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用普通刀具在立体平版印刷格式中对三角雕塑面的数控加工一个产生数字控制刀具路径的统一方法已经出现，这个刀具路径是用普通刀具在立体平版印刷格式中对三角形雕塑表面的数字控制加工产生的。这个方法的产生很重要，这是是因为一个立体平板印刷格式的应用象征着一个计算机辅助设计模型已经在很短的一段时间内被工业界广泛接受。这不仅是因为比如特别需要运用这种方法的快速设计模型的应用，而且还归结于现在可以直接用数字化和反向工程过程来制造复杂的立体平板印刷模型。虽然有很多支持立体平板印刷文件的计算机辅助设计和计算机辅助制造的软件系统，但在这些文章中只有几页纸直接涉及到从立体平板印刷文件来的数字控制加工问题。一种用普通自动编程切削刀具产生刀具路径的一般计算算法已经出现。这种算法看起来普遍，它可适用于包括球头刀、平底刀和内圆角精铣刀在内的各种各样的切削刀具。为了减少计算时间，一个用彩色小石块镶嵌产生网状询问区域高的效方法也已经产生。用模仿加工和真正的机械加工制造的实例来说明所提出的方法的效率。1、介绍目前，多数计算机辅助设计(CAD) 系统利用参数表面代表CAD 模型的几何形状。由于各种各样的设计或制造过程的不同，需要转换不同的计算机辅助设计系统（CAD）和计算机辅助制造系统（CAM）之间的模型。中立数据文件譬如最初的图表交换规格(IGES) 广泛地被用于(美国产品数据协会1996) 。最初的图表交换规格(IGES)描述在建立CAD 模型时可能被使用的信息、定义个体模型时的参数量和各个个体之间不同的关系。但是，使用最初的图表交换规格(IGES)翻译CAD 模型并不是总是容易的，这是因为大多数计算机辅助设计系统（CAD）使用不同的内部表示方法而且转换并不是总是直接的而错误又比较多。与最初的图表交换规格(IGES) 相比，立体平版印刷(STL) 格式比较简单，并且它的实施比较容易 (Jacobs 和Reid 1992 年, Kochan 1993) 。基本上，一个立体平版印刷(STL)文件只包含一个三角形和它们的法线传播媒介。立体平版印刷(STL)文件不能替换最初的图表交换规格(IGES)，最初的图表交换规格(IGES)包含更多与设计相关的信息，然而对于许多顺流制造业活动譬如迅速设计模型、数字控制(NC) 加工制造甚至有限元素分析来说，这些信息包含在立体平版印刷(STL)之中的信息是充足的。由于它的简单和在各种各样的工程学领域中的不同用途，立体平版印刷(STL)翻译受到大多数计算机辅助设计和计算机辅助制造（CAD/CAM）系统的支持。在过去，立体平版印刷(STL)文件是负担对内存分配和计算速度的任务。但是，随着中央处理单元的加速，更多力量和存储芯片逐渐变得更加便宜，这不再是转换和处理立体平版印刷(STL) 文件时的一个障碍。此外，最新的三维扫描技术也促使反向工程的应用迅速增长，在反向工程应用中，创造很大并且很复杂的模型后，将它存放在立体平版印刷(STL)文件之中(Chuang 等2002) 。人们现在已经普遍接受这样一个事实，分成三角形的表面和立体平版印刷(STL) 文件的应用将使设计和制造业应用变得越来越普遍。在过去，人们已经学到了许多三轴机械加工刀具路径的计划方法 (Dragomatz 和Mann 1997) 。刀具路径的产生方法可以分成两种类型： 解析和参数(Zeid 1991) 。前者产生于横切机械加工表面的短剖面飞机。而后者产生于沿着平面或者曲面走刀的数字控制刀具路径，而刀具切削点(CL) 通常是用计算机从机械制造加工表面的设置来进行计算的(Kishinami等1987 年，Tang 等1995 年，Choi 等1997 年，Lee 2003 年)。参数设置方法在应用精确表面信息之中有其特定的优点，但是它可能不适合应用于加工带有很多凸台的复合表面并且很容易受到凹平面的损坏(Choi 和Jerard 1998) 。在另一方面，解析的方法的优点是可以产生没有凹平面损坏现象的刀具路径，但是它的缺点的是不能产生直线的走刀路径和进行清角加工的刀具路径(Dragomatz 和Mann 1997) 。所以，在零件的现实加工之中，解析和参数路径的应用策略在互换性之中被运用。立体平版印刷(STL)加工主要是应用解析的加工策略，这是因为它不包含有完整的表面信息。在解析路径计划之中，当切削刀具接触到机械加工表面时， CLs就被计算出来。在解析路径计划之中一个最有效的方法是绘制Z轴 (Choi 等1988 年，Choi 1991 年，Saito 和Takahashi 1991 年，林和刘1998) 。绘制Z轴的方法计算栅格数据库设置中的无干涉CLs。机械加工是精确度取决于库栅格数据的密度。这通常是一个对栅格数据库大的存储空间的分配的需要。Hwang 和他的同事提出这样一个方法，就是用平底刀、球头刀和圆角精铣刀从三角形表面产生无干涉机械机械刀具路径（Hwang 1992 年，Hwang 和Chang 1998) 。但是，这种方法是相对于不同的切削刀, 而且它的算法限制可切削刀类型的开发。尽管在现实之中使用着更多不同的切削刀具类型。例如，经常用一把细而利的精铣刀具作为浅槽的标号。它会繁琐而笨拙地产生所有需要的刀具的不同算法和代码。本论文介绍一种直接产生刀具路径的统一方法，这种方法是用通用的自动编程加工刀具(APT)在立体平版印刷(STL)的三角形表面上产生的(Kral 1986)。APT切削刀的拓扑结构定义通常被用于数字控制的应用实例，但大多数刀具路径的形成方法都是为特殊的刀具类型开发的，不的通用的 (Chung等1998 年，Chiou and Lee 1999) 。在这里为所有刀具类型产生的刀具路径形成是普遍的，在这里以包括球头刀、平底刀和圆角精铣刀以及其它更多的刀具作为代表(图3 和4)。从这个研究结果来看，只有一种系统的和统一的算法是必要的, 这个算法对通用APT切削刀的原则是非常兼容的。为了减少在处理一个大立体平版印刷(STL)文件时的计算时间，一个在区域询问方面的高效方法产生了。2、在三轴机械加工中的数字控制刀具路径计划在实际的应用之中，数字控制刀具路径可以适用于各种不同的机械加工过程(Choi 等1994) (图1)：图1 数字控制刀具路径计划中不同的机械加工过程主要规程包括粗加工、半精加工和去除咬边加工(通常叫做平行加工或清角加工) 。用大尺寸的切削刀具和高的进给量进行粗加工（通常用平底精铣刀）可以高效率地去除庞大的重复材料。为得到一个更好的加工表面，在精加工之前通常通常要进行几次半精加工(通常是用圆精铣刀和球头精铣刀)。完成了半精加工之后，工件表面就留下均匀厚度的材料，这个厚度是作为精加工（通常是用小的球头精铣刀）要去除的工序余量薄层。有时完成精加工后还需要进行平行加工和清角加工，因为沿壁角边缘有一个咬边的区域 (图2)。一种更小的切削刀具是用在精加工之后的加工过程，以塑造局部角度外形或边缘并且去除未切削的材料。