Polynomial Interpolation Method Research Articles

Application of Newton Polynomial Interpolation Method in Determining the Continuity of Functions Represented by Tabulated Discrete Points

Application of Newton Polynomial Interpolation Method in Determining the Continuity of Functions Represented by Tabulated Discrete Points

Two Level Security for Key Access Using Cloud Computing

In this work, the author constructs a key access management scheme that seamlessly transitions any hierarchical-like access policy to the digital medium. The proposed scheme allows any public cloud system to be used as a private cloud. We consider the data owner an entity consisting of several organization units. We provide a secure method for each user of this entity to access the public cloud from both inside and outside the company’s network. The idea of the key access control scheme, which is based on Shamir’s secret sharing algorithm and polynomial interpolation method, is suitable especially for hierarchical organizational structures. It offers a secure, flexible, and hierarchical key access mechanism for organizations utilizing mission-critical data.

Motion state factor driven for doubly-curved shallow shell deformation reconstruction

The conventional shape sensing method, based on C0 continuity, is recognized for its computational efficiency. However, it is prone to shear locking due to low-order polynomial interpolation in shell analysis, introducing two primary challenges. Firstly, it lacks consideration for C1 continuity. Secondly, existing high-order polynomial interpolation methods significantly increase the number of degrees of freedom (DOF), reducing computational efficiency. Therefore, a brand-new real-time structural health monitoring (SHM) method driven by motion state factor instead of nodal factors is proposed. This method accommodates all deformation modes of shallow shells with C1 continuity and improves the shape function's order without additional DOF. It begins with deriving motion states from compatibility equations, leading to a polynomial shape function expression. Subsequently, conversion equations for real constants and DOF are developed, establishing a least-squares model for theoretical strains and measured values. Following that, the higher-order element formula, optimized for minimal DOF, is then inferred from varied displacement data. Finally, simulations and experiments demonstrate this method's effectiveness, accuracy, and reliability in analyzing shallow shell elements.

A Type of Interpolation between Those of Lagrange and Hermite That Uses Nodal Systems Satisfying Some Separation Properties

In this paper, we study a method of polynomial interpolation that lies in-between Lagrange and Hermite methods. The novelty is that we use very general nodal systems on the unit circle as well as on the bounded interval only characterized by a separation property. The way in which we interpolate consists in considering all the nodes for the prescribed values and only half for the derivatives. Firstly, we develop the theory on the unit circle, obtaining the main properties of the nodal polynomials and studying the convergence of the interpolation polynomials corresponding to continuous functions with some kind of modulus of continuity and with general conditions on the prescribed values for half of the derivatives. We complete this first part of the paper with the study of the convergence for smooth functions obtaining the rate of convergence, which is slightly slower than that when equidistributed nodal points are considered. The second part of the paper is devoted to solving a similar problem on the bounded interval by using nodal systems having good properties of separation, generalizing the Chebyshev–Lobatto system, and well related to the nodal systems on the unit circle studied before. We obtain an expression of the interpolation polynomials as well as results about their convergence in the case of continuous functions with a convenient modulus of continuity and, particularly, for differentiable functions. Finally, we present some numerical experiments related to the application of the method with the nodal systems dealt with.

A Kalman Filtering Algorithm for Measurement Interruption Based on Polynomial Interpolation and Taylor Expansion.

Combined SINS/GPS navigation systems have been widely used. However, when the traditional combined SINS/GPS navigation system travels between tall buildings, in the shade of trees, or through tunnels, the GPS encounters frequent signal blocking, which leads to the interruption of GPS signals, and as a result, the combined SINS/GPS-based navigation method degenerates into a pure inertial guidance system, which will lead to the accumulation of navigation errors. In this paper, an adaptive Kalman filtering algorithm based on polynomial fitting and a Taylor expansion is proposed. Through the navigation information output from the inertial guidance system, the polynomial interpolation method is used to construct the velocity equation and position equation of the carrier, and then the Taylor expansion is used to construct the virtual measurement at the moment of the GPS signal interruption, which can make up for the impact of the lack of measurement information on the combined SINS/GPS navigation system when the GPS signal is interrupted. The results of computer simulation experiments and road measurement tests based on the loosely combined SINS/GPS navigation system show that when the carrier faces a GPS signal interruption situation, compared with a combined SINS/GPS navigation algorithm that does not take any rescue measures, our proposed combined SINS/GPS navigation algorithm possesses a higher accuracy in the attitude angle estimation, a higher accuracy in the velocity estimation, and a higher accuracy in the positional localization, and the system possesses higher stability.

