Complete swept volume generation, Part I: Swept volume of a piecewise C-continuous cutter at five-axis milling via Gauss map

In this paper, we present a methodology to generate swept volume of prevailing cutting tools undergoing multi-axis motion and it is proved to be robust and amenable for practical purposes with the help of a series of tests. The exact and complete SV, which is closed from the tool bottom to the top of the shaft, is generated by stitching up envelope profiles calculated by Gauss map.The novel approach finds the swept volume boundary for five-axis milling by extending the basic idea behind Gauss map. It takes piecewise C¹-continuous tool shape into account. At first, the tool shape is transformed from Euclidean space into Tool map (T-Map) on the unit sphere and the velocity vector of a cutter is transformed into Contact map (C-Map) using Gauss map. Then, closed intersection curve is found between T-Map and C-Map on the Gaussian sphere. At last, the inverse Gauss map is exploited to get envelope profile in Euclidean space from the closed curve in the range. To demonstrate its validity, a cutting simulation kernel for five-axis machining has been implemented and applied to mold and die machining.     

An accurate and efficient approach to geometric modeling of undeformed chips in five-axis CNC milling

Many important and complex parts, such as aero-engine compressors and automotive punch dies, are often machined in five-axis computer numerically controlled (CNC) milling. To machine the parts with accurate dimensions and shapes and low machining costs it is necessary to construct 3D models of the finished parts in the geometric simulation and in-process workpiece models of the parts in the physical simulation of their five-axis milling. A kernel technique of the geometric and the physical simulations is to accurately and efficiently model the geometry of the workpiece material removed at every moment of the machining, which is the instantaneous, undeformed chip geometry. Although in the past decades much research has been conducted on modeling cutter swept volumes in CNC milling to represent the finished part geometry in the geometric simulation, it is very time consuming to calculate the instantaneous, undeformed chip geometries using the cutter swept volumes. Besides, the existing method of modeling undeformed chip geometry in three-axis milling cannot be used for that in five-axis milling. To address this problem, our work proposes an accurate and efficient approach. In this article, a generic theory about the boundary of the area covered by the instantaneous cutting edges on a workpiece layer at any moment is established, which is called the boundary theory. A simple diagram of determining the boundary is invented, which is called boundary construction diagram. This approach lays a theoretical foundation for the geometric and the physical simulations of five-axis milling and will significantly promote them for high performance machining in industry.     

Accurate tool position for five-axis ruled surface machining by swept envelope approach

This paper presents a swept envelope approach to determining tool position for five-axis ruled surface machining. The initial tool position is traditionally located to contact with two directrices of a ruled surface. The swept profile of the tool is then determined based on the tool motion. By comparing the swept profile with the ruled surface, the tool position is corrected to avoid machining errors. The cutter's swept envelope is further constructed by integrating the intermediate swept profiles, and applied to NC simulation and verification. This paper presents the explicit solution for the swept profile of a taper-end cutter in five-axis ruled surface machining. The relation of the ruled surface geometry, the tool motion and the machining errors is developed. Therefore, the error sources can be detected early and prevented during tool path planning. The explicit swept envelope indicates that the machined surface is not a ruled surface in five-axis ruled surface machining. Manufacturing industries should take extra care in high precision ruled surface machining. Computer illustrations and example demonstrations are shown in this paper. The results reveal that the developed method can accurately position tool location and reduce machining errors for five-axis ruled surface machining.     

An innovative approach to NC programming for accurate five-axis flank milling of spiral bevel or hypoid gears

To transfer power, a pair of spiral bevel or hypoid gears engages. From beginning to end of two tooth surfaces engaging with each other: for their rigid property, they contact at different points; and for their plastic property, they contact at small ellipses around the points. On each surface, the contact line (or called as contact path) by connecting these points and the contact area by joining these ellipses are critical to driving performance. Therefore, to machine these surfaces, it is important to machine the contact line and area with higher accuracy than other areas. Five-axis flank milling is efficient and is widely used in industry. However, tool paths for flank milling the gears, which are generated with the existing methods, can cause overcuts on the contact area with large machining errors. To overcome this problem, an innovative approach to NC programming for accurate and efficient five-axis flank milling of spiral bevel or hypoid gears is proposed. First, the necessary conditions of the cutter envelope surface tangent with the designed surface along a designed line are derived to address the overcut problem of five-axis milling. Second, the tooth surface including the contact line and area are represented using their machining and meshing models. Third, according to the tooth surface model, an optimization method based on the necessary conditions is proposed to plan the cutter location and orientation for flank milling the tooth surface. By using these planned tool paths, the overcut problem is eliminated and the machining errors of contact area are reduced. The proposed approach can significantly promote flank milling in the gear manufacturing industry.     