根据上述讨论，产生CLs通用切削刀具的一个统一方法的形成是不仅实用的, 而且更加容易实施和维护。因为有有效觉得方法来将三角形参数或所包含的表面的公差控制在允许的范围内，在这里开发的算法能够对普通的计算机辅助制造（CAM）产生一个核心引擎作用。图2 咬边区域3、普通几何形状的APT 切削刀根据APT 的定义，可以用如图3所示的参数来将普通几何形状的切削刀具完整地描述出来：图3 普通几何形状刀具的参量d、切削刀具直径，刀具直径是辐形距离两倍，这个辐形距离的切削刀具轴到上部和下部直线段交叉点的距离计量的；r、壁角半径； e 辐形距离，它是从切削轴到壁角圈子中心的距离，如果它的壁角和中心是在工具轴的同一边，那么它是正面的，否则它就是反面的；f、从终点到壁角圈子中心的距离，这个距离是通过平行于工具轴来测量的。切削刀具参数的值的必须与其内部的各种参数相一致，而且不能违背某些规定的约束，以便适当地描述允许范围之内加工刀具的几何形状(Kral 1986)如图4所示是切削刀具集合形状的几种选择: 图4 根据APT 定义的机械加工刀具形状的几种选择一些附属参量的方法如下，被使用帮助描述CL 点的计算。这些附属参数可用于帮助描述刀具切削点的计算。R=+（Lc-tan1）tan2-（1）那里的半径R，是切削刀具在加工零件表面上的最大伸出界限。这个界限将用于帮助寻找在伸出区域内的相交的三角形。从机械加工的切削刀具的几何学外形来看，圆环圈子的半径R1和R2 的计算方法如下：R1=（u+）/2-（2）哪里 R2=e+（vsin(22)+）/2-（3）和 V=（R-e）/tan2）-（Le-f）哪里 L=Lc-f+-（4）距离L，是用半径R2从圆环圈子的中心开始计量来进行计算的，计算的方法如下：L=Lc-f+在刀具参侧面上距离分别为R1和R2的两个不同的点从工具轴开始将机械加工切削刀具分成三个不同的区域。在上面的部分是锥体截面体其半径是R，R2高度是L，中间部分是圆环半径e和壁角圆半径r，底部是半径R1 和高度R1 圈子锥体tan_1。通常,切削刀具侧面不需要包含所有三个区域。比如在上图4中是（a）图，它的形状是一个圆筒；在图4中的(c)图，花托成为一个半圆球；在图4中的中(d)，它是一把逐渐变得尖细切削刀。4、形成刀具切削区域的算法数字控制加工中形成塑造传统的实体模型的方法需要垂直于刀具切削表面（CL）的平面。虽然它在概念是简单的，但是还是有几的缺点。首先,垂直表面的形成不是一个琐细的问题。在固体模型中，界限表示法(B Rep)模型是最普遍的代表形式。被整理的不均匀的B多槽轴(NURBS)表面的垂距是复杂的而且计算费用昂贵的操作。其次,被整理的表面的垂距可以容易地制造复杂的自身内部的交叉点和外部的交叉点(以毗邻表面)问题。此外，这个统一的垂距表面只实用于球头铣刀刀具切削区域的产生。用圆角铣刀加工形成的刀具切削区域垂距表面是一个更加困难的问题，更不用说更加通用的APT刀具。总的来说，用垂距表面在数字控制刀具路径中产生传统的实体模型在计算过程中是复杂的，而且计算的效率也很底。此外，解析机械加工中，在给定部分参数表面后，机械加工刀具路径是从垂距部分表面和平行于工具轴的一系列垂直平面的交叉点产生的。非线性等式解决的方法也许包含在要寻找的交叉点曲线中。对于一个立体平版印刷(STL)模型，因为零件表面已经被分成三角形，刀具路径的形成是从多面体表面开始计算CLs的。在许多情况下,唯一线性运算是很有必要的。如图5所示，切削刀具与零件表面接触的地方叫做刀具接触点（CC），而刀具的端点被定义为刀具切削点。在机械加工中，刀具接触点（CC）并不是固定的，而刀具切削点（CL）的x-y 坐标值可以任意确定(多数情况下是落在固定的网格点)。唯一的未知数就是刀具切削点的Z轴坐标值。因此，刀具路径通常的由很多连续的刀具切削点组成。当工具轴向二维点(xc,yc)移动时，零件表面将形成一个区域，这个区域是半径为R的二维圆组成的，这个二维圆的圆心在(xc,yc)点上。这个区域叫做刀具接触点（CC）（图5）。 图5 CC点、CL点和CC区域本文提出一种计算无推断从零件表面的那些小的三角形来的刀具切削点（CL）的算法，这些零件表面与刀具接触点（CC）区域是重叠的。当切削刀具与一个三角形多面体接触时，刀具接触点（CC）可能位于端点、小平面或者是边沿上。对于切削刀具本身，刀具接触点（CC）可能与包络线、花托区域或更低的锥体连在一起。对于这些各种各样的联系情况，刀具切削点（CL）的计算方法是不相同的。先确定接触区域和刀具接触点（CC）是非常必要的，然后从刀具接触点（CC）区域可以计算出刀具路径的刀具切削点（CL）。对于一把普通的APT 切削刀具，有九类型计算模型。实际上，并不是每把切削刀具都包含有三个区域。通常，一把切削刀具只包含有一个或者两个切削区域（图4）。刀具切削点（CL）的计算过程首先是从分成三角形小平面的刀具接触点（CC）区域开始的。这是一个节约时间的策略，因为如果刀具接触点（CC）小平面里面， 切削刀具就不接触到端点或者三角形的边沿，因此，后面二者更加费时的步骤就可能得到避免。8Numerical control machining of triangulatedsculptured surfaces in a stereo lithographyformat with a generalized cutterA unified approach to the generation of numerical control tool paths for triangulated sculptured surfaces in a stereo lithography format using a generalized cutter is presented. This is important because the use of a stereo lithography format for representing a computer-aided design model has been widely accepted in industry for quite some time. It is not only just because of an application such as rapid prototyping (RP), which specifically requires the use of it, but also it is due to the fact that complex stereo lithography models can now be created directly by the digitization and reverse engineering process. Although many computer-aided design/computer-aided