Rodrigues Vector Research Articles

Comparative Study of Methods to Present the Results of Texture Analysis of Polycrystalline Materials

It is necessary to use various methods of texture analysis at all stages of the technological processing of metals and alloys in order to select the optimal parameters for processing materials and obtain the necessary properties of final products. The article deals with the methods of presenting the results of texture analysis. Usually, the texture of a material is described by several preferred texture components, where the way of representation depends on the given task. The textural components combine the orientations of crystals having similar position in the sample volume. The results of texture analysis are the distribution of orientations relative to the base coordinate system. Orientation is described by four parameters: three coordinates in space and intensity, which is defined as the probability of occurrence of this orientation in the sample. Methods for representing the orientation of a single crystal by several descriptors: Euler angles, Miller indices, Rodrigues vectors, etc., each describing the rotation of the crystal coordinate system relative to the selected sample coordinate system, are given in the article. The formulae for the transition from one descriptor to another are given. The texture calculation algorithms depend on the way orientations are represented. Some representations are more convenient to use for calculations, others - for visualization and data processing. In particular, the use of the Rodrigue space allows not only to visualize the orientation cloud, but also to form textural components in it with a given accuracy by clustering. Variants of the analysis result presentation are shown in a model example. The textural components computed with high accuracy are displayed in the spaces of Euler angles and Rodrigues vectors. The orientation distributions are shown as a histogram and a pole figure {001}.

Texture Componets Scattering Parameter Selection for Polycrystalline Materials when Clustering Rodrigues Space

The article considers the problem of maximum scattering cluster value selection at texture components formation within Rodrigues space for texture analysis representation. Texture analysis is carried out in terms of direct pole figure. Orientation cloud is being formed within Rodrigues space by calculating Rodrigues vectors for each found orientation. Texture components describing the texture are being formed by maximin algorithm clustering. The algorithm is considered finished when a predetermined value of maximum scattering in a cluster is reached. In this case, maximum scattering in clusters will be determined automatically. The algorithm enables texture presentation with required detail level corresponding to the posed problem. Implementation of Rodrigues space to form texture components enables texture analysis results visualization. The article gives principal expressions for Rodrigues-vector operations. The calculation of mean Rodrigues vector in a cluster is shown. The algorithm that allows comparison of texture analysis results obtained by two pole figures, taken for one sample. Comparison is carried out by calculation of distances between mean Rodrigues vectors of two pole figure clusters for different scattering parameter values. Texture representation is chosen as the best correspondence of texture components obtained by two pole figures. An example of aluminum tape texture analysis is presented. It is shown that formation of eight texture components with maximum cluster scattering 11.3° by pole figure of family {111} and of six texture components with maximum cluster scattering 10.15° by pole figure {200} their correspondence is 85 %.

Strapdown Attitude Computation: Functional Iterative Integration versus Taylor Series Expansion

This paper compares two basic approaches to solving ordinary differential equations, which form the basis for attitude computation in strapdown inertial navigation systems, namely, the Taylor series expansion approach that was used in its low-order form for deriving all mainstream algorithms and the functional iterative integration approach developed recently. They are respectively applied to solve the kinematic equations of major attitude parameters, including the quaternion, the Rodrigues vector and the rotation vector. Specifically, the mainstream algorithms, which have relied on the simplified rotation vector without exception, are considerably extended by the Taylor series expansion approach using the exact rotation vector and recursive calculation of high-order derivatives. The functional iterative integration approach is respectively implemented on both the normal polynomial and the Chebyshev polynomial. Numerical results under the classical coning motion are reported to assess all derived attitude algorithms. It is revealed that in the relative frequency range when the coning to sampling frequency ratio is below 0.05-0.1 (depending on the chosen polynomial truncation order), all algorithms have the same order of accuracy if the same number of samples are used to fit the angular velocity over the iteration interval; in the range of higher relative frequency, the group of Quat/Rod/RotFIter algorithms (by the functional iterative integration approach combined with the Chebyshev polynomial) perform the best in both accuracy and robustness, thanks to the excellent numerical stability and powerful functional representation capability of the Chebyshev polynomial.

