Dwelling on the Connection Between SO(3) and Rotation Matrices in Rigid Multibody Dynamics – Part 1: Description of an Index-3 DAE Solution Approach

In rigid multibody dynamics simulation using absolute coordinates, a choice must be made in relation to how to keep track of the attitude of a body in 3D motion. The commonly used choices of Euler angles and Euler parameters each have drawbacks, e.g., singularities, and carrying along extra normalization constraint equations, respectively. This contribution revisits an approach that works directly with the orientation matrix and thus eschews the need for generalized coordinates used at each time step to produce the orientation matrix A. The approach is informed by the fact that rotation matrices belong to the SO(3) Lie matrix group. The numerical solution of the dynamics problem is anchored by an implicit first order integration method that discretizes, without index reduction, the index 3 Differential Algebraic Equations (DAEs) of multibody dynamics. The approach handles closed loops and arbitrary collections of joints. Our main contribution is the outlining of a systematic way for computing the first order variations of both the constraint equations and the reaction forces associated with arbitrary joints. These first order variations in turn anchor a Newton method that is used to solve both the Kinematics and Dynamics problems. The salient observation is that one can express the first order variation of kinematic quantities that enter the kinematic constraint equations, constraint forces, external forces, etc., in terms of Euler infinitesimal rotation vectors. This opens the door to a systematic approach to formulating a Newton method that provides at each iteration an orthonormal rotation matrix A. The Newton step calls for repeatedly solving linear systems of the form Gδ = e, yet evaluating the iteration matrix G and residuals e is inexpensive, to the point where in the Part 2 companion contribution the proposed formulation is shown to be two times faster for Kinematics and Dynamics analysis when compared to the Euler parameter and Euler angle approaches in conjunction with a set of four mechanisms.

Dwelling on the Connection Between SO(3) and Rotation Matrices in Rigid Multibody Dynamics – Part 2: Comparison Against Formulations Using Euler Parameters or Euler Angles

We compare three solution approaches that use the index 3 set of differential algebraic equations (DAEs) to solve the constrained multibody dynamics problem through straight discretization via an implicit time integrator. The first approach is described in a companion paper and dwells on the connection between the orientation matrix and the SO(3) group. Its salient point is that the orientation matrix A is a problem unknown, directly computed without resorting to the use of other position-level generalized coordinates such as Euler angles or Euler parameters. The second approach employs Euler angles as part of the position-level generalized coordinates, and uses them to subsequently evaluate the orientation matrix A. The third approach replaces the Euler angles with Euler parameters (quaternions). The numerical integration method of choice in this contribution is first order implicit Euler. We report a similar number of iterations for convergence for all solution implementations (called rA, rε, and rp); we also observed an approximately twofold speedup of rA over rp and rε. The tests were carried out in conjunction with three models: simple pendulum, slider crank, and four-link mechanism. These simulation results were obtained using two Python simulation engines that were developed independently as part of this formulation comparison undertaking. The codes are available in a GitHub public repository and were developed to provide two different perspectives on the formulation performance issue. The improvements in simulation speed are traced back to a simpler form of the equations of motion and more concise Jacobians that enter the numerical solution. It remains to investigate whether these speed gains carry to higher order integration formulas, where the underlying Lie-group structure of SO(3) brings additional complexity in the rA solution.

Int3D: A Data Reduction Software for Single Crystal Neutron Diffraction

We describe a new software package for the data reduction of single crystal neutron diffraction using large 2D detectors. The software consists of a graphical user interface from which the user can visualize, interact with and process the data. The data reduction is achieved by sequentially executing a series of programs designed for performing the following tasks: peak detection, indexing, refinement of the orientation matrix and motor offsets, and integration. The graphical tools of the software allow visualization of and interaction with the data in two and three dimensions, both in direct and reciprocal spaces. They make it easy to validate the different steps of the data reduction and will be of great help in the treatment of complex problems involving incommensurate structures, twins or diffuse scattering.

Pattern-matching indexing of Laue and monochromatic serial crystallography data for applications in materials science

Serial crystallography data can be challenging to index, as each frame is processed individually, rather than being processed as a whole like in conventional X-ray single-crystal crystallography. An algorithm has been developed to index still diffraction patterns arising from small-unit-cell samples. The algorithm is based on the matching of reciprocal-lattice vector pairs, as developed for Laue microdiffraction data indexing, combined with three-dimensional pattern matching using a nearest-neighbors approach. As a result, large-bandpass data (e.g. 5–24 keV energy range) and monochromatic data can be processed, the main requirement being prior knowledge of the unit cell. Angles calculated in the vicinity of a few theoretical and experimental reciprocal-lattice vectors are compared, and only vectors with the highest number of common angles are selected as candidates to obtain the orientation matrix. Global matching on the entire pattern is then checked. Four indexing options are available, two for the ranking of the theoretical reciprocal-lattice vectors and two for reducing the number of possible candidates. The algorithm has been used to index several data sets collected under different experimental conditions on a series of model samples. Knowing the crystallographic structure of the sample and using this information to rank the theoretical reflections based on the structure factors helps the indexing of large-bandpass data for the largest-unit-cell samples. For small-bandpass data, shortening the candidate list to determine the orientation matrix should be based on matching pairs of reciprocal-lattice vectors instead of triplet matching.

