Euler Angles

9 Key excerpts on "Euler Angles"

• 

  eBook - PDF

  Modern Classical Mechanics

  • T. M. Helliwell, V. V. Sahakian(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  483 12.4 The Euler Angles us ever since. So along with everyone else, we adhere to Euler’s convention and the corresponding set of angles known as the Euler Angles. According to Euler, a general rotation is defined through three rotations ˆ R(ϕ, θ, ψ) = ˆ R 3 (ψ) · ˆ R 2 (θ) · ˆ R 1 (ϕ) (12.23) where we define, in order: • First, rotate about the z axis by ϕ: ˆ R 1 (ϕ) = ⎛ ⎝ cos ϕ sin ϕ 0 − sin ϕ cos ϕ 0 0 0 1 ⎞ ⎠ . (12.24) • Second, rotate about the new x axis by θ: ˆ R 2 (θ) = ⎛ ⎝ 1 0 0 0 cos θ sin θ 0 − sin θ cos θ ⎞ ⎠ . (12.25) • Third, rotate about the newest z axis by ψ: ˆ R 3 (ψ) = ⎛ ⎝ cos ψ sin ψ 0 − sin ψ cos ψ 0 0 0 1 ⎞ ⎠ . (12.26) Figure 12.6 depicts three angles ϕ, θ, and ψ, known as the Euler Angles. The coordinate axes of the body frame are fixed within the rigid body and are labeled by primed coordinates x , y , and z ; the orientation of these axes can then be described with respect to the lab frame coordinates x, y, and z through the three Euler Angles. As the figure illustrates, this is a three-step prescription that takes us from x–y–z to x –y –z . The corresponding rotation matrix (12.23) relates the components of any vector between the lab and body reference frames. The lab frame is also sometimes referred to as the “lab axes” or the “space axes.” Let us then put this machinery to use. Let R be the position vector of the tagged point in the rigid body measured from the origin of the lab frame, as shown in Figure 12.7. If we pick any other point a in the rigid body we may denote its position vector by r a , measured from the origin of the lab frame. As the body moves and tumbles around, both R and r a will in general change in both direction and magnitude. Now consider the position of this second point with respect to the tagged point: we call this vector r a pointing from the tagged point.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Railroad Vehicle Dynamics

  A Computational Approach

  • Ahmed A. Shabana, Khaled E. Zaazaa, Hiroyuki Sugiyama(Authors)
  • 2007(Publication Date)
  • CRC Press

    (Publisher)

  2.2.3 E ULER A NGLES In the method of Euler Angles, three simple successive rotations are used to define the orientation of the rigid body in space. Using the three Euler Angles associated i j i z z i z z =           = −    cos sin , sin cos θ θ θ θ 0 0        =           , k i 0 0 1 A i z z z z = −           cos sin sin cos θ θ θ θ 0 0 0 0 1 A i y y y y = −           cos sin sin cos θ θ θ θ 0 0 1 0 0 A i x x x x = −           1 0 0 0 0 cos sin sin cos θ θ θ θ 42 Railroad Vehicle Dynamics: A Computational Approach with three independent axes, the orientation matrix A i given in Equation 2.1 can be defined as the product of three simple rotation matrices as follows: (2.18) where ( k = 1, 2, 3) are, respectively, the rotation matrices defined in terms of the following three Euler Angles: (2.19) One can choose an appropriate sequence of the three successive rotations to reach any orientation in space. For example, the orientation of the wheel shown in Figure 2.2 can be conveniently described using the following three successive rotations: (2.20) Let XYZ and X i Y i Z i be, respectively, the global and the wheel-body coordinate systems, which initially coincide. Rotation of the wheel-body coordinate system X i Y i Z i by an angle ψ i ( yaw ) about the Z i axis leads to the rotation matrix (2.21) A second rotation φ i ( roll ) of the wheel coordinate system X i Y i Z i about the X i axis leads to the rotation matrix (2.22) FIGURE 2.2 Euler Angles. A A A A i i i i = 1 2 3 A k i θ i i i i T = [ ] θ θ θ 1 2 3 θ i i i i T =     ψ φ θ A 1 0 0 0 0 1 i i i i i = −           cos sin sin cos ψ ψ ψ ψ A 2 1 0 0 0 0 i i i i i = −           cos sin sin cos φ φ φ φ Dynamic Formulations 43 Finally, the wheel coordinate system X i Y i Z i is rotated by an angle θ i ( pitch ) about the Y i axis.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Engineering Dynamics

