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.     

Constant cusp toolpath generation in configuration space based on offset curves

Constant cusp is a common strategy for generating tool paths in many NC machining applications. Cusps need to be regulated to ensure high precision without wasting machining efforts. Constant cusp strategies frequently operate on NURBS surfaces or triangular meshes and, thus, have to deal with the issues of patch-boundary oscillations or long, stretched triangles. To avoid these issues, one can operate in a pre computed configuration space (c-space). The c-space is given in form of a regular quadrilateral heightfield mesh, which may be adaptively subdivided, where the slope is large. This simple data structure is memory efficient and is widely used in CAD/CAM frameworks. In this paper we introduce an algorithm for creating a constant cusp tool path with the help of a given c-space. The constant cusp algorithm iteratively produces curves in the c-space by fitting a tube around the current curve and intersecting the tube with the c-space mesh to detect the subsequent curve. As tool paths are handed to the machine controller in form of point sequences, it suffices to operate on piecewise linear curves. The tube becomes a concatenation of cylinders, which we derive using geometric considerations. In each iteration of the constant cusp algorithm, intersection points of the cylinders with the not yet traversed part of the mesh are detected and checked for their validity. The validity check can efficiently remove global or local self-intersections of the new curve by just deleting the respective points. In a final step, the detected intersection points are connected to form constant cusp tool paths. Dealing with piecewise linear curves, we achieve low computation times for real-world data sets.     

基于NURBS曲线的数控机床几何误差实时补偿研究

阐述了一种基于NURBS曲线的数控机床几何误差补偿技术,利用误差模型计算预设的NURBS路径与控制点之间的位置偏移能得到NRUBS误差补偿函数控制点,通过插入新的节点产生新的函数控制点,并提高NURBS曲线表达误差补偿函数的柔度,从而得到更高的补偿精度。结果表明,在公差精度为0．00057mm的情况下,沿3种不同工作路径的误差补偿精度都小于1mm。     

基于矢量分析的数控加工轨迹设计方法研究

描述了基于矢量分析和NURBS的数控加工轨迹设计方法。基于给定的被加工曲面在其参数域上的优化走刀方向集合,并依据标量场与梯度场的转化关系,建立了精确逼近离散方向矢量的走刀矢量场拟合模型,由此给出了数控加工轨迹的矢量表达形式。以NURBS作为发生矢量场的流函数并借助其较强的局部调控能力,通过调整控制点列,可望实现数控加工轨迹拓扑形状的整体优化调控。验证实例表明该方法能够进行复杂形状数控加工轨迹的精细设计,有利于保证精度指标下曲面加工效率的最大化。     

Tool-path generation for pockets with freeform curves using Bezier convex hulls

The objective of this paper is to generate tool paths for pockets with freeform curves, i.e. Bezier curves or B-spline curves. Time efficiency and overcut avoidance are two of the most important factors in NC machining. A method, based on the convex hull property of Bezier curves, is thus developed so that the portion of the boundary defined by a Bezier curve is replaced by parts of its convex hull on the cavity side. Therefore, the new pocket boundary, redefined by only segments of straight lines, can be solved efficiently based on current algorithms without overcuts. Cutting tool paths can be generated for pockets with islands avoiding computation of higher degree curve/curve intersections. Furthermore, recursive subdivisions on Bezier curves are used to improve the accuracy of the cut with an allowance criterion based on sizes of convex hulls.The portion of the boundary defined by a B-Spline curve is transformed into piecewise Bezier curves. The tool-path generation for pockets with B-spline curves can thus be solved by reducing the problem to one of pockets with Bezier curves.     

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.     

非均匀有理B样条曲线多分辨率特征几何建模研究

NURBS曲线的几何建模,传统方法是通过模型的控制顶点对曲线进行操作、编辑和处理,繁琐费时,使NURBS曲线的设计变得十分困难。基于一种新的半正交非均匀B样条小波多分辨率技术将单一尺度几何空间变换到多尺度空域和频域空间,NURBS曲线的整体结构和局部细节被转化为曲线的多尺度整体特征和细节特征。通过对这种多尺度特征的编辑、操作和处理有效实现NURBS曲线的几何建模。     

A new approach to generating arc length parameterized NURBS tool paths for efficient three-axis machining of smooth, accurate sculptured surfaces

