An accurate NURBS curve interpolation algorithm with short spline interpolation capacity

In this paper, a novel and accurate real-time non-uniform rational B-spline curve interpolation algorithm is proposed. This algorithm not only considers chord errors, feedrate fluctuations, jerk-limited, and acceleration/deceleration (Acc/Dec) capabilities of the machine, but also optimizes the look-ahead process. In the meanwhile, it improves machining efficiency by adding the circular buffer and pre-interpolation (non-off-line) and enhances the real-time performance by removing the time-consuming calculation from the interrupt service routine. Furthermore, the proposed interpolation algorithm can interpolate both the long spline and the short spline with uniform method. The advantages of the proposed method were confirmed by the simulation results.     

An adaptive off-line NURBS interpolator for CNC machining

As a key technique in CNC machining, nonuniform rational B-spline (NURBS) interpolator has been proposed to overcome the drawbacks caused by linear and circular interpolator, such as large data size, velocity discontinuity, shocks or variations in mechanical systems, and low machining efficiency. To improve machining quality and efficiency, an adaptive off-line interpolator was developed for NURBS curves in this paper. Both the chord accuracy and the acceleration/deceleration capability of machine tool are considered in the algorithm. There are four modules in the algorithm, adaptive feed rate adjustment, acceleration/deceleration disposal, feed rate modification, and S-type velocity generation. The acceleration/deceleration around the sharp corners is carefully calculated by those modules. A case study was provided to evaluate the feasibility of the interpolator.     

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.     

一种参数预估-校正的NURBS曲线插补算法研究

提出了Milne-Simpson参数预估-校正的NURBS曲线插补算法.详细阐述了参数插补预估及校正机理.采用最大弓高误差、最大进给速度和最大法向加速度约束,以便实时调整插补进给步长,从而满足了NURBS曲线插补的高速和高精度要求.     

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.     

自适应参数曲线插补算法研究

在现有参数曲线插补算法的基础上,基于预估-校正和牛顿迭代公式提出一种新的自适应参数曲线插补快速求解算法;研究指数加减速曲线的时间常数值和系统的柔性、加减速能力的关系,提出变时间常数的指数加减速控制算法。算法改善了传统指数曲线加减速控制中速度突变的缺点,能够在保证加工精度的前提下减少速度波动,并提高曲线的加工效率。仿真实验表明,该算法计算过程简单、切实有效,满足数控系统的强实时性要求并大大提高了其加工过程的速度平稳性和插补效率。     

Real-time variable feed rate NURBS curve interpolator for CNC machining

This paper presents a real-time control algorithm based on Taylor’s expansion for implementing variable feed rate non-uniform rational B-spline (NURBS) curve interpolators using a digital signal processor for precision CNC machining. To efficiently compute the NURBS curve and its derivatives in real-time, an effective method is proposed. The variable feed rate NURBS curve interpolator can be used to realise the ACC/DEC before feed rate interpolation in which the ACC/DEC (acceleration/deceleration) planning on the feed rate command executes before the interpolation takes place, so that the path command errors caused by conventional ACC/DEC planning using the post feed rate interpolation can be effectively eliminated. To demonstrate the performance of the proposed algorithm, an X-Y table driven by two servomotors is controlled to track command paths represented by multiple blocks of NURBS curves. Experimental results verify the effectiveness of the proposed method.     

一种优化的NURBS曲线插补算法

为弥补目前非均匀有理B样条插补算法的不足,提出一种优化的非均匀有理B样条曲线插补算法。算法采用插补前S曲线加减速方法,不仅优化了前瞻过程,而且在回溯过程中考虑了短样条的情况。算法对长样条和短样条非均匀有理B样条曲线能用统一的方法进行插补,通过引入环形缓冲区和预插补(非离线),提高了加工效率,同时合理地安排插补任务增强了系统的实时性。通过MATLAB仿真,验证了算法的有效性。     

NURBS曲线插补算法的研究

针对目前制造业对数控加工精度要求越来越高的要求,介绍-种改进后的NURBS曲线插补算法,以实例表明其高速、高精度性.     

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.     