  A Comprehensive Introduction

  • N. Jeremy Kasdin, Derek A. Paley(Authors)
  • 2011(Publication Date)
  • Princeton University Press

    (Publisher)

  We then rotate by θ about the a 2 axis of frame A to reach the intermediate frame C . (This frame is the same as the spherical frame!) Finally, we rotate about the c 3 axis by φ to get to the B frame. Note that all are right-handed rotations. The set of angles (ψ, θ, φ) I B are known as orientation angles or Euler Angles. (Note the slight change in notation when the scalar coordinates describe the orientation of B in I rather than the location of a particle relative to an origin; we need to indicate both frames.) The particular ordered set of rotations described above and shown in Figure 10.8 is known as a 3-2-3 rotation, as it involves an ordered sequence of rotations about the 3-axis, the 2-axis, and the 3-axis of a series of intermediate reference frames. It is possible to describe any orientation of reference frame B in I by means of these three angles. Thus they constitute a set of scalar coordinates for describing the orientation of a reference frame. In fact, you have already seen an example of using Euler Angles in Example 10.4, where we used azimuth ( AZ ) and elevation ( EL ) to describe the orientation of a radar tracking dish. As you might have guessed, there are also many other possible sets of Euler Angles (e.g., a 3-1-3 rotation or a 1-2-3 rotation). In all, there are 12 possible sets of three 428 CHAPTER TEN Euler-angle rotations that can be used to describe the orientation of one frame with respect to another. Which set is best depends on the specific mechanical system being analyzed. We use a few different sets in the examples that follow. We begin by examining how to transform a vector from one set of unit vectors to another set in terms of the Euler Angles, just as in the planar case. That is, how do we write the unit vectors of frame B in terms of those in I ? We use the same approach as for the spherical frame and write the transformation tables between each pair of intermediate frames related by a simple rotation.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Intermediate Dynamics for Engineers

  Newton-Euler and Lagrangian Mechanics

  • Oliver M. O'Reilly(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  As discussed in [50, 308], this representation dates to works by Euler [70, 73] that he first presented in 1751. 15 In these papers, he shows how three angles can be used to parameterize a rotation and he also establishes expressions for the corotational components of the angular velocity vector. One interpretation of the Euler Angles involves a decomposition of the rotation tensor into a product of three fairly simple rotations: R = ` R  γ 1 , γ 2 , γ 3  = L  γ 3 , g 3  L  γ 2 , g 2  L  γ 1 , g 1  . (6.22) Here, {γ i } are the Euler Angles and the set of unit vectors {g i } is known as the Euler basis. The function L (ϑ , b) is defined by use of the Euler representation: L (ϑ , b) = cos(ϑ )(I − b ⊗ b) + sin(ϑ ) skwt (b) + b ⊗ b, where b is a unit vector and ϑ is the counterclockwise angle of rotation. In general, g 3 is a function of γ 2 and γ 1 and g 2 is a function of γ 1 . As we shall see shortly, there are 12 possible choices of the Euler Angles. For example, Figure 6.6 illustrates these angles 15 For discussions of these papers, and several other interesting historical facts on the development of representations for rotations, see Blanc’s introduction to parts of Euler’s collected works in [75, 76], Cheng and Gupta [50], and Wilson [308]. Although [73] dates to the eighteenth century, it was first published posthumously in 1862. 6.8 Euler Angles 225 for a set of 3–2–3 Euler Angles. Because there are three Euler Angles, the parameter- ization of a rotation tensor by use of these angles is an example of a three-parameter representation. If we assume that g 1 is constant, then the angular velocity vector associated with the Euler angle representation can be established by use of the relative angular velocity vector. In this case, there are two relative angular velocity vectors (cf. (6.19)). For the first rotation, the angular velocity vector is ˙ γ 1 g 1 (cf. (6.17)).

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Game Physics Engine Development

