Orthogonal Error Research Articles

Uniformity enhancement of flow field through optimizing nozzle structure for jet electrochemical machining

To improve the processing quality during the Jet electrochemical machining (JECM) process, a single-orifice nozzle and dual-orifice nozzles with different inclination angles were designed, where the fluid dynamic behavior was numerically simulated. Then, through the L49 (74) orthogonal design and error analysis, the influence of nozzle dimension on the flow field characteristics was analyzed. Finally, the nozzle with optimal structure and dimension was obtained and validated by experiment. Results show that the dual-orifice nozzles had better flow field uniformity than the single-orifice nozzle. In the dual-orifice nozzle, the increase of the inlet inclination angle can improve the flow field uniformity. The influence of the nozzle dimension parameters on the flow field uniformity is prioritized as follows: the outlet slit width > upper chamber diameter > the height of the inflow chamber > the height of the transition chamber. After optimization, the optimal dimension parameters were obtained under the upper chamber diameter of 16 mm, the jet outlet slit width of 0.4 mm, the height of the inflow chamber of 7 mm, and the height of the transition chamber of 23 mm. The flow velocity dispersions under the pressures of 0.3, 0.4, and 0.5 MPa can be reduced by 25.41 %, 25.17 %, and 28.16 %. The findings can help understand the flow behavior and guide the structure optimization for improving JECM quality.

Modified Universal Kriging-based clearance error optimization for orthogonal robot

Abstract Aiming at the problem that the clearance error of the traditional tire laser engraving robot is too high, and thus the resulting positioning accuracy cannot meet the accuracy requirements needed for engraving at this stage, this paper takes the homemade orthogonal robot as the object of research, and researches the optimization method of the positioning error of the orthogonal robot based on the modified Kriging. In this paper, we use spinor theory to construct a robot clearance error model, establish an objective function with orthogonal robot clearance error as the optimization objective, and use a particle swarm algorithm to modify universal kriging (MUK) to optimize the positioning error. The optimization method is simulated and analyzed, and MUK is selected for comparison with other optimization methods. The simulation results show that 52% relative reduces the number of iterations of the MUK to the universal kriging model (UK), 79% relative to the particle Swarm Optimization (PSO), and 81% relative to the ordinary kriging model (OK), which makes the algorithm better applicable in comparison. At the same time, a tire laser engraving orthogonal robot experimental platform was built for physical verification experiments, and the experimental results show that the maximum error before and after the optimization of the integrated error of orthogonal robot positioning accuracy has been reduced by 89.91%; the average positioning error has been reduced by 86.90%; and the root-mean-square error has been reduced by 83.01%, which further proves the practicability of the proposed method.

Error correction analysis of wavefront testing in quadriwave lateral shearing interferometry

Quadriwave lateral shearing interferometry (QWLSI) is based on double birefringent crystals of a beam displacer (DBCs-BD), which can generate the lateral shearing interference wavefront of four beams of overlapped replicas in the DBCs-BD orthogonal directions. When the replica waves are overlapped incident to the analyzer and the direction of the transmission axis is set as 45° or 135°, the QWLSI’s polarization interferogram can be obtained. This paper deduces the principle of QWLSI based on the DBCs-BD and presents the analysis of orthogonal error influence based on the DBCs-BD and the phase retrieval error of QWLSI when shear displacement by tilted incident on the DBCs-BD. In our investigation, we have established the correction range of the PBD’s orthogonal angle error is within −0.5∘−0.5∘, the maximum error in PV is 0.0012λ, and the maximum error in RMS is 1.3789×10−4λ in wavefront reconstruction. Moreover, when the testing light tilts to incident on the PBD in the range of −0.4∘−0.4∘, correction of the shear distance is used for wavefront reconstruction to achieve a high-precision wavefront testing result. Finally, the experiment shows that QWLSI based on the DBCs-BD exhibits feasibility and high precision.