manufacturing software systems support the translator of stereo lithography files, only a few papers have addressed the issue of numerical control machining directly from a stereo lithography file. A general computing algorithm to generate tool paths by using a generalized automaticallyprogrammed tools cutter is presented. It is general in the sense that it can be applied to various cutters including ball, flat and fillet end-mills. To reduce the computation time, an efficient method for the region query of a tessellated mesh is also presented. Simulations as well as real machining examples are given to illustrate the effectiveness of the proposed method.1. IntroductionCurrently, most computer-aided design (CAD) systems use parametric surfaces to represent the geometry of a CAD model. To transfer models between different CAD/computer-aided manufacturing (CAM) systems for various designs or manufacturing processes, neutral data files such as Initial Graphics Exchange Specification (IGES) are used extensively (US Product Data Association 1996). IGES describes the possible information to be used in building a CAD model, the parameters for the definition of model entities and the relationships between different entities. However, the translation of CAD models using IGES is not always easy because most CAD systems use different internal representations and the conversion is not alwaysstraightforward and error free. In contrast to IGES, the stereo lithography (STL) format is simple and its implementation is easy (Jacobs and Reid 1992, Kochan 1993). Basically, an STL file contains only triangles and their normal vectors. The STL file is not intended to replace IGES, which contains more design-related information, nevertheless the information contained in STL is sufficient for many downstream manufacturing activities such as rapid prototyping, numerical control (NC) machining and even finite-element analysis. Because of its simplicity and use in various engineering fields, STL translation today is supported by most CAD/CAM systems. In the past, large STL files had been a burden to memory allocation and computation speed. However, as the central processing unit cranks up more power and memory chips become less expensive, this is no longer a barrier for the transfer and processing of STL files. Furthermore, the latest three-dimensional scanning technology also helps the rapid growth of the reverse engineering application in which very large and complex models are created and stored in STL files (Chuang et al. 2002). It is generally agreed that the use of triangulated surfaces and STL files for design and manufacturing applications will become increasingly popular.In the past, many path-planning approaches for three-axis machining have been studied (Dragomatz and Mann 1997). The tool-path generation methods can be categorized into two types: Cartesian and parametric (Zeid 1991). The former is