  How to Build a Robust Commercial-Grade Physics Engine for your Game

  • Ian Millington(Author)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  Recall that a position is represented as a vector, where each component repre-sents the distance from the origin in one direction. We could use a vector to represent rotation, where each component represents the amount of rotation about the corre-sponding axis. We have a similar situation to our 2D rotation, but here rather than a single angle, we need three, or one for each axis. These three angles are called Euler Angles . This is the most obvious representation of orientation. It has been used in many graphics applications. Several of the leading graphics modeling packages use Euler Angles internally, and those that don’t still represent orientations to the user as Euler Angles. Unfortunately, Euler Angles are almost useless for our needs. We can see this by looking at some of the implications of working with them. You can follow this through 166 Chapter 9 The Mathematics of Rotations Roll Yaw Pitch F IGURE 9.5 Aircraft rotation axes. by making a set of axes with your hand (as described in Section 2.1.1), remembering that your imaginary object is facing in the same direction as your palm (along the Z axis) and your index pointer should be pointing up (i.e., the Y axis is vertical). Imagine we first perform a pitch, by 30 degrees or so, keeping your thumb pointed in the same direction. The object now has its nose up in the air. Now perform a yaw by about the same amount, keeping your first finger pointed in the same direction. Note that the yaw axis (your first finger) is no longer pointing up: when we pitched the object the yaw axis also moved. Remember where the object is pointing. Now start again, but perform the yaw first and then the pitch. Note now that the object will be in a slightly different position (if it doesn’t seem to be different, then try it again with bigger rotations, until you can see the difference—the bigger the rotation, the bigger the difference). What does this mean? If we have a rotation vector like ⎡ ⎢ ⎣ 0 .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Rigid Body Kinematics

  • Joaquim A. Batlle, Ana Barjau Condomines(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 Space and Time Orientation and Euler Angles Kinematics deals with the geometry of motion of material bodies along time – change of position in space and over time – without regard to the physical phenomena on which it depends. Such description requires mathematical models for space and time (which is the physical framework of mechanical phenomena) and mathematical models for bodies. There is a range of models of increasing complexity for material objects: mass point (or particle), rigid body, continuous media (including deformable bodies and fluids). This text deals exclusively with the first two (Chapters 2 and 3), whose mathematical complexity is much lower than that required by continuous media. The mathematical operation time derivative, to evaluate rates of change (of position, orientation, etc.), is a fundamental tool in mechanics. The time derivative of vectors is more complex than that of scalars, since it has to take into account both the change of value and of orientation. While the former is identical for all observers (or all reference frames), the latter is not. For example, a radius marked on a rotating platform (and taken as a vector) has the same value and orientation for observers fixed to the platform but variable orientation for observers fixed to the ground. Orientation is a key concept, and its mathematical description is not trivial when 3D motion is addressed. In this text, only the Euler Angles are used as they are the most appropriate procedure for a first study. However, Appendix 1B presents a brief review of some alternative rotation parameters due to its importance in engineering branches such as robotics and spacecraft dynamics. Operations on vectors can be done through their graphic representation (an arrow with value indication 1 ) or their components (projections on a vector basis).

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  Tensor Analysis for Engineers

  Transformations - Mathematics - Applications

  • Mehrzad Tabatabaian(Author)
  • 2020(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  CHAPTER 15

  RIGID BODY ROTATION: Euler Angles, QUATERNIONS, AND ROTATION MATRIX

  In Chapter 12 , we discussed Cartesian tensors and rotation of coordinate system using Rodrigues’ formula (see Equation 12.6). For practical applications, mainly rigid body rotation, in this chapter we would like to expand on this topic in relation to Euler Angles and quaternions methods.

  A rigid body transformation can be composed of two parts: a) translation and b) rotation. The translation is usually referred to a vector connecting the origin of the coordinate system to the centre of mass. Therefore, we can always refer the rotation to the instantaneous center of mass and focus on the rotation for calculation and analysis of rigid body motion.

  In general, there are at least eight methods to represent rotation [13]. Among these three inter-related methods commonly used for calculating rigid body orientation after it goes through possible rotations in a 3D space. These are: 1) rotation matrix, 2) single axis-angle, and 3) quaternions. The first two methods are discussed in Chapter 12 and expanded in this chapter with the inclusion of Euler Angles method. The third method is more robust and with less limitations compared to the other methods in terms of practical applications and is new to this second edition.

  When dealing with rotation of a rigid body, several confusions may arise in terms of definition of rotation matrix and rotation variables. To avoid these ambiguities, identifying two definitions is critical. First, the assumed transformation should be clear by the way the rotation matrix (see Equation 12.7) is meant to be, i.e., it is meant to relate the coordinates obtained after rotation to the original ones or vice versa. Second, how the rotation itself is performed, i.e., is it that the coordinates are rotated and quantities, like a vector, is kept fixed and we seek to find its new coordinates with reference to the rotated ones or is it that the quantities, like a vector, are rotated and the coordinates are kept fixed and we would like to calculate the coordinates of the rotated vector with reference to the fix-kept coordinates.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Analytical Mechanics