数控仿真技术的回顾与评述

本文在对当前国内外数控切削加工几何仿真和切削过程力学仿真的两个并行发展的研究领域的研究现状和成果的分析评述的基础上，指出了仿真的发展趋势，并提出一种几何仿真和力学仿真相结合的更精确的试切模型。     

Minimization of the kinematics error for five-axis machining

Kinematics of a particular five-axis milling machine can drastically change the machining accuracy. Therefore, the reduction of the kinematics error is an important problem associated with the tool path planning.Our new optimization method employs a closed form of the kinematics error represented as a function of the positions of the cutter contact points. The closed form is derived from the inverse kinematics associated with a particular five-axis machine and obtained through automatic symbolic calculations.The second component of the algorithm is the optimal setup of the part surface on the mounting table employed in an iterative loop with the generation of the cutter contact points.For a prescribed tolerance the proposed optimization allows for substantial reduction in the number of required cutter contact points. The reduction can be significant and may amount to long hours of machining if the machining time at the programmed feed is less than the sampling time of the controller.In turn, when the number of cutter location points is fixed, the error can be substantially reduced. However, this refers to commanded error wherein the dynamics of machine tool are not taken into account.We present an analysis, systematic numerical experiments and results of real cutting (ball nose and flat-end cutters) as an evidence of the efficiency and the accuracy increase produced by the proposed method. We also evaluate the relative contributions of the setup and the point optimization.The method is shown to work with advanced tool path generation techniques proposed earlier such as the adaptive space filling curves.The numerical and machining experiments demonstrate that the proposed procedure outperforms tool paths based on the equi-arclength principle and paths generated by MasterCam 9.     

基于压缩体素模型的鼓形刀空间扫描体构造方法及其应用

首先给出了鼓形刀空间扫描体构造公式并构造出其表面模型，利用Ray Casting方法将该表面模型进行离散，转化为压缩体素模型，该模型采用沿X，Y，Z3个坐标轴方向相互垂直的Dexel模型表示，各个Dexel模型之间按体素模型大小均匀分布．刀具空间扫描体模型和仿真工件模型之间的布尔运算转化为Dexel模型之间的一维布尔运算，简化了布尔操作并提高了操作速度．通过MarchingCubes方法提取数控加工仿真工件表面三角网格模型并进行图形显示，提高了仿真工件显示质量，NC编程人员可实时地从任意方向观察、验证仿真结果．该方法已成功地应用于基于压缩体素模型的五坐标数控加工仿真系统中，克服了现有五坐标数控加工仿真方法和商品化软件系统的不足．     

直线与刀具扫描体求交算法及其应用研究

直线与刀具扫描体的交点问题是数控图形验证，加工过程仿真技术中的核心问题，本文根据五坐标联动机床的运动特点，把刀具扫描体表达成几组简单曲面的组合，利用直线与这几类曲交点来求解直线与刀具扫描体的交点。利用该算法，我们开发了一个用于检验五坐标铣削加工精度的计算机领导具软件。加工与仿真的实践证明，该算法计算正确正确，对进一步研制开发加工过程仿真系统有重要意义。     

Arc–surface intersection method to calculate cutter–workpiece engagements for generic cutter in five-axis milling

Calculating cutter–workpiece engagements (CWEs) is essential to the physical simulation of milling process that starts with the prediction of cutting forces. As for five-axis milling of free form surfaces, the calculation of CWEs remains a challenge due to the complicated and varying engagement geometries that occur between the cutter and the in-process workpiece. In this paper, a new arc–surface intersection method (ASIM) is proposed to obtain CWEs for generic cutter in five-axis milling. The cutter rotary surface is first represented by the family of section circles which are generated by slicing the cutter with planes perpendicular to the tool axis. Based on the envelope condition, two grazing points on each section circle are analytically derived, which divide the circle into two arcs. The feasible contact arc (FCA) is then extracted to intersect with workpiece surfaces. Using arc/surface intersection and distance fields based approach, the boundary of the closed CWEs is accurately and efficiently calculated. Compared with the solid modeler based method and the discrete method, the ASIM has higher computational efficiency and accuracy. Moreover, an analytical solution for calculating CWEs can be obtained with this method in five-axis milling of the workpiece merely comprising of flat and quadric surfaces. Finally, two case tests are implemented to confirm the validity of the ASIM and comparisons have been made with a Vericut based system which utilizes the Z-buffer method. The results indicate that the ASIM is computationally efficient, accurate and robust.     

