An accurate adaptive parametric curve interpolator for NURBS curve interpolation

In this paper, an adaptive parametric curve interpolator with a real-time look-ahead function is developed for non-uniform rational B-spline curves (NURBS) interpolation, which considers the maximum acceleration/deceleration of the machine tool. In the proposed interpolator, both constant feedrate and high accuracy are achieved while the inconsistency of feedrate is reduced dramatically as well. In order to deal with the acceleration/deceleration around the feedrate sensitive corners, a look-ahead function is introduced to detect and adjust the feedrate adaptively. Two cases were respectively studied to evaluate the feasibility of the developed interpolator: one is for feedrate sensitive arc corner; the other is for feedrate sensitive sharp corner.     

Design of a real-time adaptive interpolator with parameter compensation

The interpolator is the key part of a CNC system, which has strong effect on machining accuracy, tool motion smoothness, and machining efficiency. In this paper, a real-time adaptive interpolator is developed for non-uniform rational B-spline curves (NURBS) interpolation while considering the maximum acceleration/deceleration of the machine tool. In this proposed interpolator, both constant feedrate and high accuracy are achieved while the inconsistency of feedrate is dramatically reduced as well. In order to deal with the acceleration/deceleration around the feedrate-sensitive sharp corners, a look-ahead function is introduced to detect and adjust the feedrate adaptively. Furthermore, a parameter compensation scheme is proposed to eliminate the parametric truncation error which has been analyzed by several researchers but still not incorporated into any real-time interpolator so far. A case study was conducted to evaluate the feasibility of the developed interpolator.     

Modeling of the residual kinematic errors of coordinate measuring machines using LaserTracer system

It is complex and divergent for the conventional motion controllers to process various G codes using different interpolation algorithms. This impairs programming efficiency and robustness of the controller. In this paper, we propose the universal non-uniform rational B-splines (NURBS)-based interpolator which can simplify the architecture of interpolation in spite of interpreting different kinds of inputs. Direct conversion of long G01 and G02/G03 numerical control (NC) segments to NURBS segment is first implemented. The fitting of multiple short segments into a continuous and smooth NURBS segment is then carried out. More importantly, the universal NURBS-based interpolator utilizes the Cox–de Boor algorithm which is highly efficient and can take advantage of the parallel computing scheme to accelerate the processing speed. Furthermore, due to the construction of real-time environment, the proposed algorithm enables interpretation, look-ahead functions, and motion control to work simultaneously. A 2D NC program possessing hundreds of short G01 segments and long segments (i.e., G02, G03, and long G01) is tested on an in-house developed XY-table with a PC-based motion controller. The results have shown the effectiveness and feasibility of the proposed real-time NURBS-based interpolator.     

Development of a real-time look-ahead interpolation methodology with spline-fitting technique for high-speed machining

Methodologies for converting short line segments into parametric curves were proposed in the past. However, most of the algorithms only consider the position continuity at the junctions of parametric curves. The discontinuity of the slope and curvature at the junctions of the parametric curve might cause feedrate fluctuation and velocity discontinuous. This paper proposes a look-ahead interpolation scheme for short line segments. The proposed interpolation method consists of two modules: spline-fitting and acceleration/deceleration (acc/dec) feedrate-planning modules. The spline-fitting module first looks ahead several short line segments and converts them into parametric curves. The continuities of the slope and curvature at each junctions of the spline curve are ensured. Then the acc/dec feedrate-planning module proposes a new algorithm to determine the feedrate at the junction of the fitting curve and unfitted short segments, and the corner feedrate within the fitting curve. The chord error and acceleration of the trajectory are bounded with the proposed algorithm. Simulations are performed to validate the tracking and contour accuracies of the proposed method. The computational efforts between the proposed algorithm and the non-uniform rational B-spline (NURBS)-fitting technique are compared to demonstrate the efficiency of the proposed method. Finally, experiments on a PC-based control system are conducted to demonstrate that the proposed interpolation method can achieve better accuracy and reduce machining time as compared to the approximation optimal feedrate interpolation algorithm.     

