Jacobian Of Transformation Research Articles

Definition and estimation of joint-space contour error based on generalized curve for five-axis contour following control

Five-axis contour following is one of the main tasks for five-axis CNC machine tools. The contour following accuracy directly determines the final machining precision. Therefore, control of the five-axis contour error is significant. Currently, most existing definitions of five-axis contour error are in the Cartesian task space, and this inevitably requires direct and inverse Jacobian transformations between the task-space contour-error computation and the joint-space contour-error control. Different from them, this paper defines the five-axis contour error in the ℝ5 joint space through adjusting uniform units of five joints, and accordingly proposes a third-order estimation algorithm for the defined joint-space contour error, with the help of the concept of generalized curve. Based on the joint-space contour-error definition and estimation, a joint-space five-axis cross-coupling control scheme is finally provided. Simulation and experimental results demonstrate that the presented third-order joint-space contour-error estimation algorithm has a satisfactory estimation accuracy, and the presented joint-space five-axis contour control method can decrease both of the joint-space and the task-space five-axis contour errors by more than 49%. It is also analyzed and verified that comparing with routine task-space five-axis contour control method, the presented joint-space method not only needs not the direct and inverse Jacobian transformations during error estimation, which saves the computational burden, but also generates minimum axial contour-control commands, which enhances the control stability, thus resulting in better contour-following performances.

A truly mesh‐distortion‐enabled implementation of cell‐based smoothed finite element method for incompressible fluid flows with fixed and moving boundaries

SummaryMesh‐free properties are part of the superiority of cell‐based smoothed finite element method (CS‐FEM), but have yet to be fully exploited for computational fluid dynamics. A novel implementation of CS‐FEM for incompressible viscous fluid flows in stationary and deforming domains discretized by severely distorted bilinear four‐node quadrilateral (Q4) elements is presented in this article. The negative determinant of the Jacobian transformation from the Cartesian coordinates to the natural coordinates is intentionally stipulated for the corresponding mesh over which FEM inevitably fails in practice. It is found that, without ad hoc modifications, CS‐FEM incurs unsatisfactory results and even a failure on fixed meshes. To cater for general computations on either a uniform or nonuniform mesh represented by these badly degenerated elements, four smoothing cells (SCs) are deployed in convex Q4 element whereas one SC in concave Q4 element. A simple hourglass control is introduced into those under‐integrated quadrilaterals for stabilizing the one‐SC quadrature in smoothed Galerkin weak form. Thanks to the adoption of characteristic‐based split (CBS) scheme for the fluid solution, a byproduct is the unfolded equivalence of the CBS stabilization and balancing tensor diffusivity under the incompressibility constraint. Several benchmark problems involving incompressible fluid flow and fluid‐structure interaction are solved. Numerical results show the good accuracy and robustness of the proposed approach that raises a seductive idea for resolving moving‐mesh problems.

Product and Commutators of Pseudo-differential Operators Involving Fourier–Jacobi Transform

The purpose of this paper is to define a new symbol class $$\Lambda $$ and discuss the theory of two different pseudo-differential operators (p.d.o.) involving Fourier–Jacobi transform associated with a single symbol in $$\Lambda $$. We also derive boundedness results for p.d.o.’s in Sobolev type space. A new pseudo-differential operator is developed using the product of symbols. Finally, norm inequality for commutators between two pseudo-differential operators is obtained.

