Fast parametric curve interpolation with minimal feedrate fluctuation by cubic B-spline

2016 ◽  
Vol 232 (9) ◽  
pp. 1642-1652 ◽  
Author(s):  
Lei Lu ◽  
Lei Zhang ◽  
Yan Gu ◽  
Ji Zhao
Keyword(s):  
Approximation Method ◽  
Interpolation Method ◽  
Control Points ◽  
Parametric Curve ◽  
B Spline ◽  
Computation Efficiency ◽  
Taylor Approximation ◽  
Curve Interpolation ◽  
Feedrate Fluctuation ◽  
Computation Load

Due to the reliable feedrate fluctuation and computation load of the existing parametric curve interpolation, a fast interpolation method by cubic B-spline for parametric curve is presented which results in a minimum feedrate fluctuation and light computation load. As there are many geometry implementation tools and many good properties in the B-spline compared with the polynomial, the relation between the arc length s and curve parameter u can be fitted by the cubic B-spline accurately. Because the feedrate fluctuation of the generally used Taylor approximation method is sensitive to the curvature of the toolpath, its accuracy cannot be controlled. For a given feedrate fluctuation of 0.05%, the proposed interpolation method can guarantee the error requirements by increasing the number of the control points. After the de Boor method is applied in real time, the computation load of the cubic B-spline interpolation is decreased compared with the Taylor approximation method and higher order polynomial-fitting method. In order to save the memory consumption for storing the parameters of the fitted cubic B-spline, an iterative optimization process is applied to obtain the knot vector elements and optimize the control points. Simulations and experiments show that the interpolation method can attain high accuracy and computation efficiency. According to the simulations, for most of the complex curves, the feedrate fluctuation of the proposed interpolation method is decreased by about 50% when the feedrate is scheduled and the computation load of the proposed method is decreased by about 70% compared with the second-order interpolation method.

On the Modeling and Analysis of an Improved CNC Interpolation Algorithm

2009 ◽  
Vol 626-627 ◽  
pp. 459-464 ◽  
Author(s):  
Lei Luo ◽  
L. Wang ◽  
Jun Hu
Keyword(s):  
Interpolation Method ◽  
Control Points ◽  
Interpolation Algorithm ◽  
Modeling And Analysis ◽  
B Spline ◽  
Spline Curve ◽  
Spline Surface ◽  
Cnc Interpolation ◽  
Data Points ◽  
Back Calculation

An improved interpolation method is presented based on B-spline curve back calculation which regards data points as control points. First, a B-spline surface reconstruction is done, and a favorable condition for real-time interpolation can be provided for NC machining. Then, by prejudging the trajectory feedrate, the tangent vectors of spline curve junction can be calculated, which can be used to establish the spline curve equations based on time. At last, with the equations mentioned above, the trajectory and feedrate profile can be generated simultaneously by the improved interpolation algorithm. An error analysis is also discussed and the feasibility of the improved algorithm is verified by the simulation results.

A Novel Adaptive GA-based B-spline Curve Interpolation Method

2019 ◽  
Vol 13 (3) ◽  
pp. 289-304
Author(s):  
Maozhen Shao ◽  
Liangchen Hu ◽  
Huahao Shou ◽  
Jie Shen
Keyword(s):  
Genetic Algorithm ◽  
Traditional Method ◽  
Interpolation Method ◽  
Tangent Vector ◽  
Adaptive Genetic Algorithm ◽  
B Spline ◽  
Spline Curve ◽  
Curve Interpolation ◽  
Data Points ◽  
Data Point

