Orthonormal Coordinate System Research Articles

A novel moving orthonormal coordinate-based approach for region of attraction analysis of limit cycles

<p style='text-indent:20px;'>The paper proposes a Lyapunov theory-based method to compute inner estimates of the <i>region of attraction</i> (ROA) of stable limit cycles. The approach is based on a transformation of the system to transverse coordinates, defined on a <i>moving orthonormal coordinate system</i> (MOC) for which a novel construction is presented. The proposed <i>center point MOC</i> (cp-MOC) is associated with a user-defined center point and provides flexibility to the construction of the transverse coordinates. In particular, compared to the standard approach based on hyperplanes orthogonal to the flow, the new construction allows the analyst to obtain larger regions of the state space where the well-definedness property of the transformation is satisfied. This has important benefits when using transverse coordinates to compute inner estimates of the ROA. To demonstrate these improvements, a sum-of-squares optimization-based formulation is proposed for computing inner estimates of the ROA of limit cycles for polynomial dynamics described in transverse coordinates. Different algorithmic options are explored, taking into account computational and accuracy aspects. Results are shown for three different systems exhibiting increasing complexity. The presented algorithms are extensively compared, and the newly cp-MOC is shown to markedly outperform existing approaches.</p>

A Dynamic Anatomic Study of Parasacral Perforators: Mapping, Perforasome, and Applications for Buttock Enhancement.

Due to the lack of knowledge about parasacral artery perforators, flaps from this region cannot be used with complete confidence in their security and effectiveness. Knowledge of the clusters and perforasome of these perforators could help in the design of more reliable flaps and extend the range of applications. This study aimed to identify the location, number, and density of perforators, and to subsequently analyze the perfusion flow and linking vessel distribution. Five fresh cadavers were harvested and dissected. For the mapping, after injecting lateral sacral arteries with colored latex, perforators with a diameter of >0.5 cm were examined in 5 sacral regions. All data were collected on the suprafascial plane, with an orthonormal coordinate system placed on iliac crests and median lines. For perforasome analysis, 5 perforators and 3 three sacral flaps were injected with radiopaque dye. A dynamic (4-dimensional) computed tomographic angiography completed the analysis. A mean [standard deviation] of 8.4 [1.36] perforators per corpse, with a mean diameter of 0.72 [0.14] mm, were identified. There was a higher density of parasacral perforators close to the median line and 7.6 cm above the iliac crests. This pattern was not a random distribution (P < 0.05). The perfusion area was preferentially in the superior gluteal region. Perfusion flow was permitted by the dominant direct-linking vessels towards adjacent lumbar perforators, oriented diagonally upward and outward to the midline. Parasacral perforator flaps appear to be a useful procedure in reconstruction and in aesthetic surgery, especially in gluteal augmentation. Their reliability depends on sound anatomic knowledge, with accurate preoperative perforator mapping.

Analysis and control of a Stewart platform as base motion compensators - Part I: Kinematics using moving frames

Kinematics and its control application are presented for a Stewart platform whose base plate is installed on a floor in a moving ship or a vehicle. With a manipulator or a sensitive equipment mounted on the top plate, a Stewart platform is utilized to mitigate the undesirable motion of its base plate by controlling actuated translational joints on six legs. To reveal closed loops, a directed graph is utilized to express the joint connections. Then, kinematics begins by attaching an orthonormal coordinate system to each body at its center of mass and to each joint to define moving coordinate frames. Using the moving frames, each body in the configuration space is represented by an inertial position vector of its center of mass in the three-dimensional vector space ℝ3, and a rotation matrix of the body-attached coordinate axes. The set of differentiable rotation matrices forms a Lie group: the special orthogonal group, SO(3). The connections of body-attached moving frames are mathematically expressed by using frame connection matrices, which belong to another Lie group: the special Euclidean group, SE(3). The employment of SO(3) and SE(3) facilitates effective matrix computations of velocities of body-attached coordinate frames. Loop closure constrains are expressed in matrix form and solved analytically for inverse kinematics. Finally, experimental results of an inverse kinematics control are presented for a scale model of a base-moving Stewart platform. Dynamics and a control application of inverse dynamics are presented in the part II-paper.