Design and Test of Seed–Fertilizer Replenishment Device for Wheat Seeder

In view of existing problems, such as the seed and fertilizer supply link for wheat seeders still relying on manual installation and the lack of practical application equipment, a seed–fertilizer replenishment device based on the three-degree-of-freedom mechanical arm and screw conveying principle is designed using the seed box installation and supply as the operation scenario to replace the manual installation process. Combined with the requirements of the seed box replenishment operation, the key parameters of the replenishment robot arm and the screw conveyor auger are determined. Then, the kinematic model of the replenishment robot arm is established based on the modified D-H method, forward and inverse kinematics calculations are performed, and the workspace is analyzed using the Monte Carlo method. Based on this, the robotic arm task path is designed, the fifth-degree polynomial interpolation method is used to complete the trajectory planning, and MATLAB R2016a software is used to simulate the motion trajectories of each joint, verifying the feasibility of the trajectory planning solution. Finally, a prototype is trial-produced and quadratic regression orthogonal testing and response surface analyses are conducted to obtain the optimal working parameters of the replenishment device. The verification test shows that when the angular velocity of the lumbar joint of the replenishment device is 4°/s, the speed of screw conveyor is 90 r/min, and the angle of the big arm is 12°, the conveying loss rate is 3.98%, and the conveying efficiency is 0.833 kg/s. The relative errors with the theoretical optimal values are 4.2% and 2.4%, respectively, both less than 5%. The supply trajectory is reasonable, and the robot arm runs smoothly. This study can provide reference for the design of seed–fertilizer replenishment device for wheat seeders.

Kinematic-driven human-robot interaction system with deep learning for flexible acupuncture needling manipulations

Acupuncture is an important therapeutic method in traditional Chinese medicine and essentially characterized by flexible needling manipulations that are complicated for repetitions. In this work, we designed a bio-inspired acupuncture robot combined with human–machine interaction system that can implement flexible and precise needle manipulations as manual acupuncture. In the proposed framework, the manual needling process was recorded with online imaging techniques. With a deep learning-based acupuncture estimator, kinetic parameters of lift insertion (LI) and twist rotation (TR) manipulations were extracted in three-dimensional space and the precent of correct key points for operator’s hand pose estimation achieved 94.03%. It is founded that the variation of operator’s finger vector angle for LI manipulation was much weaker than TR manipulation. Furthermore, inspired by kinematic characteristics of manual acupuncture, a six-degree-of-freedom robot was used to simulate the coordinated motion of hand joints in needling manipulations through robot kinematics analysis and trajectory planning with polynomial interpolation method. Numerical and experiential results showed that the acupuncture robot can mimic various needling manipulations as manual acupuncture. The flexibility and stability of the robotic acupuncture framework were evaluated for a wide range of needling parameters. It was demonstrated that the robot can follow the periodic dynamics of LI and TR manipulations with various frequencies and amplitudes, and perform accurate acupuncture operations with smooth needling trajectories as joint jitters during needle manipulations were completely eliminated. The presented results provide a new strategy for modern applications of acupuncture treatment with robotic and flexible needling manipulations to improve the therapeutic efficacy.

Two classes of third-order weighted compact nonlinear schemes for Hamilton-Jacobi equations

The solutions of Hamilton-Jacobi (HJ) equations may contain discontinuous derivatives, which brings numerical difficulties in capturing sharply these derivatives. This paper presents a third-order weighted compact nonlinear scheme (WCNS) and a third-order perturbed WCNS (WCNS-P) for solving HJ equations. A linear combination of hybrid cell-edge and cell-node values of the function is applied to approximate the spatial derivatives of the function. For the WCNS, the upwind-biased nonlinear weighted interpolation is used for the unknown cell-edge values, while for the WCNS-P perturbations with a free parameter is introduced to the upwind-biased linear interpolation. The free parameter can be optimized to reduce numerical errors in smooth regions. To enhance the numerical stability, a monotone polynomial interpolation method is designed to switch the WCNS-P to the WCNS. Several numerical experiments are performed to test the order of accuracy and the discontinuity-capturing ability of the two schemes.

