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。     

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 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.     

基于速度指令波动的NURBS曲线插补算法研究

由于NURBS曲线的弧长是曲线参数的非解析积分形式,因而曲线弧长的准确参数化是插补计算的关键。通常,基于泰勒展开或曲线弧长多项式拟合的近似求值方法被用于插补计算;然而,截断误差的引入最终会导致速度指令波动,影响数控加工精度。针对具有最小速度指令波动的NURBS曲线插补算法研究,首先提出了曲线参数-弧长的三阶拟合多项式方法;然后以该方法计算结果为初值,并提出了求取精确曲线参数的弦截速度校正方法;最后进行了仿真验证。仿真结果表明,所提出的方法计算量小,提高了曲线参数的计算精度,相比于现有的方法,速度指令波动有很大减小。     

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.     

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.     

Real-time NURBS interpolator: application to short linear segments

This study proposes the use of a real-time non-uniform rational B-spline (NURBS) interpolator with a look-ahead function to handle numerous short linear segments. The short linear segments conforming to the continuous short block (CSB) criterion can be fitted into NURBS curves in real time. A modified maximum feedrate equation based on the geometric characteristics of the fitting curves and the dynamics of the servo control system has been derived in this paper. Taking advantage of the multi-thread design and the look-ahead function, the real-time NURBS interpolator can process enough G01 block information and complete feedrate planning before interpolation. In addition, the S-shaped jerk-limited acceleration method is adopted for smoother feedrate profiles. Two part shapes, which possess more than 1,000 short linear segments, are tested on our PC-based real-time control system. Both simulation and experimental results verify the feasibility and precision of the proposed interpolation algorithm.     

NURBS曲线泰勒展开插补法的平稳性与改进研究

泰勒1阶或2阶级数展开是NURBS曲线迭代插补的常用方法,这种方法会导致弦长误差,从而导致速度波动与运动冲击。为此,分析了泰勒展开插补法的3种误差来源,提出了在1阶泰勒展开法基础上通过界定搜索邻域并采用二分插值搜索方法来精确求取插补点参数的改进方法。该方法充分利用了泰勒展开法的初始精度和NURBS曲线的局部线性特征。试验表明,平均位移相对误差降低至6.46×10-11,且每个插补周期的搜索不多于3次,从而有效抑制了速度波动,实现了高速平稳加工。     

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.     

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.     

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.     

A strain energy minimization method for generating continuous NURBS-based motion curves in free-form surface machining

In spline-based free-form surface machining, segmented toolpaths are inevitable, and the discontinuities between segments cause large feedrate errors in the generated motion commands. Methods of the geometric design to create smooth composite curves may introduce new error sources in the original curve, and thus they are not suitable for motion planning in high precision free-form surface machining. In this paper, a strain energy minimization model (SEMM) and a non-uniform rational B-spline (NURBS)-based motion planning algorithm is proposed for free-form surface machining. The proposed SEMM and motion planning algorithm are capable of creating composite NURBS-based motion curves with up to C2 integral continuity without oscillations. Results from simulation show that with the proposed energy minimization model and motion curve generation algorithm, the feedrate errors caused by discontinuities in toolpaths can be effectively removed.     

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.     

NURBS曲线S形加减速双向寻优插补算法研究

由于非均匀有理B样条(Non-uniform rational B-splines,NURBS)曲线的弧长与参数之间无精确解析关系,并且进给速度总是受到非线性变化的曲线曲率的约束,因此基于S形加减速进行NURBS曲线插补时,减速点难以准确预测。传统算法通常是沿曲线单方向插补,不仅未考虑曲率对进给速度的持续限制,而且加减速分类与计算公式复杂。为此,提出运动路程未知情况下不依赖于弧长精确计算的正向和反向同步加速的插补新算法,实时动态地求解曲线段内最大进给速度和正反向插补会合点,从而实现处处满足全部速度约束条件的最优插补。该算法无需求解高次方程与繁琐的加减速模式分类,并可保证以确定的速度通过曲率极值点和曲线终点。通过两个插补实例证明算法简明高效,适应性好,能够满足高速高精度数控要求。     

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.     

基于进给速度敏感点识别的NURBS曲线平滑插补算法研究

提出了一种基于进给速度敏感点识别的NURBS曲线插补算法,该方法对于兼容NURBS形式的高档数控系统至关重要。粗插补计算造成的轮廓误差与插补经过该点时的进给速度大小有关,敏感点则可根据插补微段逼近时的弓高误差来界定。进而,根据相邻敏感点之间的距离,通过增设安全缓冲区等方法,进行速度曲线自适应规划。整体进给速度曲线可以由各部分进给速度曲线连接而成。为评价算法的有效性,采用3次NURBS曲线在三种不同进给速度指令下进行仿真计算。仿真结果证明,该算法很好地将轮廓精度和进给速度的平滑性进行了系统考虑,能在相邻危险点复杂分布的情况下执行柔性的插补控制。     

实时快速NURBS直接插补技术

为了解决基于泰勒展开式的NURBS插补算法存在的速度波动问题,提高NURBS插补实时性,深入研究了NURBS曲线直接插补方法。根据插补原理,提出了一种不同于泰勒展开式的插补计算方法,并研究了一种NURBS快速计算方法。在满足插补过程精度要求的前提下,由进给速度直接计算插补点坐标,并采用递推矩阵对NURBS进行快速求值求导计算,有效地减小了速度波动,而且提高了计算速度和插补实时性。仿真结果证明了该方法的可行性和有效性。     

An improved feedrate scheduling method for NURBS interpolation in five-axis machining

Five-axis machining plays an important role in manufacturing by dint of its high efficiency and accuracy. While two rotation axes benefit the flexibility of machining, it also brings limitations and challenges. In order to further balance machining precision and efficiency, an improved feedrate scheduling method is presented considering geometric error and kinematic constraints for the Non Uniform Rational B-Spline (NURBS) interpolation in five-axis machining. A simplification method is proposed to calculate the geometric error which describes the deviation between the ideal tool path and the real tool path induced by the non-linear movement. A linear relation between geometric error and feedrate is built to limit the feedrate. The constraints determined by single axis kinematic performance and tangential kinematic performance are also considered. Under these constraints, a constrained feedrate profile is determined. Aiming to get more constant feedrate in the difficult-to-machine areas, this work proposes a scheduling method which combines morphological filtering and S-shape acceleration/deceleration (acc/dec) mode. Simulations and experiments are performed to compare the proposed feedrate scheduling method with two previous feedrate scheduling method and the results prove that the proposed feedrate scheduling method is reliable and effective.     

临界曲率值分割曲线尖角的NURBS曲线插补

为兼顾插补含尖角NURBS曲线的精度与速度,提出尖角分割且速度修正插补算法。由插补弦高误差限、法向加速度及其导数约束,得满足插补精度及机床动力学性能的临界曲率;用大于临界曲率的局部极大曲率及临界曲率分割NURBS曲线为是否包含尖角的若干子段;用S曲线加减速算法规划各子段进给速度,并用段间速度及位移协调关系修正各段加速度及其导数,使各段加减速时间为整数倍插补周期。在相同约束条件下,分别用曲率单调无速度修正、尖角分割无速度修正及尖角分割有速度修正算法,规划一条含大曲率尖角NURBS曲线插补速度,并用一阶泰勒级数展开算法插补该曲线。对比结果表明尖角分割且有速度修正算法可稳定得到较高插补精度,因此该算法可用于含大曲率尖角NURBS曲线高速度高精度加工。