Interpolation Interval Research Articles

Exponential fitting for interpolation of oscillatory functions. A numerical approach

We develop an entirely numerical approach for the interpolation of the first derivative of an oscillatory function y(x) with frequency ω. Two versions for the number of data at equidistant points xk, k=1,2,…,K are considered: single y(xk) and pair y(xk),y′′(xk), identified by parameter nd with values 1 and 2, respectively. Experimental results show that the accuracies from the new ω-dependent procedure are better than from the polynomial fitting, with a gain which increases with ω. They also show that an approximate value for ω continues to give advantageous results. A special test is reported with respect to the evolution of the error along the interpolation interval. It is found that when nd=1 the errors have the tendency of increasing in amplitude towards the ends, thus following the Runge effect for the interpolation of y(x) but, quite unexpected, when nd=2 the error amplitude is unchanged over the whole interval.

A Double Interpolation and Mutation Interval Reconstruction LMD and Its Application in Fault Diagnosis of Reciprocating Compressor

The accuracy and stability of the envelope estimation function are enduring issues throughout the research process of LMD. This paper presents double interpolation and mutation interval reconstruction local mean decomposition (DIMIRLMD) to improve the stability of the demodulation process and the accuracy of PF components. DIMIRLMD first proposes a mutation interval reconstruction envelope algorithm using extreme symmetry points to suppress the demodulation mutation phenomenon, which disturbs the stability of the demodulation process, and then selects the optimal PF component from a double interpolation PF component library based on the index of orthogonality (IO) for a better hierarchical property. DIMIRLMD was employed to analyze the simulation signal and vibration signal of a reciprocating compressor in an oversized bearing clearance state, and the results illustrate its performances are more excellent than those of three other LMD methods. Furthermore, the envelope frequency spectrum obtained from the proposed LMD presents a clear double rotation fault frequency and lower noise disturbance.

Efficient buffeting analysis of long-span bridges under non-stationary winds: A 2D interpolation enhanced approach

The buffeting analysis of long-span bridges under non-stationary winds is a crucial consideration in the context of exceptional winds recurring in frequent and violent patterns. The time–frequency representation is widely used for the solution of structural non-stationary buffeting responses due to its popularity and simplicity. However, the traditional approach consumes considerable computational resources to achieve high accuracy, which renders the approach computationally inefficient. In such a case, a 2D interpolation enhanced approach is developed for the efficient buffeting analysis of long-span bridges under non-stationary wind actions. Central to this approach is the 2D interpolation applied to the reconstruction of structural buffeting responses, accompanied by the well-designed non-uniformly distributed interpolation knots. For the enhanced approach, the computations in solving governing equations are only required at limited interpolation knots instead of at each frequency and time instant, which yields a dramatic reduction of computational resources and efforts. By taking a long-span suspension bridge as an example, a parametric analysis is conducted to investigate the effect of interpolation intervals on the simulation efficiency and accuracy, and the preferred suggestions are made accordingly. Then, the effectiveness of the enhanced approach is further verified via a case study by focusing on the evolutionary spectra and non-stationary buffeting responses. The analytical results indicate that the enhanced approach is particularly efficient in simulating non-stationary buffeting responses of long-span bridges with few losses of accuracy.

Efficient analytical modeling for pulsed eddy current signals using adaptive interpolation-based Fourier transform

ABSTRACT In the analytical modelling for pulsed eddy current (PEC) signals based on the Fourier transform, there is a computational burden caused by calculating hundreds of harmonic impedance changes. In this paper, an adaptive interpolation method that takes inflection points as the initial interpolation nodes is proposed for calculating harmonic impedance changes. The inflection points of impedance change with frequency are deduced from the transformer model to divide the initial interval of interpolation. In the process of the adaptive interpolation method, it is necessary to divide the subintervals which fail to meet the requirements of interpolation accuracy, and then calculate the harmonic impedance changes by the cubic spline interpolation method. The adaptive interpolation method was verified by the simulation and experimental results. The results show that the adaptive interpolation method interpolates hundreds of harmonic impedance changes with dozens of interpolation nodes. There is a good agreement between the experimental PEC signals and the PEC signals obtained by the adaptive interpolation method, and the relative errors of the peak values are about 0.1%.