Real-time PH-based interpolation algorithm for high speed CNC machining

Various methods for parametric interpolation of non-uniform rational B-spline (NURBS) curves have been proposed in the past. However, the errors caused by the approximate nature of the NURBS interpolator were rarely taken into account. This paper proposes an integrated look-ahead algorithm for parametric interpolation along NURBS curves. The algorithm interpolates the sharp corners on the curve with the Pythagorean-hodograph (PH) interpolation. This will minimize the geometric and interpolator approximation errors simultaneously. The algorithm consists of four different modules: a sharp corner detection module, a PH construction module, a feedrate planning module, and a dynamics module. Simulations are performed to show correctness of the proposed algorithm. Experiments on an X?CY table confirm that the developed method improves tracking and contour accuracies significantly compared to previously proposed algorithms.     

基于进给速度引导曲线的参数曲线实时插补器

设计了一种基于进给速度引导曲线的参数曲线实时插补器,给出了进给速度引导曲线的B-样条曲线生成方法。该方法在数控加工之前,通过建立进给速度引导曲线,来取代参数曲线实时插补进给速度规划的前瞻处理。实验表明,采用这种基于速度引导曲线的参数曲线实时插补器后,实时计算任务少,可以获得良好的伺服轴进给速度平滑效果和实时插补性能。     

Performance Evaluation of a Real-Time Interpolation Algorithm for NURBS Curves

This paper presents a real-time interpolation algorithm for NURBS curves. In contrast to the existing linear and circular interpolators, the proposed interpolator can maintain small contour errors and feedrate fluctuations. Feedrate components, acceleration components and the servomotor driving force for each axis are precalculated from the given curve shape and themachine tool dynamic properties. As a sudden change in the geometrical properties of the tool path can increase contour errors and cause a sudden change of driving force of each servomotor, a new strategy of variable feedrate machining based on the geometrical properties of tool path is suggested. Real-time performance measurement of this interpolator is performed to demonstrate its practical feasibility.     

Precision NURBS interpolator based on recursive characteristics of NURBS

In NURBS interpolation, real-time parameter update is an indispensable step which affects not only feedrate fluctuation but also contour error. Using Taylor approximation interpolation method to find the next interpolation point causes a large feedrate fluctuation due to the accumulation and truncation errors. This paper presents a new, simple, and precise NURBS interpolator for CNC systems. The proposed interpolation algorithm does not use Taylor’s expansion, but the recursive equation of the NURBS formula. A simulation study is conducted to demonstrate the advantages of this proposed interpolator compared with those using Taylor’s equation. It is readily seen that this interpolator using the new concept of interpolation for modern CNC systems is simple and precise. The proposed method can be used for interpolating a continuous NURBS curve.     

Development of integrated acceleration/deceleration look-ahead interpolation technique for multi-blocks NURBS curves

Modern motion control adopts acceleration/deceleration before interpolation (ADBI) motion planning to eliminate path command errors. However, the individual velocity profiles might not be continuous at the junction of the blocks. Acceleration/deceleration after interpolation (ADAI) method may provide an alternative for solving the discontinuous problems, but it causes path command errors. In this paper, an integrated acceleration/deceleration interpolation (IAD) scheme which integrates the ADBI and ADAI modules is proposed. The ADBI provides a look-ahead function which plans the feedrate profiles based on chord errors, command errors, curvatures, and acceleration limits. Within the look-ahead function, the command error equation is utilized to determine the feedrate at the junction of adjacent blocks. Then the ADBI performs non-uniform rational B-spline (NURBS) interpolation using the planned feedrate profile and outputs the position points to the ADAI module. The ADAI module processes the points by a digital convolution technique such that the continuity of the block junction velocity is ensured. Finally, the IAD is applied to the multi-block NURBS interpolation to validate its effectiveness. Simulations and experiments are conducted to demonstrate the IAD scheme. It is shown that the IAD scheme can reduce the acceleration significantly at the junctions of the blocks under the given tolerance of the command error. Furthermore, the proposed algorithm can improve tracking and contour accuracies as compared to the hybrid multi-blocks look-ahead approach.     

