Fast real-time NURBS path interpolation for CNC machine tools

In this paper, a novel fast real-time non-uniform rational B-spline (NURBS) path interpolation method is presented. This method efficiently integrates the data processing of a NURBS path in a CNC controller, from pre-processing to real-time interpolation. In the calculation of the total length of the NURBS path, the numerical adaptive quadrature method adapts to the integrand, i.e. the first derivative of the length function, automatically, dividing the parameter interval into subintervals with fine or coarse spacing according to the varying condition of the integrand. This new method takes full advantage of the subdivision scheme. The key point is to generate inverse length functions (ILF) for each resulting subinterval. In the real-time NURBS path interpolation, the new setting path parameter can be calculated directly using the ILF without the need for any time-consuming computation of NURBS derivatives and iteration. The proposed method is extremely fast, accurate and suitable for real-time implementation, and simulations and practical tests proved its effectiveness.     

基于弧长的椭圆插补新算法

根据不同变量对所用近似公式精度的影响,将曲线划分为不同区域.先在各区域内使用不同变量表示高斯公式来近似弧长,再利用分段三次多项式插值拟合建立弧长与坐标的函数关系,最后求出各个插补点坐标和各步的进给增量,完成插补过程.     

Guide curve based interpolation scheme of parametric curves for precision CNC machining

Real-time parametric interpolation has played a key role in the computer control of machine tools. To achieve highest quality parts, generated trajectories not only describe the desired toolpath accurately, but also have smooth dynamics profiles. This paper presents a novel parametric interpolator based on guide curve. The relationships between geometric properties and kinematic properties are firstly discussed. Then, with a consideration of the important effect of the curvature of curvilinear path on the machining dynamics, a corresponding formula, which describes the relation of the maximum allowed feed acceleration/deceleration and the maximum allowed rate of change of curvature radius of paths, is built. Thus, based on a near arc parameterization and through modifying the curvature radius curve to deal with corners, key regions and other cases, adaptive feedrate schedule is completed according to the reconstructed smooth curvature radius curve. Consequently, confined chord errors, corners on the path and the acceleration/deceleration capabilities of the machine tool are simultaneously considered and incorporated into the guide curve based parametric interpolation system without using look-ahead scheme. Simulation results indicate the feasibility and precision of the proposed interpolation method.     

Robust real-time NURBS path interpolators

In a review of the real-time non-uniform rational B-splines (NURBS) path interpolation method in CNC controllers, it was found that none of the NURBS interpolators described in the literature has the necessary robustness against an extreme knot distribution. The problems begin with the calculation of the total length of the NURBS path: most interpolators handle knots as a global curve parameter and may deliver incorrect results if the knots are of an extreme distribution. Further, Taylor's-expansion-based NURBS interpolators may overlook the path portion of extremely small knot spans. To solve these problems, methods for improving the robustness of Taylor's-expansion-based NURBS interpolators are proposed in this study, and an improved robust fast NURBS path interpolator is described. These new methods process NURBS data on the basis of knot spans: the adaptive quadrature method is applied to each knot span, and the calculated lengths of all knot spans are summed to build the total path length. Further, the inverse length functions are also generated based on the knot spans, and a control mechanism is introduced to prove the validity of the resulting inverse length functions. Experimental results have proved the effectiveness of the proposed robust fast NURBS path interpolator.     

基于机床动力学特性的NURBS曲线直接插补

提出一种基于机床动力学特性的NURBS曲线实时插补方法。陔方法利用NURBS曲线在小范围内的相似持性,采用合理的近似计算,保证了计算的实时性;通过建立低速点的参数链表．实现了满足机床动力学特性的NURBS曲线自适应插补控制。     

A parametric interpolator with minimal feed fluctuation for CNC machine tools using arc-length compensation and feedback correction

Traditionally, approximation methods are utilized in the parametric interpolation because of the nonanalytic relationship between the spline parameter and the arc length. The approximation error has been considered as the source of the feedrate fluctuation. This paper shows that the discrepancy between the desired tool path and the target trajectory of the motion system is another primary source, and presents a feedback interpolator to eliminate the feedrate fluctuation. To evaluate the initial parameter value for the interpolator, an arc-length based Taylor's expansion with arc-length compensation is proposed, which alleviates greatly the feedrate command error caused by the trajectory deviation. Then, a feedback correction scheme is developed to further reduce the feedrate command error that results from the approximation error. Both computational load analysis and numerical simulations are conducted, and the results show that the present interpolator has very good performance in both efficiency and accuracy, thus is a good choice for high speed and high precision CNC machines.     

The feedrate scheduling of parametric interpolator with geometry,process and drive constraints for multi-axis CNC machine tools

Parametric interpolator has been widely adopted in machining sculptured parts. Accordingly, the feedrate scheduling of parametric interpolator plays a role in CNC machine tools especially for multi-axis machines with linear and rotary axes, since a smooth movement is beneficial for achieving better surface geometry as well as shorter machining time. This paper presents a new feedrate scheduling method for the five-axis machining of geometrically complex part with geometry, process and drive constraints. The satisfaction conditions of constraints are first built and the proportional adjustment of feedrate sensitive regions is proved to be suitable for simultaneously reducing the magnitudes of constraints such as angular acceleration, linear acceleration, axis accelerations and jerks. The initial feedrate profile is first constructed with confined chord error, angular velocity and axis velocities owing to the independence of these constraints. Then, for each iterative adjustment a curve evolution strategy is used to deform the target feedrate profile to the adjusted positions instead of the re-interpolation of feedrate profile, until the final desired feedrate profile is achieved without violated constraints. Simulations and experiments are conducted and the results validate the performances of the proposed method.     