With the development of a new function of computer numerical control controllers, nonuniform rational B-spline (NURBS) interpolation, NURBS tool path generation for sculptured surface machining is under extensive research. The common procedures of the current NURBS tool path planning methods are as follows: first, to find a group of cutter contact points on a sculptured surface; second, to calculate their corresponding cutter locations (CLs); then, to fit a NURBS tool path to the CLs within a prescribed tolerance; and finally, to inspect the tool path for possible gouge by the tool and delete the invalid path segments, if any. However, the NURBS tool path has the following problems: (a) although it passes through the discrete CLs of the theoretical CL path, the deviation along the two paths could be larger than the tolerance; (b) its parameter is not the arc length of the path; and (c) it is difficult to detect gouge along the NURBS path and to remove the invalid segments from it. Consequently, NURBS tool paths generated with the current methods of commercial computer-aided design/computer-aided manufacturing (CAD/CAM) software cannot be used to make smooth and accurate surfaces. To address these problems, this work proposes a new approach to generating arc length parameterized NURBS tool paths with high accuracy in terms of the theoretical CL paths and without gouge and interference. Two practical examples in this work clearly demonstrate the feasibility of this approach and the advantages of the generated NURBS tool paths. Therefore, this approach can be implemented into the CAD/CAM software to promote NURBS machining in industry.     

Generating Tool Paths for Free-Form Pocket Machining Using z-Buffer-Based Voronoi Diagrams

Voronoi diagrams for closed shapes have many practical applications, ranging from numerical control machining to mesh generation. Curve offsetting based on Voronoi diagrams avoids the topological problems encountered in traditional offsetting algorithms. In this paper, we present a new procedure for generating tool paths using z-buffer-based Voronoi diagrams specially to deal with free-form shaped pockets. Using a z-buffer, a proposed algorithm effectively extracts the topological information on the Voronoi diagram, and generates the geometric information on the Voronoi edges approximately. These Voronoi edges are refined using a numerical algorithm. This method is independent of curve type and is applicable to any pockets with parametric curve boundaries that are twice differentiable.     

Fast and accurate NURBS fitting for reverse engineering

Fast and accurate fitting of non-uniform rational B-spline (NURBS) curves and surfaces through large sets of measured data is an important problem in applications such as reverse engineering and geometric modelling. This paper presents a method for realising significant improvements in the computational efficiency of this task. The basic idea is that the sparsity structures of the relevant matrices that are specific to the problem of NURBS fitting can be precisely defined and that full exploitation of these structures leads to significant savings in both computational and storage requirements. These savings allow for a large number of control points to be used in order to define the surface and consequently to improve the accuracy of shape representation. The achieved computational complexity is linear in both the number of measured points and the number of control points while the storage requirements of the algorithm are linear with the number of control points only. The complexity analysis, as well as the analysis of actual running times is presented. The results demonstrate that, using this approach, highly complex shapes may be modelled accurately with a single NURBS surface.     

自由曲线轮廓加工的圆弧样条刀具路径研究

给出了一种平面曲线轮廓的圆弧样条拟合及刀具路径生成算法，该算法面向零件轮廓的光顺性刀具路径生成，通过应用曲线的曲率关系，对以NURBS表示的被拟合自由曲线按参数递增的方向用G^1连续的圆弧或直线段逐段拟舍，并生成相应的圆弧样条刀具路径，从而实现零件曲线轮廓表面的光顺加工。实例计算和分析结果证明了该算法具有计算稳定性好、计算效率高的优点，易于在数控加工中实现。     

Cutting temperature,tool wear,and tool life in heat-pipe-assisted end-milling operations

In this paper, using the specifications of nodal points on a nonuniform rational B-spline (NURBS) curve of three degrees with respect to NURBS curve parameter and defining the coefficients for the velocity and acceleration vectors on these points, a new method is presented to design a tool path via C² PH spline curves. Values of the velocity/acceleration vector coefficients corresponding to the nodal points on the original NURBS curve are computed by pattern search algorithm. To this end, the normal distance between the constructed C² PH spline curve and its corresponding original NURBS curve is considered as the objective function. Using combination of the time-dependent feed rate interpolation in the acceleration/deceleration phase of the motion and the constant feed rate interpolation in the middle region of the motion, the position commands of the designed NURBS-based C² PH spline curve are generated. Several improved NURBS-based C² PH spline curve following tasks were implemented with pseudo-derivative feedback feed forward (PDFF) controller. The experimental and simulation results confirm that the devised interpolator with designed PDFF controller is not only feasible for machining the complicated tool path represented in the improved NURBS-based C² PH spline form but also yields satisfactory contouring performance under variable feed rate.     