Process development and validation of next generation 3D calibration standards for application in optical microscopy

Abstract A scalable wafer-based fabrication process for a new generation of 3D standards enabling the 3D calibration of optical microscopes is presented and validated. The 3D standards are based on step pyramids with several layers in the µm range and a system of cylindrical knops distributed across the layers as marks for coordinate based calibration. This enables calibration for the three coordinate axes and the orthogonality error between them in a single measurement step. The requirements necessary for such a calibration, as optical non-transparency, reproducible flatness of the pyramid step heights and the lowest possible deviations of the lateral marks coordinates, are met by optimizing the manufacturing process: The deviation of the height steps distributed over the wafer is ±3.6 nm and is primarily caused by the layer deposition processes. The lateral manufacturing accuracy was determined using calibrated scanning electron microscope (SEM) and show a mean deviation of 20 or 60 nm, depending on the lateral size of the structures. The electron beam lithography process and the level of inaccuracy of the SEM standard have an influence on the lateral scaling accuracy. Based on the tactilely generated height values and the coordinates of the mark determined by a calibrated SEM, an example calibration of a confocal laser scanning microscope was successfully performed and showed good conformity to conventional calibration techniques.

Research and Experimentation on the Orthogonal Error Correction Method for Horizontal Displacement Monitoring

Research and Experimentation on the Orthogonal Error Correction Method for Horizontal Displacement Monitoring

Addressing Advanced IC Substrate Deformation and Pattern Distortion Using an Extremely Large Exposure Field Fine Fine-Resolution Lithography System

The More than Moore era is upon us, as manufacturers increasingly turn to back-end advances to meet the next-generation device performance gains of today and tomorrow. In the advanced packaging space, heterogeneous integration combines multiple chips with different functionalities and from different silicon nodes inside one package, ranging in size from 75mm x 75mm to 175mm x 175mm. But as with any new technology, heterogeneous integration comes with its own set of unique challenges. For starters, package sizes are expected to grow significantly due to the number of components making up each integrated package. The problem: these significantly larger packages would typically require multiple exposure shots to complete the lithography steps for the package. Furthermore, the process of adding multiple redistribution layers (RDL) may cause stress to both the surface and inside of the substrate. Formation changes experienced by the panel, as a result of thermal, high-pressure and other fan-out processes, shift the design location from its nominal coordinates; this causes inaccurate overlay and low-overlay yield in the lithography process. And then there is the matter of tightening resolution requirements. RDL layers in advanced integrated circuit substrates (AICS) will require a resolution of 3µm and will eventually move toward a resolution of 1µm. To address these challenges, and better enable the heterogenous integration process, an extremely large exposure field, fine-resolution lithography solution was proposed to enable packages over 250mm x 250mm without the need for image stitching, while exceeding the overlay and critical dimension uniformity requirements for these packages. In this paper, a 515mm x 510mm Ajinomoto build-up film (ABF)+copper clad laminate (CCL) substrate is selected as the test vehicle. We will analyze the pattern distortion of an ABF+CCL substrate to understand the distribution of translation, rotation, scale, magnification, trap, orthogonality and other errors in the substrate, and then use an extremely large exposure field, fine-resolution lithography system to address the pattern distortion of the substrate. This demonstration will provide an analysis of panel distortion and detail how the extremely large exposure field, fine-resolution lithography solution addresses panel distortion to achieve an aggressive overlay number.