Kinematic Comparison of Exponential and Cayley Maps in the Planar Motion Group

In this work, the exponential and the Cayley maps, from the Lie algebra $\mathfrak{se(2)}$ of the planar motion group $SE(2)$, to the group itself are studied. The comparison between these maps on $SE(2)$ is given by using the Rodrigues vector. A three joint mechanism is discussed as an application.

Attitude Reconstruction From Inertial Measurements: QuatFIter and Its Comparison with RodFIter

RodFIter is a promising method of attitude reconstruction from inertial measurements based on the functional iterative integration of Rodrigues vector. The Rodrigues vector is used to encode the attitude in place of the popular rotation vector because it has a polynomial-like rate equation and could be cast into theoretically sound and exact integration. This paper further applies the approach of RodFIter to the unity-norm quaternion for attitude reconstruction, named QuatFIter, and shows that it is identical to the previous Picard-type quaternion method. The Chebyshev polynomial approximation and truncation techniques from the RodFIter are exploited to speed up its implementation. Numerical results demonstrate that the QuatFIter is comparable in accuracy to the RodFIter, although its convergence rate is relatively slower with respect to the number of iterations. Notably, the QuatFIter has about two times better computational efficiency, thanks to the linear quaternion kinematic equation.

Fast RodFIter for Attitude Reconstruction From Inertial Measurements

Attitude computation is of vital importance for a variety of applications. Based on the functional iteration of the Rodrigues vector integration equation, the RodFIter method can be advantageously applied to analytically reconstruct the attitude from discrete gyroscope measurements over the time interval of interest. It is promising to produce ultra-accurate attitude reconstruction. However, the RodFIter method imposes high computational load and does not lend itself to onboard implementation. In this paper, a fast approach to significantly reduce RodFIter's computation complexity is presented while maintaining almost the same accuracy of attitude reconstruction. It reformulates the Rodrigues vector iterative integration in terms of the Chebyshev polynomial iteration. Due to the excellent property of Chebyshev polynomials, the fast RodFIter is achieved by means of appropriate truncation of Chebyshev polynomials, with provably guaranteed convergence. Moreover, simulation results validate the speed and accuracy of the proposed method.

RodFIter: Attitude Reconstruction From Inertial Measurement by Functional Iteration

Rigid motion computation or estimation is a cornerstone in numerous fields. Attitude computation can be achieved by integrating the angular velocity measured by gyroscopes, the accuracy of which is crucially important for the dead-reckoning inertial navigation. The state-of-the-art attitude algorithms have unexceptionally relied on the simplified differential equation of the rotation vector to obtain the attitude. This paper proposes a functional iteration technique with the Rodrigues vector (named the RodFIter method) to analytically reconstruct the attitude from gyroscope measurements. The RodFIter method is provably exact in reconstructing the incremental attitude as long as the angular velocity is exact. Notably, the Rodrigues vector is analytically obtained and can be used to update the attitude over the considered time interval. The proposed method gives birth to an ultimate attitude algorithm scheme that can be naturally extended to the general rigid motion computation. It is extensively evaluated under the attitude coning motion and compares favorably in accuracy with the mainstream attitude algorithms. This paper is believed having eliminated the long-standing theoretical barrier in exact motion integration from inertial measurements.

Visualizing rotations and composition of rotations with the Rodrigues vector

The purpose of this paper is to show that the mathematical treatment of three-dimensional rotations can be simplified, and its geometrical understanding improved, using the Rodrigues vector representation. We present a novel geometrical interpretation of the Rodrigues vector. Based on this interpretation and simple geometrical considerations, we derive the Euler–Rodrigues formula, Cayley’s rotation formula and the composition law for finite rotations. The level of this discussion should be suitable for undergraduate physics or engineering courses where rotations are discussed.

