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# Euler angles

About: Euler angles is a research topic. Over the lifetime, 1861 publications have been published within this topic receiving 32298 citations.

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TL;DR: Two computationally efficient algorithms are presented for determining three-axis attitude from two or more vector observations that are useful to the mission analyst or spacecraft engineer for the evaluation of launch-window constraints or of attitude accuracies for different attitude sensor configurations.

Abstract: Two computationally efficient algorithms are presented for determining three-axis attitude from two or more vector observations. The first of these, the TRIAD algorithm, provides a deterministic (i.e., nonoptimal) solution for the attitude based on two vector observations. The second, the QUEST algorithm, is an optimal algorithm which determines the attitude that achieves the best weighted overlap of an arbitrary number of reference and observation vectors. Analytical expressions are given for the covariance matrices for the two algorithms using a fairly realistic model for the measurement errors. The mathematical relationship of the two algorithms and their relative merits are discussed and numerical examples are given. The advantage of computing the covariance matrix in the body frame rather than in the inertial frame (e.g., in terms of Euler angles) is emphasized. These results are valuable when a single-frame attitude must be computed frequently. They will also be useful to the mission analyst or spacecraft engineer for the evaluation of launch-window constraints or of attitude accuracies for different attitude sensor configurations.

1,281 citations

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07 Aug 2000

TL;DR: A Guide to the Book Descriptors of Orientation Crystal Structures and Crystal Symmetries Transformation between Coordinate Systems: The Rotation matrix The "Ideal Orientation" (Miller or Miller-Bravais Indices) Notation The Reference Sphere, Pole Figure, and Inverse Pole Figure The Euler Angles and Euler Space The angle/axis of Rotation and Cylindrical Angle/Axis of Rosters Space The Rodrigues Vector and Rodrigues Space Application of Diffraction to Texture Analysis Diffraction of Radiation and Bragg's

Abstract: Part I: Fundamental Issues Introduction The Classical Approach to Texture The Modern Approach to Texture: Microtexture A Guide to the Book Descriptors of Orientation Crystal Structures and Crystal Symmetries Transformation between Coordinate Systems: The Rotation Matrix The "Ideal Orientation" (Miller or Miller-Bravais Indices) Notation The Reference Sphere, Pole Figure, and Inverse Pole Figure The Euler Angles and Euler Space The Angle/Axis of Rotation and Cylindrical Angle/Axis Space The Rodrigues Vector and Rodrigues Space Application of Diffraction to Texture Analysis Diffraction of Radiation and Bragg's Law Structure Factor Laue and Debye-Scherrer Methods Absorption and Depth of Penetration Characteristics of Radiations Used for Texture Analysis Part II: Macrotexture Analysis Macrotexture Measurements Principle of Pole Figure Measurement X-Ray Diffraction Methods Neutron Diffraction Methods Texture Measurements in Low-Symmetry and Multiphase Materials Sample Preparation Evaluation and Representation of Macrotexture Data Pole Figure and Inverse Pole Figure Determination of the Orientation Distribution Function from Pole Figure Data Representation and Display of Texture in Euler Space Examples of Typical Textures in Metals Part III: Microtexture Analysis The Kikuchi Diffraction Pattern The Kikuchi Diffraction Pattern Quantitative Evaluation of the Kikuchi Pattern Pattern Quality Scanning Electron Microscopy-Based Techniques Micro-Kossel Technique Electron Channeling Diffraction and Selected-Area Channeling Evolution of Electron Backscatter Diffraction EBSD Specimen Preparation Experimental Considerations for EBSD Calibration of an EBSD System Operation of an EBSD System and Primary Data Output Transmission Electron Microscopy-Based Techniques High-Resolution Electron Microscopy Selected Area Diffraction Kikuchi Patterns, Microdiffraction, and Convergent Beam Electron Diffraction Evaluation and Representation of Microtexture Data Representation of Orientations in a Pole Figure or Inverse Pole Figure Representation of Orientations in Euler Space Representation of Orientations in Rodrigues Space General Representation of Misorientation Data Representation of Misorientations in Three-Dimensional Spaces Normalization and Evaluation of the Misorientation Distribution Function Extraction of Quantified Data Orientation Microscopy and Orientation Mapping Historical Evolution Orientation Microscopy Orientation Mapping and Its Applications Orientation Microscopy in the TEM Crystallographic Analysis of Interfaces, Surfaces, and Connectivity Crystallographic Analysis of Grain Boundaries Crystallographic Analysis of Surfaces Orientation Connectivity and Spatial Distribution Orientation Relationships between Phases Synchrotron Radiation, Nondiffraction Techniques, and Comparisons between Methods Texture Analysis by Synchrotron Radiation Texture Analysis by Nondiffraction Techniques Appendices Glossary References General Bibliography Index

1,062 citations

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TL;DR: In this article, a variational formulation and computational aspects of a three-dimensional finite-strain rod model, considered in Part I, are presented, which bypasses the singularity typically associated with the use of Euler angles.