On-chip full bridge bipolar linear spin valve sensors through modified synthetic antiferromagnetic layers

Realizing the rising demand for high-sensitivity magneto-resistive (MR) sensors in the industry, an attempt was made towards the fabrication of a spin valve (SV) based bipolar sensor on chip in the form of a full Wheatstone bridge with push–pull configuration. For a sensor with a push–pull configuration, the opposite arms are configured with opposite sense layers. To achieve this, either odd or even layers of synthetic antiferromagnetic (SAF) structure were introduced in the conventional spin valve structure, through which the magnetization orientation of the sensing layer was regulated by either negative or positive sensitivity to the applied field direction respectively. A detailed theoretical simulation was carried out to achieve the stable antiparallel magnetic state around zero fields by varying individual layer thicknesses in the case of single and double-layer SAF structures. With optimized conditions, experimental data were presented, and a bipolar device with a sensitivity of the order of 0.7 mV/V/Oe in the linear field range of ± 15 Oe was fabricated. The performance of the device was characterized by a wider temperature range and it was observed that the device has high thermal stability in the range of - 40 °C to 125 °C with a thermal coefficient of the voltage around 12 μV/V/∘C. Angular performance of the fabricated sensor with minimized orthogonality error was presented and it was observed that the sensor can measure the angle with an accuracy of +/-2 ° .

Online Calibration of RIMU Based on Multistage EKF

Redundant inertial measurement units (RIMUs) are more reliable than normal IMUs, but they are suffering from the calibration of non-orthogonal gyroscopes and accelerometers. In this paper, aiming at the problem of relying on the orthogonal axis IMU information and error term coupling in the online calibration of RIMUs, a method based on the multi-stage EKF algorism is proposed. In this method, the calibration model of the inertial system is established, which does not depend on the orthogonality with the inclined axis inertial system, meanwhile, the attitude, velocity, and position information is taken as the measurement, and a filtering model is established based on the EKF method. Based on this model, the observability of the calibration method is proved, and the window and threshold for secondary filtering are set to reduce the coupling effect of scale factor error and installation error. The digital simulation indicates that the proposed method can provide high-precision online calibration results for the inclined axis scale factor error and installation error of RIMUs, and the calibration error is below 10%.

Fast-Tracking Optical Coherent Receiver Tolerating Transmitter Component Distortion

In this paper, we propose a receiver architecture capable of compensating for a time-varying signal distortion induced by the transmitter analog components. In the proposed architecture, the novel transmitter distortion compensating (TDC) block is implemented outside of the feedback control loop of the adaptive equalizer to suppress the feedback delay by optimizing the parameter used in the least mean square (LMS) algorithm. The proposed TDC block is designed for compensating for the DC offset, the IQ amplitude imbalance, the IQ orthogonality error, and the IQ timing skew. Our simulation results show that the proposed scheme significantly reduces the required signal-to-noise ratio penalty from the theoretical limit, which is imposed due to the transmitter components distortion. Furthermore, our theoretical analysis confirms that the delay induced in the feedback loop of the adaptive equalizer determines the upper bound of the LMS step size under the stable condition, hence allowing us to maximize the tracking speed of our receiver.

Tensor coaxial calibration of magnetic sensor array

In order to effectively solve the influence of the magnetic sensor array error caused by axial inconsistency on the measurement accuracy, an array composed of four fluxgate sensors is built by using the Euler rotation matrix, and the consistency correction is carried out. Firstly, aiming at the non orthogonal, scale factor, zero bias and other error factors of the sensor itself, a 9-PARAMETER correction coefficient matrix is constructed to compensate the non orthogonal magnetic field measurement of the sensor, and the ideal orthogonal three component magnetic field measurement value is obtained. On the above-mentioned basis, the reference sensor in the magnetometry array is selected, and the roll, pitch and azimuth of the sensor to be calibrated are transformed to establish the Euler rotation matrix, and the model for 9-PARAMETER consistency correction is constructed. The reference sensor and the sensor to be calibrated are placed on the magnetic gradiometer for experimental verification. The experimental results show that the root mean square error of magnetic field measurement is within 10 nT after the sensor measurement error compensation correction, and the angle error of each axis measurement is within 0.01åfter the consistency correction (4.22 nT). It shows that the magnetic field measurement values of different measuring points in the sensor array can be well projected on the coordinate axis of the reference sensor after correction, and the magnetic field information of the measuring points can be correctly projected, which has high reliability.

