Non-uniform Rational B-spline Interpolation Research Articles

Impact of smoothing interpolations on the non-conventional machining of pockets

Smoothing curves have been used in computer aided design software for years. This is the reason why machinists very early wondered about the possibilities of creating toolpaths directly generated from this type of mathematical curves. In addition, in recent research work, we have shown that the non-uniform rational Bspline interpolation is the most favorable towards high-speed machining of complex shapes. As a result, we are thinking of simulating by this type of interpolation the pocket recess machining strategies proposed by a computer aided manufacturing software, currently developed in linear and circular interpolations only. In this article we propose a study of the influence of the parameters of Bspline curves on the hollowing out of pockets by smoothing initial cutting methods. To do this, we chose the smoothing of the zigzag method by Bspline and NURBS while varying the order and the discretization parameters to know the impact on the dynamics of the high-speed machining mechanism and on the non-conventional machining precision. This work has shown that parametric curves are favored for this type of machining provided that this tool is mastered.

Correction to: A novel feed rate scheduling method with acc-jerk-continuity and round-off error elimination for non-uniform rational B-spline interpolation

Correction to: A novel feed rate scheduling method with acc-jerk-continuity and round-off error elimination for non-uniform rational B-spline interpolation

Transmission performance evaluation of non-circular spur bevel gear based on a novel isochronal measurement method

This paper aims to study the transmission performance evaluation of non-circular spur bevel gear (NCSB-gear) based on a novel isochronal measurement method. Firstly, the mathematical model of NCSB-gear is established. Then a novel isochronal measurement method is proposed, and the actual tooth surfaces are reconstructed by non-uniform rational B-spline interpolation. The advantages of the proposed method are discussed based on the geometric characteristics of tooth surface and measurement fitting error. A pair of forged NCSB-gears is measured through a coordinate measurement machine and the maximum tooth surface fitting error by the proposed method reduces 2μm compared with the traditional method. Moreover, deviation tooth contact analysis (DTCA) and load tooth contact analysis (LTCA) are conducted to evaluate the transmission performance. Finally, the transmission performance of measured NCSB-gear is obtained by a gear rolling tester. By comparing the TCA calculation of different measurement methods, it is found that the relative error of the maximum transmission error based on the isochronal measurement method reduces 4.719% compared with the traditional measurement method, thus verifying the validity of the proposed measurement method.

Ship hull surface reconstruction from scattered points cloud using an RBF neural network mapping technology

The ship hull surface reconstruction based on three-dimensional scattered points cloud is vital to Computer-Aided Design modeling of ship reverse engineering. There are several problems with traditional methods: the preprocessing of a large-scale scattered points cloud is complicated, and the result is susceptible to noise. Because of the “black box” characteristic, the surface reconstruction based on neural networks cannot apply to the practical engineering. To address these issues, combining the Radial Basis Function neural network and Non-Uniform Rational B-Spline interpolation algorithm, a new ship hull surface reconstruction method was proposed, which can satisfy the standard geometric model description in Computer-Aided Design system. Firstly, the Radial Basis Function neural network was used to pre-fit the three-dimensional scattered points cloud. Then the mathematical model of the surface was mapped. Finally, based on the bilinear interpolation algorithm, the mathematical model was transformed to a Non-Uniform Rational B-Spline surface to apply to the ship practical engineering. In addition, by comparing our method with the traditional method, the advantages of our method in surface reconstruction quality and surface repair ability were verified, which provided a new way for the application of Radial Basis Function neural network in ship reverse engineering.

A novel feed rate scheduling method with acc-jerk-continuity and round-off error elimination for non-uniform rational B-spline interpolation

