Piecewise Cubic Research Articles

Geometric Piecewise Cubic Bézier Interpolating Polynomial with C2 Continuity

Bézier curve is a parametric polynomial that is applied to produce good piecewise interpolation methods with more advantage over the other piecewise polynomials. It is, therefore, crucial to construct Bézier curves that are smooth and able to increase the accuracy of the solutions. Most of the known strategies for determining internal control points for piecewise Bezier curves achieve only partial smoothness, satisfying the first order of continuity. Some solutions allow you to construct interpolation polynomials with smoothness in width along the approximating curve. However, they are still unable to handle the locations of the inner control points. The partial smoothness and non-controlling locations of inner control points may affect the accuracy of the approximate curve of the dataset. In order to improve the smoothness and accuracy of the previous strategies, а new piecewise cubic Bézier polynomial with second-order of continuity C2 is proposed in this study to estimate missing values. The proposed method employs geometric construction to find the inner control points for each adjacent subinterval of the given dataset. Not only the proposed method preserves stability and smoothness, the error analysis of numerical results also indicates that the resultant interpolating polynomial is more accurate than the ones produced by the existing methods.

Hybrid genetic algorithm based smooth global-path planning for a mobile robot

The goal of the global path planning task is to produce an optimal safe path for robot respect to the given map. There are many global path planning methods that have been studied. In this paper, we propose a novel Hybrid Genetic Algorithm (HGA) that is used to generating smooth paths for differential wheeled robots. The main idea of HGA is to provide the dynamic mutation rate and switchable global-local search method to the mutation operator of ordinary genetic algorithm. By deploying these modifications, the premature convergence of the generic genetic algorithm and the high time-consuming fitness calculation of the memetic algorithm are reduced. HGA also takes care of chromosome length (defined by the size of a set of points that construct a path) by applying the population replacement method. Our algorithm satisfies the important criteria in the path planning task: safe and minimum-length. Based on continuous-curvature Piecewise Cubic Bezier Curve, HGA directly provides the smoothed paths for differential wheeled robots. Therefore, our proposed algorithm does not need any third-party algorithm for smoothing planned path. Several experiments regarding the proposed algorithm on our robot and its results are analyzed and presented.

Electromagnetic Wave Propagation From Low-Earth Orbit Satellite to Ground Station Considering Interpolated Atmospheric Environments

We propose a novel method to calculate the electromagnetic (EM) wave propagation from low-earth orbit satellite (LEO) to a ground station based on the physical optics (PO), ray tracing technique, and geometric optics (GO) considering interpolated atmospheric environments. Our method includes the reflector antenna analysis using PO, the interpolation of the meteorological data using PCHIP and Kriging interpolation, transmission analysis using ray tracing and geometrical optics. Tropospheric and stratospheric environments are modeled using meteorological data-air pressure and temperature, relative humidity, and rain rate-measured at 9 different radiosonde observatories in and around South Korea. Furthermore, we utilize Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) and Kriging-exponential methods for vertical and horizontal interpolations of the raw meteorological data, respectively. Hence, the interpolated atmospheric environments are amenable to the best use of ray tracing technique and GO. Subsequently, effective refractive indices of the stratified media can be extracted via millimeter-propagation-model93. The simplified Appleton-Hartree equation characterizes the ionospheric environment. Considering a sun-synchronous orbit satellite passing through South Korea, we calculate atmospheric attenuation, boresight error, received power, and compensation angle of satellite antenna for various conditions.

Improved EMD Based on Piecewise Cubic Hermite Interpolation and Mirror Extension

This paper proposes an improved Empirical mode decomposition (EMD) method piecewise cubic hermite interpolation and mirror extension. Firstly, mirror expansion method is used to expand preliminary signal endpoint; then the piecewise cubic Hermite interpolation algorithm is applied to obtain the envelopes. Finally Intrinsic mode functions (IMFs) are obtained through improved EMD (MPC-HEMD). Through the analysis on simulation signals and Electroencephalogram (EEG) signals, we found that the new method was better than the traditional method.