Signal Subspace Reconstruction for DOA Detection Using Quantum-Behaved Particle Swarm Optimization

Spatial spectrum estimation, also known as direction of arrival (DOA) detection, is a popular issue in many fields, including remote sensing, radar, communication, sonar, seismic exploration, radio astronomy, and biomedical engineering. MUltiple SIgnal Classification (MUSIC) and Estimation Signal Parameter via Rotational Invariance Technique (ESPRIT), which are well-known for their high-resolution capability for detecting DOA, are two examples of an eigen-subspace algorithm. However, missed detection and estimation accuracy reduction often occur due to the low signal-to-noise ratio (SNR) and snapshot deficiency (small time-domain samples of the observed signal), especially for sources with different SNRs. To avoid the above problems, in this study, we develop a DOA detection approach through signal subspace reconstruction using Quantum-Behaved Particle Swarm Optimization (QPSO). In the developed scheme, according to received data, a noise subspace is established through performing an eigen-decomposition operation on a sampling covariance matrix. Then, a collection of angles randomly selected from the observation space are used to build a potential signal subspace on the basis of the steering matrix of the array. Afterwards, making use of the fact that the signal space is orthogonal to the noise subspace, a cost function, which contains the desired DOA information, is designed. Thus, the problem of capturing the DOA information can be transformed into the optimization of the already constructed cost function. In this respect, the DOA finding of multiple signal sources—that is, the multi-objective optimization problem—can be regarded as a single objective optimization problem, which can effectively reduce the probability of missed detection of the signals. Subsequently, the QPSO is employed to determine an optimal signal subspace by minimizing the orthogonality error so as to obtain the DOA. Ultimately, the performance of DOA detection is improved. An explicit analysis and derivation of the developed scheme are provided. The results of computer simulation show that the proposed scheme has superior estimation performance when detecting signals with very different SNR levels and small snapshots.

Calibration of the empirical fundamental relationship using very large databases

This paper describes a procedure for fitting traffic stream models using very large traffic databases. The proposed approach consists of four steps: (1) an initial treatment to eliminate noisy, inaccurate data and to homogenize the information over the density range; (2) a first fitting of the model, based on the sum of squared orthogonal errors; (3) a second filter, to eliminate outliers that survived the initial data treatment; and (4) a second fitting of the model. The proposed approach was tested by fitting the Van Aerde traffic stream model to 104 thousand observations collected by a permanent traffic monitoring station on a freeway in the metropolitan region of São Paulo, Brazil. The model fitting used a genetic algorithm to search for the best values of the model parameters. The results demonstrate the effectiveness of the proposed approach.

Error Analysis of Accelerometer- and Magnetometer-Based Stationary Alignment

This paper revisits the stationary attitude initialization problem, i.e., the stationary alignment, of Attitude and Heading Reference Systems (AHRSs). A detailed and comprehensive error analysis is proposed for four of the most representative accelerometer- and magnetometer-based stationary attitude determination methods, namely, the Three-Axis Attitude Determination (TRIAD), the QUaternion ESTimator (QUEST), the Factored Quaternion Algorithm (FQA), and the Arc-TANgent (ATAN). For the purpose of the error analysis, constant biases in the accelerometer and magnetometer measurements are considered (encompassing, hence, the effect of hard-iron magnetism), in addition to systematic errors in the local gravity and Earth magnetic field models (flux density magnitude, declination angle, and inclination angle). The contributions of this paper are novel closed-form formulae for the residual errors (normality, orthogonality, and alignment errors) developed in the computed Direction Cosine Matrices (DCM). As a consequence, analytical insight is provided into the problem, allowing us to properly compare the performance of the investigated alignment formulations (in terms of ultimate accuracy), as well as to remove some misleading conclusions reported in previous works. The adequacy of the proposed error analysis is validated through simulation and experimental results.

