal axes of the XYZ coordinates, taken in a precise order. So any **rotation** could be decomposed into the sequence of three elementary matrices. **AN-1005 Understanding Euler Angles** Document rev. 1.0, updated 10/19/2012 - 1 - 1. Introduction Attitude and Heading Sensors from CH Robotics can provide orientation information using both **Euler Angles** and Quaternions. Compared to quaternions, **Euler Angles** are simple and intuitive and they lend themselves well to simple analysis and control. On. **Angles** are given in degrees (0 to 360) and counted counter-clockwise. For **rotations** in three dimensions the so-called **Euler angles** are used: pan is the horizontal **angle** (0..360) about the upright Z axis, tilt is the vertical **angle** (-90..+90) about the **rotated** Y axis, and roll is the **angle** (0..360) about the **rotated** and tilted X axis (see image). This **matrix** power calculator can help you raise a square **matrix** of 2x2, 3x3 or 4x4 to a specific number. The **Matrix** of a Linear Transformation We have seen that any **matrix** transformation x Ax is a linear transformation Increases the **rotation** step by 1 While there are several forms to specify a **rotation** , for instance with a **rotation** center, this. . The attitude of a rigid-body in the three dimensional space has a unique and global definition on the Special **Orthogonal Group** SO (3). This paper gives an overview of the **rotation matrix**, attitude kinematics and parameterization. The four most frequently used methods of attitude representations are discussed with detailed derivations, namely **Euler angles**, **angle**. **Euler** **angles** are a common way of defining a **rotation** by combining 3 successive **rotations** around different axes. use the convention of Bunge which is to rotate first around Z then around the new X and finally around the new Z. This example will show how the 3 successive **rotations** are carried out and that they indeed bring the laboratory frame (XYZ). **EulerMatrix** is also known as **Euler** **rotation** **matrix** or **Euler** **rotation**, and the **angles** α, β, and γ are often referred to as **Euler** **angles**. **EulerMatrix** is typically used to specify a **rotation** as a sequence of basic rotations around coordinate axes where each subsequent **rotation** is referring to the current or intrinsic coordinate frame.. The **rotation matrix** (') is used to transform the stresses from the machine coordinate system (global) to the slip plane (local) coordinate system based on **euler angles**.'. For **Euler** **angles**, a "gimbal lock" occurs iff the **Euler** **angle** representation for a given **rotation** **matrix** is not unique, i.e. there are infinite solutions. At the same time, the mapping from the **rotation** **matrix** **to** **Euler** **angles** is non-smooth. In the case of z-x-z extrinsic **Euler** **angles**, this special case is for R [3] [3]==0. expressed in terms of **Euler** **angles**. If we have the full direction cosine **matrix**, we can convert to **Euler** **angles** from the last row and the first column of the **matrix**: ( ) () yx xx zy zz zx r r r r r atan2, atan2 , arcsin = = =− ψ φ θ Eqn. 3 The pitch **angle** is between -90 degrees and +90 degrees. Note that we must use atan2 in order to get a. Similar **matrices** are used **to rotate** around Y and Z (then the Y (or Z) row and column values are set to 0 except the diagonal that is 1) Now, if you want the **rotation matrix** ( **Euler angle** ) you need to multiply the 3 individual **rotation matrices** in the good order. omega phi kappa. orientation based on Miller indices to **matrices** to <b>**Euler**</b> <b>**angles**</b>. Orientation: an orientation in 3D space, that can be specified as **Euler angles**, a quaternion, or a **rotation matrix**. The orientation can be absolute (i.e. relative to the world frame), or relative to a specific frame. Reference frame: also referred to as simply transformation or frame, it represents a position and orientation in 3D space. rotm =** eul2rotm** (eul) converts a set of Euler angles, eul, to the corresponding rotation matrix, rotm. When using the rotation matrix, premultiply it with the coordinates to be rotated (as opposed to postmultiplying). The default order for Euler angle rotations is "ZYX". example. The simplest approach to extract correctly **Euler** **angles** from a **rotation** **matrix** for any sequence of **angles** is using the $\mathrm{atan2}$ function. In the end, it is done in the same way (and maybe also explained why) in the text you linked. create link telegram; pdf magazine download. **Angles** are given in degrees (0 to 360) and counted counter-clockwise. For **rotations** in three dimensions the so-called **Euler angles** are used: pan is the horizontal **angle** (0..360) about the upright Z axis, tilt is the vertical **angle** (-90..+90) about the **rotated** Y axis, and roll is the **angle** (0..360) about the **rotated** and tilted X axis (see image). This proves Euler's theorem. Equivalence of an orthogonal matrix to a rotation matrix A proper orthogonal matrix is equivalent to If R has more than one invariant vector then φ = 0 and R = E. Any vector is an invariant vector of E. Excursion into matrix theory In order to prove the previous equation some facts from matrix theory must be recalled. As to the **rotation** order, it determines which axes **to rotate** around in which order. You need this because there are several **Euler angle** "solutions" for one quaternoin. I would expect them to be defined in EulerAngles.h, in something like: Code: 0: x,y,z 1: x,z,y 2: y,x,z ...etc. Where each of those is which **rotations** to do first. When using the **rotation matrix** , premultiply it with the coordinates to be **rotated** (as opposed to postmultiplying). The default order for **Euler angle rotations** is "ZYX". example. rotm = eul2rotm. **Rotation** **matrix**, Quaternion, Axis **angle**, **Euler** **angles** and Rodrigues' **rotation** explained.