On D-brane interaction & its related properties

We compute the closed-string cylinder amplitude between one Dp brane and the other Dp′ brane, placed parallel at a separation, with each carrying a general constant worldvolume flux and with p−p′=0,2,4,6 and p≤6. For the p=p′, we show that the main part of the amplitude for p=p′<5 is a special case of that for p=p′=5 or 6 case. For all other p−p′=2,4,6 cases, we show that the amplitude is just a special case of the corresponding one for p=p′ case. Combined both, we obtain the general formula for the amplitude, which is valid for each of the cases considered and for arbitrary constant worldvolume fluxes. The corresponding general open string one-loop annulus amplitude is also obtained by a Jacobi transformation of the general cylinder one. We give also the general open string pair production rate. We study the properties of the amplitude such as the nature of the interaction, the open string tachyonic instability, and the possible open string pair production and its potential enhancement. In particular, in the presence of pure magnetic fluxes or magnetic-like fluxes, we find that the nature of interaction is correlated with the existence of potential open string tachyonic instability. When the interaction is attractive, there always exists an open string tachyonic instability when the brane separation reaches the minimum determined by the so-called tachyonic shift. When the interaction is repulsive, there is no such instability for any brane separation. We also find that the enhancement of open string pair production, in the presence of pure electric fluxes, can occur only for the p−p′=2 case.

Minimal Soft Lattice Theta Functions

We study the minimality properties of a new type of “soft” theta functions. For a lattice Lsubset {mathbb {R}}^d, an L-periodic distribution of mass mu _L, and another mass nu _z centered at zin {mathbb {R}}^d, we define, for all scaling parameters alpha >0, the translated lattice theta function theta _{mu _L+nu _z}(alpha ) as the Gaussian interaction energy between nu _z and mu _L. We show that any strict local or global minimality result that is true in the point case mu =nu =delta _0 also holds for Lmapsto theta _{mu _L+nu _0}(alpha ) and zmapsto theta _{mu _L+nu _z}(alpha ) when the measures are radially symmetric with respect to the points of Lcup {z} and sufficiently rescaled around them (i.e., at a low scale). The minimality at all scales is also proved when the radially symmetric measures are generated by a completely monotone kernel. The method is based on a generalized Jacobi transformation formula, some standard integral representations for lattice energies, and an approximation argument. Furthermore, for the honeycomb lattice {mathsf {H}}, the center of any primitive honeycomb is shown to minimize zmapsto theta _{mu _{{mathsf {H}}}+nu _z}(alpha ), and many applications are stated for other particular physically relevant lattices including the triangular, square, cubic, orthorhombic, body-centered-cubic, and face-centered-cubic lattices.

A Method to Quantify Multi-Degree-of-Freedom Lower Limb Isometric Joint Torques in Children with Hemiplegia.

Pediatric hemiplegia, caused by a unilateral brain injury during childhood, can lead to motor deficits such as weakness and abnormal joint torque coupling patterns which may result in a loss of independent joint control. It is hypothesized that these motor impairments are present in the paretic lower extremity, especially at the hip joint where extension may be abnormally coupled with adduction. Previous studies investigating lower extremity isometric joint torques in children with spastic cerebral palsy used tools that limited data collection to one degree of freedom, making it impossible to quantify these coupling patterns. We describe the adaptation of a multi-joint lower extremity isometric torque measurement device to allow for quantification of weakness and abnormal joint torque coupling patterns at the hip in the pediatric population. We also present preliminary data in three children without hemiplegia to highlight how the presence of atypical femoral bony geometry, often observed in childhood hemiplegia, can be accounted for in the Jacobian transformations and affect joint torque measurements at the hip.

Magnetic properties in three electrons under Rashba spin‐orbit interaction and magnetic field

AbstractIn this work, we study three‐electron magnetic susceptibility in quantum dots under Rashba spin‐orbit interaction (SOI) and magnetic field by an analytical methodology. The Hamiltonian of the system is separated to center of mass and relative terms using the Jacobi transformations and the hyperspherical coordinates. By solving Schrodinger equation, energy levels and thereby the susceptibility are calculated using canonical ensemble. At zero temperature, the magnetization reduces with increasing magnetic field with and without Rashba SOI in three‐electron‐quantum dot without electron‐electron (e‐e) interaction. Also, SOI slightly changes the magnetization for three‐electron‐quantum dot without e‐e interaction. At nonzero temperature, the magnetization shows a paramagnetic peak when the magnetic field increases. This peak position changes under the SOI. In the presence of e‐e interaction, the susceptibility enhances with raising magnetic field and it shows a maximum. The susceptibility at low magnetic field is negative and then it becomes positive. The susceptibility with e‐e interaction and without SOI is always diamagnetic and its magnitude reduces with enhancing magnetic field. The susceptibility shows a transition between diamagnetic and paramagnetic with e‐e interaction and SOI.