NURBS曲线曲面重构的方法

CAD/CAM中用有理多项式函数表示曲面越来越广泛。由于非均匀有理B样条(NURBS)可以精确表示解析形状和自由曲线曲面,国际标准组织(ISO)与1991年把NURBS作为表示工业产品几何形状的工业标准。这里主要讨论了NURBS曲线曲面重构的方法。     

采用Voronoi图生成任意形状凹槽的刀具轨迹

为了自动生成任意形状凹槽的刀具轨迹 ,采用Voronoi图为基础的曲线置偏方法 ,避免传统的置偏算法所遇到的拓扑问题 ,提供了一种以Voronoi图为基础的简便有效的算法 ,可用来处理带“岛屿”的任意形状凹槽的刀具轨迹生成。     

Contour-parallel offset machining for trimmed surfaces based on conformal mapping with free boundary

The parametric surfaces of some manufactured parts are often subjected to the Boolean operation of other objects; generating suitable NC tool paths from such trimmed surface remains a challenge. This paper presents a new planar development-based method to generate contour-parallel offset paths of trimmed surfaces. To avoid direct frequent identifications and removals of interferences of offset curves on 3D-trimmed surface possibly with multiple holes or restricted regions, the original surface is flattened in the plane domain using a two-stage approach which consists of conformal mapping with free boundary and further nonlinear accuracy improvement. Then, sequent offsets of the boundary curves of the planar region are generated, and the global interferences are detected and removed using an efficient and robust divide-and-conquer strategy. Based on a tree data structure, a tool-path linking algorithm is also given with less or no tool retractions, and subsequently, the resulting planar paths are inversely mapped to the physical space. Illustrated examples have been conducted to testify the affectivity and the applicability of the proposed contour-parallel offset machining method.     

Mesh-based tool path generation for constant scallop-height machining

This paper presents a new approach to mesh-based tool path generation for obtaining constant scallop heights. The mesh surface has recently become the focus of considerable interest, because its geometric computation is simpler and more robust than that of the parametric surface. These advantages make it easy to check and remove interference in the process of tool path generation. The previous tool path generation method based on the mesh surface, however, can generate only one topology of iso-planar type where tool paths have evenly spaced tool path intervals. As constant scallop heights cannot be obtained from evenly spaced tool path intervals, unevenly spaced tool paths based on offset meshes are necessary for reducing the machining time and for easy interference removal. This paper proposes and compares four methods to estimate curvatures from the mesh surface; the curvature is essential for calculating unevenly spaced tool path intervals. This paper also proposes an improved drive surface method to propagate CL-paths unevenly and to generate tool paths with various topologies.     

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.     

等几何分析中Dirichlet边界条件的配点施加方法

相对于传统有限元,等几何分析使用NURBS基函数统一表示几何和分析模型,它能消除传统有限元的网格离散误差,容易构造高阶协调单元。但是由于NURBS基函数缺乏插值性,控制顶点也不一定位于几何边界上,使得难以直接施加Dirichlet边界条件。针对这一问题,提出一种基于样条拟合的Dirichlet边界条件施加方法,通过引入一组边界配点来拟合边界条件。注意到不合适的配点策略会使得边界插值方程组奇异或者条件数过大,详细讨论配点选取的基本准则,提出两种鲁棒的配点方案:Greville横标和Chebyshev插值点法。并且将配点方法扩展到最小二乘形式,设计一种快速的场变量消去算法。通过实例验证上述方法的可行性和有效性。试验结果表明,各种配点策略的收敛率基本保持在一个量级,因此配点法的关键是选择稳定的配点方案。     

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.     

A novel approach for computing C2-continuous offset of NURBS curves

Computing offset curves is an important geometric operation in areas of CAD/CAM, robotics, cam design and many industrial applications. In this paper, an algorithm for computing offsets of NURBS curves using C²-continuous B-spline curves is presented. The progenitor curve in database is initially approximated by a line-fitting curve, and then the exact offset of this line-fitting curve is introduced as an initial offset. Based on the initial offset and a set of selected knots, an intended C²-continuous B-spline curve is subsequently constructed. The method uses a new error-measuring scheme, which is based on the convex hull property of Bézier curves and the idea of cumulative errors, to calculate the global error bound of offset approximation. The method obtains offset curves with C² continuity and guarantees that the actual error bound is precisely within the prescribed tolerance. In addition, it also allows one to selectively parametrize the offset curve.