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.     

NURBS-based fast geometric error compensation for CNC machine tools

In this paper, a novel method for the compensation of geometric errors of CNC machine tools is presented. The key idea is to use the basis functions of the setting NURBS path to approximate its error compensation function and to generate a new compensated NURBS path. In this way, both the setting and the compensated NURBS path have the same NURBS form. More importantly, the control points of the error compensation function can be obtained by simply calculating the positioning deviations of the control points of the setting NURBS path using the error model. A high compensation accuracy can be achieved through the systematic insertion of new knots, which creates new control points and raises the flexibility of NURBS in representing the error compensation function. The real-time interpolation of the compensated NURBS path completes the error compensation automatically. Simulations and experiments have shown that the new method delivers the same positioning accuracy as a model-based real-time geometric error compensation method does, but without additional real-time CPU loading. The proposed method can also be implemented in the post-processor of a CAM system for off-line compensation.     

NURBS曲线插补算法及加减速控制研究

针对复杂零件高速高精密加工的需求,提出了一种基于阿当姆斯微分方程的NURBS曲线实时插补算法。通过对算法的合理简化与近似,保证了算法的实时性。此算法基于轮廓误差和法向进给加速度控制,使进给速度能随曲线曲率自适应调整。与之相适应,配合此插补算法,利用NURBS曲线的对称性预测减速点,提出了一种新的插补前抛物线-直线-抛物线S形加减速控制方法。该方法具有位置精度高、速度无突变、过渡平滑、计算简便等优点。通过采用MATLAB对插补轨迹仿真和实例分析,证明了插补算法和加减速控制方法的正确、合理、有效性。     

Development and implementation of a NURBS interpolator with smooth feedrate scheduling for CNC machine tools

Parametric interpolation for Non-Uniform Rational B-Spline (NURBS) curve has become more important than ever before in the control of CNC machine tools. An effective NURBS interpolator not only can obtain accurate contour trajectories, but also have smooth dynamics performance. This paper proposes a numerically efficient NURBS interpolation scheme which consists of two stages namely preprocessing and interpolating. In the stage of pre-processing, the parameter interval is split into several blocks at breakpoints and an iterative numerical quadrature method is applied for each block. By means of the iterative quadrature method, the initial parameter intervals of each block are divided into several subintervals according to the arc length approximation error. Meanwhile, the curvature of each knot and the cubic polynomial coefficients of each subinterval are obtained. Then the critical points with large curvature of each block are found from the candidate points and the tolerated speed of each critical point is calculated according to the constraints of chord error and centripetal acceleration. Hence, the feedrate scheduling based on the S-shaped acceleration profile for each block can be preplanned via the approximate arc length of each subinterval, the tolerated speed of each critical point and kinematics characteristics such as acceleration/deceleration and jerk limits of the machine tools. In the stage of interpolating, the parameter of the next interpolation point can be calculated directly using the cumulative arc length and the cubic polynomial coefficients of each subinterval. Finally, a series of numerical simulations and real machining experiments are conducted, and the simulation and experimental results have showed the good performance of the proposed NURBS interpolator both in efficiency and accuracy.     

A novel adaptive-feedrate interpolation method for NURBS tool path with drive constraints

Feedrate scheduling is crucial for CNC systems to generate a smooth movement which is able to satisfy increasing requirements on machining quality and efficiency. In this paper, a novel adaptive feedrate interpolation method is proposed for NURBS tool path with drive constraints. The tool path is first expressed in NURBS form, and then the satisfaction conditions of drive constraints are derived according to the kinematic and geometric characteristics of the NURBS tool path. On this base, a proportional adjustment algorithm, which can quantitatively reduce the accelerations and jerks of drive axes at the sensitive regions of feed profile, is proposed to achieve the new positions of violated sampling points. After each adjustment, a curve evolution strategy is used to ensure the feed profile is locally or globally deformed to the target positions with a good smoothness of path curve and the avoidance of re-interpolation. Through the iterative adjustment, a smooth feed profile with limited velocities, accelerations and jerks of drive axes is thus yielded along the entire tool path. Finally, performances of the proposed method are validated by performing both simulations and experiments on two freeform NURBS curves. The results show the effectiveness and reliability of the proposed feedrate interpolation method.     

高档数控机床NURBS曲线Newton Rapson迭代的插补算法

为了解决NURBS曲线在Taylor一阶、二阶展开式求解曲线插补坐标点ui+1,存在计算量大、插补时间长等问题,提出一种Newton-Rapson迭代法的NURBS插补算法。该插补算法可以减少插补计算量和插补时间,提高插补精度,并提高曲线加工的效率。仿真实验结果表明,Newton-Rapson迭代法的NURBS插补算法计算过程简单、切实有效,具有可行性和实用性,满足数控系统实时、高速、高效率插补的要求,在其他数控插补的过程中具有很强的借鉴意义。     

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。     

五自由度机器人高效均匀曲面喷涂方法

针对被喷涂对象是平缓光滑的曲面,给出了一种高效均匀的喷涂方法。使用五自由度机器人,采用特定算法在喷涂过程中保持喷头垂直于喷涂面,且喷涂距离恒定。合理设计喷涂路径,进行等距喷涂,从而提高喷涂的效率和均匀度。以NURBS作为曲面的描述方法,采用"之"字形或者"回"字形路径规划,计算喷头运动插补点,采用泰勒展开式保证步长恒定和喷涂间距相等。在喷涂机器人运动模型基础上,通过对喷涂机器人的正逆运动特性进行分析,建立各关节求解方法。仿真结果表明,这种等速等间距插补方法,能够保持喷头以恒定的速度快速行进以及喷涂间距的稳定。     