Adaptive Filters With Robust Augmented Space Linear Model: A Weighted $k$-NN Method

As a new member of convex universal learning machines (CULMs), an augmented space linear model (ASLM) demonstrates strong learning and tracking capabilities in the fields of signal process and machine learning. Unfortunately, the ASLM is only suitable for offline learning, and its estimated accuracy and noise immunity highly depend on the estimated error learned from a generated error table. To address these two issues and apply ASLM into adaptive filtering, online ASLM algorithms with a weighted k nearest neighbor (k-NN) method, based on an updated orthogonal error table, are therefore proposed in this paper. To further improve the robustness of ASLM to heavy-tail noises, a robust ASLM (RASLM) algorithm is proposed with the help of kernel function, and thus a novel robust adaptive filtering algorithm named maximum correntropy criterion RASLM (MCC-RASLM) is developed based the maximum correntropy criterion. However, the linear growth error table existing in MCC-RASLM incurs an ever-growing computational burden. Therefore, to reduce the computational complexity of MCC-RASLM, another novel online variable-centroid clustering is proposed to generate a maximum correntropy criterion cluster RASLM (MCC-C-RASLM) algorithm. Finally, the theoretical analyses regarding robustness and the error bound of RASLM with weighted k-NN regression are performed to verify the effectiveness of MCC-RASLM. Simulation results based on synthetic and real-world data validate the superiorities of the proposed algorithms from aspects of filtering accuracy and robustness.

An accuracy analysis of Galerkin meshfree methods accounting for numerical integration

Presently, it has been widely recognized that numerical integration plays a crucial role on the solution accuracy of Galerkin meshfree methods due to the rational nature of meshfree shape functions and the misalignment of shape function supports with integration cells. Nonetheless, the corresponding underlying cause is still not quite clear. To address this issue, an accuracy analysis of Galerkin meshfree methods is presented in this study with a special focus on the contribution of numerical integration to the error estimates. One important fact is that the integration difficulty of meshfree methods leads to a loss of the Galerkin orthogonality condition which usually serves as a basis to develop error bounds. In order to enable the accuracy analysis, an error measure is proposed to evaluate the loss of Galerkin orthogonality condition, which is actually related to the order of integration consistency for Galerkin meshfree formulation. With the aid of this orthogonality error measure, both H1 and L2 error estimates are established for Galerkin meshfree methods. It turns out that these errors essentially consist of two parts, one comes from the standard interpolation error, and the other one attributes to the numerical integration error. In accordance with the proposed error estimates, it is evident that for the conventional Gauss integration schemes violating the integration consistency, the solution errors eventually are controlled by the integration error and optimal convergence cannot be achieved even with high order quadrature rules. On the other hand, for the consistent integration approaches, e.g., the stabilized conforming nodal integration and the more recent reproducing kernel gradient smoothing integration, the integration and interpolation errors have the same accuracy order and thus an optimal convergence of Galerkin meshfree methods is ensured. The proposed theoretical error estimates for Galerkin meshfree methods are well demonstrated by numerical results.

ПОСТРОЕНИЕ ОРТОГОНАЛЬНЫХ КРИВОЛИНЕЙНЫХ СЕТОК ДЛЯ РЕГИОНАЛЬНОГО МОДЕЛИРОВАНИЯ ОКЕАНА: МАТЕМАТИЧЕСКИЙ АЛГОРИТМ И РУКОВОДСТВО ПОЛЬЗОВАТЕЛЯ

A new algorithm for constructing orthogonal curvilinear grids on a sphere for a fairly general geometric shape of the modeling region is implemented as a “compile-once - use forever” software package. It is based on the numerical solution of the inverse problem by an iterative procedure -- finding such distribution of grid points along its perimeter, so that the conformal transformation of the perimeter into a rectangle turns this distribution into uniform one. The iterative procedure itself turns out to be multilevel - i.e. an iterative loop built around another, internal iterative procedure. Thereafter, knowing this distribution, the grid nodes inside the region are obtained solving an elliptic problem. It is shown that it was possible to obtain the exact orthogonality of the perimeter at the corners of the grid, to achieve very small, previously unattainable level of orthogonality errors, as well as make it isotropic -- local distances between grid nodes about both directions are equal to each other.