Abstract Feed rate scheduling is a critical step in computer numerical control machining, as it has a close relationship with machining time and surface quality. It has now become a hot issue in both industry and academia. In this article, we present a novel and complete S-shape-based feed rate scheduling method for three-axis non-uniform rational B-spline (NURBS) tool paths, which can reduce high chord errors and round-off errors, and generate continuous velocity, acceleration, and jerk profile. The proposed feed rate scheduling method consists of three modules: a bidirectional scanning module, a velocity scheduling module, and a round-off error elimination module. The bidirectional scanning module aims to guarantee the continuity of the feed rate at the junctions between successive NURBS blocks, where the chord error, tangential acceleration, and tangential jerk limitations are considered. After the NURBS blocks have been classified into two cases by the previous module, the velocity scheduling module first calculates the actual maximum feed rate. It then generates the feed rate profiles of all NURBS blocks according to the proposed velocity profile. Later, the round-off error elimination module is applied to adjust the actual maximum feed rate so that the total interpolation time becomes an integer multiple of the interpolation period, which leads to the elimination of round-off errors. Finally, benchmarks are conducted to verify the applicability of the proposed method. Compared with the traditional method, the proposed method can save the interpolation time by $4.67$ to $14.26\% $.

A NURBS Curve Interpolator Based on Double-Step Signal and Finite Impulse Response Filters

Abstract In order to solve the problem that the traditional trajectory planning method cannot flexibly control the velocities and the arrival time of critical points in the trajectory, this paper proposes a non-uniform rational B-spline interpolator based on double-step signal and finite impulse response filters. The interpolator consists of two modules: a pre-processing module and a real-time interpolation module. To reduce the contour error, first, the pre-processing module scans the curve for critical points and sets their velocities according to the chord error. Then, the curve is split, and for each segment, the length and inverse length functions are calculated to eliminate the length error and enhance the real-time performance. Moreover, to control the arrival time and smooth the velocity, the real-time interpolation module generates the trajectory by processing the double-step signal with three finite impulse response filters, which satisfies the constraints of velocity, acceleration, jerk, and displacement. Simulation results indicate that the proposed interpolator can accurately reach the critical points at the set arrival time with smooth velocity and low chord error, and experimental results on a SCARA robot show that the proposed interpolator is feasible and effective.

Numerical simulation of fracture in layered and sandwich structures: A systematic literature review

A systematic literature review on numerical strategies suitable to simulate defects and discontinuities in layered and sandwich structures is presented here. Particular care is given to studies whose scope concerns the so-called Moving Mesh (MM) or Isogeometric Analysis (IGA) methods. A general overview of the peculiarities of each approach is also provided. A total of fifteen and twenty-six journal/conference articles were summarised and categorised for MM and IGA methods, respectively. Based on the available literature, it can be stated that MM approaches generally allow a lower computational burden due to the reduced number of re-meshing steps required when compared to standard finite element approaches. Conversely, IGA approaches bring strong advantages in the geometric description of curved shell structures and, due to a non-uniform rational b-spline interpolation function, ensure higher numerical accuracy in stress analysis problems.

A high-precision non-uniform rational B-spline interpolator based on S-shaped feedrate scheduling

Parametric interpolation has been widely used in robot control because of its advantages over traditional linear or circular interpolation. The NURBS interpolation based on S-shaped feedrate scheduling can realize the smooth continuous motion of robots, but it also faces two challenges. One challenge is that when the NURBS curve is split into multiple blocks, if one or more blocks are too short, the final feedrate profile is often discontinuous. The other challenge is that the NURBS curve cannot be obtained entirely due to calculation errors. In this paper, a high-precision NURBS interpolator based on S-shaped feedrate scheduling is proposed. In this interpolator, a bidirectional velocity scanning scheme is developed to effectively handle the too short split blocks of the NURBS curve. Meanwhile, an optimization algorithm is proposed to ensure the precision of NURBS curves. The feasibility and advantages of the proposed method are verified through case studies.

Precision Trajectory Tracking on XY Motion Stage Using Robust Interval Type-2 Fuzzy PI Sliding Mode Control Method

Precision contour tracking is one of the most important factors used to determine product quality in a machining tool. An interval type-2 fuzzy proportional–integral (PI) sliding mode control (IT2FPISMC) system is proposed herein to control the mover position of the two-axis motion stage with optical encoder sensors for trajectory feedback. A type-2 fuzzy method that can handle rule uncertainties is developed to approach the unknown nonlinear systems. The PI term is used to approximate the discontinuous control signal and mitigate the chattering phenomenon in the presence of unmodeled system dynamics and external disturbances. The adaptive control laws are derived based on the Lyapunov theorem, such that the closed-loop stability is guaranteed, and the output tracking errors of the system asymptotically converge to zero. A non-uniform rational B-spline interpolator with high accuracy is adopted in the biaxial linear stage. Moreover, typical circular, bowknot, heart, and star reference contours are tested. The results on the average tracking error and the tracking error standard deviation are experimented and compared to illustrate the performance of our proposed method. The tracking performance obtained from the IT2FPISMC method is better than that of the conventional method. Furthermore, the proposed method can achieve robustness for tracking different reference contours in industrial applications.

