Orientation Matrix Research Articles

On the boundedness of outer polyhedral estimates for reachable sets of linear systems

New properties of outer polyhedral (parallelepipedal) estimates for reachable sets of linear differential systems are studied. For systems with a stable matrix, it is determined what the orientation matrices are for which the estimates possessing the generalized semigroup property are bounded/unbounded on an infinite time interval. In particular, criteria are found (formulated in terms of the eigenvalues of the system’s matrix and the properties of bounding sets) that guarantee for previously mentioned tangent estimates and estimates with a constant orientation matrix that either there are initial orientation matrices for which the corresponding estimate tubes are bounded or all these tubes are unbounded. For linear stationary systems, a system of ordinary differential equations and algebraic relations is derived that determines estimates with constant orientation matrices for reachable sets that have no generalized semigroup property but are tangent and also bounded if the matrix of the system is stable.

Protein crystals can be incommensurately modulated

For a normal periodic crystal, the X-ray diffraction pattern can be described by an orientation matrix and a set of three integers that indicate the reciprocal lattice points. Those integers determine the spacing along the reciprocal lattice directions. In aperiodic crystals, the diffraction pattern is modulated and the standard periodic main reflections are surrounded by satellite reflections. The successful indexing and refinement of the main unit cell andqvector usingTWINSOLVE, developed by Svensson [(2003). Lund University, Sweden], are reported here for an incommensurately modulated, aperiodic crystal of a profilin:actin complex. The indexing showed that the modulation is along thebdirection in the crystal, which corresponds to an `actin ribbon' formed by the crystal lattice. Interestingly, the transition to the aperiodic state was shown to be reversible and the diffraction pattern returned to the periodic state during data collection. It is likely that the protein underwent a conformational change that affected the neighbouring profilin:actin molecules in such a way as to produce the observed modulation in the diffraction pattern. Future work will aim to trap the incommensurately modulated crystal state, for example using cryocooling or chemical crosslinking, thus allowing complete X-ray data to be collected.

On the Role of Universities in Building Knowledge and Human-centred Societies

This article presents an enlightened and egalitarian framework that evaluates the impact of universities in the infrastructure, governance, talent and culture dimensions of enlightened, neo-humanist and knowledge societies. We will expound the above stated role of a university by first developing the concept of an enlightened knowledge society. The quality of life and its sustainability is to a large extent fashioned by the society, while society is itself a collectivity of human thought and ideals. What is implied then is that human talent and society influence one another. How then can this influence be positive or progressive? We need to know what constitutes positive or progressive influence, and hence what role the university has in inculcating this progressive influence. This is a fundamental enquiry, which constitutes the very foundation of the purpose, orientation, course and educational matrix of a university. This enquiry will take us through a panoramic survey of an enlightened knowledge society, of the different sectors of the society, of how these sectors provide creative and/or fulfilling expression for its people, and thereby enrich human life. More specifically, we look at dimensions covering the infrastructure, talent, cultural and governance facets of such a society. The enquiry will also involve a study of how the themes and structures of the societal sectors are to be developed at the university, to be subsequently implemented through its educational and research programs. We will develop this enquiry in stepwise fashion proceeding from our social roots to our climactic desideratum, as we embark on a romantic odyssey of the neo-humanist knowledge society and the role of the university in enabling this exalted progression.

Rudimentary Finite Element Thermal Modeling of Platelet-Filled Polymer-Ceramic Composites

In order to improve the thermal performance of polymeric materials, they can be filled with intrinsically high thermal conductivity fillers that provide heat-conducting paths through the resulting composite. The thermal performance of polymers loaded with platelet-shaped fillers was modeled using finite element analysis in order to provide a prediction of thermal conductivity as a function of variables such as filler thermal conductivity, orientation, and polymer matrix thermal conductivity. Modeling results were compared to experimental data. An unexpectedly strong effect that the matrix conductivity has on the conductivity of the polymer-ceramic composite was predicted by modeling and confirmed experimentally.