An Efficient FPGA Parallel Implementation for 2-D MUSIC Algorithm

The multi signal classification (MUSIC) algorithm is a classical one in the direction estimation of radar. An highly efficient implementation scheme is proposed in this paper. In order to solve the problem that computing two-dimensional(2-D) MUSIC algorithm is not fast enough in engineering implementation and the resource consumption is large, and we have completed the simulation implementation of the algorithm on field programmable gate array (FPGA). Through the analysis of the steps of Jacobi transformation, a high efficient serial control scheme is proposed to realize Jacobi transformation, and the characteristic value decomposition is realized by using cyclic Jacobi iterative algorithm. This design avoids finding the most value through threshold comparison, reduces the hardware complexity and improves the decomposition speed. In spectral peak search, a parallel computing method is presented, which realizes parallel 2-D spectral peak search through spatial division, this method can effectively improve the effectiveness of search, and also can achieve the estimation of the target at the junction of airspace. Compared with the traditional implementation scheme, the design scheme has the advantages of high precision, high real time and saving hardware resources. Theoretical analysis and MATLAB simulation test show that this method and its FPGA design have good performance and have high application value in the practical process.

Robust CT ventilation from the integral formulation of the Jacobian.

Computed tomography (CT) derived ventilation algorithms estimate the apparent voxel volume changes within an inhale/exhale CT image pair. Transformation‐based methods compute these estimates solely from the spatial transformation acquired by applying a deformable image registration (DIR) algorithm to the image pair. However, approaches based on finite difference approximations of the transformation's Jacobian have been shown to be numerically unstable. As a result, transformation‐based CT ventilation is poorly reproducible with respect to both DIR algorithm and CT acquisition method.PurposeWe introduce a novel Integrated Jacobian Formulation (IJF) method for estimating voxel volume changes under a DIR‐recovered spatial transformation. The method is based on computing volume estimates of DIR mapped subregions using the hit‐or‐miss sampling algorithm for integral approximation. The novel approach allows for regional volume change estimates that (a) respect the resolution of the digital grid and (b) are based on approximations with quantitatively characterized and controllable levels of uncertainty. As such, the IJF method is designed to be robust to variations in DIR solutions and thus overall more reproducible.MethodsNumerically, Jacobian estimates are recovered by solving a simple constrained linear least squares problem that guarantees the recovered global volume change is equal to the global volume change obtained from the inhale and exhale lung segmentation masks. Reproducibility of the IJF method with respect to DIR solution was assessed using the expert‐determined landmark point pairs and inhale/exhale phases from 10 four‐dimensional computed tomographies (4DCTs) available on http://www.dir-lab.com. Reproducibility with respect to CT acquisition was assessed on the 4DCT and 4D cone beam CT (4DCBCT) images acquired for five lung cancer patients prior to radiotherapy.ResultsThe ten Dir‐Lab 4DCT cases were registered twice with the same DIR algorithm, but with different smoothing parameter. Finite difference Jacobian (FDJ) and IFJ images were computed for both solutions. The average spatial errors (300 landmarks per case) for the two DIR solution methods were 0.98 (1.10) and 1.02 (1.11). The average Pearson correlation between the FDJ images computed from the two DIR solutions was 0.83 (0.03), while for the IJF images it was 1.00 (0.00). For intermodality assessment, the IJF and FDJ images were computed from the 4DCT and 4DCBCT of five patients. The average Pearson correlation of the spatially aligned FDJ images was 0.27 (0.11), while it was 0.77 (0.13) for the IFJ method.ConclusionThe mathematical theory underpinning the IJF method allows for the generation of ventilation images that are (a) computed with respect to DIR spatial accuracy on the digital voxel grid and (b) based on DIR‐measured subregional volume change estimates acquired with quantifiable and controllable levels of uncertainty. Analyses of the experiments are consistent with the mathematical theory and indicate that IJF ventilation imaging has a higher reproducibility with respect to both DIR algorithm and CT acquisition method, in comparison to the standard finite difference approach.