Application of the Gram–Schmidt factorization of the deformation gradient to a cone and plate rheometer

In this paper, we study the cone and plate rheometer using the Gram–Schmidt factorization of the deformation gradient. This new solution has several advantages over the traditional approach. It is shown that with the use of these kinematics, one can avoid the need for using a convected, curvilinear, coordinate system, which often leads to cumbersome calculations. Here, the use of a convected coordinate system has been replaced with a certain orthonormal coordinate system that arises from the Gram–Schmidt factorization of the deformation gradient. Moreover, by using this solution procedure, it is possible to obtain the normal stress differences and shear stress explicitly. Therefore, this solution procedure opens up a possibility for characterizing material properties by using only a cone and plate rheometer.

The specific branches leading to the genioglossus muscle: three-dimensional localisation using skin reference points.

Supra-selective stimulation of the branches destined for the horizontal part of genioglossus muscle (GGh) could be a target of choice in the treatment of mild-to-severe obstructive sleep apnea syndrome. The main aim of our study was to assess a percutaneous method for the three-dimensional localisation of the terminal branches destined to GGh. Twenty cadaveric hypoglossal nerves were dissected and included in the injection protocol. The distance between the posterior edge of the mandibular symphysis and the hyoid bone on the sagittal midline as the approximated distance of the geniohyoid muscle (dGH) was measured before any dissection. Methylene blue mixed with a thickening agent, was injected. The injection point was defined in relation to dGH, in an orthonormal coordinate system. For each dissection, we recorded the theoretical and the real (X, Y, Z) coordinates of GGh motor points and measured their distance to each other. X was accurately estimated. Y and Z were overestimated by +5.34 ± 5.21mm ([Formula: see text]) and +4.79 ± 3.99mm ([Formula: see text]) on average, respectively. We found a more significant difference between the theoretical and real Y and Z coordinates in the subgroup BMI < 25kg/m2 (8.6 ± 4.5mm and 6.9 ± 2.5mm, respectively, p = 0.0009), and of Z in subgroup with dGH ≥ 50mm (6.89 ± 3.26mm, p = 0.0494). X can be estimated accurately using the relationship [Formula: see text]. Y seems to be related to BMI and Z may be estimated with the relationship [Formula: see text]. This three-dimensional localisation could be very helpful to facilitate placement of cuff electrodes to manage refractory sleep apnea.

Parameterization of rotations in reference frames with redundant crystallographic axes

Parameterizations of rotations in three-dimensional space are fundamental for description of crystallographic textures. Such parameterizations (e.g., Rodrigues parameters) are usually specified using orthonormal coordinate systems, whereas bases of crystal lattices are generally non-orthogonal. In the case of crystals with hexagonal or rhombohedral lattices, the reference frames involve redundant crystallographic axes, and hence a question arises about feasibility of the generalization of the rotation parameterizations to such frames. The definition of Rodrigues parameters can be extended so they are directly linked to non-Cartesian bases of crystal lattices. The new Rodrigues parameters are contra- or covariant components of vectors specified with respect to exactly the same lattice basis as atomic positions in a unit cell. The generalized formalism allows for using redundant crystallographic axes. Also the orientation matrices can be represented in such frames. The Rodrigues parameterization in non-Cartesian coordinate frames is convenient for crystallographic applications because the generalized parameters are directly related to indices of rotation-invariant lattice directions and to Miller indices of rotation-invariant lattice planes. In the case of the hexagonal and rhombohedral lattices, the frames with redundant axes are used to account for lattice symmetry, but one may apply such frames for other reasons. They can be convenient for handling arbitrary symmetries, in particular symmetries arising in description of some phenomena or symmetries of physical processes. The practicality of frames is illustrated by an alternative description of body-centered lattices, a formula for lattice rotation in deformation by slip, and a new interpretation of indexing of single crystal diffraction patterns.