Nonuniform sampling pitch acquisition method in myocardial single-photon emission computed tomography.

Single-photon emission computed tomography is a tomographic imaging method that acquires a projection image by rotating a gamma camera around by 380° or 180°. For myocardial single-photon emission computed tomography, 180° acquisition is common, but it has limitations including an incomplete reconstruction, which can distort the resulting image. It is possible to produce a complete reconstruction using 360° acquisition, but the testing time is long and is burdensome to patients. The nonuniform sampling pitch acquisition (NUSPA) method devised in this study involves reducing the total sampling count using NUSPA that reduces the sampling pitch in the range in which the gamma cameras are closer to the myocardium (RAO45-LPO45) and increases it elsewhere. The NUSPA-1 method based on a 6° sampling pitch had 20 views fewer than 360° acquisition. In addition, the NUSPA-2 method based on a 3.75° sampling pitch had 60 views fewer than 360° acquisition, considerably reducing the testing time. The acquired sinograms from the NUSPA methods were subjected to nonuniform rational B-spline surface interpolation processing, producing data with a uniform sampling pitch, after which image reconstruction was performed. The images after nonuniform rational B-spline interpolation for both the line sources and heart-liver phantom investigated in this study were not found to have the distortion observed from 180° acquisition or a count decrease at the center, resulting in image quality nearly equivalent to 360° acquisition. This method enabled a reduction in testing time without impacting image quality.

NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells

An investigation into the postbuckling and geometrically nonlinear behaviors of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells is carried out in this study. The discrete nonlinear equation system is established based on non-uniform rational B-Spline (NURBS) basis functions and the first-order shear deformation shell theory (FSDT). The nonlinearity of shells is formed in the Total Lagrangian approach considering the von Karman assumption. The incremental solutions are obtained by using a modified Riks method. In the present formulation, the rule of mixture is used to estimate the effective material properties of FG-CNTRC shells. Effects of CNTs distribution, volume fraction and CNTs orientation on the postbuckling behavior of FG-CNTRC shells are particularly investigated. Exact geometries of shells are modeled by using NURBS interpolation. Several verifications are given to show the high reliability of the proposed formulation. Especially, some complex postbuckling curves of FG-CNTRC panels and cylinders are first provided that could be useful for future references.

Acceleration smoothing algorithm for global motion in high-speed machining

When machining a series of short linear segments, the computer numerical control system interpolates the machining path with the pre-specified spline curve to obtain continuous feed motion. However, to obtain adequate smooth tool-paths to smooth the feed velocity and acceleration, the computer-aided manufacturing system needs to use a higher order spline curve, which will be limited by some technical bottlenecks. In this article, a new real-time interpolation algorithm is proposed from the perspective of kinematics to achieve uninterrupted feed motion throughout the global tool-path. At the same time, the acceleration profile achieves G2 continuous to avoid unnecessary feed frustration and inertial impact, reaching the balance between time-optimal and motion performance. First, the jounce-limited acceleration curve is blended at the corner of the machining path, and the optimal cornering transition velocity is obtained by adding the velocity, acceleration and contour error constraints to the corner transition motion. Then, according to the linear segments with different lengths between the corners, combined with the feed motion around the corner, a look-ahead interpolation algorithm is proposed to calculate the maximum feed rate with the constraint of the linear segment length and kinematic boundary conditions. At last, for the linear segments whose corner contours overlap with each other after interpolation, the smooth transition between the two corners can be realized by mixing the feed motion of the next corner. Compared with non-uniform rational B-spline interpolation algorithm, the proposed algorithm reduces the total machining time by 14% and the computer numerical control system improves the computational efficiency by 11%. It proves that the proposed algorithm has better application value in the manufacturing of complex parts.

