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.     

Non-circular parametric curve and curved surface interpolation and tool compensation for WEDM based on unit arc length increment method

This paper proposes a novel non-circular parametric curve interpolation algorithm based on the unit arc length increment method (UALI). In this method, axial coordinates are taken as functions of arc length. The reference point moves by 1 basic length unit (BLU) along a curve in each interpolation period, and the corresponding projections on each coordinate are added into accumulators of each axis respectively. An axis will then move by 1 BLU when its accumulator exceeds 1 BLU. Theoretically, based on the UALI method, an interpolation algorithm is derived for parametric curves such as involutes, Archimedes spirals, cycloids, and parabolas. This interpolation algorithm is applicable for ruled surfaces of a workpiece with different parametric curves in the upper and lower planes. Tool compensation is considered for both non-tapered workpieces as well as tapered workpieces. Simulation results and experiment results demonstrate the advantages of this interpolation algorithm, and the accuracy of this interpolation algorithm is analyzed.     

参数曲线的自适应实时前瞻插补算法

为实现数控加工中进给速度的平滑过渡,减少速度急剧变化时对机床的冲击,提出了一种参数曲线的实时前瞻插补算法。该算法根据加工弓高误差要求自适应地调整进给速度,同时找出速度敏感点。通过把前瞻距离分成两部分的方法,分析速度敏感点,找出最佳的加减速控制点,避免相邻速度敏感点间加减速过程的互相干涉,提早进行加减速控制,防止速度的急剧变化,从而在满足加工精度的同时也满足了机床的加减速性能。通过RT-Linux软数控下的实例,表明该算法能够适应曲线的各种变化,验证了其可行性。     

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.     

Simultaneous 3D machining with real-time NURBS interpolation

Increasing demand on precision machining using computerized numerical control (CNC) machines have necessitated that the tool move not only with the smallest possible position error but also with smoothly varying feedrates in 3-dimensional (3D) space. This paper presents the simultaneous 3D machining process investigated using a retrofitted PC-NC milling machine. To achieve the simultaneous 3-axis motions, a new precision interpolation algorithm for 3D Non-Uniform Rational B-Spline (NURBS) curve is proposed. With this accurate and efficient algorithm for the generation of complex 3D shapes, a real-time NURBS interpolator was developed using a PC and the simultaneous 3D machining was accomplished satisfactorily.     

A local smoothing interpolation method for short line segments to realize continuous motion of tool axis acceleration

In traditional processing, a large number of G01 blocks are adopted to discretize free surface or curve for NC machining. But, the continuity of G01 line segments is only C⁰, which may lead to discontinuity of axis acceleration, resulting in the frequent fluctuation of tool motion at the junctions in high-speed machining, deteriorating the quality of work piece, and reducing processing efficiency. To solve this problem, a local smoothing interpolation method is proposed in this paper. At first, the analytic relationship between the continuity of the trajectory and the continuity of the axes motion is first systematically described by formula. Based on this relationship, a local smoothing algorithm and a feed-rate scheduling method are proposed to generate a C² continuous tool path motion with axis-acceleration continuity. The local smoothing algorithm smoothes the corners of G01 blocks by the cubic B-spline according to the cornering error tolerance specified by the user. After the feed rate at critical points of smoothed tool path was determined by a modified bidirectional scanning algorithm by considering constrains of chord error and kinematic property, an iterative S-shape feed rate scheduling is employed to minimize residual distance caused by round of time while ensuring the continuity of feed rate and acceleration. Then, a look-ahead interpolation strategy combined with smoothing algorithm and feed-rate scheduling as mentioned is proposed for real-time interpolation of short line segments. At last, simulations are conducted to verify the effectiveness of the proposed methods. Compared with the traditional G01 interpolation, it can significantly improve the processing efficiency and shorten the processing time within error tolerance.     

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。     

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.     

基于Newton-Rapson迭代法的NURBS插补算法的研究

一阶、二阶Taylor展开式求解NURBS曲线插补坐标点ui+1,存在计算量大、插补时间长等问题,该文提出了一种基于Newton-Rapson迭代法的NURBS插补算法。仿真和实验结果表明:该算法计算过程简单有效,具有可行性和实用性,可以减少插补计算量和插补时间,提高插补精度和加工效率,满足了数控系统实时、高速、高效率插补的要求,对其他数控插补的过程具有很强的借鉴意义。     

Correcting and compressing interpolation algorithm for free-form surface machining

The existing interpolation algorithm cannot meet the need of high-speed and high-accuracy machining of a free-form surface. So this paper proposed a correcting and compressing interpolation algorithm. Depending on the distance and angle evaluated from the adjacent command points, the machining path of free form can be divided into two machining types. For those regions where the accurate figure is critical such as corners, the convention linear interpolation is performed exactly between the adjacent command points. For those regions having a large radius of curvature where the smooth figure is critical, firstly, the interior point selection method based on circle transition is derived to reduce the tolerance between the machining path and the original surface; secondly, the interior point correction method based on the least-square method is proposed to reduce the calculation error and round-off error in the interior point and estimate the first- and second-order derivative vectors of the interior point; thirdly, the shape-defining point is selected by the bend direction of the machining path and fitted to a quintic spline curve which has the C2 continuity; fourthly, the fitting accuracy controlling method is proposed to ensure the machining accuracy; lastly, the curve interpolation is performed on the fitted smooth curve. Machining tests carried out on a vertical machining center show that the proposed algorithm can improve the machining efficiency and machining quality of a free-form surface.     

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.     