Transformation Optics-Based Finite Difference Time Domain Algorithm for Scattering From Object With Thin Dielectric Coating

A proper dielectric coating can reduce the electromagnetic scattering of the conducting object significantly so that it cannot be detected by the radar. However, when the object contains a thin coating, fine grid size is needed to discretize the thin coating in the conventional finite-difference time-domain (FDTD) algorithm, which increases the amount of memory and computational time significantly. To overcome this dilemma, we present a transformation optics-based FDTD (TO-FDTD) algorithm to accelerate the solution of electromagnetic scattering from objects with thin dielectric coatings. Two kinds of novel TO-FDTD models are proposed in this paper for a coated cylinder and a coated arbitrary polygonal cylinder, respectively. Through coordinate transformation, the size of the object remains unchanged while its thin coating is enlarged to a thicker one, meaning that it can be simulated by the FDTD algorithm with uniform coarse grids instead of fine grids. The transformed material parameters become inhomogeneous and anisotropic in the transformed region, which can be obtained by solving a Jacobian transformation matrix. We then develop a stable FDTD algorithm for solving anisotropic Maxwell’s equations. Bistatic scatterings of coated cylinders and a coated polygonal cylinder are solved by the TO-FDTD algorithm proposed in this paper, respectively. The result of the TO-FDTD algorithm matches well with the exact value and the result of the commercial software Comsol. The computational efficiency and accuracy of the proposed TO-FDTD algorithm are validated by numerical experiments. Numerical results show that the TO-FDTD algorithm has higher computational accuracy than the conventional FDTD algorithm that fails to simulate the absorbing property of the coating, when the same coarse grid size is used in the simulation. Under the same level of accuracy, the proposed TO-FDTD method can improve the computational efficiency by 62-63 times than the conventional FDTD method with fine grids in the simulations in the paper.

Invertibility of the dual energy x-ray data transform.

The Alvarez-Macovski method extracts the x-ray energy-dependent information by expanding the attenuation coefficient as a linear combination of functions of energy multiplied by basis set coefficients. Since the basis functions are known a priori, the coefficients represent all the energy-dependent information. The method then computes the line integrals of these coefficients, summarized as a vector A, from measurements with multiple x-ray spectra, summarized as a vector L. The purpose of this paper is to determine the factors that affect the invertibility of the L(A) transformation with a two function basis set and two spectral measurements, the dual energy transformation. A general invertibility theorem is applied that requires testing for zero values of the Jacobian of the transformation in its input domain. General conditions for invertibility are proved. It is shown that the generalized A vector noise variance is proportional to the generalized measurement noise variance divided by the square of the Jacobian. The relationship between the zero Jacobian values and ambiguous sets of A vector points with the same L values is determined. The effect of zero Jacobian values on an iterative algorithm that inverts L(A) is simulated. The choice of a particular valid basis set does not affect invertibility. Nonoverlapping measurement spectra such as those from photon counting detectors with perfect pulse height analysis are invertible. The widely used x-ray tube spectra with different voltages are shown to be invertible. Spectra with the same maximum energy, such as those from layered detectors, approach noninvertibility with small absolute value Jacobian for large object thicknesses. The zero Jacobian values fall on curves in A vector space that, except for a simple artificial case, are close to but not exactly straight lines. With noninvertible spectra, pairs of ambiguous points are located on opposite sides of the zero Jacobian curve. The iterative algorithm has large convergence errors near zero Jacobian curves and converges to the closest ambiguous point to the initial estimate for other points. The invertibility of dual energy systems is determined by the presence of zero values of the Jacobian of the dual x-ray energy data transformation L(A) in the input domain.