  • Nivaldo A. Lemos(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  Angular Velocity in Terms of the Euler Angles Some important problems in rigid body dynamics require the angular velocity vector to be expressed in terms of the Euler Angles. An infinitesimal rotation of the rigid body can be thought of as the result of three sucessive infinitesimal rotations whose angular velocities have, respectively, magnitudes ˙ φ, ˙ θ , ˙ ψ . Let ω φ , ω θ and ω ψ be the corresponding angular velocity vectors. The angular velocity vector ω is simply given by ω = ω φ + ω θ + ω ψ (3.82) in virtue of Eq. (3.58). One can obtain the components of ω either along inertial axes (x, y, z) or along axes (x  , y  , z  ) attached to the body. Because of its greater utility, we will consider the latter case. Clearly, the angular velocity ω depends linearly on ˙ φ, ˙ θ , ˙ ψ . Therefore, in order to find the general form taken by ω we can fix a pair of Euler Angles at a time, determine the angular velocity associated with the variation of the third angle and then add the results (Epstein, 1982). Fixing θ and ψ , the z-axis, which is fixed in space, also becomes fixed in the body (see Fig. 3.4). So, z is the rotation axis and ω φ is a vector along the z-axis with component ˙ φ, and from (3.50) we have ⎛ ⎝ (ω φ ) x  (ω φ ) y  (ω φ ) z  ⎞ ⎠ = A E ⎛ ⎝ 0 0 ˙ φ ⎞ ⎠ = ⎛ ⎝ ˙ φ sin θ sin ψ ˙ φ sin θ cos ψ ˙ φ cos θ ⎞ ⎠ . (3.83) Fixing now φ and ψ , the line of nodes (the ξ  -axis direction in Fig. 3.4) becomes fixed both in space and in the body, so it is the rotation axis. As a consequence, ω θ is a vector with the single component ˙ θ along the ξ  -axis and with the help of (3.49) we find ⎛ ⎝ (ω θ ) x  (ω θ ) y  (ω θ ) z  ⎞ ⎠ = B ⎛ ⎝ ˙ θ 0 0 ⎞ ⎠ = ⎛ ⎝ ˙ θ cos ψ − ˙ θ sin ψ 0 ⎞ ⎠ . (3.84) Finally, with θ and φ fixed the z  -axis, which is fixed in the body, becomes fixed in space. Therefore, z  is the rotation axis and ω ψ is a vector along the z  -axis with component ˙ ψ , there being no need to apply any transformation matrix to ω ψ .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Legacy Of Leonhard Euler, The: A Tricentennial Tribute

  A Tricentennial Tribute

  • Lokenath Debnath(Author)
  • 2009(Publication Date)
  • ICP

    (Publisher)

  (4.6.2) Euler first introduced the transformation from the 0 xyz -system to the 0 x y z system whose equations are represented (see Figure 4.18) in terms 128 The Legacy of Leonhard Euler — A Tricentennial Tribute of the Euler Angles φ , ψ and θ . These angles are considered as the angles through which the former must be successively rotated about the axes of the latter so that in the end the two systems coincide. The angle φ is measured in the xy -plane from the x -axis to the line of nodes which is the line of intersection of the planes 0 xy and 0 x y . The Cartesian coordinates x , y , z and x , y , z are related by the relations x = x (cos ψ cos φ − cos θ sin ψ sin φ ) − y (cos ψ sin φ + cos θ sin ψ sin φ ) + z sin θ sin φ, (4.6.3) y = x (sin ψ cos φ + cos θ cos ψ sin φ ) − y (sin ψ sin φ − cos θ cos ψ sin φ ) − z sin θ sin φ, (4.6.4) z = x sin θ sin φ + y sin θ cos φ + z cos θ. (4.6.5) Euler used this transformation to transform (4.6.2) to cannonical forms and obtained seven different cases: cone, cylinder, ellipsoid, hyperboloid of one and two sheets, parabolic cylinder and hyperbolic paraboloid -the last of these was his own discovery. He also discovered that the degree of a curve is invariant under a linear transformation. Isaac Newton proved that the general third degree algebraic equation representing cubic curves in the form ax 3 + bx 3 y + cxy 2 + dy 3 + ex 2 + fxy + gy 2 + hx + jy + k = 0 , (4.6.6) can be transformed by a change of axis, into one of the following four forms such as (i) xy 2 + ey = f ( x ), (ii) xy = f ( x ), (iii) y 2 = f ( x ), and (iv) y = f ( x ), where f ( x ) = ax 3 + bx 2 + cx + d . Newton’s work on third degree plane curves stimulated much other work on higher degree plane curves. The classification of third and fourth degree curves also continued to interest mathematicians and physicists of the eighteenth and nineteenth centuries.

  Sign up to read

  Learn more about book