Augmented Taylor's expansion method for B-spline curve interpolation for CNC machine tools

In computer numerical control (CNC) systems, parametric curves can be used instead of a large amount of linear blocks to describe tool paths for freeform surface or curve machining. However, existing parametric curve interpolation methods may cause large feed fluctuations or even a failure of the machining process near the sharp corners of a parametric curve. Therefore, a parametric curve interpolation method with an error correction and failure prevention scheme is required. In this paper, the augmented Taylor's expansion (ATE) method for computing B-spline curve parameters is proposed. A group of calibrators consisting of the knots and the arc lengths between adjacent knots are pre-computed before the interpolation starts. The parameter is computed based on Heun's method in a prediction–correction manner, and the accumulated errors caused by the cut-off errors of Taylor's expansion are eliminated by the calibrators at the knots. To cope with the extreme cases that usually occur near the sharp corners of a curve, a linear parametric interpolation between the previous parameter and its next calibrator is carried out when Heun's method fails to obtain a parameter in the domain. Simulation and experimental results show that, when the arc length increments are kept small enough near the sharp corners, the ATE method attains high accuracy and robust computation. The proposed method is also applicable to the NURBS curves.     

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.     

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.     

Real-time generation and control of cutter path for 5-axis CNC machining

This paper presents a new approach to real-time generation and control of the cutter path for 5-axis machining applications. The cutter path generation method comprises real-time algorithms for cutter-contact path interpolation, cutter offsetting, and coordinate conversion. In addition, a global feedback loop is closed by the CNC interpolator so as to augment the controlled accuracy in practical cutter path generation. An error compensation algorithm and a feedrate adaptation algorithm for the control loop are developed, respectively.     

Trajectory generation in high-speed,high-precision micromilling using subdivision curves

Motion control in high-speed micromilling processes requires fast, accurate following of a specified curvilinear path. The accuracy with which the path can be followed is determined by the speed at which individual trajectories can be generated and sent to the control system. The time required to generate the trajectory is dependent on the representations used for the curvilinear trajectory path. In this study, we introduce the use of subdivision curves as a method for generating high-speed micromilling trajectories. Subdivision curves are discretized curves which are specified as a series of recursive refinements of a coarse mesh. By applying these recursive properties, machining trajectories can be computed very efficiently. Using a set of representative test curves, we show that with subdivision curves, trajectories can be generated significantly faster than with NURBS curves, which is the most common method currently used in generating high-speed machining trajectories. Trajectories are computed efficiently with subdivision curves as they are natively discretized, and do not require additional evaluation steps, unlike in the case of NURBS curves. The reduced trajectory generation time allows for improved performance in high-speed, high-precision micromilling. We discuss the use of several metrics to quantify the quality of the subdivision interpolation, and apply them in calculating the error during trajectory generation for the test curves.     

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。     

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。     

Fast Bezier interpolator with real-time lookahead function for high-accuracy machining

This study presents a real-time fast Bezier interpolation method that solves linearly segmented contour problems that occur during milling using conventional CNC machines. Depending on the length and the CSB criterion, these linearized segments can be regarded as noise, continuous short blocks (CSBs), or G01 blocks. The CSB criterion proposed in this paper is employed to identify CSBs during the NC code interpreting stage. The CSBs are fitted into cubic Bezier curves and interpolated to produce smoother contours in real-time machining. Two different NC programs possessing a large number of short blocks were tested on our PC-based real-time control system. Simulations and experimental results demonstrate that the proposed fast Bezier interpolator (FBI) with real-time lookahead function provides satisfactory performance.     

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.     

A real-time NURBS surface interpolator for precision three-axis CNC machining

Due to the fact that the cutting occurs around the cutter contact (CC) point, the efficiency and quality of CNC machining can be improved significantly if the CC velocity along the surface is kept costant. Conventional approaches to machining mainly maintain a constant cutter location (CL) velocity, so that the CC velocity along the surface is often not constant and usually results in non-uniform machining and unsatisfactory quality. To overcome this difficulty, this paper presents a novel NURBS surface interpolator that is capable of real-time generation of CL motion command for ball-end milling of NURBS surfaces and maintaining a constant CC velocity along the CC path and its intervals. For performance evaluation, a three-axis servomechanism driven by three servomotors is controlled to track segments represented by NURBS surfaces. Experimental results verify the effectiveness of the proposed method.     

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。     

涂胶机器人离线编程系统设计与应用

为提高涂胶的质量与效率,借助SolidWorks二次开发接口,设计一种涂胶机器人离线编程系统。针对具有非均匀有理B样条曲线(NURBS)的工件轮廓的涂胶,研究一种NURBS插补算法,与粗精路径点提取方法相结合,能够实现涂胶过程规避障碍物;利用C#.NET结合OpenGL开放图形库实现了仿真环境;以ZZRT-608六自由度机器人作为机械系统、ZMC406作为控制系统搭建了实验平台。以典型的鞋样涂胶工件为测试对象进行实验验证,与KUKA机器人手动示教编程作对比,结果表明:该离线编程系统在效率上提高了30%～50%,涂胶轨迹更加均匀,能够更好地满足高精度生产需求。