基于压缩Voxel模型的五坐标数控加工仿真新方法

文章介绍了一种基于压缩Voxel模型的五坐标数控加工仿真新方法。该方法采用压缩Voxel模型表示数控加工工件模型和刀具空间扫描体模型,计算机内部存贮空间小,布尔操作简单、速度快。通过MarchingCubes方法提取数控加工仿真工件表面三角网格模型并进行图形显,提高了仿真工件显示质量。文章同时提出了一种基于压缩Voxel模型的刀具空间扫描体构造新方法,通过构造基本几何体球体、圆柱体、圆环体、锥形体和鼓形体的空间扫描体,完成各种加工刀具棒刀、球头刀、环形刀、锥形刀和鼓形刀空间扫描体压缩Voxel模型。该方法在《基于压缩Voxel模型的五坐标数控加工仿真系统》中得到了应用,仿真结果三维信息完备,操作人员可从任意方向观察、验证仿真结果,克服了现有五坐标数控加工仿真方法和商品化软件系统的不足,该方法是虚拟数控加工的关键技术。     

Geometric parameter optimization in multi-axis machining

This paper presents a systematic method for the determination of optimal geometric machining parameters in multi-axis machining. Machining accuracy is considered to be determined by a set of geometric parameters: the design parameters of the cutter, the positioning of the cutter, the orientation of the cutter etc. First, we formulate the general nonlinear constrained optimization model of the machining process. The optimal machining result is expected to produce the least deviation between the designed surface and the actual surface. This objective is accomplished by minimizing the deviation between the designed surface and the actual surface during machining. The details of how to characterize and calculate the deviation is then discussed for both ruled surface milling and general free-form surface milling. The swept surface is developed based on robotic manipulation and is used to model the actual surface. A signed distance function is constructed to perform the comparison which returns the signed distance from each sampled point to the designed surface. The direct search algorithm (Nelder-Mead simplex algorithm and pattern search algorithm in this paper) is used to solve our optimization problems due to possible discontinuity of the objective function and large nonlinearity of the problem. Three numerical examples and necessary comparisons are given to demonstrate the effectiveness of our method. The first example shows the generation of the swept volume of a filled-end cutter. The second example employs the swept surface generation method to solve a parameter optimization problem. Sensitivity analysis is performed for the parameters critical to machining accuracy. The third example optimizes the cutter orientation relative to the part surface to minimize the kinematics error caused by kinematics transformation and interpolation of multi-axis machines.     

Five-axis NC cylindrical milling of sculptured surfaces

In theory, the five-axis numerical control (NC) machining of sculptured surfaces can be classified into facing milling and cylindrical milling (or side milling). In general, the first one, using flat-end cutter, is suitable for the machining of large sculptured surfaces, e.g. the blade of hydraulic turbine, whose binding relations with drive surface (DS) and check surface (CS) are simple, and the second one, using cylindrical cutter, has wide applications for the milling of small and middle dimensional surfaces whose binding relations with DS and CS are more complex, such as the milling of integral turbine wheels. In practice, the second one suffers more difficulties than the first one, which are mostly related to gouge avoidance, interference avoidance and tool strength. This paper, on the basis of the theories of differential geometry and analytical geometry, describes research on algorithms for the toolpath generation of five-axis cylindrical milling of sculptured surfaces with cylindrical cutter. The approach includes (a) single point offset (SPO) algorithm, and (b) double point offset (DPO) algorithm for the cutter location data (CLDATA) calculation of five-axis cylindrical milling.     

Global optimization of tool path for five-axis flank milling with a conical cutter

In this paper, optimum positioning of the conical cutter for five-axis flank milling of slender surfaces is addressed from the perspective of approximating the tool envelope surface to the data points on the design surface following the minimum zone criterion recommended by ANSI and ISO standards for tolerance evaluation. Based on the observation that a conical surface can be treated as a canal surface, i.e. envelope surface of one-parameter family of spheres, the swept envelope of a conical cutter is represented as a sphere-swept surface. Then, an approach is presented to efficiently compute the signed distance between a point in space and the swept surface without constructing the swept surface itself. The first order differential increment of the signed point-to-surface distance with respect to the differential deformation of the tool axis trajectory surface is derived. By using the distance function, tool path optimizations for semi-finish and finish millings are formulated as two constrained optimization problems in a unified framework, and a sequential approximation algorithm along with a hierarchical algorithmic structure is developed for the optimization. Numerical examples are given to confirm the validity and efficiency of the proposed approach. Comparing with the existing approaches, the present one improves the machining accuracy greatly. The rationale developed applies to general rotary cutters.     