On representing rotations by Rodrigues parameters in non-orthonormal reference systems.

A Rodrigues vector is a triplet of real numbers used for parameterizing rotations or orientations in three-dimensional space. Because of its properties (e.g. simplicity of fundamental regions for misorientations) this parameterization is frequently applied in analysis of orientation maps of polycrystalline materials. By conventional definition, the Rodrigues parameters are specified in orthonormal coordinate systems, whereas the bases of crystal lattices are generally non-orthogonal. Therefore, the definition of Rodrigues parameters is extended so they can be directly linked to non-Cartesian bases of a crystal. The new parameters are co- or contravariant components of vectors specified with respect to the same basis as atomic positions in a unit cell. The generalized formalism allows for redundant crystallographic axes. The formulas for rotation composition and the relationship to the rotation matrix are similar to those used in the Cartesian case, but they have a wider range of applicability: calculations can be performed with an arbitrary metric tensor of the crystal lattice. The parameterization in oblique coordinate frames of lattices is convenient for crystallographic applications because the generalized parameters are directly related to indices of rotation-invariant lattice directions and rotation-invariant lattice planes.

The five-dimensional parameter space of grain boundaries

To specify a grain boundary at a macroscopic length scale requires the specification of five degrees of freedom. We use a specification in which three degrees of freedom associated with the boundary misorientation are in an orthogonal subspace from two associated with the mean boundary plane. By using Rodrigues vectors to describe rotations, we show how paths through these subspaces may be characterized. Some of these paths correspond to physical processes involving grain boundaries during microstructural evolution. Exploiting the orthogonality of the subspaces, a metric to measure ‘distance’ between two boundaries is defined in terms of the minimum set of rotations required to map one boundary on to the other. We compare our metric with others that have appeared. The existence of rotational symmetry in face-centred cubic crystals leads to as many as 2304 equivalent specifications of a boundary. We illustrate this multiplicity of descriptions for the (111) twin and a more general boundary. We present an algorithm to evaluate the geodesic distance between two boundaries, and apply it to identify the path along which the distance between these two boundaries is minimized. In general, the shortest path does not involve descriptions of boundary misorientations with the smallest misorientation angles.

Double-couple earthquake source: symmetry and rotation

We consider statistical analysis of double couple (DC) earthquake focal mechanism orientation. The symmetry of DC changes with its geometrical properties, and the number of 3-D rotations one DC source can be transformed into another depends on its symmetry. Four rotations exist in a general case of DC with the nodal-plane ambiguity, two transformations if the fault plane is known, and one rotation if the sides of the fault plane are known. The symmetry of rotated objects is extensively analyzed in statistical material texture studies, and we apply their results to analyzing DC orientation. We consider theoretical probability distributions which can be used to approximate observational patterns of focal mechanisms. Uniform random rotation distributions for various DC sources are discussed, as well as two non-uniform distributions: the rotational Cauchy and von Mises-Fisher. We discuss how parameters of these rotations can be estimated by a statistical analysis of earthquake source properties in global seismicity. We also show how earthquake focal mechanism orientations can be displayed on the Rodrigues vector space.