Nonlinear attitude and gyroscope's bias estimation for a VTOL UAV

This paper addresses the problem of attitude and heading restitution for a VTOL UAV (Vertical Take Off and Landing Unmanned Aerial Vehicle). In a first step, we propose an observation strategy to restore the attitude starting from a measure of the orientation matrix and angular rate readings polluted by an unknown bias vector. In the second step, we detail the implementation of this algorithm in the HoverEye, a ducted fan VTOL UAV developed by Bertin Technologies, equipped with an inertial measurement unit (IMU) and magnetometers. A measured orientation matrix is calculated from the gravity and the earth's magnetic field. Then, an estimated orientation is built by integration of gyroscopic readings, and corrected by the measured ones. The proposed observer, directly designed in the special orthogonal group SO(3), is as efficient as classical Extended Kalman Filtering based observers, but easier to implement in real time. Experiments achieved on the HoverEye illustrate the concept.

OS5-1-6 The effect of film thickness on crystal rotation behavior with fatigue crack propagation in copper films

Using a fatigue testing method by which fatigue cracks can be initiated and propagated in a film adhered to cover an elliptical through-hole in a base plate subjected to push-pull cyclic loads, annealed copper films with the thickness of 100μm and those reduced the thickness from the 100μm to 50μm by an electro-polishing were fatigued under a constant stress amplitude with a stress ratio of zero. The crystal rotation behavior with the fatigue crack propagation was investigated by measuring the crystal orientation around the fatigue crack initiated from the notch root before and after fatigue testing, using EBSD (Electron Back-scatter Diffraction) method. Then, the change of crystal orientation with fatigue testing was evaluated quantitatively from the misorientation between the crystal orientation matrix on the same point obtained before and after fatigue testing. As a result, the angle of the crystal rotation obtained from the region showing the high fatigue crack propagation rate was larger than that obtained from the region showing the low fatigue crack propagation rate for the film with the thickness of 100μm, while the fatigue crack propagated faster in the film with the thickness of 50μm than that with the thickness of 100μm regardless of the small crystal rotation angles with the fatigue testing for the film with the thickness of 50μm.

Revisiting the Sexual Orientation Matrix for Supervision

This article addresses the use of the Sexual Orientation Matrix for Supervision (SOMS) in preparing supervisees to work with GLBTQ families. Originally published as a tool to assist supervisors and educators to help their supervisees work more effectively with gay, lesbian, and bisexual (GLB) couples, the SOMS is founded on two core concepts: (1) degree of heterosexual bias; and (2) degree of acceptance of GLBTQ orientations and behavior. In this revision, we have expanded heterosexual bias to include sexual identities, sexual orientations, and gender identities. Supervisors can employ the Matrix to explore their own and their trainees' levels of comfort, knowledge, and experience. We also provide specific suggestions and tasks.

Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions

If the three moments of inertia are distinct, the solution to the Euler equations for the free rigid body is given in terms of Jacobi elliptic functions. Using the arithmetic–geometric mean algorithm (Abramowitz and Stegun 1992 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover)), these functions can be calculated efficiently and accurately. Compared to standard numerical ODE and Lie–Poisson solvers, the overall approach yields a faster and more accurate numerical solution to the Euler equations. This approach is designed for mass asymmetric rigid bodies. In the case of symmetric bodies, the exact solution is available in terms of trigonometric functions, see Dullweber et al (1997 J. Chem. Phys. 107 5840–51), Reich (1996 Fields Inst. Commun. 10 181–91) and Benettin et al (2001 SIAM J. Sci. Comp. 23 1189–203) for details. In this paper, we consider the case of asymmetric rigid bodies subject to external forces. We consider a strategy similar to the symplectic splitting method proposed in Reich (1996 Fields Inst. Commun. 10 181–91) and Dullweber et al (1997 J. Chem. Phys. 107 5840–51). The method proposed here is time-symmetric. We decompose the vector field of our problem into a free rigid body (FRB) problem and another completely integrable vector field. The FRB problem consists of the Euler equations and a differential equation for the 3 × 3 orientation matrix. The Euler equations are integrated exactly while the matrix equation is approximated using a truncated Magnus series. In our experiments, we observe that the overall numerical solution benefits greatly from the very accurate solution of the Euler equations. We apply the method to the heavy top and the simulation of artificial satellite attitude dynamics.