基于De_Boor递推算法的速度自适应NURBS曲线分段插补算法研究

针对现有大多数NURBS曲线插补算法不能同时满足插补精度和机床加减速性能要求的问题,提出一种基于De_Boor递推算法的速度自适应NURBS分段插补算法。通过前瞻预处理完成插补分段和减速点确定,通过速度修正完成减速段最终插补。MATLAB仿真结果表明,该算法能够同时满足插补精度和机床加减速性能要求。     

基于NURBS曲面的五轴联动插补算法

文章提出了一种基于NURBS曲面的五轴联动插补算法。对于NURBS曲面,确定一个参数方向的一系列参数值,就可确定一系列的另一个参数方向的NURBS曲线。沿这些曲线逐条进行插补,就可实现对整个曲面的插补。在每个插补点,求出两个参数方向的切矢,确定出该点的曲面法矢,在假设刀控方向垂直于插补曲面的情况下求出两个旋转轴的运动,实现五轴联动插补。同时,利用NURBS曲线的局部性质来保证插补的实时性。     

涡轮叶片造型方法的研究

对涡轮片进行精确的数学描述是对其进行数控加工的前提，本文讨论了在给定叶片沿叶高方向上截面数据的情况下，采用双三次B样条拟合法和三次NURBS曲面插值法构造叶片曲面的具体算法，分析了两种方法的优缺点，并给出了具体的算法和公式，深入探讨了叶片构造过程中遇到的部分网格确定，插值曲面构造，圆弧的二次NURBS曲线精确表示，NURBS曲线节点插入算法及其升降阶等关键性技术问题。     

NURBS曲线自适应插值拟合算法

针对从DXF文件中构造的NURBS曲线,提出了一种曲线的自适应分解算法.该算法结合了NURBS曲线的形状信息,通过计算比较插补点的曲率半径与给定阈值的大小,以决定当前插补类型为直线段或是圆弧段,在满足加工精度的前提下,采用伸缩步长法使得拟合的区间尽可能大.经实际运行表明,此方法拟合的加工程序段少,能简化数值计算和编程,减少数控机床的预处理时间.该算法具有通用性,为其他曲线的插值拟合也提供了一种有效途径.     

Development of a dynamics-based NURBS interpolator with real-time look-ahead algorithm

In this paper, a dynamics-based interpolator with real-time look-ahead (DBLA) algorithm is proposed to generate a smooth and jerk-limited acceleration/deceleration (ACC/DEC) feedrate profile. The interpolator consists of three modules: geometric, dynamics-based, and jerk-limited modules. The geometric module can detect the local maximum/minimum (max/min) curvatures, and divide a NURBS curve into small segments according to the information of sharp corners. The feedrates at the sharp corners are determined based on confined chord errors and curvatures of the curve. The dynamics-based module utilizes a dynamics feedrate modification equation (DFME) to estimate contour errors at the sharp corners and adjusts the feedrates at the locations of the sharp corners. The jerk-limited module plans the feedrate profile of the curve according to the segments’ length and the given jerk limit. Simulations are performed to verify real-time performance of the look-ahead algorithm. Experiments using a PC-based motion controller and an X–Y table are conducted to demonstrate that high-accuracy can be achieved with the proposed dynamics-based interpolator as compared to the adaptive-feedrate and the curvature-based feedrate interpolation algorithms.     

基于NURBS曲线自适应实时前瞻插补算法研究

为了提高数控机床的插补精度,在算法中引入轮廓误差和法向加速度作为约束条件,采用ADAMS微分方程迭代计算下一个插补点,通过预估—校正法校正迭代精度。采用前S型加减速控制进行速度规划,利用前瞻模块预测速度敏感点和计算减速点的位置,通过回溯法保证进给速度变化量符合机床加速度的要求。最后利用MATLAB对算法编写相应程序,仿真结果验证了该差补算法的有效性。     

基于Cox-de Boor递推的任意次NURBS曲线插补算法的研究与仿真

论文基于Cox-de Boor递推算法实现了任意次NURBS曲线的插补,利用差分插补方法来预估参数,结合机床实际加工过程中所必需满足的条件,将进给速度,机床最大加速度,最大弓高误差分别约束的参数进行比较,优化出最佳参数值,实现了速度自适应控制。论述了控制顶点、节点矢量、权因子对NURBS的影响,利用二分法线性搜索节点区间,给出了系统生成的NURBS曲线插补的NC代码形式,列出了整个插补算法的流程框图,在C++builder开发环境下完成了对任意次NURBS曲线的插补仿真,验证了算法的可行性。     