Stability analysis of Cohen-Grossberg neural networks with time-varying delay by flexible terminal interpolation method

<abstract><p>In the paper, the existence and uniqueness of the equilibrium point in the Cohen-Grossberg neural network (CGNN) are first studied. Additionally, a switched Cohen-Grossberg neural network (SCGNN) model with time-varying delay is established by introducing a switched system to the CGNN. Based on reducing the conservativeness of the system, a flexible terminal interpolation method is proposed. Using an adjustable parameter to divide the invariant time-delay interval into multiple adjustable terminal interpolation intervals $ (2^{\imath +1}-3) $, more moments when signals are transmitted slowly can be captured. To this end, a new Lyapunov-Krasovskii functional (LKF) is constructed, and the stability of SCGNN can be estimated. Using the LKF method, a quadratic convex inequality, linear matrix inequalities (LMIs) and ordinary differential equation theory, a new form of stability criterion is obtained and specific instances are given to prove the applicability of the new stability criterion.</p></abstract>

The use of the rapid unequal interval interpolation method (RUIIM) for a non-RTCP NC program and five-axis machining

In industries such as electronics 3C, medical, aviation, and energy, multi-axis machining machines are an important element of modern production systems, especially for post-processing, which necessitates the safe operation of control equipment. Some controller and software for five-axis simultaneous machining only interpolate equidistantly between cutter location (CL) points and do not support post-processing using a rotation tool center point (RTCP) so the machining path of the tool has manufacturing errors (overcutting or undercutting) and the workpiece fails to be machined to the correct size. Depending on the maximum allowable error (δMax), this study uses rapid unequal interval sub-point interpolation to perform sub-point interpolation on the original path of the five-axis machining machine, and the sub-point values are output in the computer numerical control (CNC) machining program in C# programming language. The tool tip is then moved to the desired machining position. Finally, an example involving helical milling of a curved surface is used to verify the feasibility of the rapid unequal interval interpolation method (RUIIM). The experimental results show that rapid unequal interval sub-point interpolation gives a more accurate original path that is generated in the CAM and a faster processing speed for the entire controller so the operation speed is increased.

S-Velocity Profile of Industrial Robot Based on NURBS Curve and Slerp Interpolation

This paper presents a novel algorithm for industrial robot trajectory planning based on the NURBS(Non-Uniform Rational B-Spline) curve and Slerp interpolation aiming at the problems that the trajectory of a six-axis industrial robot is not smooth enough in the operation process, the posture planning process is non-uniform, and the six-axis industrial robot starts and stops frequently. Firstly, aiming at the first problem, the trajectory planning algorithm based on the NURBS curve is presented to improve the smoothness of the trajectory curve. Combined with Slerp posture planning based on quaternion description, which realizes the uniform change of posture on the robot’s end-effector. Secondly, aiming at the second problem, the S-velocity planning algorithm is presented in the interpolation interval of the robot, which realizes the operation process of complex curves continuously, and improves the operation quality. Finally, this paper uses Bernoulli’s lemniscate as the incentive trajectory, and the contrast experiment of trajectory planning between two incentive profiles is designed, which are the NURBS curve and the five-order polynomial curve. The result of the experiment indicates that the planning algorithm proposed in this paper could effectively improve the smoothness of trajectory in a Cartesian workspace, decrease the impact and tremulous in a Cartesian workspace, and effectively improve the performance of the robot working process. The results drawn from this paper lay a certain foundation for the future high-precision control of industrial robots.

An enhanced N-point interpolation method to eliminate average precision distortion

• Existing N -point interpolation method to calculate average precision raises three anomalies. • The average precision distortion problem is commonly present in the N -point interpolation method. • The proposed enhanced N -point interpolation method reduces more than 90% of the average precision distortion. Existing N -point interpolation methods generate large errors in the average precision calculation for object detection. These errors lead to average precision distortion, which makes it impossible to accurately evaluate the performance of the model. We investigate the reason for the average precision distortion and propose an enhanced N -point interpolation method. These improvements are based on the N -point interpolation method and can be summarized in two parts: (1) The interpolation point position is changed to the middle interpolation. (2) Dynamic selection of parameters for calculating the area of the interpolation interval. Experiments verify the existence of severe average precision distortion in the N -point interpolation method. Furthermore, the proposed enhanced N -point interpolation method reduces the average precision distortion by more than 90% to only 0.04%. In this way, the enhanced N -point interpolation method is able to replace the all-point interpolation method for fast and accurate evaluation of object detection model.