Local Polynomial Interpolation Method Optimization in the Process of Digital Terrain Model Creation Based on Data Collected From a Multibeam Echosounder

Local Polynomial Interpolation Method Optimization in the Process of Digital Terrain Model Creation Based on Data Collected From a Multibeam Echosounder

MODERNIZATION OF THE ELECTRIC DRIVE OF PUMPING UNITS OF THE DISTRICT HEATING NETWORK IN THE CONTEXT OF INCREASING THEIR ENERGY EFFICIENCY

Quantitative and qualitative indicators of the modernization of the electric drive of a pumping unit of a network with throttle control of the coolant supply of a centralized heat supply system are determined. The possibility of modernizing the electric drive by replacing outdated electric motors of energy efficiency class IE1, which are still used in electric drives of pumping units, with electric motors of energy efficiency classes IE2 and IE3 from different manufacturers, has been substantiated. The energy consumption was calculated based on the technical data of the pump and electric motors provided by the manufacturers and includes the calculation of the energy consumption of pumping units in a typical operating cycle typical for district heating networks, which involves operation at reduced load for a long time. Comparative calculated data on the daily and annual energy consumption of the considered electric motors in a typical operating cycle of a pumping unit, with respect to the least energy-efficient electric motor among those considered, were obtained. The practical significance of the results obtained is to understand that the selection of an electric motor by efficiency at variable load, i.e., actually on the basis of the IE energy efficiency class assigned in accordance with the IEC 60034-30-1 standard, is incorrect, It is substantiated that the choice should be made using the efficiency interpolated by the method of polynomial interpolation of the motor loss curve, while taking into account the economic feasibility of choosing an electric motor between energy efficiency classes, in this case IE2 and IE3, and different manufacturers.

Qualitative Analysis of RLC Circuit Described by Hilfer Derivative with Numerical Treatment Using the Lagrange Polynomial Method

This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω∈(1,2) for the RLC model. Based on Schaefer’s fixed point theorem and Banach’s contraction principle, the existence and uniqueness results are established. Furthermore, Ulam–Hyers and Ulam–Hyers–Rassias stability results for the boundary value problem of the RLC model are discussed. To showcase the practicality and efficacy of our theoretical findings, a two-step Lagrange polynomial interpolation method is applied to solve some numerical examples.

Gold Price Fluctuation Forecasting Based on Newton and Lagrange Polynomial Interpolation

Gold is a highly valuable commodity and an investment opportunity for many people. However, thereare often significant fluctuations in gold prices that affect investment decisions. Various mathematicalforecasting methods have been developed to anticipate gold price fluctuations. This study uses historicaldaily data of gold prices during January-May 2023. The method used in this study is the Newton andLagrange polynomial interpolation method with several orders to analyze data and forecast gold pricefluctuations. The purpose of this study is to compare the performance and accuracy of the order levels ofthe Newton and Lagrange polynomial interpolation forecasting models. In this study, the test data pointsand orders are selected so that a range is formed that matches the amount of data available. The testorders used in this study include orders 2, 3, 5, 6, and 10. This study found that the 2nd order polynomialinterpolation method is more effective and accurate in forecasting gold price fluctuations compared tothe higher orders tested. This is indicated by the results of the calculation of MAE, RMSE, and MAPEvalues in 2nd order polynomial interpolation which are smaller than in 3rd, 5th, 6th, and 10th orderpolynomial interpolation. This suggests that a polynomial of 2nd order has been able to model andforecast gold price fluctuations well. However, it is important to remember that these conclusions arebased on the data and methods used in this study. Variability in forecasting results can occur dependingon the quality of the data, the time period used, and the interpolation method applied, among others.Therefore, further research and wider testing needs to be conducted to validate these conclusions.

Temporal and Spatial Analysis of Rainfall and Evapotranspiration in Erbil Plain and the Peripheral Areas

Since analysis of spatial events is region specific and cannot be generalized, local studies are of vital importance to identify the most accurate interpolation methods. Furthermore, the trend analysis of climatic parameters at different time scales is helpful for making a better climate change adaptation and mitigation plan to overcome water scarcity. Accordingly, the current study was proposed to detect trends in rainfall and evapotranspiration at monthly and annual time scales over the Erbil plain and the surrounding area using parametric and non-parametric tests. Moreover, four deterministic and five geostatistical methods were evaluated for searching the best interpolation method to generate a continuous surface for the indicated climatic variables. The results revealed that the majority of data sets are categorized useful class and serially independent. Furthermore, it was found both rainfall and potential evapotranspiration have a mix of upward and downward trends and most of them are insignificant at 5% level of significance. Further, the interpolation analysis indicated that local polynomial interpolation (LPI) method was proven to be best interpolator for generating continuous surfaces for rainfall and ETo over the area under study followed by the empirical Bayesian Kriging method.