Conversion method study from cutter-location points to nonuniform rational B-spline toolpath NC file for high-speed machining

Parts with complex curved surfaces are widely applied and the demands for the machining quality and the machining efficiency of such complex parts become increasingly higher. In order to realize high-speed and high-quality machining, the nonuniform rational B-spline interpolator is widely researched and demonstrated to be superior to the conventional linear or circular interpolators. However, the nonuniform rational B-spline toolpath NC files cannot be directly generated from the computer-aided design (CAD) models by using commercial computer-aided manufacturing (CAM) software. To deal with this problem, a conversion method from the cutter-location points to the nonuniform rational B-spline toolpath numerical control (NC) file is presented. To avoid the bad curve-fitting effect at sharp corners of the toolpath and to meanwhile reduce the computational burden, the cutter-location pre-processing method, consisting data segmentation and data simplification, is provided first. Then, the least-square method is employed to fit the cutter locations to the nonuniform rational B-spline curves, and an iterative fitting approach is proposed for linear/nonuniform rational B-spline hybrid toolpath generation. Finally, a user interface is designed for displaying the fitting results and outputting the NC file with nonuniform rational B-spline toolpaths. By using this method, nonuniform rational B-spline and linear toolpaths hybrid interpolation NC program can be generated for the high-speed machining of complex curved surface parts with the utilization of the nonuniform rational B-spline interpolator. The experimental results demonstrate the feasibility and the advantages of the presented method.

Feedrate Scheduling of NURBS Interpolation Based on a Novel Jerk-Continuous ACC/DEC Algorithm

In order to improve the motion smoothness while maintaining the machining efficiency of the non-uniform rational B-spline (NURBS) interpolation, a novel jerk-continuous acceleration/deceleration (ACC/DEC) algorithm and the corresponding feedrate scheduling method are proposed in this paper. The polynomial and trigonometric functions are combined to construct the jerk-continuous ACC/DEC profile. Meanwhile, to reduce the computational load of feedrate scheduling, a proportional control method is proposed to determine the form of the jerk profile, which also improves the flexibility of feedrate control. Therefore, the maximum jerk and acceleration can be maintained to achieve higher machining efficiency compared with the conventional trigonometric methods, while the feedrate scheduling process is far simpler than the polynomial methods with continuous jerk. In addition, to improve the machining accuracy and motion smoothness, the round-off error caused by cycle sampling is also considered in feedrate scheduling and real-time interpolation. The proposed ACC/DEC algorithm is simplified to achieve the round-off error compensation, which can guarantee the continuity of jerk profile. A series of simulations and experiments for two NURBS curves is conducted to verify the good performance and applicability of the proposed method.

Interpolation using non-uniform rational B-spline for the smooth milling of ruled-surface impeller blades

The integral impeller is a complex part characterized by highly curved surfaces. Modeling and interpolation methods for manufacturing integral impellers are currently the subject of intensive research. In this study, to represent the shape of the ruled-surface blades, third-order non-uniform rational B-splines are constructed from data points for the hub and shoulder on the pressure and suction sides of the blade. Various methods for determining the tool axis vector offset and the location of the tool center for a cylindrical ball cutter are developed and verified with simulations of machining operations. To obtain a smooth interpolation and high accuracy in machining, a third-order non-uniform rational B-spline interpolation algorithm for ruled-surface blades is developed. This method can create a smooth transition between the hub and the blade shoulders. The method is implemented in an open-architecture computer numerical control controller. Simulations and experimental tests of machining tasks are performed to verify the non-uniform rational B-spline interpolation method. The test results show that this interpolation technique can produce a smooth and continuous path along the axes of motion. The new technique has the advantages of requiring fewer lines of numerical control code and producing higher machining accuracy.