Reliability of Color Doppler Ultrasound Imaging for the Assessment of Anterolateral Thigh Flap Perforators: A Prospective Study of 30 Perforators.

Diagnostic, II.

An adaptive approach for the reconstruction and modeling of as-built 3D pipelines from point clouds

Abstract Automated extraction of 3D geometric shapes such as planes, spheres, cylinders, cones, and tori in laser-scanned point clouds is a challenging problem and a tedious process, especially when using cluttered data. This paper describes a modification of the existing Hough transform for the automatic detection of cylinder parameters in point clouds. Careful analysis reveals that the existing method still has excessive space and time complexity or yields imprecise outcomes. The approach described here modifies the orientation estimation with an area-based adaptive method that utilizes a small accumulator to detect significant peaks in the Hough space in the presence of single or multiple cylinders in the point cloud data. After orientation estimation, the position and radius are estimated using an orthonormal coordinate system with a circle fitting algorithm. These modifications are tested with extensive sets of real point cloud data, and experimental results show that the presented approach minimizes the space and time complexity. After detection, the relationship between cylinders is reconstructed to form a continuous axis network by tracking cylinder parameters obtained from earlier steps. Using the axis network of cylinders obtained from point clouds, models of entire pipelines that include straight pipes, elbow joints, and T-junctions are determinately defined, and output data is reconstructed in Smart Plant 3D (SP3D). The presented results show that the proposed approach indeed improves the computational complexity by reducing the space and time, and yields methods that can be employed in the automation of 3D pipeline model reconstruction.

Productivity Index for Arbitrary Well Trajectories in Laterally Isotropic, Spatially Anisotropic Porous Media

SummaryFor an anisotropic medium, the permeability is a second-order tensor. As such, permeability is an entity that is dependent upon the coordinate system used to describe the flow quantitatively, although the flow itself is independent of this system. In particular, this means that when a well is producing fluids from a porous medium, the coordinate system must be specified for descriptive purposes and, consequently, the permeability tensor relative to that coordinate system is determined. It is impossible to simultaneously align a 3D orthonormal coordinate system and a cylindrical well segment defined by a single axis except for special cases. Because the permeability tensor will introduce different formulations depending on the defined coordinate system, its complexity will vary accordingly. The objective of this paper is to define an optimal coordinate system with respect to the simplicity of the flow description into the well with an arbitrary trajectory for the special situation of a laterally isotropic, spatially anisotropic medium. The derivation for this optimization strategy is dependent on a sequence of flow-rate-preserving geometric transformations. The resulting virtual medium has isotropic attributes in the transformed 2D plane. Under this transform, known closed-form models are applicable by use of a single permeability value.

Inflation with teleparallelism: Can torsion generate primordial fluctuations without local Lorentz symmetry?

Arbitrary generalization to the teleparallel equivalent of general relativity loses local Lorentz invariance to reparametrize the orthonormal coordinate system and gives rise to asymmetry field equations. We investigate consequences of local Lorentz violation to primordial fluctuations in extended single field inflationary models based on the scalar-tensor formulation of the torsion scalar $T$ that effectively includes $f(T)$ gravity as a special case. We show that despite some asymmetry part of the field equations are removed in a spatially homogeneous and isotropic cosmic background, no subhorizon scalar-perturbation mode can survive by the time of horizon crossing. As a result, any scalar field mediated in torsion cannot generate enough primordial density inhomogeneity alone, even if it brings some de Sitter background solutions in generalized teleparallel gravity.

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.