Introduction A **rotation matrix**, \({\bf R}\), describes the **rotation** of an object in 3-D space. It was introduced on the previous two pages covering deformation gradients and polar decompositions. The **rotation matrix** is closely related to, though different from, coordinate system transformation **matrices**, \({\bf Q}\), discussed on this coordinate transformation page and on this. EulerMatrix [ { α, β, γ }] gives the Euler 3D rotation matrix formed by rotating by α around the current axis, then by β around the current axis, and then by γ around the current axis. EulerMatrix [ { α, β, γ }, { a, b, c }]. Hmm, looks like we don’t have any results for this search term. Try searching for a related term below. The **Rotation Matrices** of elemental **rotations** along x-axis, y-axis and z-axis by **angle** α, β and ɣ respectively are given as below, Now to represent the orientation of the target frame w.r.t reference frame in **Rotation matrix** form, we need to multiply the three **Euler angles** (α β, ɣ) **rotation matrices** in order. Given a **rotationmatrix** R, we can compute the **Euler** **angles**, ψ, θ, and φ by equating each element in Rwith the corresponding element in the **matrix** product R z(φ)R y(θ)R x(ψ). This results in nine equations that can be used to ﬁnd the **Euler** **angles**. Finding two possible **angles** for θ Starting with R 31, we ﬁnd R 31 = −sinθ. The simplest approach to extract correctly **Euler** **angles** from a **rotation** **matrix** for any sequence of **angles** is using the $\mathrm{atan2}$ function. In the end, it is done in the same way (and maybe also explained why) in the text you linked. create link telegram; pdf magazine download. Calculate matrix 3x3 rotation X To perform the calculation, enter the rotation angle. Then click the button 'Calculate' The unit of measurement for the angle can be switched between degrees or radians The active rotation (rotate object) or the passive rotation (rotate coordinates) can be calculated Description of the matrix X axis rotation. . A **rotation matrix** from **Euler angles** is formed by combining **rotations** around the x-, y-, and z-axes. For instance, **rotating** θ degrees around Z can be done with the **matrix** ┌ cosθ -sinθ 0 ┐ Rz = │ sinθ cosθ 0 │ └ 0 0 1 ┘ Similar **matrices** exist for **rotating** about the X and Y axes:. **Euler Angles** to/from Direction Cosine **Matrix**. ... For the special case where the attitude consists entirely of small-**angle rotations**, where small is defined as <5 °, the DCM only differs from the. import math import numpy as np # RPY/**Euler angles to Rotation** Vector def **euler**_to_rotVec(yaw, pitch, roll): # compute the **rotation** matr... Level up your programming skills with exercises across 52 languages, and insightful discussion with our dedicated team of welcoming mentors. rotm = eul2rotm(eul) converts a set of **Euler angles**, eul, to the corresponding **rotation matrix**, rotm. When using the **rotation matrix**, premultiply it with the coordinates to be **rotated** (as opposed to postmultiplying). The default order for **Euler angle rotations** is "ZYX". var mirrorNormalQuat = new Quaternion ( plane.x, plane.y, plane.z, 0); var reflectedQuat = mirrorNormalQuat * objectQuat * mirrorNormalQuat; Click to expand... Although this does mostly work well, for some reason it kept giving me a 180 degree **rotation** of mirrored **rotation**, if the normal was equal to Z+ or Z- axis. A more rigorous explanation of the Euler angles would define each angle rotation as an intermediate reference frame. So for the rotation shown above you would have 4 total reference frames. But for our modeling right now we are only really interested in the inertial frame and final body-fixed frame after all 3 Euler angle rotations. Description. rotm = eul2rotm (eul) converts a set of **Euler angles**, eul, to the corresponding **rotation matrix**, rotm. When using the **rotation matrix**, premultiply it with the coordinates to be. The order of **matrix** multiplication of **rotational matrices** is of extreme importance. **Euler angles** are used to specify