Attitude and gyro bias estimation for a VTOL UAV

In this paper, a nonlinear complementary filter ( x-estimator) is presented to estimate the attitude of a vertical take off and landing unmanned aerial vehicle (VTOL UAV). The measurements are taken from a low-cost IMU (inertial measurement unit) which consists of 3-axis accelerometers and 3-axis gyroscopes. The gyro bias are estimated online. A second nonlinear complementary filter ( z-estimator) which combines 3-axis gyroscope readings with 3-axis magnetometer measurements, is also designed. Based on the proposed estimators, the full orientation matrix R will be retrieved. Note that R evolves on the special orthogonal group SO ( 3 ) , since it is a transformation matrix between two orthogonal frames. This representation of orientation by the rotation matrix allows to overcome the problem of singularities that appear in local parametrization such as Euler angles. Therefore, both estimators may be used to describe any type of 3D motion. The convergence of the two observers is theoretically and experimentally proven; simulations and experiments are conducted on a real platform in hovering flight conditions.

INVERSE KINEMATICS OF 2R POSITIONER BASED ON WELDING POSTION REPRESENTATION

An algorithm of general 2R(rotation) positioner inverse kinematics is presented. According to the model of the welding position, giving a weld seam point, the weld orientation then can be resolved from the weld rotation angle and the weld slope angle. Symbolic are used to represent the positioner's link parameters and the orientation matrix between the robot and the positioner. The kinematics coordinate systems are established , the formula of 2R positioner inverse kinematics is derived. The simulation experiment results show that this algorithm can improve the ability of robotic arc welding off-line programming system to program all types of welding positions . This algorithm can also be used when the positioner is placed differently.

Preliminary observations on the calcaneal trabecular microarchitecture of extant large‐bodied hominoids

In this pilot study, we point out potential differences between calcaneal trabecular microarchitecture in humans and nonhuman large apes, such as increased degree of anisotropy, reduced bone volume fraction, and very stereotypical orientation of the trabeculae. Even though sample size does not permit us to investigate the issue statistically, the observed differences between humans and other hominoids warrants further in-depth investigation. We also show that some measurements of the trabecular network might be dependent on sampling density, which can be difficult to deal with in the case of animals of different body masses. We also present a new visualization technique that summarizes the trabecular network orientation, and makes it more readily interpretable than the summary statistics of the underlying fabric tensor of the orientation matrix.

Determination of crystal orientation by an area-detector image for surface X-ray diffraction

A new method of calculating the crystal orientation matrix (Umatrix) of a specified sample using two-dimensional X-ray diffraction spots that are recorded on an area detector is presented. In this way, theUmatrix is calculated using at least three two-dimensional diffraction spots of known two-dimensional indices, which provide the projection of the crystal truncation rod onto the detector. The method, in the case of surface X-ray diffraction measurements with an area detector, enables easier and faster sample alignment than the conventional method to determine theUmatrix.