generated from cross-section planes that intersect the machined surfaces. The latter generates NC tool paths along constant u or v surface curves and the cutter location (CL) point is usually computed from the offset of the machined surface (Kishinami et al. 1987, Tang et al. 1995, Choi et al. 1997, Lee 2003). The parametric method has the advantage of utilizing accurate surface information, but it might not be suitable for machining a compound surface consisting of surface patches and is susceptible to concave gouging (Choi and Jerard 1998). On the other hand, the Cartesian method is good at generating gouging-free tool paths but it lacks the ability to generate pencil cuts or cornering cuts (Dragomatz and Mann 1997).Therefore, in the machining of a real-world part, both Cartesian and parametric path generation strategies are used interchangeably. STL machining primarily employs the Cartesian machining strategy since it does not contain the full surface information. In Cartesian path planning, CLs are calculated when the cutter touches the machined surface. One of the most robust methods in Cartesian path planning is the Z-map method (Choi et al. 1988, Choi 1991, Saito and Takahashi 1991, Lin and Liu 1998). The Z-map method computes the interference-free CLs from a grid data set. The precision of machining is dependent on the density of the grid data. There is usually a need for a large memory space to be allocated for the grid data. Hwang and colleagues presented a method to generate interference-free tool pathsfrom tessellated surfaces by using flat, ball and fillet end-mills (Hwang 1992, Hwang and Chang 1998). However, the method treats each cutter separately, and algorithms were developed for limited cutter types. In practice, however, there are more different types of cutter that are being used. For example, a tapered and sharp end mill is often used for marking thin grooves. It would be tedious and cumbersome to develop separate algorithms and codes for all the needed cutters.The present paper presents a unified approach to the tool path generation directly from the triangulated surface of an STL model by using a generalized automatically programmed tools (APT) cutter (Kral 1986). The topology definition of an APT cutter is usually used for NC verification, but most tool path generation approaches are developed for specific types of cutters and are not general (Chung et al. 1998, Chiou and Lee 1999). The method presented here is general in that it can be applied to all types of cutter represented by the APT cutter, which includes the frequently-used ball, flat and fillet end-mills, and more (figure 3 and 4). From this research result, only one systematic and unified algorithm is needed, which is very compatible to the principle of the APT generalized cutter. To reduce the computation time when dealing with a large STL file, an efficient method for the region query has presented。