the orientation of one reference frame relative to another. It can be three **Euler angles** or it could be three roll, pitch, yaw **angles**. So if we consider this vector of **angles** as the symbol Gamma, then we can introduce the initial **angles**, the final **angles**, and then we can interpolate the intermediate **angles**. ... The result is a **rotation matrix**, but the **matrix** has got dimensions of 3 x 3 x 100. So we can. Given a **rotationmatrix** R, we can compute the **Euler** **angles**, ψ, θ, and φ by equating each element in Rwith the corresponding element in the **matrix** product R z(φ)R y(θ)R x(ψ). This results in nine equations that can be used to ﬁnd the **Euler** **angles**. Finding two possible **angles** for θ Starting with R 31, we ﬁnd R 31 = −sinθ. . rotm = eul2rotm(eul) converts a set of **Euler angles**, eul, to the corresponding **rotation matrix**, rotm. When using the **rotation matrix**, premultiply it with the coordinates to be **rotated** (as opposed to postmultiplying). The default order for **Euler angle rotations** is "ZYX". I'm using the opencv function cv::recoverpose, after obtaining the Essential **matrix**, to get the **Rotation** and translation **matrices**. Is there any standard method to get **Euler angles** from the 3x3 **Rotation matrix**? I've seen a couple of set of equations for deriving the **Euler angles** and even these have multiple solutions. **Rotation** about x0 of **angle** γ + **Rotation** about y0 of **angle** β + **Rotation** about z0 of **angle** α All **rotations** are about fixed frame (x0, y0, z0) base vectors Homogeneous **Matrix** and **Angles** are identical between these two conventions: Roll Pitch Yaw XYZ ( γ,β,α) ⇔ **Euler** ZYX (α,β,γ) = − − − =. To retrieve the axis-**angle** representation of a **rotation matrix**, calculate the **angle** of **rotation** from the trace of the **rotation matrix** θ = arccos ( Tr ( R) − 1 2) and then use that to. Sep 18, 2021 · import math import numpy as np # RPY/**Euler** **angles** **to Rotation** Vector def **euler**_to_rotVec(yaw, pitch, roll): # compute the **rotation** matr... Level up your programming skills with exercises across 52 languages, and insightful discussion with our dedicated team of welcoming mentors.. When using the **rotation** **matrix**, premultiply it with the coordinates to be rotated (as opposed to postmultiplying).The default order for **Euler** **angle** **rotations** is "ZYX". example rotm = eul2rotm (eul,sequence) converts **Euler** **angles** **to** a **rotation** **matrix**, rotm.The **Euler** **angles** are specified in the axis **rotation** sequence, sequence. **Rotation** **Matrix** **to** **Euler** **Angles**. The code to be considered takes into account 3 input parameters: the **rotation** **matrix**, expressed as a multidimensional array of doubles; the **rotation** sequence of the axes with which we wish to carry out the transformation; the units of the **Euler** **angles** we wish to calculate. Also in this version of the code, we. For **Euler** **angles**, a "gimbal lock" occurs iff the **Euler** **angle** representation for a given **rotation** **matrix** is not unique, i.e. there are infinite solutions. ... At the same time, the mapping from the **rotation** **matrix** **to** **Euler** **angles** is non-smooth. In the case of z-x-z extrinsic **Euler** **angles**, this special case is for R [3] [3]==0. **Rotation** **matrix**. The orientation of a rigid body i can be specified by a transformation **matrix**, the elements of which may be expressed in terms of suitable sets of coordinates, such as **Euler angles**, **Bryant angles** or **Euler** parameters. Since. Robotics System Toolbox™ provides functions for transforming coordinates and units into the format required for your applications. To learn more about the different coordinate systems, see Coordinate **Transformations** in Robotics. You can also generate trajectories using polynomial equations, B-splines, **rotation matrices**, homogeneous. The **rotation matrix** R is obtained by looking at the **rotations** of the individual pairs of cans, ﬁgure 3. The ﬁrst pair of cans Figure 3. **Euler angle** sequence with ‘cans’ in series. describe the **rotation** about the~ez axis by an **angle** φas in r =Rφρ; with Rφ= 0 @ cosφ¡sinφ 0 sinφcosφ 0 0 0 1 1 A; (2). The **rotation matrix** (') is used to transform the stresses from the machine coordinate system (global) to the slip plane (local) coordinate system based on **euler angles**.'