Near orthogonal discrete quaternion Fourier transform components via an optimal frequency rescaling approach

The quaternion-valued signals consist of four signal components. The discrete quaternion Fourier transform is to map these four signal components in the time domain to that in the frequency domain. These four signal components in the frequency domain are called the discrete quaternion Fourier transform components. There are a total of 16 inner products among any two discrete quaternion Fourier transform components. The total orthogonal error among the discrete quaternion Fourier transform components is defined based on these 16 inner products. This study aims to find the optimal quaternion number in the discrete quaternion Fourier transforms so that the total orthogonal errors among the discrete quaternion Fourier transform components are minimised. It is worth noting that finding the optimal quaternion number in the discrete quaternion Fourier transform is equivalent to finding the optimal rescaling factors. Since the discrete quaternion Fourier transform components are expressed in terms of the high-order polynomials of the trigonometric functions of the rescaling factors, this optimisation problem is non-convex. To address this problem, a two-stage approach is employed for finding the solution to the optimisation problem. The comparison results show that the authors proposed method outperforms the existing methods in terms of achieving the low total orthogonal error among the discrete quaternion Fourier transform components.

Drift of the sensitive direction of Hall-effect devices in (1 0 0)-silicon caused by mechanical shear stress

Conventional Hall effect devices are designed to respond to a single magnetic field component in 3D space. However, if plastic encapsulated Hall effect devices in cubic crystals are exposed to poorly defined and unstable mechanical shear stress, small portions of unwanted, perpendicular magnetic field components will show up in the Hall output signals. Mathematically, any such crosstalk – may it originate from mechanical stress or not – can be expressed as a Taylor series in powers of the magnetic field. The even powers are the well known planar Hall effect at zero mechanical stress. In practice powers of even order are irrelevant, because they are eliminated by the widely used spinning current schemes. In this paper we address the odd powers of the magnetic crosstalk and their drift versus mechanical stress. The effect is small for Hall plates but large for Vertical Hall devices in (100)-silicon. It is fully described by piezo-resistance and piezo-Hall tensors. We present results of numerical simulations and measurements. Thin devices are less affected than thick devices. If magnetic angle sensors are made of Vertical Hall devices, in-plane shear stress leads to small orthogonality errors. This article lays the foundations for compensation circuits to eliminate such shear-stress induced orthogonality errors.

Polarization characteristic analysis of orthogonal reflectors with a side-scanning galvanometer in a laser communication terminal system.

An alignment tolerance allocation and alignment method of orthogonal reflectors with high polarization-maintaining characteristics is presented. A geometric optics vector model of orthogonal reflectors is established to derive the propagation vector of the output beam, including pitch angle error of the first mirror, the side-scanning angle of the galvanometer, and rotation angle error of the secondary mirror. The polarization transmission matrix of the orthogonal reflectors is derived based on the 3D polarization ray-tracing algorithm to present the relationship between assembly errors and diattenuation or retardance of the orthogonal reflectors. A calculation example of certain polarization ellipticity error is given to quantitatively distribute the orthogonal mirror assembly error. The optimal side-sway angle selection method of the galvanometer side-sway angle in a laser communication system is presented; further, effective measures to minimize the polarization ellipticity error is put forward.

Orthogonal polynomial projection error in Dunkl–Sobolev norms in the ball

We study approximation properties of weighted L2-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the reflection-invariant form (1−‖x‖2)α∏i=1d|xi|γi, α,γ1,…,γd>−1. Said properties are measured in Dunkl–Sobolev-type norms in which the same weighted L2 norm is used to control all the involved differential–difference Dunkl operators, such as those appearing in the Sturm–Liouville characterization of similarly weighted L2-orthogonal polynomials, as opposed to the partial derivatives of Sobolev-type norms.