Étude de la normalité du bandeau orbito-naso-frontal

Craniosynostoses are cranio-facial malformations affecting about 1/2100 newborns in France. The involvement of anterior sutures (coronal and metopic) leads to orbito-frontal deformities. The treatment calls upon surgery the goal being, on an esthetic point of view, to restore a normal anatomy. The purpose of our work was to establish if some facial and/or frontal measures easy to perform on a CT are correlated to the global shape of the normal orbito-naso-frontal bandeau (ONFB). Cranial CTs of 123 consecutive non-malformed children aged between 4 and 12 months were selected in the database of Department of Radiology of the University Hospital of Besançon - France. The CTs were all relocated by rigid transformation in an orthonormal coordinate system. On each of the 123 CTs, 21 reproducible measures representative of the global shape of the ONFB were made. Statistical analyses of these measures were achieved, considering age and gender, in order to determine the correlation between each measure and the ONFB shape. The only measure statistically correlated to the ONFB shape was the distance between the fronto-zygomatic sutures (FZD). The FZD was independent from age (in an interval of 4 months) and from gender. The 20 other measures did not show any correlation with age or gender. The FZD allows in itself to determine the ONFB global shape. This measure, easily available on a CT, can help the surgeon to perform a customized reshaping of the ONFB. The development of a surgical template using this measure is ongoing.

A method for characterisation of the static elastic properties of the porous frame of orthotropic open-cell foams

This paper proposes a method to identify the static, fully relaxed elastic Hooke’s matrix of a porous open-cell material. The moduli are estimated through an inverse estimation method, by performing a fit of a numerical model on the measured displacements on the faces of the porous material. These displacements are obtained from a static compression along each of the three coordinate axes. The material is modelled as an orthotropic equivalent solid, of which the principal directions are not necessarily aligned with the orthonormal coordinate system in which the experiments are conducted. The angles of relative orientation accounting for the misalignment are among the properties to be estimated. The focus in this paper is on the methodology itself, and its validity is verified by applying the method to four artificial materials with different levels of anisotropy. In addition, the robustness of the method to perturbations on the input data is investigated.

Tensors: A guide for undergraduate students

A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. We show that once orthonormality is relaxed, a dual basis, together with the contravariant and covariant components, naturally emerges. Manipulating these components requires some skill that can be acquired more easily and quickly once a new notation is adopted. This notation distinguishes multi-component quantities in different coordinate systems by a differentiating sign on the index labelling the component rather than on the label of the quantity itself. This tiny stratagem, together with simple rules openly stated at the beginning of this guide, allows an almost automatic, easy-to-pursue procedure for what is otherwise a cumbersome algebra. By the end of the paper, the reader will be skillful enough to tackle many applications involving tensors of any rank in any coordinate system, without index-manipulation obstacles standing in the way.

Non coplanar positioning evaluation on stereotactic cranial treatments with a TruebeamTM on board imaging

Purpose/Objective Cranial stereotactic treatments are planned with Volumetric Arctherapy (VMAT) and delivered with an accelerator TrueBeam (STX-HD)/Varian. Several arcs are defined with three iscocentric treatment table rotations. The images used to control the positioning are obtained only with the kV imager OBI (On Board Imaging). The accelerator room does not contain any additional inroom imagers. Each table rotation is validated with specific planar kV images. Material/methods The patient positioning is assessed with the help of a CB-CT compared to the reference treatment planning CT. An automatic table displacement is performed when set-up errors appear. This position being defined as a reference; two additional anterior and posterior planar images are performed. Three markers are placed on the tabletop (MT), on the mask at the front isocenter projection (MI) and on the patient's skin (MP) through a hole above an orbit. Positions of the 3 markers in the orthonormal coordinate system are recorded (L–R and H–F directions). The markers positions are then compared to those measured at the other treatment table rotations. Results The evaluation involved 1094 measurements of 360 images and 15 different patients. The images comparison is carried out for three markers in the two orthogonal directions. The measurement error, due to the use of a graphic rule, is estimated to be 0.2 mm. The evaluation is performed for each marker (MT, MI, MP). The maximum deviation is, respectively, 0.9, 0.9, 4.2 mm. The average deviation is, respectively, 0.4, 0.3, 0.4 mm. The standard deviation is, respectively, 0.3, 0.2, 0.5 mm. Conclusion Knowing that no additional in-room images are used, comparing the position of every marker between the images obtained at table rotations, relative to the reference position, ensures the traceability of each non-coplanar position. The three markers provide additional information on the three elements (table, patient, mask) related to the target. The results are consistent with the mechanical precision of the TrueBeam accelerator and its treatment table. We finally validate the safety margin PTV set to 3 mm.