NURBS曲线实时插补算法研究

提出了一种包含插补误差和进给加速度实时监控的NURBS曲线实时插补算法,该算法有效的避免了曲线求导和曲率的复杂计算。运用参数的对分法预估下一插补点,极大限度简化了插补的实时计算,保证了算法的实时性。     

NURBS曲线修正插补算法的研究

NURBS曲线插补参数递推一阶、两阶求解比较复杂,加工误差较大。本文介绍一种NURBS曲线修正的插补算法,该算法不仅满足加工对精度方面的要求,同时也满足实时性的要求。最后,通过在matlab7.0上验证该算法是正确的,符合NURBS曲线插补的要求。     

Design of a real-time adaptive NURBS interpolator with axis acceleration limit

With recent advances in high-speed and high-accuracy machining, the NURBS interpolator has shown significant effect on dealing with the free-form curves and surfaces. The existing study aims at developing the adaptive interpolator which confines the chord error, the tangent acceleration, and jerk. However, the excessive axis acceleration is still unavoidable at the sharp corners and will deteriorate the contour accuracy. In this paper, a real-time adaptive NURBS interpolator considering the acc/dec capacity for each individual axis is developed to confine both the chord error and the axis acceleration. The maximum feasible feed and tangent acceleration range are deduced, respecting the given axis acc/dec limit. A two-stage feed determination scheme is applied to calculate the adaptive feed rate for each sampling period. A look-ahead window is utilized to improve the calculation performance for real-time application. Simulations and experiments are performed to verify the resulting feed rate, acc/dec profiles, and the real-time performance of the proposed interpolator.     

NURBS曲线实时插补进给速度控制的研究

数控技术标志着现代制造业的核心,数控插补模块是数控技术中最为重要的模块之一。NURBS曲线是自由曲线的一种,由于其NURBS曲线的诸多优越性,使其能够很好的应用到数控领域中。NURBS曲线插补及加减速控制的精度和速度是CNC系统的重要指标。通过分析NURBS曲线的插补原理,提出了基于Taylor展开公式逼近NURBS样条参数。同时考虑速度波动于曲率的关系,弦误差与插补周期的关系,根据弓高误差调节进给速度,能够将进给速度波动控制在理想水平。     

NURBS曲线插补技术研究

基于一种等弧长插补思想,综合考虑插补精度要求,实现进给速度随曲线曲率变化的调整.     

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.     

Fast NURBS interpolation based on the biarc guide curve

In parametric spline interpolation, the real-time parameter update is a crucial step which will directly affect the processing performance such as the feed rate fluctuation, the contour error, the online computational effort, etc. The use of Taylor approximation interpolation method to identify the next interpolate point will cause large feed rate fluctuation due to the accumulation error and the truncation error, which will affect the machining quality. As there is no accurate analytic expression between the parameter u and arc length S and the mapping between them is nonlinear, and in order to reduce the feed rate fluctuation and light computation requirement for online interpolation, the paper first samples the tool path with step parameter and Gauss integration, with the sampled points being in the coordinate system defined by parameter u and arc length S. Then, the sampled points are fitted into the guide curve with the use of the biarc fitting method, and the analytic expression between parameter u and arc length S is established. The biarc so derived can be used to realize a fast NURBS interpolation and the simulation results validate the reliability and effectiveness of the proposed method.     

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.     

自动调节进给速度的NURBS插补算法的研究与实现

现代计算机数控系统中已经普遍使用非均匀有理B样条插补,但大多数非均匀有理B样条插补算法都致力于取得恒定的进给速度而不是轮廓精度。对此,提出了一种限定弓高误差的自动调节进给速度的空间非均匀有理B样条曲线插补算法,它在通常加工时,是泰勒展开式2阶近似插补,而在小曲率半径零件的高速加工时,可以根据曲率半径和限定的弓高误差自动地调整进给速度,保证了轮廓加工精度。加工实例证实了这种插补算法的实时性和实际应用的可行性。     

一种简化计算的S型加减速NURBS插补算法

针对目前NURBS曲线插补中加减速控制方法不足的问题,实现了加工过程中进给速度的平滑过渡,提出了一种新的NURBS曲线插补方法,包括速度规划和实时插补两个方面。速度规划采用了一种基于曲率自适应的简化计算的S型加减速方法,并结合"双向插补"的思想实时预测减速点,防止产生过大的弓高误差;实时插补则利用Muller插值和Newton迭代法计算了下一周期的插补参数,进而求出了下一时刻到达的空间坐标点。最后与已有插补方法进行了仿真分析比较。研究结果表明,该方法能保证加速度连续和加加速度有界,有效减少弓高误差和进给速度波动,提高机床运行的平稳性。