Development of FPGA based NURBS interpolator and motion controller with multiprocessor technique

The high-speed computational performance is gained at the cost of huge hardware resource, which restricts the application of high-accuracy algorithms because of the limited hardware cost in practical use. To solve the problem, a novel method for designing the field programmable gate array(FPGA)-based non-uniform rational B-spline(NURBS) interpolator and motion controller, which adopts the embedded multiprocessor technique, is proposed in this study. The hardware and software design for the multiprocessor, one of which is for NURBS interpolation and the other for position servo control, is presented. Performance analysis and experiments on an X-Y table are carried out, hardware cost as well as consuming time for interpolation and motion control is compared with the existing methods. The experimental and comparing results indicate that, compared with the existing methods, the proposed method can reduce the hardware cost by 97.5% using higher-accuracy interpolation algorithm within the period of 0.5 ms. A method which ensures the real-time performance and interpolation accuracy, and reduces the hardware cost significantly is proposed, and it’s practical in the use of industrial application.

Design of a real-time NURBS interpolator with constant segment length for milling EDM

The utilization of the non-uniform rational B-spline (NURBS) toolpaths becomes more important than ever before. However, in traditional milling electrical discharge machining (EDM) of parametric curves, there commonly exist problems such as speed loss and an overincreased sampling period, which directly cause a decrease in machining efficiency. Moreover, traditional approaches normally suffer from a complex toolpath planning. In this paper, a real-time interpolator with a constant segment length is proposed to improve the milling EDM of NURBS curves. The proposed interpolator can directly process NURBS curves with their definition information. The toolpath planning can thus become simpler. A new two-stage interpolation method is adopted such that both a high speed accuracy and a relatively short sampling period can be achieved, and the magnitude of chord errors can also be reduced. While the first-stage interpolation samples a NURBS curve with a constant-length segment, the second-stage interpolation, or the re-interpolation, executes multiple interpolations in a sampling period to generate the required feed rate. A loop buffer is designed for the implementation of the real-time interpolator. Experimental results show that the proposed interpolator demonstrates a superior machining performance to that by traditional interpolators, especially in terms of chord errors and erosion rate.

A Postprocessor for Generating Multi-Axis Polynomial Interpolation NC Machining Program

Multi-axis CNC machining shows great advantages in manufacturing parts with geometries being defined by complex surfaces. NURBS (Non-Uniform Rational B-Spline) interpolation and polynomial interpolation are two new types of interpolation to improve productivity. Compared to the NURBS interpolation, the polynomial interpolation can gain higher productivity. However, no commercial software tool is available to generate polynomial interpolation CNC programs. This paper proposes a postprocessor to generate polynomial interpolation multi-axis CNC program. A case study with simulations demonstrated that the polynomial interpolation program generated by the proposed postprocessor has higher machining efficiency than liner interpolation and NURBS interpolation program.

Smooth non-uniform rational B-spline (NURBS) machining with kinematic limit for short linear segments

Set of methods based on non-uniform rational B-spline (NURBS) are proposed in this article, which are adopted to improve the smoothness during short linear numerical control (NC) codes machining. The smoothness refers to two aspects: the contour fairness of CL trajectory and the machining stability of the machine tool. To obtain contour fairness, the optimised knots combination strategy (OKCS) is proposed in least square NURBS fitting from short linear segments, which generates NURBS curve for the next interpolating process. Sequentially, an axis-based look-ahead NURBS interpolator (ALANI) is designed for NURBS interpolation in order to improve the machining stability according to the kinematic limit. Both OKCS and ALANI are simulated and compared with congener algorithms, respectively. Furthermore, algorithms are implemented on digital signal processor (DSP) platform, and actual machining examples are presented at last.

Realization of NURBS for Cranium in a Laser Sintering Machine

It is an issue in the application of Non-Uniform Rational B-Spline (NURBS) how to achieve the characteristic parameters based on a given curve. In this work, the parameterization method of accumulated chord length is used to construct the knot vector. The tangent vector is chosen as the boundary condition. The characteristic parameters including the control points, the knot vector and the weights are calculated according to the key points and corresponding weights of a given curve. An example of NURBS interpolation for cranium is given based on a selective laser sintering (SLS) machine. The characteristic parameters are obtained with Rhino software. The NURBS interpolation for cranium is realized using these characteristic parameters in the SLS machine. The results show that the practical sintering curve is coincident with the given curve.