Background: Curve interpolation is very important in engineering such as computer aided design, image analysis and NC machining. Many patents on curve interpolation have been invented. Objective: Since different knot vector configuration and data point parameterization can generate different shapes of an interpolated B-spline curve, the goal of this paper is to propose a novel adaptive genetic algorithm (GA) based interpolation method of B-spline curve. Method: Relying on geometric features owned by the data points and the idea of genetic algorithm which liberalizes the knots of B-spline curve and the data point parameters, a new interpolation method of B-spline curve is proposed. In addition, the constraint of a tangent vector is also added to ensure that the obtained B-spline curve can approximately satisfy the tangential constraint while ensuring strict interpolation. Results: Compared with the traditional method, this method realizes the adaptive knot vector selection and data point parameterization. Therefore, the interpolation result was better than the traditional method to some extent, and the obtained curve was more natural. Conclusion: The proposed method is effective for the curve reconstruction of any scanned data point set under tangent constraints. Meanwhile, this paper put forward a kind of tangent calculation method of discrete data points, where users can also set the tangent of each data point in order to get more perfect interpolation results.

Using the Combination of GM(1,1) and Taylor Approximation Method to Predict the Academic Achievement of Student

2014 ◽  
Vol 1 (2) ◽  
pp. 55-69 ◽  
Author(s):  
Phuoc-Hai Nguyen ◽  
◽  
Tian-Wei Sheu ◽  
Phung-Tuyen Nguyen ◽  
Duc-Hieu Pham ◽  
...  
Keyword(s):  
Academic Achievement ◽  
Approximation Method ◽  
Taylor Approximation

scholarly journals Complex Uncertainty of Surface Data Modeling via the Type-2 Fuzzy B-Spline Model

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1054
Author(s):  
Rozaimi Zakaria ◽  
Abd. Fatah Wahab ◽  
Isfarita Ismail ◽  
Mohammad Izat Emir Zulkifly
Keyword(s):  
Complex Surface ◽  
Complex Data ◽  
Fuzzy Data ◽  
Control Points ◽  
B Spline ◽  
Spline Surface ◽  
Data Control ◽  
Surface Data ◽  
Model Complex

This paper discusses the construction of a type-2 fuzzy B-spline model to model complex uncertainty of surface data. To construct this model, the type-2 fuzzy set theory, which includes type-2 fuzzy number concepts and type-2 fuzzy relation, is used to define the complex uncertainty of surface data in type-2 fuzzy data/control points. These type-2 fuzzy data/control points are blended with the B-spline surface function to produce the proposed model, which can be visualized and analyzed further. Various processes, namely fuzzification, type-reduction and defuzzification are defined to achieve a crisp, type-2 fuzzy B-spline surface, representing uncertainty complex surface data. This paper ends with a numerical example of terrain modeling, which shows the effectiveness of handling the uncertainty complex data.

scholarly journals On the BIC for determining the number of control points in B-spline surface approximation in case of correlated observations

2020 ◽  
Vol 10 (1) ◽  
pp. 110-123
Author(s):  
Gaël Kermarrec ◽  
Hamza Alkhatib
Keyword(s):  
Shape Control ◽  
Simple Procedure ◽  
Spline Approximation ◽  
Autoregressive Process ◽  
Optimal Number ◽  
Information Criteria ◽  
General Effect ◽  
Control Points ◽  
B Spline ◽  
Correlated Observations

Abstract B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi- likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.

Sequential B-Spline Surface Construction Using Multiresolution Data Clouds

2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Yuan Yuan ◽  
Shiyu Zhou
Keyword(s):  
Control Points ◽  
Finite Sample ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Filtering Technique ◽  
Engineering Practices ◽  
Asymptotical Convergence ◽  
Surface Construction ◽  
Fitting Error

B-spline surfaces are widely used in engineering practices as a flexible and efficient mathematical model for product design, analysis, and assessment. In this paper, we propose a new sequential B-spline surface construction procedure using multiresolution measurements. At each iterative step of the proposed procedure, we first update knots vectors based on bias and variance decomposition of the fitting error and then incorporate new data into the current surface approximation to fit the control points using Kalman filtering technique. The asymptotical convergence property of the proposed procedure is proved under the framework of sieves method. Using numerical case studies, the effectiveness of the method under finite sample is tested and demonstrated.

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