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.     

Parameter u and arc S modeling of B-spline for NC machining with CL points preprocessing algorithm

This paper proposes a novel B-spline interpolation based on CL points preprocessing algorithm and on u-S modeling, aiming to overcome the shortages of linear and circular interpolations. A B-spline toolpath trajectory is generated by fitting a given set of CL points. CL points are preprocessed by chord distance to reduce computation load, thus using fewer feature CL points to generate a B-spline toolpath trajectory. As for disadvantages due to parameter updating for B-spline interpolation, a fifth order polynomial based on u-S modeling is constructed to map the desired curve length S to the corresponding spline parameter u, cutting the real-time computation load. Simulation results are provided to verify the feasibility and effectiveness of the proposed interpolation algorithm.     

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.     

一种三次插值样条曲线的插补方法研究

针对复杂轮廓零件数控加工的实际需求,基于数据采样插补原理,对一种三次插值样条曲线的插补算法进行了研究.样条曲线进行参数化运算插补,先使用一系列首尾相接的微小直线段来逼近给定的插值样条曲线轮廓,再利用轨迹空间与参变量空间的对应关系,控制参变量,求取插补点坐标,得到整个离散化插补轨迹.此算法加工速度稳定性高,有利于高速高精地生成插补轨迹.     

Efficient algorithm for time-optimal feedrate planning and smoothing with confined chord error and acceleration

In this paper, an efficient algorithm is proposed to generate a smooth near-time-optimal feedrate function along a parametric tool path for three-axis CNC machining under a feedrate bound, an acceleration bound for each axis, and a chord error bound. The algorithm first gives a discrete and computationally efficient algorithm to find a sequence of globally optimal velocity points under the feedrate, acceleration, and chord error bounds. A linear programming strategy is proposed to smooth the velocity points sequence. Finally, the velocity points sequence is fitted into a cubic spline to obtain a near-time-optimal and smooth feedrate function. Simulation and experiment results for Non-uniform rational basis spline curves are presented to illustrate the feasibility of the algorithm.     

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。     

参数曲线的恒速通用插补原理与实例分析

阐述了参数曲线恒速插补的通用原理,同时讨论了廓误差近似计算方法。以摆线为例推导了相庆的插补公式和轮廓误差近似计算公式。最后通过VISUAL LISP编程进行插补模拟,表明该插补原理具有无累计误差、恒速进给以及轮廓误差易于控制的特点,而且该算法插补功能的开发和应用前景十分广阔。     

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。     

2GSPS数字存储示波器插值算法

数字存储示波器在快时基档位工作时,由于实时采样率不够,仅能采集信号的少数样点,为了重构信号,必须进行插值。通过编程和仿真比较了三次样条插值、分段线性插值、三次插值、最邻近插值对几种不同波形的恢复结果,并计算和分析了相关的均方误差。考虑实际待测的信号的不可知性和复杂性,并权衡精度与速度,当待插信号的点数小于100点时,选用三次样条插值;当待插信号的点数不小于100点时,采用三次插值法。Matlab仿真表明,设计结果的各项性能指标均达到指定要求,设计过程简便易行。     

CNC参数曲线实时插补进给速度控制方法

通过分析数控加工中参数曲线的实时插补原理和进给速度控制原理,指出了Taylor展开算法和“参数－弧长”分段拟合算法用于参数曲线的实时插补具有理论上的局限性,并对CNC实时插补进给速度控制问题给出了本质描述,提出了基于牛顿迭代法的解决方案,给出了算法流程。仿真对比实验表明,该算法具有稳定性好、收敛快、运算量小、精度高等特点,满足实时运算要求,能够将进给速度波动率控制在理想水平。     

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。     

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.     

Efficient high-speed cornering motions based on continuously-variable feedrates. I. Real-time interpolator algorithms

The problem of high-speed traversal of sharp toolpath corners, within a prescribed geometrical tolerance ??, is addressed. Each sharp corner is replaced by a quintic Pythagorean–hodograph (PH) curve that meets the incoming/outgoing path segments with G² continuity, and deviates from the exact corner by no more than the prescribed tolerance ??. The deviation and extremum curvature admit closed-form expressions in terms of the corner angle ?? and side-length L, allowing precise control over these quantities. The PH curves also permit a smooth modulation of feedrate around the corner by analytic reduction of the interpolation integral. To demonstrate this, real-time interpolator algorithms are developed for three model feedrate functions. Specifying the feedrate as a quintic polynomial in the curve parameter accommodates precise acceleration continuity, but has no obvious geometrical interpretation. An inverse linear dependence on curvature offers a purely geometrical specification, but incurs slight initial and final tangential acceleration discontinuities. As an alternative, a hybrid form that incorporates the main advantages of these two approaches is proposed. In each case, the ratio \(f=V_{\min }/V_{0}\) of the minimum and nominal feedrates is a free parameter, and the improved cornering time is analyzed. This paper develops the basic cornering algorithms—their implementation and performance analysis are described in detail in a companion paper.     

隐曲线的线性和旋转插补

基于数控五轴刀具运动分析,提出了旋转插补的概念,给出了旋转插补的相关定义,在此基础上提出了隐曲线的线性和旋转插补原理和方法,并进一步提出了改进的插补方法。     

基于冗余误差控制的非均匀有理B样条曲线插补算法研究

提出了一种能有效控制冗余误差的非均匀有理B样条曲线插补算法.该算法综合了等弓高误差插补算法和恒定进给速度插补算法的优点,小曲率情形时在保证加工精度的前提下,通过引入进给倍率因子,增大进给速度以改善误差过度冗余;同时在大曲率情形下,可控制弓高误差在限定的误差范围以保证轮廓精度.这样既可保证轮廓精度,又可提高加工效率.仿真实例证实了该插补算法的有效性和可行性.     

Approximate tool posture collision-free area generation for five-axis CNC finishing process using admissible area interpolation

This paper proposes an efficient algorithm to generate tool posture collision-free area for the whole free-form surface during five-axis CNC finishing period. The algorithm is consisted of two phases: sampling and interpolation. In the first phase, a few points are picked on the surface and the admissible area of tool posture is calculated at each point. The admissible area is a two-dimensional figure on the plane. Base on the assumption that the admissible area of adjacent surface points should transform continuously, so in the second phase, the admissible area of the sampling points are interpolated with cubic B-surface interpolation technique, forming an expression as ?(u, v), through which when the parameters u and v are assigned, the global collision-free area for the corresponding surface point can be easily calculated. The proposed algorithm is efficient because only a few surface points are needed to get the expression. The proposed algorithm is programmed and tested. Results show that the algorithm is relatively precise and efficient while considering the whole surface.     

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.