New kinematic parameters of finite rotation of a solid

A new family of kinematic (global and local) orientation parameters of a solid is presented and described. All kinematic parameters are obtained by the method of mapping the variables onto the corresponding oriented subspace (hyperplane). In particular, the method of stereographic projection of a point of a five-dimensional sphere S6 ⊂ R6 onto the oriented hyperplane R5 is presented for the classical direction cosines of the angles determining the orientation of two coordinate systems. A family of global kinematic parameters obtained by the method of mapping of five-dimensional kinematic Hopf parameters given in the space R5 onto the four-dimensional oriented subspace R4 is described. The theorem of the homeomorphism of two topological spaces (the four-dimensional sphere S4 ⊂R5 with one deleted point and the oriented hyperplane R4) is used to establish the correspondence between five- and four-dimensional kinematic parameters defined in the corresponding spaces. It is also shown what global four-dimensional orientation parameters, quaternions defined in the subspace R4 are associated with the classical local parameters, i.e., the Rodrigues and Gibbs three-dimensional finite rotation vectors. The projection method is used to obtain the kinematic differential equations (KDE) of rotation corresponding to the five- and four-dimensional orientation parameters. All above-introduced kinematic orientation parameters of a solid permit efficiently solving the classical Darboux problem by using the corresponding KDE, i.e., determining the body current angular position in the space R3 from the known (measured) angular velocity of the object rotation and its initial position in space.

New kinematic parameters of the finite rotation of a rigid body

A new family of kinematic parameters for the orientation of a rigid body (global and local) is presented and described. All the kinematic parameters are obtained by mapping the variables onto a corresponding orientated subspace (hyperplane). In particular, a method of stereographically projecting a point belonging to a five-dimensional sphere S5⊂R6 onto an orientated hyperplane R5 is demonstrated in the case of the classical direction cosines of the angles specifying the orientation of two systems of coordinates. A family of global kinematic parameters is described, obtained by mapping the Hopf five-dimensional kinematic parameters defined in the space R5 onto a four-dimensional orientated subspace R4. A correspondence between the five-dimensional and four-dimensional kinematic parameters defined in the corresponding spaces is established on the basis of a theorem on the homeomorphism of two topological spaces (a four-dimensional sphere S4⊂R5 with one deleted point and an orientated hyperplane in R4). It is also shown to which global four-dimensional orientation parameters–quaternions defined in the space R4 the classical local parameters, that is, the three-dimensional Rodrigues and Gibbs finite rotation vectors, correspond. The kinematic differential rotational equations corresponding to the five-dimensional and four-dimensional orientation parameters are obtained by the projection method. All the rigid body kinematic orientation parameters enable one, using the kinematic equations corresponding to them, to solve the classical Darboux problem, that is, to determine the actual angular position of a body in a three-dimensional space using the known (measured) angular velocity of rotation of the object and its specified initial position.

Codf‐evolutions on polycrystalline orientation continua obtained by fast geometric estimates of plastic slip

AbstractThis paper presents a framework for the modeling of evolving crystal orientation distribution functions (codf) and orientation averages of polycrystalline aggregates undergoing finite deformations. The method is designed for fast estimates of the polycrystalline deformation textures in metals. It bases on the Taylor approach of rigid polycrystal plasticity, where the decision of slip‐activity in the local crystalline grains is based on a geometric argument. This approach allows a purely kinematic estimate of codf evolution completely independent of the hardening response of the single crystal grains. The investigation starts with a precise description of a deforming orientation continuum. This concerns the general geometric setup as well as details of the parameterization of rotations in terms of Rodrigues vectors. We then outline Lagrangian and Eulerian geometric settings of crystal fraction conservation laws and discuss the details of their discretization. An important aspect is the definition of orientation averages. This homogenization over the orientation space is an important ingredient of a multiscale modeling of evolving anisotropy. It is shown that the discrete evaluation of orientation averages does not need the explicit computation of codf. The core of the proposed method is a fast update algorithm for rate‐independent rigid crystal plasticity that directly determines the current Rodrigues vectors. We show for several deformation modes of polycrystalline aggregates that the proposed geometric estimates predict the codf evolution and overall anisotropy very well in the range of large strains. The new method is extremely robust and suitable for multiscale methods for the description of evolving anisotropy in polycrystals. Copyright © 2010 John Wiley & Sons, Ltd.