Realtime PPG based respiration rate estimation for remote health monitoring applications

Among the many vital parameters of human body, breath rate monitoring is paramount to detect the symptoms of several respiratory diseases such as sleep apnea syndrome, chronic obstructive pulmonary disease (COPD), and asthma. Hence, this paper puts forward a novel framework to measure the respiration rate (RR) using the reflective type photoplethysmogram (PPG) signals acquired using an inexpensive and easy-to-use wearable device. Extracting the respiration induced amplitude variations (RIAV) from PPG signal using the incremental merge segmentation (IMS) algorithm, the proposed approach shows that robust RR estimation in realtime is feasible through low cost Cortex-M4 microcontroller. Using the sliding window approach augmented with adaptive thresholding technique that can deal with motion artifacts, we show that robust RR estimation is viable using reflectance type PPG, which is largely unexplored. Moreover, to handle the non-uniform nature of RIAV signal, this work employs uniform interval interpolation technique and removes the non-respiratory frequencies using finite impulse response (FIR) band-pass filter. Through the fast Fourier transform (FFT) analysis of regular interval RIAV sequence, the dominant frequency corresponding to the RR is extracted. The performance of the proposed scheme is validated not only on two publicly available PPG datasets but also on a custom designed experimental setup integrated with the smartphone. The experimental results highlight that our approach can achieve a RR estimation accuracy of 1 bpm deviation against the ground truth.

Environment-adaptive learning from demonstration for proactive assistance in human–robot collaborative tasks

Proactive assistance in human–robot collaboration remains a challenging objective, as the spatial–temporal coordination of the human–robot motion must be considered in conjunction with the object and environmental context. In this paper, we propose an environment-adaptive probabilistic interaction primitive method using learning-from-demonstration. In particular, we propose a novel phase estimation algorithm called Single-axis Uniform Interval Interpolation, which alleviates the restriction of Gaussian or uniform distribution of phase variables. In addition, the environmental constraints in human–robot interactive skills are learned via the regression between environmental parameters and the weight vectors. The proposed method is implemented in a proactive robotic system for typical industrial-motivated human–robot collaborative scenarios, such as assistive push-button assembly and human–robot collaborative object covering. The experimental result validates the effectiveness of the proposed approach.

Numerical Simulation and Application of Zero‐Thickness Contact Surface Element with Variable Shear Stiffness on Pile Foundation

In this study, a contact surface constitutive model with zero‐thickness unit and variable shear stiffness was improved based on the statistical damage constitutive model. The model parameters were derived by means of the shear stress‐shear displacement curve of the pile‐soil contact surface, and the model parameters under different normal stresses were obtained by the linear interpolation method. At the same time, the influence of the interpolation interval range on the model parameters was explored. The shear stiffness adjustment factor was introduced, and the improved pile‐soil contact surface constitutive model was applied to the numerical simulation of pile‐soil contact surface shear calculations using the fish language embedded in FLAC 3D, and the variation of monopile bearing characteristics and ultimate bearing capacity were investigated and analyzed. The results show that the improved contact surface constitutive model is able to reflect the nonlinear variation of shear stiffness, and that different normal stresses correspond to different fitting parameters, demonstrating the depth effect of the model. The accuracy of the model parameters reduces as the interpolation interval increases, and the interpolation results are more accurate when the interval range is smaller. The numerical model accurately simulates the pile‐soil contact surface shear calculation and the monopile bearing calculation, and the simulation results of the ultimate pile bearing capacity are closer to the results computed by the equations in the Chinese code. At the same time, the variation law of pile axial force and pile lateral frictional resistance along the depth direction and the variation of pile ultimate bearing capacity under different working conditions are reasonable, which shows the validity of the contact surface principal structure model and the reasonableness of numerical calculation in this study.

Newton’s divided gH-difference interpolation formula with unequal spacing for interval-valued function using Hukuhara difference

Interval interpolation formulae play a significant role to find the value of an unknown function at some points under interval uncertainty. The objective of this paper is to establish Newton’s divided interpolation formula for interval-valued functions using generalized Hukuhara difference of intervals. For this purpose, arithmetic of intervals, Hukuhara difference and its some properties and concept of interval-valued function have been discussed briefly. Using Hukuhara difference of intervals, the definition of Newton’s divided gH-difference for interval-valued function has been introduced. Then Newton’s divided gH-differences interpolation formula has been derived. Finally, with the help of some numerical examples, the proposed interpolation formula has been illustrated.

Optimization of Virtual Shack-Hartmann Wavefront Sensing.