复杂曲面五轴加工局部干涉处理技术研究

针对复杂曲面环形刀五轴数控加工中的局部干涉问题,提出了一种基于曲 率匹配及网格点的干涉处理技术。首先,利用曲率匹配原则选出合理的刀具半径,以保证在 切触点处沿任何方向上刀具与被加工曲面之间不会发生干涉,然后在各个切触点处通过比较 刀具曲面最小主曲率与加工曲面最大主曲率确定出刀具的初始倾角。为了判断切触点邻近区 域是否存在干涉问题,采用了网格点来快速自动生成检测区域及初始检测点。文中对有效检 测点的筛选以及干涉的判断和处理技术分别进行了详细论述。最后,以非均匀有理B 样条 曲面为加工实例,对上述算法进行了测试和验证。     

基于四点偏置法的非可展直纹面侧铣刀位计算

针对非可展直纹面五轴侧铣加工的问题,分析了非可展直纹面几何特点,根据等距 映射下的极差不变性,提出了一种计算非可展直纹面叶片五轴侧铣刀位数据的新方法。以刀具包 络面与设计曲面之间的整体误差为优化目标,建立了圆柱铣刀侧铣非可展直纹面的刀位计算方 法,运用四点偏置法确定初始刀位,采用最小二乘法对初始刀位进行优化,建立刀轴矢量偏转模 型进一步修正刀位以减小过切误差。通过实例计算分析,表明该方法可以在一定程度上减小加工 误差。     

Feature conversion based on decomposition and combination of swept volumes

In this paper, an approach is proposed for converting design features into machining features based on decomposition and combination of swept volumes. In the approach, delta volumes between a negative swept volume and a positive swept volume, which are directly generated from original design features, are decomposed into a set of smaller negative swept volumes first. Then, the negative swept volumes are combined to form maximal swept features, each of which represents a set of machining features with the same cutting-paths. Since sweeping is a common characteristic of different type features, the maximal swept features can serve as an intermediate representation for the different types of feature-based models; in particular, they can be conveniently converted into milling features or turning features, common features or user-specific features, and their alternatives.     

Simultaneous optimization of tool path and shape for five-axis flank milling

By representing the swept envelope of a generic rotary cutter as a sphere-swept surface, our previous work on distance function based tool path optimization is extended to develop the model and algorithm for simultaneous optimization of the tool path and shape for five-axis flank milling. If the tool path is fixed, a novel tool shape optimization method is obtained. If the tool shape is fixed, a tool path optimization method applicable to any rotary cutter is obtained. The approach applies to non-ruled surfaces, and also finds applications in cutter dimension optimization and flank millable surface design. Numerical examples are given to confirm its validity.     

基于DEXEL和离散点阵法的数控加工仿真研究

本文提出了基于DEXEL模型实现数控铣削过程中毛坯体的实时建模方法,减少了内存存储空间,支持观察视角的变化和仿真结果的缩放;根据三轴数控铣床加工的特点,将刀具扫描体看成离散刀位点之间的单个扫描体的组合,采用离散点阵法对刀具扫描体建模.使用局部搜索算法,用其搜索发生切削关系的点并动态更新仿真工件的高度值.在VC++ 6....     

设计特征向加工特征转换的扫体重构法

针对同时需要铣削和车削加工的零件,提出一种基于扫体的从设计特征向加工特征转换的统一方法.采用扫体方式重构零件和毛坯模型,并将毛坯与零件之间的差体沿着这些扫体扫掠路径拆分出各种新的加工扫体,最后将这些加工扫体依照它们之间的邻接关系和加工特征形状结构选择合并成加工特征.该方法实现了同一零件中直扫体和回转扫体2种特征的统一转换,能够产生对应各种不同加工方法和方向的加工特征,有利于工艺优化.文中方法已应用于一个CAPP系统,其结果证明了该方法的可靠性.     

复杂曲面加工精度检验中曲面法矢与刀具扫描体求交算法的研究

提出了一种用于实现矢量与扫描体（代表五轴数控加工铣刀的运动）求交的有效算法,该算法首先对曲面法矢和刀具扫描体进行预处理,将曲面法矢与刀具扫描体之间的求交问题转化为有向线段与三角网格之间的求交计算,然后建立有向线段和三角网格的求交子集,减少了不必要的求交计算,提高了乍法的效率。文中举例说明了该算法在复杂曲面NC精度检验中的应用。