. The angles of rotation are called Euler angles, denoted as phi (ϕ), theta (θ), and psi (ѱ). Relative to aircraft control, these angles are roll, pitch, and yaw, as shown in Figure 2. The roll, pitch, and yaw rotation sequence is an E123 sequence, as the Euler angles are about the 1st, 2nd, and 3rd axis of the successive frames in that order. I need the inverse **rotation** (working on coordinate system transforms). What I do now is transforming these **angle** to a **rotation** **matrix** (using Rodrigues formula implemented in OpenCV) then calculate the inverse **rotation** **matrix** and finally use Rodrigues formula again to get the inverse **angles**. With an **angle** input of [0; -0.34906585; 3.14159265]. Similar **matrices** are used **to rotate** around Y and Z (then the Y (or Z) row and column values are set to 0 except the diagonal that is 1) Now, if you want the **rotation matrix** ( **Euler angle** ) you need to multiply the 3 individual **rotation matrices** in the good order. omega phi kappa. orientation based on Miller indices to **matrices** to <b>**Euler**</b> <b>**angles**</b>. Similar **matrices** are used **to rotate** around Y and Z (then the Y (or Z) row and column values are set to 0 except the diagonal that is 1) Now, if you want the **rotation matrix** ( **Euler angle** ) you need to multiply the 3 individual **rotation matrices** in the good order. omega phi kappa. orientation based on Miller indices to **matrices** to <b>**Euler**</b> <b>**angles**</b>. **Rotation Matrices**. **Rotation** Vectors. Modified Rodrigues Parameters. **Euler Angles**. The following operations on **rotations** are supported: Application on vectors. **Rotation** Composition. **Rotation**. Where α, β, and γ are angles of rotation about the X Y, and Z axes respectively. For the Euler angles a sequence might be rotations about the Z, Y, and Z axes respectively, {R}_ {ZXZ}= {R}_Z\left (\gamma \right) {R}_X\left (\beta \right) {R}_Z\left (\alpha \right) (9). The order of **matrix** multiplication of **rotational matrices** is of extreme importance. **Euler angles** are used to specify the orientation of one reference frame relative to another. **EulerMatrix** is also known as **Euler** **rotation** **matrix** or **Euler** **rotation**, and the **angles** α, β, and γ are often referred to as **Euler** **angles**. **EulerMatrix** is typically used to specify a **rotation** as a sequence of basic rotations around coordinate axes where each subsequent **rotation** is referring to the current or intrinsic coordinate frame.. A **rotation** of y; Question: (a) ZYZ-**Euler Angles** → **Rotation Matrix**: Background Information: In the class, we discussed that the composite **rotation matrix** corresponding to the ZYZ-**Euler angle** transformations is: Any **rotation** can be obtained by three successive **rotations** as follows: 1. A **rotation** of o about the current z-axis 2. **Euler angles** are a terrible set of coordinates for the **rotation** group. Compared to the other three standard presentations of **rotations** (**rotation matrices**, axis-**angle** form, and the closely related unit quaternions), **Euler angles** present no advantages and many severe disadvantages. Composition of **rotations** is complicated, numerically slow and. If →v is the axis of **rotation matrix** R, then we have both R→v = →v and RT→v = →v because RT is just the inverse **rotation**. Therefore, with a ≡ RT − R as above, we get. a→v = →0. Now the skew-symmetric property aT = − a, which can be seen from its definition, means there are exactly three independent **matrix** element in a. Oct 06, 2015 · For **Euler **angles, a "gimbal lock" occurs iff the **Euler angle **representation for a given **rotation matrix **is not unique, i.e. there