Real-time NURBS curve interpolator based on section

Parameter interpolation is more capable of modern computer numerical control (CNC) than traditional linear/circular interpolation with higher speed and higher precision. Most of non-uniform rational B-spline (NURBS) interpolation algorithms were developed based on the chord error and machines capability, where interpolation points are calculated beforehand to overcome acceleration/deceleration (acc/dec) and jerk problem, which needs large memory. In this paper, a NURBS interpolator based on feedrate section is proposed. Instead of a single interpolation point, this interpolator aims to feedrate section, which makes it possible to run on a digital signal processor or field programmable gate array (FPGA) whose memory is limited. Experiment on FPGA showed the performance of interpolation. A mould experiment verifies the feasibility of application.     

The design of a NURBS pre-interpolator for five-axis machining

A non-uniform, rational B-spline (NURBS) interpolator has great advantages for free-form surface machining compared to the conventional linear/circular interpolator. However, the existing NURBS interpolators can only handle the NURBS trajectory given in a customized NURBS G code. Also, it is limited to three-axis applications. In this paper, a NURBS pre-interpolator with three function options is proposed for a computer numerical control (CNC) system so that the NURBS interpolator can be thoroughly applied for five-axis machining. The first function is called the NURBS converter function, which is used to convert a series of linear/circular segments exactly into a NURBS curve. The second function is the NURBS smoother function, by which, a series of linear segments are fitted to a NURBS curve. The third option provides two kinds of NURBS G codes definition, by which, the NURBS trajectory with five axes can be represented directly. Upon using the three options of the NURBS pre-interpolator, a unified NURBS curve can be obtained for further interpolation. Two actual machining cases are conducted to evaluate the feasibility of the proposed pre-interpolator.     

LOOK-AHEAD ALGORITHM FOR VELOCITY CONTROL BASED ON PARAMETERIZED CURVE INTERPOLATOR

To avoid suffering gouge and transient overshooting in high speed cutting machining, a novel parametefized curve interpolator model with velocity look-ahead algorithm is proposed. Based on a prearrangement step interpolation algorithm for parameterized curves and considering high curvature points, parameterized curve tool path is divided into acceleration segments and deceleration segments by look-ahead algorithm. Under condition of characteristics of acceleration and deceleration stored in control system, deceleration before high curvature points and acceleration after high curvature points are realized in real-time in high speed cutting machining. Based on new parameterized curve interpolator model with velocity look-ahead algorithm, a real cubic spline is machined simulativly. The simulation results show that velocity look-ahead algorithm improves velocity changing more smoothly.     

An iterative feedrate optimization method for real-time NURBS interpolator

Non-uniform rational B-spline (NURBS) interpolator has been widely used in modern manufacturing systems to machine arbitrary geometries with great relief of the data flow bottleneck and feedrate fluctuation. However, in practice, real-time feedrate does not always meet the computer numerical control (CNC) command exactly subject to the system dynamics. To solve this problem, we present a real-time NURBS interpolator with feedrate optimization for CNC machining tools in this work. The parametric curve is first approximated with the Adams–Bashforth method which provides uniform feedrate in each sampling period in the interpolation process. And then, a feedback scheme is introduced to adjust the feedrate in real time so as to guarantee a specified deviation between the measured and the desired feedrate. The convergence condition for this closed-loop algorithm is presented and analyzed. Simulation and real experiments on an X–Y table are employed to verify its feasibility. And the comparisons with traditional interpolators based on Taylor's expansion are also provided to demonstrate its improvement in accuracy.     

An adaptive feedrate scheduling method of dual NURBS curve interpolator for precision five-axis CNC machining

Non-uniform rational b-spline (NURBS) tool path is becoming more and more important due to the increasing requirement for machining geometrically complex parts. However, NURBS interpolators, particularly related to five-axis machining, are quite limited and still keep challenging. In this paper, an adaptive feedrate scheduling method of dual NURBS curve interpolator with geometric and kinematic constraints is proposed for precision five-axis machining. A surface expressed by dual NURBS curves, which can continuously and accurately describe cutter tip position and cutter axis orientation, is first used to define five-axis tool path. For the given machine configuration, the calculation formulas of angular feedrate and geometric error aroused by interpolation are given, and then, the adaptive feedrate along the tool path is scheduled with confined nonlinear geometric error and angular feedrate. Combined with the analytical relations of feed acceleration with respect to the arc length parameter and feedrate, the feed profiles of linear and angular feed acceleration sensitive regions are readjusted with corresponding formulas and bi-directional scan algorithm, respectively. Simulations are performed to validate the feasibility of the proposed feed scheduling method of dual NURBS curve interpolator. It shows that the proposed method is able to ensure the geometric accuracy and good machining performances in five-axis machining especially in flank machining.     