Universal mock theta functions as quantum Jacobi forms

Quantum Jacobi forms were defined in 2016, naturally combining Zagier’s definition of a quantum modular form with that of a Jacobi form. To date, just three examples of such functions exist in the literature. Here, we prove that the universal mock theta function $$g_2$$ , as well as the universal mock theta functions $$K, K_1, K_2,$$ and $$\kappa $$ , gives rise to an infinite family of quantum Jacobi forms $$G_{a,b}(z;\tau )$$ of weight 1 / 2 in dense subsets $${{\mathscr {Q}}}_{a,b} \subseteq {\mathbb {Q}} \times {\mathbb {Q}}$$ . We then use these quantum Jacobi transformation properties to establish polynomial expressions for $$G_{a,b}$$ at pairs of rational numbers, as well as simple closed-form expressions for sums of Mordell integrals.

Some Improvements on Relativistic Positioning Systems

Abstract We make some considerations about Relativistic Positioning Systems (RPS). Four satellites are needed to position a user. First of all we define the main concepts. Errors should be taken into account. Errors depend on the Jacobian transformation matrix. Its Jacobian is proportional to the tetrahedron volume whose vertexes are the four tips of the receiver-satellite unit vectors. If the four satellites are seen by the user on a circumference in the sky, then, the Jacobian and the tetrahedron volume vanish. The users we consider are spacecraft. Spacecraft to be positioned cannot be close to a null Jacobian satellites-user configuration. These regions have to be avoided choosing an appropriate set of four satellites which are not seen too close to the same circumference in the sky. Errors also increase as the user spacecraft separates from the emission satellite region, since the tetrahedron volume decreases.We propose a method to autonomously potion a user-spacecraft which can test our method. This positioning should be compared with those obtained by current methods. Finally, a proposal to position a user-spacecraft moving far from Earth, with suitable devices (autonomous), is presented.

Diffusion-model analysis of pPb and PbPb collisions at LHC energies

We present an analysis of centrality-dependent pseudorapidity distributions of produced charged hadrons in pPb and PbPb collisions at the Large Hadron Collider (LHC) energy of [Formula: see text] = 5.02 TeV, and of minimum-bias pPb collisions at 8.16 TeV within the non-equilibrium-statistical relativistic diffusion model (RDM). In a three-source approach, the role of the fragmentation sources is emphasized. Together with the Jacobian transformation from rapidity to pseudorapidity and the limiting fragmentation conjecture, these are essential for modeling the centrality dependence. For central PbPb collisions, a prediction at the projected FCC energy of [Formula: see text] = 39 TeV is made.

Quantum Jacobi forms and balanced unimodal sequences

The notion of a quantum Jacobi form was defined in 2016 by Bringmann and the second author in [1], marrying Zagier's notion of a quantum modular form [12] with that of a Jacobi form. Only one example of such a function has been given to-date (see [1]). Here, we prove that two combinatorial rank generating functions for certain balanced unimodal sequences, studied previously by Kim, Lim and Lovejoy [8], are also natural examples of quantum Jacobi forms. These two combinatorial functions are also duals to partial theta functions studied by Ramanujan. Additionally, we prove that these two functions have the stronger property that they exhibit mock Jacobi transformations in C×H as well as quantum Jacobi transformations in Q×Q. As corollaries to these results, we use quantum Jacobi properties to establish new, simpler expressions for these functions as simple Laurent polynomials when evaluated at pairs of rational numbers.