are infinite solutions. At the same time, the mapping from the **rotation matrix to Euler **angles is non-smooth. In the case of z-x-z extrinsic **Euler **angles, this special case is for R [3] [3]==0.. Given a **rotation** **matrix** R, we can compute the **Euler** **angles**, ψ, θ, and φ by equating each element in Rwith the corresponding element in the **matrix** product R z(φ)R y(θ)R x(ψ). This results in nine equations that can be used to ﬁnd the **Euler** **angles**. Finding two possible **angles** for θ Starting with R 31, we ﬁnd R 31 = −sinθ.. Introduction A **rotation matrix**, \({\bf R}\), describes the **rotation** of an object in 3-D space. It was introduced on the previous two pages covering deformation gradients and polar decompositions. The **rotation matrix** is closely related to, though different from, coordinate system transformation **matrices**, \({\bf Q}\), discussed on this coordinate transformation page and on this. **Euler angles** provide a way to represent the 3D orientation of an object using a combination of three **rotations** about different axes. ... The **rotation matrix** for moving from the inertial frame to the vehicle‐2 frame consists simply of the yaw **matrix** multiplied by the pitch **matrix**: 5.The Body Frame (Yaw, Pitch, and Roll **Rotation**).Your **rotation matrices** will be in the.

Type 2 Rotations (proper Euler angles): xyx - xzx - yxy - yzy - zxz - zyz Singular if second rotation angle is 0 or 180 degrees. Euler angles [psi, theta, phi] range from -90 to 90 degrees. Tait-Bryan angles [psi, theta, phi] range from 0 to 180 degrees. Angles about Euler vectors range from 0 to 180 degrees. Value. """**Euler**-**angle** to **matrix** and **matrix** to **Euler**-**angle** utility routines: This module provides utility routines to convert between 4x4 homogeneous: transformation **matrices** and **Euler angles** (and. name: str = 'rotation_matrix_2d_from_euler_angle' ) -> tf.Tensor Converts an angle θ to a 2d rotation matrix following the equation R = [ cos ( θ) − sin ( θ) sin ( θ) cos ( θ)]. Note The. Euler Angles Euler angles are the easiest way to think of an orientation. You basically store three rotations around the X, Y and Z axes. It’s a very simple concept to grasp. You can use a vec3 to store it: vec3 EulerAngles( RotationAroundXInRadians, RotationAroundYInRadians, RotationAroundZInRadians);. Euler Angles and Quaternions Euler angles are a set of three angles that represent orientation in space. Each angle represents a rotation around one of three orthogonal axes (for example, the x, y, and z axes). The order of rotations is important because matrix multiplication is non-commutative (see below). As every **rotation** **matrix** can be written the above correspondence associates such a **matrix** with the complex number (this last equality is **Euler's** formula ). In three dimensions [ edit] A positive 90° **rotation** around the y -axis (left) after one around the z -axis (middle) gives a 120° **rotation** around the main diagonal (right). name: str = 'rotation_matrix_2d_from_euler_angle' ) -> tf.Tensor Converts an angle θ to a 2d rotation matrix following the equation R = [ cos ( θ) − sin ( θ) sin ( θ) cos ( θ)]. Note The.

Euler anglesor it could be three roll, pitch, yawangles. So if we consider this vector ofanglesas the symbol Gamma, then we can introduce the initialangles, the finalangles, and then we can interpolate the intermediateangles. ... The result is arotation matrix, but thematrixhas got dimensions of 3 x 3 x 100. So we can ...rotation matrixA is chosen randomly by theEuler angles: yaw, pitch, and roll within (-90, 90) x (-90, 90] x [0, 360) degree. Computationally efficient iterative pose estimation for space robot based on visionRotationmatrix, Quaternion, Axisangle,Euleranglesand Rodrigues'rotationexplainedEuleror Tait–Bryanangles(α, β, γ) are the amplitudes of these elementalrotations. For instance, the target orientation can be reached as follows: The XYZ system rotates about the z axis by γ. The X axis is now atangleγ with respect to the x axis. The XYZ system rotates again about the x axis by β.