基于遗传算法的机器人轨迹插值方法的研究

为了降低工业机器人在运动过程中关节的最大加速度,提升运动的平稳性。通过对机器人运动学分析,求出起始和终止关节角度,分别采用三次多项式和高阶多项式插值算法进行运动插值。分析两种方法的特点,提出一种新型多项式插值方法为三段多项式“414”插值算法。根据运动过程加速度连续且最小原则,采用遗传算法给出最优的两个时间节点。通过MATLAB进行仿真实验证明了“414”多项式插值方法能够使关节加速度变化更加平稳,解决了三次多项式插值算法不能约束加速度和五次多项式插值算法加速度过大的问题,降低了传动机构的冲击。     

NURBS interpolator for constant material removal rate in open NC machine tools

Conventionally used linear or circular interpolators are undesirable for the precision machining of 3D free-form surfaces for the following reasons: the transmission errors due to the huge number of point data, discontinuity of curve segmentation, and unsmooth motion speed. In this regard, modern CNC machine tools are designed with a function for machining arbitrary parametric curves. However, these systems do not consider controlling feedrate adaptively, which dominates the quality of the machining process. This paper proposes a NURBS interpolator based on the adaptive feedrate control for the constant material removal rate. This is accomplished by varying feedrate using the curvature of a surface. The curvature-compensated feedrate system has important potential applications in ensuring part accuracy and protecting the cutting tool. The simulated and experimental results show it is applicable to real machining.     

Machineability in NURBS interpolator considering constant material removal rate

Increasing demands on precision machining of three-dimensional free-form surfaces have necessitated that the tool move smoothly and at varying feedrates. To achieve this, parametric interpolators, such as the Non-Uniform Rational B-Spline (NURBS) interpolator, have been introduced in CNC machining systems. Such interpolators reduce the data burden in the Numerical Control (NC) code, increase data transfer rate into the NC controller, and finally give smooth motion while machining. In this research, a new concept to control cutting load in a NURBS interpolator based on the degree of curvature was tried. This protects the cutting tool and improves machineability. To prove the system, cutting force and surface topography were evaluated. From the experimental results, the interpolator is adequate for machining a free-form surface.     

Bézier polygons for the linearization of dual NURBS curve in five-axis sculptured surface machining

Motivated by the excellent performance of three-axis NURBS interpolation, this paper presents a numerically efficient and accuracy controllable five-axis sculptured surface machining method with dual NURBS curve. Unlike the traditional three-axis NURBS interpolation, a dual NURBS format of the five-axis toolpath is developed to accurately and smoothly describe the tool movement in the part coordinate system. Different from the subdivision methods using the Taylor series expansion or inverse function, a piece-wise Bézier curve method is implemented to fast subdivide the NURBS curve within the user-defined tolerance. A generic rotation tool center point management module is also designed to realize the coordinate transformation and adaptive nonlinear error control for major five-axis machine tools. The overall effectiveness of the proposed five-axis NURBS machining scheme is demonstrated by the five-axis machining of an impeller’s flow channel.     

Adaptive interpolation scheme for NURBS curves with the integration of machining dynamics

This paper develops a comprehensive interpolation scheme for non-uniform rational B-spline (NURBS) curves, which does not only simultaneously meet the requirements of both constant feedrate and chord accuracy, but also real-time integrates machining dynamics in the interpolation stage. Although the existing work in this regard has realized the importance to simultaneously consider chord error and machining dynamics, none has really incorporated these in one complete interpolation scheme. In this paper, machining dynamics is considered for three aspects: sharp corners or feedrate sensitive corners on the curves, components with high frequencies or frequencies matching machine natural ones and high jerks. A look-ahead module was developed for detecting and adaptively adjusting the feedrate at the sharp corners. By Fast Fourier Transform (FFT) analysis with a moving window in the interpolation stage identified were some special frequency components such as those containing high frequencies or with frequencies matching machine natural ones. Then, the notch filtering or time spacing method was used to eliminate these components. To more completely reduce feedrate and acceleration fluctuations, the jerk-limited algorithm was also developed. Finally, the interpolated feedrate was further smoothened with B-spline fitting method and the NURBS curves were re-interpolated with the smoothened feedrate. During the interpolation, the chord error was repeatedly checked and confined in the prescribed tolerance. Two NURBS curves were used as examples to test the feasibility of the developed interpolation scheme.     

基于NURBS曲面的汽轮机叶片激光熔覆再制造路径规划

针对汽轮机叶片的复杂造型,提出一种基于NURBS曲面拟合的几何模型重构方法,得到了一种获取稳定的激光功率密度的路径规划方法。通过计算点在曲面上的曲率,将局部NURBS曲面拟合为曲率球,对激光束扫描面积进行简化计算。根据熔覆加工精度要求,调整激光头姿态沿曲面的法矢方向,约束激光束扫描面积,得到熔覆加工插补点,保证了基材表面获取稳定的激光功率密度。根据插补点的法向矢量计算得到机器人对应的位姿,利用CIMOCO软件验证熔覆轨迹的正确性;通过试验得到较好的熔覆层,证明了方法的可靠性和可行性,为复杂曲面的激光熔覆再制造提供了一种新方法。