Fast estimates of evolving orientation microstructures in textured bcc polycrystals at finite plastic strains

The paper presents a framework for a fast computational estimate of the texture evolution in bcc polycrystals. The underlying basis is a Taylor-type approach to rigid plasticity in bcc crystals, where the decision of slip activity is based on a purely geometric argument in terms of the rate of total deformation. The core part of the method is a new algorithm for rate-independent rigid crystal plasticity, which uses lattice rotations parametrized by Rodrigues vectors as history variables. We show that such a geometric estimate of the orientation microstructure provides very good approximations when compared with experimental results of bcc crystals. It is shown for several deformation modes of polycrystalline aggregates, that the geometric approach predicts the texture evolution very well at large strains. In contrast to standard approaches for the analysis of texture, the proposed new method is extremely robust and fast even for a high number of discrete grain orientations. As a consequence, such an approach provides a perfect base for large scale computations of texture development in polycrystals.

Error Analysis of the Rodrigues Vector Estimators Using Two Direction Measurements

Two attitude determination algorithms developed by Paul Davenport in 1965 are examined. The algorithms use two direction measurements to estimate attitude using the Rodrigues vector representation. A complete error, covariance and singularity analysis of both algorithms is performed. The results are verified numerically and compared to other attitude determination algorithms.

Optimal Linear Attitude Estimator

An optimal linear attitude estimator is presented for the case of a single-point real-time estimation of spacecraft attitude using the minimum-element attitude parameterization: Rodrigues (or Gibbs) vector g. The optimality criterion, which does not coincide with Wahba's constrained criterion, is rigorously quadratic and unconstrained. The singularity, which occurs when the principal angle is π, can easily be avoided by one rotation. The attitude accuracy tests show that the proposed method provides a precision comparable with those fully complying with the Wahba optimality definition. Finally, computational speed tests demonstrate that the proposed method belongs to the class of the fastest optimal attitude estimation algorithms.

MRAD: Modified rodrigues vector attitude determination

This paper presents a novel single-point optimal attitude determination algorithm in terms of the Modified Rodrigues Vector. The proposed technique has been derived from the Second Estimator of the Optimal Quaternion algorithm replacing the quaternion with the Modified Rodrigues Vector. Simulation results illustrate that the proposed attitude determination algorithm can be an efficient alternative over the quaternion-based algorithms in terms of computational efficiency for optimal attitude representation since the Modified Rodrigues Vector is a minimal attitude representation. The use of the Modified Rodrigues Vector has proven useful in the development of Extended Kalman Filters for attitude estimation, because it reduces the length of the state vector and there is no need to enforce the unit length constraint as for the quaternion.

A six-dimensional approach to microtexture analysis

A mathematical framework is presented for microtexture derivation and analysis. A six-dimensional space is introduced, defined as the product of direct space (x, y, z) and orientation space, parametrized by the Rodrigues vector (r 1, r 2, r 3). The framework is adapted to diffraction experiments, where orientation information from many grains or deformation microstructures is recorded in parallel, with the prime aim of allowing fast measurements and thus the ability to study microstructure dynamics. Assuming the use of a monochromatic beam the geometry of the projection surfaces is deduced. The derivation of the microtexture is formulated as an inverse problem similar to reconstruction problems in absorption or emission tomography. The applicability of the two major types of reconstruction algorithm, namely transform methods and algebraic methods, is discussed. An explicit solution in terms of the iterative algebraic reconstruction technique algorithm is given. The choice of basis function is discussed. Furthermore, the framework is applied to a number of important simplifying cases. These include grain-by-grain reconstruction, classical pole figure inversion and the generation of three-dimensional maps of non-deformed grains.

Estimation of Rodrigues parameters from vector observations

A recursive minimum variance algorithm for attitude determination is presented. The attitude is expressed by the Rodrigues vector, where only three parameters are needed to describe orientation. The advantage of using the Rodrigues parameters is computational. Attitude calculations involve only relatively simple algebraic expressions, which is advantageous especially in real-time applications. The efficiency of the derived algorithm is demonstrated using Monte-Carlo computer simulation results and in an initial alignment aerospace application.