3D inverse dynamics in non-orthonormal segment coordinate system

The net joint forces and moments may be computed by several 3D inverse dynamic methods. To do so, an orthonormal segment coordinate system (SCS) is generally mandatory. However, the segment axes ought to be selected following anatomical, functional, and inertial requirements that are hardly compatible with orthogonal axes. An alternative method based on generalized coordinates allows computing inverse dynamics using directly a set of basic points and unitary vectors. A segment definition is put forward in order to follow all of the anatomical, functional, and inertial requirements and the inverse dynamics is performed in a non orthonormal segment coordinate system (NSCS). The NSCS seems a convenient definition in biomechanics as far as anatomical, functional and inertial axes are concerned, but providing that the 3D joint forces and moments are still computable. The inverse dynamic method in NSCS is applied to the gait of a knee valgus subject and compared to a classical inverse dynamic method. The inverse dynamic method in NSCS shows comparable results but implies further clinical interpretations.

Rotations of three-joint fingers: a radiological study.

The aim of the current study was to test a protocol of quantification of phalangeal three-dimensional (3D) rotations during flexion of three-joint digits. Three-dimensional-specific software was developed to analyze CT reconstruction images. A protocol was carried out with six fresh-frozen upper limbs from human cadavers free from any visible pathology (three females, three males). CT millimetric slices were done for reconstruction of hand bone units. Orthonormal coordinate systems of inertia were calculated for each unit. Three-dimensional phalangeal rotations were estimated between two static positions (fingers in extension and in a fist position). Results were displayed for the joints of each three-joint finger with calculation of 3D rotations. Mean longitudinal axial rotations of metacarpophalangeal (MCP), proximal interphalangeal (PIP) and distal interphalangeal (DIP) joints ranged from 14 degrees pronation to 19 degrees supination. The index finger was in a global pronation position (4/6 specimens). The fourth and fifth fingers were in a global supination position in every case. The third finger was in a more variable global rotation (pronation in 2/6 specimens). MCP, PIP and DIP flexion angles ranged respectively from 71 degrees to 89 degrees, 65 degrees to 87 degrees, and 41degrees to 77 degrees. Lateral angles ranged from 19 degrees (ulnar angulation) to 23 degrees (radial angulation). The study of phalangeal rotations was possible in spite of a heavy protocol. This protocol could be partially automatated to speed up the analyses. Longitudinal axial rotations could be analyzed, in addition to flexion/extension or abduction/adduction rotations. CT scan reconstructions would be helpful for investigating pathological fingers. Abnormal rotations of digits could be quantified more precisely than during a current clinical examination of the hand.

Helical Collapse of a Whirling Elastic Rod Forced to Lie on a Cylinder

Helical Collapse of a Whirling Elastic Rod Forced to Lie on a Cylinder

Two Reflected Analyses of Lights Out

(2001). Two Reflected Analyses of Lights Out. Mathematics Magazine: Vol. 74, No. 4, pp. 295-304.

Hyper space vectors: A new four‐quantity extension of space‐vector theory

AbstractThe space‐vector theory, which allows to describe three linearly dependent quantities (e. g. three voltages or three currents) using only two linearly independent quantities represented in an orthonormal coordinate system, is widely used in the fields of AC machine theory, power definitions and active filtering. It can be used very effectively in case of circuits with only three terminals. Problems occur in case of four terminals, when four linearly dependent quantities (e. g. four currents, the sum of which is always zero) exist. In this case a zero‐sequence quantity can be additionally introduced, which has to be treated separately with special equations. Vector operations like cross product or dot product, which are very useful to calculate e. g. power quantities, can no longer be used in the general case. This paper presents a method transforming any system represented by four linearly dependent quantities or by three linearly independent quantities into a three‐dimensional orthonormal coordinate system. All four members of a set of linearly dependent quantities are treated equally. We suggest to call this representation Hyper Space Vector (HSV). A lot of vector operations can be applied without problems to HSV, making calculations much easier and more graphic.