具有限定的弦差和加减速度的插补器

作者提出了一种数控机床在插补过程中通过调整进给速度从而将弦差限定在一定的误差范围内的插补算法。提出的插补器是一种变进给速度插补器,在保证弦差位于一定误差范围内的同时,增加了数控机床的效率。本文给出了仿真结果以证明该插补器的可行性。     

珩磨机运动控制卡的三次B样条曲线插补算法

数控珩磨机在加工复杂曲线曲面时,控制系统通常以TMS320F2812芯片为核心,设计DSP运动控制卡的定时器控制中断控制伺服电动机来完成插补。介绍了一种利用DSP运动控制卡进行三次B样条曲线的插补方法,该方法不仅可以缩短插补时间,也可以通过控制运动控制卡定时器中断来调整插补轨迹。最后实例表明,该插补算法不仅满足运动控制的要求,而且也满足了高速高精度插补的要求。     

基于埃特金逐步插值滤波器的 TIADC 时间误差校准方法

拉格朗日插值滤波器可以应用于TIADC时间误差校准,并且该方法有较好的校准效果。但是拉格朗日插值滤波器的阶数不易调节,这使得该滤波器在实际应用时不灵活。本文针对此问题,提出了基于埃特金逐步插值滤波器的TIADC时间误差校准方法。本文化简了埃特金插值多项式的计算公式,并设计了一种新增插值结点的选取方法,从而将埃特金逐步插值算法应用于时间误差的校准。本文方法能根据TIADC采样系统的输入信号频率和设定的精度要求自动调节埃特金逐步插值滤波器的阶数。仿真结果表明,在90%的奈奎斯特频带范围内,本校准方法能有效抑制时间误差引起的杂散分量,且能自动调节滤波器的阶数,当输入信号的归一化频率为0.35时,校准后的SNDR提高了40.37 dB。并通过实验验证本校准方法的效果。     

可配置型五坐标B样条插补控制器的研制

针对采用五轴联动数控机床的线性插补功能进行数控加工存在的不足,提出了一种B样条插补控制策略用于五轴联动数控机床以实现复杂曲面零部件的高速高精数控加工。参考开放式、模块化体系结构控制器(OMAC)标准,开发了具有B样条插补功能的五轴联动数控机床运动控制器。该控制器将控制任务按照实时性要求进行划分。人机交互、代码解析及参数映射关系构造等过程离线完成,插补运算、离散逻辑控制及逆运动学变换等过程由实时线程执行,保证了数控系统的硬实时性。为简化NC程序的编制过程,控制器设计为接收工件坐标系下的加工信息。通过开发适应各种形式数控机床的逆运动学变换模块,并将机床参数设计为可用户定制,使得控制器具有良好的通用性。在控制器内部建立NC程序文件中位置曲线和方位曲线间的参数映射关系,使得机床平动轴与转动轴间的运动规划符合实际加工要求,并可保证加工精度。实际加工实验中,在采用B样条插补算法的NC程序量降低为线性插补NC程序量15%倍时,其插补误差为线性插补误差的45%,控制器插补精度为0.68,表明该B样条插补控制器可以满足五坐标数控加工的要求。     

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.     

基于速度平滑控制的高效非均匀有理B样条曲线插补算法

为提高曲面加工的质量和效率,提出了一种基于冗余误差和速度平滑控制的非均匀有理B样条曲线插补算法.该算法先按照曲率大小将非均匀有理B样条曲线分段,再根据数控机床最大加速度调整分段.消除了曲率突变对加工的负面影响.同时为每个分段分别设置合适的编程进给速度,提高了加工效率.最后提出两段速度平滑控制方法,使速度过渡更为平滑.模拟试验证明了该算法的有效性.     

Three-Dimensional Implicit Curve Interpolation

An approach to 3D implicit curve interpolation in computer numerical systems (CNC) is presented. A 3D implicit curve is the intersection of two implicit surfaces. Geometric entities of 3D implicit curves are related to motion entities along the curve. The results are then used to develop real-time 3D implicit curve interpolation. An improved interpolation scheme has been proposed to augment the initial interpolation scheme to eliminate the cumulative point deviations. Simulations of implicit curve interpolation have been carried out to verify the effectiveness of the proposed algorithm. They also demonstrate that under typical machining conditions, the interpolation error for the improved scheme is well within the accuracy requirements of typical machines. The example has an application potential for tool-path interpolation when tool paths are obtained by intersecting an implicit design surface by drive cylinders. These show that the proposed algorithm is potentially useful for the machining of implicit surfaces.     

六轴义齿加工机床线性插补实现及误差分析

多轴控制技术是六轴义齿加工机床的关键技术,由于现有六轴义齿加工机床结构的特殊性,一般机床多轴控制技术的数控算法很难得到直接应用.基于现有机床的固有结构,通过结合齐次变换,提出了一种新的数控算法,实现了直线轴和圆弧轴的两轴联动.并根据其特殊的加工轨迹,分析了其联动插补运动过程中的误差.结合使用MATLAB工具进行数据运算,得出了对任意一条二维平面上的函数,插补过程中的运动控制参数及其单位加工周期过程中的误差.     

曲线的三参数路径规划车削插补算法研究

为了保证数控加工中复杂曲线零件的车削加工精度,在普通二参数车削加工的基础上,提出了一种三参数路径规划车削插补算法。该算法将复杂曲面数控加工插补过程的路径规划分解为若干插补点之间的最优路径搜索和选取问题,根据不同加工要求设置决策变量,并进行取值判别,通过3个参数联动对曲线进行插补运算,优化加工过程中刀具的路径,并应用于相关的实例分析。研究结果表明,该优化算法能保证加工曲线的光滑性,有利于提高曲线的加工质量,并且该算法还具有计算规模小、求解迅速、数控加工插补误差小等优点。