Approximating Solutions of Fredholm Integral Equations via a General Spline Maximum Entropy Method

In this paper, we describe a general spline maximum entropy method for the approximation of solutions for solving Fredholm integral equations. The general spline maximum entropy method allows to extract piecewise linear, piecewise quadratic, piecewise cubic, piecewise quartic, piecewise quintic and other maximum entropy methods for solving Fredholm integral equations. We present numerical examples of piecewise linear, piecewise quadratic, piecewise cubic, piecewise quartic, piecewise quintic and other piecewise maximum entropy methods for solving Fredholm integral equations of second kind and first kind respectively. A proof of convergence for the piecewise spline maximum entropy method is also presented.

Two Interpolation Methods Using Multiply-Rotated Piecewise Cubic Hermite Interpolating Polynomials

AbstractTwo interpolation methods are presented, both of which use multiple Piecewise Cubic Hermite Interpolating Polynomials (PCHIPs). The first method is based on performing 16 PCHIPs on 8 rotated versions of the plot of the data versus an independent variable (such as pressure or time). These 16 PCHIPs are then used to form 8 interpolations of the original data, and finally, these 8 are averaged. When the original data are unevenly spaced with respect to the independent variable, we show that it is best to perform the Multiply-Rotated PCHIP (MR-PCHIP) method using the “data index” as the independent variable, and then to subsequently perform one last PCHIP of the data index with respect to the original independent variable. This MR-PCHIP method avoids the flat spots that are a feature of the PCHIP method when the data have multiple values approximately equal to a local extreme value. The MR-PCHIP interpolated data have continuous first derivatives at the data points. This method also avoids the unrealistic overshoots that can occur when using the standard cubic spline interpolation procedure. The second interpolation method is designed specifically for hydrographic data with the aim of minimizing the formation of unrealistic water masses by the interpolation procedure. This is achieved by applying a Piecewise Cubic Hermite Interpolating Polynomial to each of 8 rotations of the salinity versus temperature plot (Multiply-Rotated Salinity–Temperature PCHIP, MRST-PCHIP) with bottle number (that is, data index) as the vertical interpolating coordinate, thereby making the MRST-PCHIP method independent of the heave of a water column. This method is equivalent to interpolating in the salinity–temperature diagram, and MRST-PCHIP proves very effective at avoiding the production of anomalous water masses that otherwise occur when interpolating temperature and salinity separately.

Computing the L∞-Induced Norm of LTI Systems: Generalization of Piecewise Quadratic and Cubic Approximations

This paper is concerned with performance analysis for bounded persistent disturbances of continuous-time linear time-invariant (LTI) systems. Such an analysis can be done by computing the L ∞ -induced norm of continuous-time LTI systems since it corresponds to the worst maximum magnitude of the output for the worst persistent external input with a unit magnitude. In our preceding study, piecewise constant and linear approximation schemes for analyzing this norm have been developed through two alternative approximation approaches, one for the input and the other for the relevant kernel function, via the fast-lifting technique. The approximation errors in these approximation schemes have been shown to converge to 0 at the rates of 1/N and 1/N 2 , respectively, as the fast-lifting parameter N is increased. Along this line, this paper aims at developing generalized techniques that offer improved accuracy named the piecewise quadratic and cubic approximation schemes. The generalization and the associated accuracy improvement discussed in this paper are not limited to the increased orders of approximation but are extended further to taking advantage of the freedom in the point around which relevant functions are expanded to Taylor series. The approximation errors in the piecewise quadratic and cubic approximation schemes are shown to converge to 0 at the rates of 1/N 3 and 1/N 4 , respectively, regardless of the point at which the Taylor expansion is applied. Finally, effectiveness of the developed computation methods is confirmed through a numerical example.