Abstract: The variational formulation and computational aspects of a three-dimensional finite-strain rod model, considered in Part I, are presented. A particular parametrization is employed that bypasses the singularity typically associated with the use of Euler angles. As in the classical Kirchhoff-Love model, rotations have the standard interpretation of orthogonal, generally noncommutative, transformations. This is in contrast with alternative formulations proposed by Argyris et al. [5–8], based on the notion of semitangential rotation. Emphasis is placed on a geometric approach, which proves essential in the formulation of algorithms. In particular, the configuration update procedure becomes the algorithmic counterpart of the exponential map. The computational implementation relies on the formula for the exponential of a skew-symmetric matrix. Consistent linearization procedures are employed to obtain linearized weak forms of the balance equations. The geometric stiffness then becomes generally nonsymmetric as a result of the non-Euclidean character of the configuration space. However, complete symmetry is recovered at an equilibrium configuration, provided that the loading is conservative. An explicit condition for this to be the case is obtained. Numerical simulations including postbuckling behavior and nonconservative loading are also presented. Details pertaining to the implementation of the present formulation are also discussed.

909 citations

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14 Dec 1998

TL;DR: In this paper, the quaternion rotation operator is introduced and defined, and a brief introduction to its properties and algebra is given, as well as its primary application, which is to compete with the conventional matrix rotation operator in a variety of rotation sequences.

Abstract: In this paper we introduce and define the quaternion; we give a brief introduction to its properties and algebra, and we show (what appears to be) its primary application—the quaternion rotation operator. The quaternion rotation operator competes with the conventional matrix rotation operator in a variety of rotation sequences.

887 citations

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TL;DR: The International Terrestrial Reference Frame (TRF2000) as discussed by the authors combines unconstrained space geodesy solutions that are free from any tectonic plate motion model, and its orientation time evolution follows, conventionally, that of the no-net-rotation NNR-NUVEL-1A model.

Abstract: [1] For the first time in the history of the International Terrestrial Reference Frame, the ITRF2000 combines unconstrained space geodesy solutions that are free from any tectonic plate motion model. Minimum constraints are applied to these solutions solely in order to define the underlying terrestrial reference frame (TRF). The ITRF2000 origin is defined by the Earth center of mass sensed by satellite laser ranging (SLR) and its scale by SLR and very long baseline interferometry. Its orientation is aligned to the ITRF97 at epoch 1997.0, and its orientation time evolution follows, conventionally, that of the no-net-rotation NNR-NUVEL-1A model. The ITRF2000 orientation and its rate are implemented using a consistent geodetic method, anchored over a selection of ITRF sites of high geodetic quality, ensuring a datum definition at the 1 mm level. This new frame is the most extensive and accurate one ever developed, containing about 800 stations located at about 500 sites, with better distribution over the globe compared to past ITRF versions but still with more site concentration in western Europe and North America. About 50% of station positions are determined to better than 1 cm, and about 100 sites have their velocity estimated to at (or better than) 1 mm/yr level. The ITRF2000 velocity field was used to estimate relative rotation poles for six major tectonic plates that are independent of the TRF orientation rate. A comparison to relative rotation poles of the NUVEL-1A plate motion model shows vector differences ranging between 0.03° and 0.08°/m.y. (equivalent to approximately 1–7 mm/yr over the Earth's surface). ITRF2000 angular velocities for four plates, relative to the Pacific plate, appear to be faster than those predicted by the NUVEL-1A model. The two most populated plates in terms of space geodetic sites, North America and Eurasia, exhibit a relative Euler rotation pole of about 0.056 (±0.005)°/m.y. faster than the pole predicted by NUVEL-1A and located about (10°N, 7°E) more to the northwest, compared to that model.

859 citations