Normal modes of synchronous rotation

The dynamics of synchronous rotation and physical librations are revisited in order to establish a conceptually simple and general theoretical framework applicable to a variety of problems. Our motivation comes from disagreements between the results of numerical simulations and those of previous theoretical studies, and also because different theoretical studies disagree on basic features of the dynamics. We approach the problem by decomposing the orientation matrix of the body into perfectly synchronous rotation and deviation from the equilibrium state. The normal modes of the linearized equations are computed in the case of a circular satellite orbit, yielding both the periods and the eigenspaces of three librations. Libration in longitude decouples from the other two, vertical modes. There is a fast vertical mode with a period very close to the average rotational period. It corresponds to tilting the body around a horizontal axis while retaining nearly principal-axis rotation. In the inertial frame, this mode appears as nutation and free precession. The other vertical mode, a slow one, is the free wobble. The effects of the nodal precession of the orbit are investigated from the point of view of Cassini states. We test our theory using numerical simulations of the full equations of the dynamics and discuss the disagreements among our study and previous ones. The numerical simulations also reveal that in the case of eccentric orbits large departures from principal-axis rotation are possible due to a resonance between free precession and wobble. We also revisit the history of the Moon's rotational state and show that it switched from one Cassini state to another when it was at 46.2 Earth radii. This number disagrees with the value 34.2 derived in a previous study.

NONLINEAR ATTITUDE AND GYROSCOPE'S BIAS ESTIMATION FOR A VTOL UAV

This paper addresses the problem of attitude and heading restitution for a VTOL UAV. We describe an observation strategy to restitute the complete attitude matrix of the vehicle starting from Inertial Measurement Unit (IMU) and magnetometers. This study is a part of the development of the ducted fan UAV designed by Bertin Technologies. First, a measured orientation matrix is calculated from both inertial vectors which are the gravity and the earth magnetic field. Then, an estimated orientation is built by integrating gyroscopic readings, and corrected by the measured one. Nonlinear observation techniques are used to design a nonlinear estimator of the orientation matrix and an adaptive filter of the gyroscope's bias which ensure the convergence of the observer. Such an observer is as efficient as classical Extended Kalman Filtering based observers, and easier to implement in real time. Simulations are proposed to illustrate the concept.

The Sexual Orientation Matrix for Supervision

SUMMARY The Sexual Orientation Matrix for Supervision (SOMS) was created in order to assist supervisors and trainers in preparing supervisees to work with lesbian, gay, and bisexual (LGB) clients. The SOMS was developed around two core concepts: (1) degree of heterosexual bias, and (2) degree of acceptance of LGB orientations and behavior. Supervisors can employ the matrix to explore both their own and their supervisee's levels of comfort, knowledge, and experience in working with LGB clients including same-sex couples. This article describes the development of the matrix, an exploration of the concepts underlying this tool, and an explanation of how to use the matrix, including suggested tasks for supervision of therapists working with same-sex couples.

Force Analysis of a Vibratory Bowl Feeder for Automatic Assembly

This paper investigates the vibratory bowl feeder for automatic assembly, presents a geometric model of the feeder, and develops force analysis, leading to dynamical modeling of the vibratory feeder. Based on the leaf-spring modeling of the three legs of the symmetrically arranged bowl of the feeder, and equating the vibratory feeder to a three-legged parallel mechanism, the paper reveals the geometric property of the feeder. The effects of the leaf-spring legs are transformed to forces and moments acting on the base and bowl of the feeder. Resultant forces are obtained based upon the coordinate transformation, and the moment analysis is produced based upon the orthogonality of the orientation matrix. This reveals the characteristics of the feeder, that the resultant force is along the z-axis and the resultant moment is about the z direction and further generates the closed-form motion equation. The analysis presents a dynamic model that integrates the angular displacement of the bowl with the displacement of the leaf-spring legs. Both Newtonian and Lagrangian approaches are used to verify the model, and an industrial case-based simulation is used to demonstrate the results.

Synthesis and Structural Characterization of an Air-stable {[HC(pyrazolyl)3]Cu(PPh3)}PF6

(λ = 0.71073 A) radiation sourceand a CCD detector at 173(2) K. The 45 frames of twodimensional diffraction images were collected and processedto obtain the cell parameters and orientation matrix. The first50 frames were retaken after complete data collection. Thecrystal showed no significant decay and no correction wasapplied for absorption or decay. The frame data wereprocessed to give structure factors, using the programSAINT.