RRT-A*-BT approach for optimal collision-free path planning for mobile robots

This paper deals with the problem of optimal collision-free path planning for mobile robots evolving inside indoor cluttered environments. Addressing this challenge, a hybrid approach is proposed combining Rapidly-exploring Random Trees (RRT), A-Star (A*) and Back-Tracking (BT) algorithms (RRT-A*-BT). Thus, a vision system is used for a nearly-exact modeling of the environment through image processing. Moreover, each iteration of the basic RRT approach is guided by A* algorithm while trying to take the shortest path linking the robot current position to target . In case of a blockage, BT algorithm is used to get out the robot from this situation. Finally, Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) is used to smooth the generated optimal path. RRT-A*-BT approach is validated through different scenarios; obtained results are compared with previous works on same environments with same conditions. The results prove that RRT-A*-BT is better and faster than other algorithms of the literature, such as Genetic Algorithms and Conventional RRT, in terms of (i) computation time,(ii) path length and (iii) transfer time..

A comparison of piecewise cubic Hermite interpolating polynomials, cubic splines and piecewise linear functions for the approximation of projectile aerodynamics

Abstract Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions. An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics. The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions: they are continuous, differentiable at least once, and have a relatively low degree. The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools, and then compares Piecewise Cubic Hermite Interpolating Polynomial (PCHIP), cubic splines, and piecewise linear functions, and their variant, as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile. A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics, and its evaluation against a set of criteria. Finally, the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.

High-Frequency Asymptotic Compression of Dense BEM Matrices for General Geometries Without Ray Tracing

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green’s function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green’s function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the sparsifying modification of the Green’s function with other accelerating schemes, such as the fast multipole method, appears possible in principle and is a future research topic.

A compound interpolation envelope local mean decomposition and its application for fault diagnosis of reciprocating compressors

Abstract Local mean decomposition (LMD) is a self-adaptive analysis method which can decompose a signal into a set of product function (PF) components, and the construction of envelope function plays an important role in the accuracy of its PF components. According to the local nonstationary characteristics of vibration signals, a compound interpolation envelope (CIE) LMD was proposed through a novel envelope construction method. By defining an nonstationary coefficient to evaluate the local nonstationary characteristics of vibration signals, an compound envelope construction method which use Monotonic Piecewise Cubic Hermite Interpolation (MPCHI) for nonstationary part and Cubic spline interpolation (CSI) for smooth part was proposed, and an CIE LMD algorithm was give based on the novel envelope construction method. A numerical example simulation was conducted to verify the performance of CIE LMD, results indicated that CIE LMD outperforms three other methods. The CIE LMD was employed to diagnose the oversized bearing clearance fault in reciprocating compressor, and the envelope frequency spectrum of PF component gives a more significant peak of fault frequency than that of original signal, which further indicates that this proposed method is competent for the diagnosis of reciprocating compressor oversized bearing clearance fault.

Fitting Rainfall Data by Using Cubic Spline Interpolation

This study discusses the application of two cubic spline i.e. natural and not-a-knot end boundary conditions to visualize and predict the rainfall data. The interpolation and the analysis of the rainfall data will be done on a monthly basis by using the MATLAB software. The rainfall data is obtained from Malaysia Meteorology Department for Ipoh and Petaling Jaya in year 2014 and 2015. The interpolating curves are then being compared and if there is any negative value on the interpolating curve on some sub-interval, that part will be replaced by using the Piecewise Cubic Hermite Interpolating Polynomial (PCHIP). We discuss the missing data imputation by using both splines.