Highly secured EHR management system based on blockchain technology with digitally signed authentication using data sanitization and polynomial interpolation

With the increased specifications of healthcare services, diagnosis become very common in most of the healthcare sector to provide better treatment for patients. Here, the person’s medical information can be tracked with the help of an Electronic Health Record (EHR). The EHR system maintains the digital medical records of medical organizations. Enterprise-wide data systems, along with other networking activities, are used to exchange medical documents in the current environment where the patient seeks rapid access to their medical records. Moreover, the EHR system faces many problems regarding integrity, management and security. Still, there are some concerns regarding the security and privacy of patient medical records. However, it suffers from managing the large volume of electronically generated healthcare data. Thus, this work adapts blockchain technology, which effectively stores healthcare data. Therefore, decentralized technology was just recently introduced in order to offer an innovative viewpoint on data security and system effectiveness. In this work, a blockchain-related system is developed to secure patients’ health data with high verifiability. A total of two mechanisms are carried out in the developed model to encrypt the privacy of the health data that is Data Sanitization and Polynomial Interpolation-based Cryptography (DSPIC). In the data sanitization mechanism, the optimal key is generated through the Modernized Position-based Coot and Penguins Search Optimization Algorithm (MP-CPeSOA). The polynomial interpolation method is carried out through the estimation of values between two data points to handle the heterogeneity. The objective function is considered for securing the patient's health data with Euclidian distance, hiding ratio, preservation ratio and correlation among the original data and the restored data. In this developed model, the generated key is digitally signed to provide high security for the patient’s health data. At last, the effectiveness of the suggested blockchain-adopted EHR preservation system is enhanced by conducting experiments on different EHR management schemes.

The structural design and simulation of the tubing handling manipulator

The size of the oil pipes in the factory is relatively large, making them inconvenient to handle. There are few existing oil pipe handling devices, and a 2P2R four degree of freedom manipulator device has been innovatively designed to complete the oil pipe handling work. First, the three-dimensional model of the manipulator was established using SolidWorks software, and then the improved D-H method was used to carry out the kinematics modeling of the manipulator system, and the forward and Inverse kinematics equations were derived. Then, the MATLAB software was used to carry out the kinematics analysis of the manipulator, and the manipulator workspace was calculated. The movement trajectory, as well as the displacement, speed, and acceleration curves under the trajectory were obtained using the fifth order Polynomial interpolation method, Finally, the 3D model was imported into Adams software, and an Adams virtual prototype was established for dynamic analysis. The force and torque curves of each joint were obtained, laying the foundation for further design and research in the future.

Research on Structural Parameter Optimization and Springback Compensation Methods of a Steel Bar Hoop Bending Machine

After being processed into a certain shape by a stirrup bending machine, steel bars are used as the skeleton of reinforced concrete in construction. The accuracy of their bending directly affects the bearing capacity of reinforced concrete. In order to improve the accuracy of the springback angle of steel bars after bending, this paper focuses on optimizing the structural parameters of the steel bar bending machine and controlling the springback angle of steel bars. The influence of the support shaft diameter, bending crankshaft diameter, and bending crankshaft radius on the springback angle of steel bars is analyzed using ANSYS explicit dynamic simulation and experimental analysis. By combining orthogonal experimental methods, the bending structural parameters were optimized, including a support shaft diameter of 20 mm, a crankshaft bending diameter of 40 mm, and a crankshaft bending radius of 80 mm. This can effectively reduce the springback angle of the steel bars and improve the accuracy of the bending angle after bending. In addition, in order to improve the control accuracy of the steel bar springback angle, a cubic spline interpolation method was proposed, and the collected steel bar springback angle data were fitted using Matlab. At the same time, comparisons were made between the least squares method, polynomial interpolation method, and cubic spline interpolation method. The results show that the compensation formula obtained by using the cubic spline interpolation method can effectively improve the control accuracy of the steel bar bending springback angle.