Virtual Shack–Hartmann wavefront sensing (vSHWS) can flexibly adjust parameters to meet different requirements without changing the system, and it is a promising means for aberration measurement. However, how to optimize its parameters to achieve the best performance is rarely discussed. In this work, the data processing procedure and methods of vSHWS were demonstrated by using a set of normal human ocular aberrations as an example. The shapes (round and square) of a virtual lenslet, the zero-padding of the sub-aperture electric field, sub-aperture number, as well as the sequences (before and after diffraction calculation), algorithms, and interval of data interpolation, were analyzed to find the optimal configuration. The effect of the above optimizations on its anti-noise performance was also studied. The Zernike coefficient errors and the root mean square of the wavefront error between the reconstructed and preset wavefronts were used for performance evaluation. The performance of the optimized vSHWS could be significantly improved compared to that of a non-optimized one, which was also verified with 20 sets of clinical human ocular aberrations. This work makes the vSHWS’s implementation clearer, and the optimization methods and the obtained results are of great significance for its applications.

Shape preserving rational cubic trigonometric fractal interpolation functions

This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of C1-rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form pi(θ)/qi(θ), where pi(θ) and qi(θ) are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in C3 is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the C1-RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data.

An adaptive Voronoi diagrams Algorithm for matching multi-scale areal Residential Areas

For the difficult problem of balanced control of the computational efficiency and boundary position accuracy of the existing Voronoi algorithms based on the idea of figure discretization, we proposed an adaptive Voronoi diagram algorithm for multi-scale areal residential area matching based on interpolation strategy according to neighbor pair type——AVARA. Firstly, it uses the Delaunay triangle network composed of the centroids of the residential areas to calculate the neighbor pairs of the residential areas. Secondly, it classifies neighbor pairs according to the size relationship between the minimum distance between the neighbor pairs and the minimum side length of their minimum area bounding rectangles. Then it adaptively interpolates points on the boundary of the residential area according to the neighbor pair type. Finally, it constructs the Voronoi diagram for residential areas based on the interpolated point set and the vertex set of the residential area. We conducted the experiments of creating Voronoi diagram by using the residential area dataset which map scale is 1:10,000 and 1:50,000. The results showed that:compared with the intervisibility point method, the 3m and 6m interval interpolation point method of 1:10,000 data, AVARA outperformed the local position accuracy and time performance, compared with the 30m interval interpolation point method of 1:50,000 data, AVARA achieved higher local position accuracy, and compared with the 15m interval interpolation point method of 1:50,000 data, AVARA time performance has been improved by 34.6%.The problem of balance control between computational efficiency and boundary position accuracy of Voronoi diagram was alleviated effectively.

Repetitive Control Scheme of Robotic Manipulators Based on Improved B-Spline Function

In this paper, a repetitive control scheme of a 2-DOF robotic manipulator based on the improved cubic B-spline curve is proposed. Firstly, a repetitive controller for robotic manipulator is designed, which is composed of an iterative controller and disturbance observer. Then, an improved B-spline optimization scheme is introduced to divide the task of the robotic manipulator into three intervals. A correction function is added to each interval of cubic spline interpolation. Finally, a variety of cases are designed and simulated by MATLAB. The experimental results show that, compared with the conventional B-spline, the improved B-spline has better performance in tracking accuracy and smoothness of motion trajectory. By changing the mechanism of the manipulator, the cases of different weights and lengths are designed. The experimental results in these cases show that the proposed scheme can be applied to most of the 2-DOF robotic manipulator control systems.

Contact Stiffness Model of Joint Surface Considering Continuous Smooth Characteristics and Asperity Interaction

A new model to accurately obtain the contact stiffness of the mechanical joint surface is established, which satisfies the continuous smooth contact characteristics of the asperity and also considers the influence of the asperity interaction. In order to satisfy the continuous and smooth contact characteristics of the asperity, the interpolation interval of the Hermite polynomial function is improved to make the asperity contact stiffness curve smoothly transition in the elastic, elastoplastic and completely plastic stages. Further, the contact situation of the joint surface considering the asperity interaction is explored through micro-contact analysis and mechanism research. Finally, using statistical principles, the microscopic contact model is extended to the entire joint surface, and the contact stiffness model of the joint surface is established. The simulation results show that the asperity stiffness model satisfies the continuous smooth contact characteristics. Surface roughness is the main factor that affects the contact stiffness, and the influence of the asperity interaction cannot be ignored.