Geometrical isomerism in uranyl chromates II. Crystal structures of Mg2[(UO2)3(CrO4)5](H2O)17 and Ca2[(UO2)3(CrO4)5](H2O)19

Abstract Geometrical isomerism of uranyl chromate sheets occurs in the structures of Mg2[(UO2)3(CrO4)5](H2O)17 and Ca2[(UO2)3(CrO4)5](H2O)19. Both compounds were prepared by evaporation from aqueous solutions. Compound 1 is orthorhombic, Pbca, a = 19.9206(10), b = 21.0526(10), c = 18.4966(9) Å, V = 7757.1(7) Å3; R 1 = 0.050 for 6332 unique reflections with |F o| ≥ 4σF . Compound 2 is monoclinic, P21/m, a = 11.0359(5), b = 17.5364(7), c = 11.5056(5) Å, β = 118.178(1)°, V = 1962.8(2) Å3; R 1 = 0.037 for 6209 unique reflections with |F o| = 4σ F. Both structures contain sheets of composition [(UO2)3(CrO4)5] involving corner-sharing UO7 pentagonal bipyramids and CrO4 tetrahedra. The sheets are parallel to (100) in both structures. Mg2+ and Ca2+ cations are coordinated by H2O molecules and are located in the interlayer. The topology of the [(UO2)3(CrO4)5] sheets in both structures is the same, whereas their geometrical structures are different, and they are geometrical isomers. The sheets differ in the orientations of non-shared corners of chromate tetrahedra relative to the plane of the sheet. In order to describe the geometrical structure of the isomers, the concept of orientation matrix of tetrahedra is introduced. It has been shown that there are at least three geometrical isomers of the [(UO2)3(CrO4)5] sheet in uranyl chromates that differ in their orientation matrices of the tetrahedra.

Orientation of O(3) and SU(2)⊗ CI representations in cubic point groups ( Oh, Td) for application to molecular spectroscopy

We propose a detailed method for the symmetrization of the standard O(3) or SU(2)⊗ C I basis | j τ , m〉 ( τ= g or u) into the O h or T d point group. This is realized by means of an orientation matrix called G. The oriented basis obtained in this way allows matrix element calculations for rovibronic spectroscopic problems concerning octahedral or tetrahedral molecules. Particular attention has been put on careful phase choices. A numerical calculation of all the G matrix elements for both integer and half-integer j values up to 399/2 has been performed. Such high angular momentum values are necessary for the case of heavy molecules with high rotational excitation. To calculate the G coefficients with high precision at high j values we pre-calculated the necessary Wigner functions using symbolic MAPLE software and made then the numerical calculations with quadruple precision. The complete list of these coefficients can be obtained freely at the URL: http://www.u-bourgogne.fr/LPUB/group.html. As an illustration, we also present briefly an application to two typical spectroscopic calculations: the pure rotational levels of SF 6 in its ground vibrational state and the ν 3 band of ReF 6 (an open-shell molecule with an odd number of electrons and a fourfold degenerate electronic ground state).

OBSIFRAC: database-supported software for 3D modeling of rock mass fragmentation

Under stress, fractures in rock masses tend to form fully connected networks. The mass can thus be thought of as a 3D series of blocks produced by fragmentation processes. A numerical model has been developed that uses a relational database to describe such a mass. The model, which assumes the fractures to be plane, allows data from natural networks to test theories concerning fragmentation processes. In the model, blocks are bordered by faces that are composed of edges and vertices. A fracture can originate from a seed point, its orientation being controlled by the stress field specified by an orientation matrix. Alternatively, it can be generated from a discrete set of given orientations and positions. Both kinds of fracture can occur together in a model. From an original simple block, a given fracture produces two simple polyhedral blocks, and the original block becomes compound. Compound and simple blocks created throughout fragmentation are stored in the database. Several fragmentation processes have been studied. In one scenario, a constant proportion of blocks is fragmented at each step of the process. The resulting distribution appears to be fractal, although seed points are random in each fragmented block. In a second scenario, division affects only one random block at each stage of the process, and gives a Weibull volume distribution law. This software can be used for a large number of other applications.