On the applicability of numerical image mapping for PIV image analysis near curved interfaces

This paper scrutinises the general suitability of image mapping for particle image velocimetry (PIV) applications. Image mapping can improve PIV measurement accuracy by eliminating overlap between the PIV interrogation windows and an interface, as illustrated by some examples in the literature. Image mapping transforms the PIV images using a curvilinear interface-fitted mesh prior to performing the PIV cross correlation. However, degrading effects due to particle image deformation and the Jacobian transformation inherent in the mapping along curvilinear grid lines have never been deeply investigated. Here, the implementation of image mapping from mesh generation to image resampling is presented in detail, and related error sources are analysed. Systematic comparison with standard PIV approaches shows that image mapping is effective only in a very limited set of flow conditions and geometries, and depends strongly on a priori knowledge of the boundary shape and streamlines. In particular, with strongly curved geometries or streamlines that are not parallel to the interface, the image-mapping approach is easily outperformed by more traditional image analysis methodologies invoking suitable spatial relocation of the obtained displacement vector.

Evaluation and design of a large-scale cloaking device by the Hamiltonian-based ray-tracing method Part II: design of the distribution of constitutive parameters

Improvement in cloaking performance by the design of distributions of the constitutive parameters is investigated. The distributions are changed by employing Navier’s equation with various distributions of Young’s modulus as the partial differential equation for the solution of the Jacobian transformation matrix used in transformation optics. The cloaking performance is evaluated by Hamiltonian-based ray tracing. The numerical results show the proposed design method can improve the cloaking performance. The design method can contribute to the realization of large-scale cloaking devices by taking account of the finite resolution of the manufacture.

Flat control of industrial robotic manipulators

A new approach to tracking control of industrial robot manipulators is presented in this paper. The highly coupled nonlinear dynamics of a six degrees of freedom (6-DOF) serial robot is decoupled by expressing its variables as a function of a flat output and a finite number of its derivatives. Hence the derivation of the flat output for the 6-DOF robot is presented. With the flat output, trajectories for each of the generalized coordinates are easily designed and open loop control is made possible. Using MATLAB/Simulink S-functions combined with the differential flatness property of the robot, trajectory tracking is carried out in closed loop by using a linear flat controller. The merit of this approach reduces the computational complexity of the robot dynamics by allowing online computation of a high order system at a lower computational cost. Using the same processor, the run time for tracking arbitrary trajectories is reduced significantly to about 10 s as compared to 30 min in the original study (Hoifodt, 2011). The design is taken further by including a Jacobian transformation for tracking of trajectories in cartesian space. Simulations using the ABB IRB140 industrial robot with full dynamics are used to validate the study.

A combined Lagrangian method for simulation of axisymmetric gas-particle vortex flows

A combined fully Lagrangian approach for meshless modeling of unsteady axisymmetric vortex flows of a gas-particle mixture with an incompressible carrier phase is developed. The approach proposed is based on the combination of a meshless vortex method for calculating axisymmetric flows of the carrier phase described by the Navier–Stokes (or Euler) equations and the full Lagrangian method for calculating the parameters of the dispersed phase. The combination of these methods reduces the problem of modeling the two phase flows to the solution of a high-order system of ordinary differential equations for the coordinates of toroidal vortex elements in the carrier phase and the particle trajectories, the velocity components, and the components of the Jacobian of transformation from the Eulerian to the Lagrangian variables in the dispersed phase. The application of the method is illustrated by modeling the behavior of an admixture of inertial Stokes particles with a small mass concentration in unsteady flows like solitary vortex rings in a viscous carrier phase and groups of vortex rings in an effectively inviscid carrier phase.

Application of a discontinuous Galerkin method on the compressible flow in the transonic axial compressor

The discontinuous Galerkin method (DGM) has been extended to simulate the compressible flow in the transonic axial compressor in a rotating coordinate system. The equations are solved in the physical space instead of being in the computational space to avoid the Jacobian transformation. A compact vertex-based limiter is proposed to prevent the spurious oscillations of the numerical results. The grid independence of the method is investigated and the results show it is weakly dependent on the mesh type. The in-house code based on the method has been applied to simulations of two transonic axial compressors. The results show that the predicted flow field, including overall performance, spanwise distributions of several parameters, shock structures and wakes, are in better agreement with the measurement. It indicates that the DGM has low numerical dissipation and could produce more reliable results.