Optimal path planning and execution for mobile robots using genetic algorithm and adaptive fuzzy-logic control

This paper presents preliminary results of the application of two-Kinect cameras system on a two-wheeled indoor mobile robot for off-line optimal path planning and execution. In our approach, the robot makes use of depth information delivered by the vision system to accurately model its surrounding environment through image processing techniques. In addition, a Genetic Algorithm is implemented to generate a collision-free optimal path linking an initial configuration of the mobile robot (Source) to a final configuration (Target). After that, Piecewise Cubic Hermite Interpolating Polynomial is used to smooth the generated optimal path. Finally, an Adaptive Fuzzy-Logic controller is designed to keep track of a mobile robot on the desired smoothed path (by transmitting the appropriate right and left velocities using wireless communication). In parallel, sensor fusion (odometry sensors and Kinect sensors) is used to estimate the current position and orientation of the robot using Kalman filter. The validation of the proposed solution is carried out using the differentially-driven mobile robot, RobuTER, to successfully achieve safe motion (without colliding with obstacles) in an indoor environment.

Cuckoo search-designated fractal interpolation functions with winner combination for estimating missing values in time series

Reliable data have always played a vital role in time series analysis and research. Nevertheless, missing data, which can bias the original properties of the time series if the pattern of missed data is systematic, is also a very common phenomenon in observed processes. Thus, how to address missing values is a very important challenge, especially in the upcoming big data era. This paper proposes the cuckoo search-designated fractal interpolation functions (CS-DFIFs) method and the CS-DFIFs-winners combining (CS-DFIFs-WC) method for estimating missing values. The former method skillfully transforms Fractal Interpolation Functions (FIFs) to make it possible to calculate a specified point's missing value, which is difficult to obtain with the traditional approach. Then, to optimize the parameters that transforming process generates, the cuckoo search algorithm (CS) and DFIFs are synthesized into a novel model, CS-DFIFs. Considering classical interpolation methods, such as Linear, Cubic Spline and Piecewise Cubic Hermite Interpolation Polynomial (PCHIP), having some born advantages, the inspiration of winners combining is sparked. CS algorithm is used to obtain the best weights of winners in this combined model, CS-DFIFs-WC. Two databases, electricity demands and prices, are chosen to be the numerical testing object at 7 missing levels in this paper. The results show that CS-DFIFs-WC overcomes the deficiency of FIFs that cannot calculate a specified point's missing value, and outperforms benchmarks at almost every level by four criteria.

An improved local mean decomposition method and its application for fault diagnosis of reciprocating compressor

Local mean decomposition (LMD) is a new time–frequency analysis method which can decompose a signal adaptively into a set of product function (PF) components, and the construction of local mean function and envelope function plays an important role in the accuracy of its PF components. Aiming at the strong nonstationarity, nonlinearity and multi-component coupling characteristics of reciprocating compressor vibration signals, an improved LMD was proposed by a novel construction method of local mean function and envelope function. By introducing an extreme symmetrical point between two extreme points and using the Monotone Piecewise Cubic Hermite Interpolation (MPCHI) instead of Cubic Spline Interpolation (CSI) to construct the envelopes, a novel construction method of local mean function and envelope function was proposed, and then the improved LMD algorithm was given based on this novel construction method. The improved LMD was applied to decompose the vibration signals of reciprocating compressor fault states, and the comparison of details between different LMD decomposition results verified the superiority of this improved method. The envelope frequency spectrum of PF component gives a more significant peak of fault frequency than that of original signal, which further indicates that this proposed method is competent for the diagnosis of reciprocating compressor oversized bearing clearance fault.

Advanced Two-State Compressing Algorithm: A Versatile, Reliable and Low-Cost Computational Method for ECG Wireless Applications