Methods for Assessing and Optimizing Solar Orientation by Non-Planar Sensor Arrays

Non-planar sensor arrays are used to determine solar orientation based on the orientation matrix formed by orientation vectors of the sensor planes. Solar panels or existing photodiodes can be directly used without increasing the size or mass of the spacecraft. However, a limiting factor for the improvement of the accuracy of orientation lies with the lack of an assessment-based approach. A formulation was developed for the supremum (i.e., the least upper bound) of orientation error of an arbitrary orientation matrix in terms of its influencing factors. The new formulation offers a way to evaluate the supremum of orientation error considering interference with finite energy and interference with infinite energy but finite average energy. For a given non-planar sensor array, a sub-matrix of the full orientation matrix would reach the optimal accuracy of orientation if its supremum of orientation error is the least. Principles for designing an optimal sensor array relate to the configuration of the orientation matrix, which can be pre-determined for a given number of sensors. Simulations and field experiment tested and validated the methods, showing that our sensor array optimization method outperforms the existing methods, while providing a way of assessment and optimization.

The ontology of person-centered healthcare: Theoretical essentials to reground medicine within its humanistic framework - Part III - The logic of subjectivity: wrapped in one‘s individual history

To ‘embrace’ focused parts of an addressed environment is the way enclosure of outside foci may be described. Here, the opening of the transfer institution called logical lock (LL) in the previous series of articles points, toward the outside of the individual, selects finite parts of it and either rejects them or utilizes them to achieve the corresponding embodiment. The different layers of the intermediating zone that have in total been described as an individual’s orientation matrix (OM) are described. They consist of mostly invisible, but emotionally perceptible and later intellectually discernible layers, such as the one formed by the personal history, present mood and present feelings, anticipations and expectations. To address a person not as an assembly of discernible organs, but as a person, is hence more demanding than addressing the person only as performing a role, a function. In establishing a logical basis for person-centered healthcare approaches, we introduce further logical and descriptive tools to take the invisible layers into account. This clearly hermeneutical approach is opposed to a method that would hypostasise what in this article are termed ‘naked objects’, abbreviated as NOs. We argue that such NOs exist only as mathematical extrapolations. As abstract extrapolations and, as far as individuality is concerned, they cannot be applied in a meaningful way.

The ontology of person-centered healthcare: Theoretical essentials to reground medicine within its humanistic framework - Part II - Nascent time and space: rooted functioning

This article presents further considerations of the inside-outside communication of human beings. The concept of a rooted time and space, introduced in the previous article (see below), is applied to conceptualise a basic structure that, in being centric itself, transforms incoming processes to be contained in centric sub-structures. These are described through a discussion of their main function of a ‘lock’ provided for ‘keys’ that may fit, or not fit. As finite structures, they also impose at least transient finitude on the incoming processes and manage to decelerate them. The slowing down allows comparative procedures to take place. If the processing of the incoming processes is not stopped, they will interact with inside processes to be continued in an altered way. Following an introduction of the concept of nascent time and space, as well as the concept of a logical lock, we define the intermediating structure responsible for providing the logical locks as active within an orientation matrix. Finally, we discuss the form in which this orientation matrix is applied to both construe and also embody coherent verbal and symbolising utterances such as words.

Orientation Uncertainty Characteristics of Some Pose Measuring Systems

We investigate the performance of pose measuring systems which determine an object’s pose from measurement of a few fiducial markers attached to the object. Such systems use point-based, rigid body registration to get the orientation matrix. Uncertainty in the fiducials’ measurement propagates to the uncertainty of the orientation matrix. This orientation uncertainty then propagates to points on the object’s surface. This propagation is anisotropic, and the direction along which the uncertainty is the smallest is determined by the eigenvector associated with the largest eigenvalue of the orientation data’s covariance matrix. This eigenvector in the coordinate frame defined by the fiducials remains almost fixed for any rotation of the object. However, the remaining two eigenvectors vary widely and the direction along which the propagated uncertainty is the largest cannot be determined from the object’s pose. Conditions that result in such a behavior and practical consequences of it are presented.

OrientXplot: a program to analyse and display relative crystal orientations

Orientations of single crystals are usually determined by diffraction experiments. Indexing of a diffraction pattern from one crystal leads to the determination of its `orientation matrix', which defines the orientation of its crystallographic axes relative to a set of reference axes associated with the diffractometer. Crystal orientations can also be described in terms of Euler angles, especially from electron backscattered diffraction measurements.OrientXplotis a Windows program that reads all common types of orientation matrices, as well as orientation data such as Euler angles. The program calculates and displays the relative orientations of pairs of crystals, such as twins or inclusion crystals trapped inside host crystals.OrientXplotcan manipulate (under user control) the orientation matrices to allow for ambiguities in indexing that arise from crystal symmetries. Orientation data can be displayed on a stereogram or output in numerical form for plotting in external programs.