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.     

五轴NURBS插补中的速度前瞻控制方法

为提高五轴非均匀有理B样条的插补精度,减轻段间转接时对机床的机械冲击,提出了一种速度前瞻控制处理方法.采用曲率圆逼近的方法建立了插补误差计算模型,并推导出满足插补误差要求的最小曲率半径与进给速度间的关系.采用数值分析方法预先计算出减速点的参数值,对样条插补进行了加减速前瞻规划处理.实时插补时对样条曲线上曲率半径较小的地方进行减速处理,以提高插补曲线的逼近精度.实例证明,该方法能有效降低插补误差和段间转接时的加速度,具有良好的实时性,可以应用到实际的数控系统中.     

On the development of a parametric interpolator with confined chord error, feedrate, acceleration and jerk

A parametric interpolator for the trajectory planning of non-uniform rational basis spline (NURBS) curves is proposed in this study. The input constraints include the chord error, speed limit, acceleration limit and jerk limit. Since there is no unique representation to relate these constraints, appropriate models are developed to determine several feedrates in terms of different kinematic conditions, the minimum one of which is selected as the desired feedrate. A look-ahead stage is developed to plan a series of segment points from the curves, where a segment point represents the change in the acceleration across zero. The segment points, which enable the appropriate design of kinematic profiles, are used in real-time sampling for determining the sampling points. Several examples are presented to demonstrate the feasibility of the proposed algorithm.     

High accurate interpolation of NURBS tool path for CNC machine tools

Feedrate fluctuation caused by approximation errors of interpolation methods has great effects on machining quality in NURBS interpolation, but few methods can efficiently eliminate or reduce it to a satisfying level without sacrificing the computing efficiency at present. In order to solve this problem, a high accurate interpolation method for NURBS tool path is proposed. The proposed method can efficiently reduce the feedrate fluctuation by forming a quartic equation with respect to the curve parameter increment, which can be efficiently solved by analytic methods in real-time. Theoretically, the proposed method can totally eliminate the feedrate fluctuation for any 2nd degree NURBS curves and can interpolate 3rd degree NURBS curves with minimal feedrate fluctuation. Moreover, a smooth feedrate planning algorithm is also proposed to generate smooth tool motion with considering multiple constraints and scheduling errors by an efficient planning strategy. Experiments are conducted to verify the feasibility and applicability of the proposed method. This research presents a novel NURBS interpolation method with not only high accuracy but also satisfying computing efficiency.     

Adaptive parametric interpolation scheme with limited acceleration and jerk values for NC machining

Parametric interpolation has many advantages over the traditional linear or circular interpolation in computer numerical control (CNC) machining. The existing work in this regard is reported to have achieved constant feedrate, confined chord error and limited acceleration/deceleration in one interpolator. However, the excessive jerk still exists due to abrupt change in acceleration profile, which will cause shock to the machine as well as deteriorate the surface accuracy. In this paper, an adaptive interpolation scheme incorporating machine’s dynamics capability consideration is proposed and illustrated in details. In the proposed algorithm, the commanded feedrate is maintained at most of the time and adaptively reduced in large curvature areas to meet the demand of the machining accuracy requirement, while at the same time, the acceleration and jerk values are limited within the machine’s capabilities during the whole interpolation process. It ensures a high machining accuracy, eliminates the phenomenon of overshoot/undershoot and reduces mechanical shock to the machine tools. The real-time performance of this interpolator is also measured to demonstrate its practical application. Two non-uniform rational B-spline (NURBS) curve interpolation experiments are provided to verify the feasibility and advantages of the proposed scheme.