Specific absorbed fractions for a revised series of the UF/NCI pediatric reference phantoms: internal photon sources

Assessment of radiation absorbed dose to internal organs of the body from the intake of radionuclides, or in the medical setting through the injection of radiopharmaceuticals, is generally performed based upon reference biokinetic models or patient imaging data, respectively. Biokinetic models estimate the time course of activity localized to source organs. The time-integration of these organ activity profiles are then scaled by the radionuclide S-value, which defines the absorbed dose to a target tissue per nuclear transformation in various source tissues. S-values are computed using established nuclear decay information (particle energies and yields), and a parameter termed the specific absorbed fraction (SAF). The SAF is the ratio of the absorbed fraction—fraction of particle energy emitted in the source tissue that is deposited in the target tissue—and the target organ mass. While values of the SAF may be computed using patient-specific or individual-specific anatomic models, they have been more widely available through the use of computational reference phantoms. In this study, we report on an extensive series of photon SAFs computed in a revised series of the University of Florida and the National Cancer Institute pediatric reference phantoms which have been modified to conform to the specifications embodied in the ICRP reference adult phantoms of Publication 110 (e.g. organs modeled, organ ID numbers, blood contribution to elemental compositions). Following phantom anatomical revisions, photon radiation transport simulations were performed using MCNPX v2.7 in each of the ten phantoms of the series—male and female newborn, 1 year old, 5 year old, 10 year old, and 15 year old—for 60 different tissues serving as source and/or target regions. A total of 25 photon energies were considered from 10 keV to 10 MeV along a logarithm energy grid. Detailed analyses were conducted of the relative statistical errors in the Monte Carlo target tissue energy deposition tallies at low photon energies and over all energies for source–target combinations at large intra-organ separation distances. Based on these analyses, various data smoothing algorithms were employed, including multi-point weighted data smoothing, and log–log interpolation at low energies (1 keV and 5 keV) using limiting SAF values based upon target organ mass to bound the interpolation interval. The final dataset is provided in a series of ten electronic supplemental files in MS Excel format. The results of this study were further used as the basis for assessing the radiative component of internal electron source SAFs as described in our companion paper (Schwarz et al 2021) for this same pediatric phantom series.

Spline adaptive filters based on real-time over-sampling strategy for nonlinear system identification

In this paper, a new spline adaptive filter (SAF) used for identifying the nonlinear systems is presented, in which a real-time over-sampling (RTOS) strategy is proposed in order to solve the performance degradation caused by the imbalanced distribution of the input data in SAF during the adaptive process. Based on the invariance property of nonlinear system, an initialization for the linear part in SAF is implemented in advance. The number of data falling into each spline interpolation interval is counted, and the historical input and output data of each interval are recorded in real-time. In each adaption loop, a new sample is generated through the existing recorded data in a certain interval according to the data distribution in all intervals, so that the distribution of input data could be relatively evened. Several numerical experiments were implemented, and the results show that the proposed RTOS strategy can significantly improve the convergence performance compared with the existing algorithms.

Hermite Interpolation Based Interval Shannon-Cosine Wavelet and Its Application in Sparse Representation of Curve

Using the wavelet transform defined in the infinite domain to process the signal defined in finite interval, the wavelet transform coefficients at the boundary are usually very large. It will bring severe boundary effect, which reduces the calculation accuracy. The construction of interval wavelet is the most common method to reduce the boundary effect. By studying the properties of Shannon-Cosine interpolation wavelet, an improved version of the wavelet function is proposed, and the corresponding interval interpolation wavelet based on Hermite interpolation extension and variational principle is designed, which possesses almost all of the excellent properties such as interpolation, smoothness, compact support and normalization. Then, the multi-scale interpolation operator is constructed, which can be applied to select the sparse feature points and reconstruct signal based on these sparse points adaptively. To validate the effectiveness of the proposed method, we compare the proposed method with Shannon-Cosine interpolation wavelet method, Akima method, Bezier method and cubic spline method by taking infinitesimal derivable function cos(x) and irregular piecewise function as an example. In the reconstruction of cos(x) and piecewise function, the proposed method reduces the boundary effect at the endpoints. When the interpolation points are the same, the maximum error, average absolute error, mean square error and running time are 1.20 × 10−4, 2.52 × 10−3, 2.76 × 10−5, 1.68 × 10−2 and 4.02 × 10−3, 4.94 × 10−4, 1.11 × 10−3, 9.27 × 10−3, respectively. The four indicators mentioned above are all lower than the other three methods. When reconstructing an infinitely derivable function, the curve reconstructed by our method is smoother, and it satisfies C2 and G2 continuity. Therefore, the proposed method can better realize the reconstruction of smooth curves, improve the reconstruction efficiency and provide new ideas to the curve reconstruction method.