Multi-objective trajectory optimization of the 2-redundancy planar feeding manipulator based on pseudo-attractor and radial basis function neural network

The establishment and solution of the inverse kinematic model is the key to improve the efficiency of trajectory optimization. To improve the trajectory smoothness and reduce energy consumption of multi-degree-of-freedom (MDOF) robots, this article presents the time-, jitter-, and energy-optimal trajectory optimization method based on pseudo-attractor and radial basis function neural network. Based on the geometric method, the forward kinematic model of MDOF robots is firstly established. The diversity of inverse kinematic solutions is reduced by determining redundant joints. Combined with the attractor theory, the time-adaptive allocation strategy can automatically endow time information with path points. On this basis, the 7-time polynomial interpolation method is used to fit discrete trajectory points and generate the initial trajectory without singularity points. Affected by the pseudo-attractor, radial basis function neural network is transformed into the improved radial basis function neural network (I-RBFNN) to optimize the initial trajectory. The 2-redundancy planar feeding manipulator (2-RPFM) is introduced to verify the effectiveness of the proposed method. Experiment and simulation results show that the proposed method is available in generating high-performance trajectories, which is beneficial to improve the production efficiency of the auto-body-out-panel stamping line.

Excavating Trajectory Planning of a Mining Rope Shovel Based on Material Surface Perception.

The mining rope shovel (MRS) is one of the core pieces of equipment for open-pit mining, and is currently moving towards intelligent and unmanned transformation, replacing traditional manual operations with intelligent mining. Aiming at the demand for online planning of an intelligent shovel excavation trajectory, an MRS excavating trajectory planning method based on material surface perception is proposed here. First, point cloud data of the material stacking surface are obtained through laser radar to perceive the excavation environment and these point cloud data are horizontally calibrated and filtered to reconstruct the surface morphology of the material surface to provide a material surface model for calculation of the mining volume in the subsequent trajectory planning. Second, kinematics and dynamics analysis of the MRS excavation device are carried out using the Product of Exponentials (PoE) and Lagrange equation, providing a theoretical basis for calculating the excavation energy consumption in trajectory planning. Then, the trajectory model of the bucket tooth tip is established by the method of sixth-order polynomial interpolation. The unit mass excavation energy consumption and unit mass excavation time are taken as the objective function, and the motor performance and the geometric size of the MRS are taken as constraints. The grey wolf optimizer is used for iterative optimization to realize efficient and energy-saving excavation trajectory planning of the MRS. Finally, trajectory planning is carried out for material surfaces with four different shapes (typical, convex, concave, and convex-concave). The results of experimental validation show that the actual hoist and crowd forces are essentially consistent with the planned hoist and crowd forces in terms of the peak value and trend variations, verifying the accuracy of the calculation model and confirming that the full bucket rate and various parameters meet the constraints. Therefore, the trajectory planning method based on material surface perception are feasible for application to different excavation conditions.

Approximate solutions of fuzzy delay integral equations with weakly singular kernels by piecewise fuzzy polynomial interpolation

In this paper, a piecewise fuzzy polynomial interpolation method is proposed to approximate the solutions of fuzzy delay integral equations with weakly singular kernels. Firstly, the existence and uniqueness of the solutions of these fuzzy delay integral equations are proved by using the Gronwall inequality and an iterative method. Then, the fuzzy weakly singular delay integral equations are discretized into related algebraic equations by using piecewise fuzzy polynomial interpolation and the fuzzy Gauss-Jacobi quadrature formula. Moreover, the convergence of the numerical method is investigated and an error estimate is provided. Finally, the accuracy and efficiency of the proposed method are verified by some numerical experiments.

Comparison of Prediction of the Number of People Exposed to Covid 19 Using the Lagrange Interpolation Method with the Newton Gregory Maju Polynomial Interpolation Method

In March 2020 the World Health Organization stated that the Corona Virus pandemic (Covid-19) was due to its massive spread and hit all countries in the world. Academics and practitioners are called upon to carry out research activities in order to obtain a mathematical model that can be used to predict the number of people exposed to Covid-19 or other diseases. The researchers previously tried research to predict the number of people exposed to Covid-19 from early 2021 using the Monte Karlo method, the Hybrid Nonlinear Regression Logistic– Double Exponential Smoothing method, the Arima method, the BackPropagation and Fuzzy Tsukamoto methods, the K-Nearest method. Neighbors, Time Series Analysis method, Winter Method and Long Short Time Memory (LSTM) Artificial Neural Network method.