Compressing the ECG signal is considered a feasible solution for supporting a system to manipulate the package size, a major factor leading to congestion in an ECG wireless network. Hence, this paper proposes a compression algorithm, called the advanced two-state algorithm, which achieves three necessary characteristics: a) flexibility towards all ECG signal conditions, b) the ability to adapt to each requirement of the package size and c) be simple enough. In this algorithm, the ECG pattern is divided into two categories: “complex” durations such as QRS complexes, are labeled as low-state durations, and “plain” duratio ns such P or T waves, are labeled as high-state durations. Each duration type can be compressed at different compression ratios, and Piecewise Cubic Spline can be used for reconstructing the signal. For evaluation, the algorithm was applied to 48 records of the MIT-BIH arrhythmia database (clear PQRST complexes) and 9 records of the CU ventricular tachyarrhythmia database (unclear PQRST complexes). Parameters including Compression Ratio (CR), Percentage Root mean square Difference (PRD), Percentage Root mean square Difference, Normalized (PRDN), root mean square (RMS), Signal-to-noise Ratio (SNR) and a new proposed index called Peak Maximum Absolute Error (PMAE) were used to comprehensively evaluate the performance of the algorithm. Eventually, the results obtained were positive with low PRD, PRDN and PMAE at different compression ratios compared to many other loss-type compressing methods, proving the high efficiency of the proposed algorithm. All in all, with its extremely low-cost computation, versatility and good-quality reconstruction, this algorithm could be applied to a number of wireless applications to control package size and overcome congested situations.

Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation

For the approximation in Lp-norm, we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots. For p = 1, ∞, we obtain its values. By these results we know that for the Sobolev classes, the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths. At the same time, the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.

Fin design for conjugate heat transfer optimization in double pipe

Optimal design of longitudinal fins augmented to the outer surface of the inner pipe in a double pipe, is investigated for maximizing conjugate heat transfer coefficient. Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) has been used for fin-surface representation at each step of the optimization process with control points as the design variables. Genetic algorithm has been employed as the optimizer together with the Discontinuous Galarkin Finite Element Method (DG-FEM) as the solver of the governing equations. The results show that the optimal fin design is greatly influenced by the characteristic length, the number of fins, the conductivity of the material of heated surface and the number of control points. Optimal designs based on the equivalent diameter give upto 289% improvement in the heat transfer coefficient and those based on the hydraulic diameter render such improvement upto 70%. Optimal fin shape has also out-performed the conventional fin shapes present in the literature and shown upto 203%, 263% and 227% increase in the heat transfer coefficient relative to the equivalent diameter for trapezoidal, triangular and parabolic fins respectively. For the case of hydraulic diameter, these figures are respectively 482%, 70% and 117%. The optimal designs based on the equivalent diameter corresponding to Ω = 500 have proven to be the best in view of cost, frictional loss and heat transfer coefficient. These give upto 39% higher heat transfer coefficient than the corresponding increase in frictional loss due to augmentation of the fins to the double pipe. The validity and accuracy of the present results has been shown by comparison with the available literature results.

A Quadratic Model with Nonpolynomial Terms for Remote Colorimetric Calibration of 3D Laser Scanner Data Based on Piecewise Cubic Hermite Polynomials

The processing of intensity data from terrestrial laser scanners has attracted considerable attention in recent years. Accurate calibrated intensity could give added value for laser scanning campaigns, for example, in producing faithful 3D colour models of real targets and classifying easier and more reliable automatic tools. In cultural heritage area, the purely geometric information provided by the vast majority of currently available scanners is not enough for most applications, where indeed accurate colorimetric data is needed. This paper presents a remote calibration method for self-registered RGB colour data provided by a 3D tristimulus laser scanner prototype. Such distinguishing colour information opens new scenarios and problems for remote colorimetry. Using piecewise cubic Hermite polynomials, a quadratic model with nonpolynomial terms for reducing inaccuracies occurring in remote colour measurement is implemented. Colorimetric data recorded by the prototype on certified diffusive targets is processed for generating a remote Lambertian model used for assessing the accuracy of the proposed algorithm. Results concerning laser scanner digitizations of artworks are reported to confirm the effectiveness of the method.

A Load Scheduling Algorithm for the Smart Home Using Customer Preferences and Real Time Residential Prices

A load scheduling method in the Smart Home using a combination of customer preferences and the price of electricity is presented. To translate the customer preference of loads into a time-varying priority curve, the Analytical Hierarchy Process (AHP) and Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) methods are used. The resulting curves of customer priority are combined with the available time-varying pricing information for determining the schedule for each load. An example with four loads is presented to demonstrate the effectiveness of the algorithm.