2. Numerical control-path planning in three-axis machiningIn practical application, NC paths are generated for different machining procedures (Choi et al. 1994) (figure 1)：Figure 1. Different machining procedures in NC path planning.The main procedures include rough cut, semifinish cut, finish cut and undercut removal (often called pencil cut or corner cut). With large size cutter and high feed rate, the rough cut (usually with a flat end-mill) is designed to remove efficiently bulky redundant material. For a better cutting result, there are usually several semifinish cuts (usually with fillet end-mills or ball end-mills) preceding the finish cut. After the semifinish cut, a uniform thickness of material remains on the final surface before the finish cut (usually with a small ball end-mill) is used to remove this thin layer. At times there is a need to generate a pencil cut or corner cut after the finish cut because there is an undercut region along the corner edge (figure 2). An even smaller cutter is used in this clean-up machining procedure to contour around corners or edges to remove uncut material. Based on the above discussion, a unified approach that can generate CLs for a generalized cutter is not only practical, but is also easier to implement and maintain. Since there are robust algorithms to triangulate parametric or implicit surfaces under a controlled tolerance, the algorithm developed here can serve as a core engine for a general CAM package.Figure 2. Undercut region.3. Generalized geometry of an APT cutterAccording to the definition of APT, the generalized cutter geometry shown in figure 3 can be described fully by the following parameters:Figure 3. Parameters for a generalized cutter geometry.d cutter diameter, which is twice the radial distance measured from the cutter axis to the intersection of the lower and upper line segments,r corner radius, e radial distance from the cutter axis to the centre of a corner circle; it is positive if its corner and centre are on the same side of the tool axis, otherwise it is negative, f distance from the endpoint to the centre of corner circle measured parallel to the tool axis.The cutter parameter values must be consistent among themselves and not violate certain restrictions so that permissible geometries are properly described (Kral 1986).Several selections of cutter shapes are shown in figure 4.Figure 4. Some selections of cutter shape based on APT definition.Some dependent parameters are derived as follows. They are used to help escribe the computation of CL points.R=+（Lc-tan1）tan2-（1）where the radius, R, is the maximum boundary of the cutter projecting on the part surface. The boundary will be used to find the intersected triangles in the projected region. From the geometric profile of the cutter, the radius of ring circles R1 and R2 can be computed as follows:R1=（u+）/2-（2）WhereR2=e+（vsin(22)+）/2-（3）AndV=（R-e）/tan2）-（Le-f）where L=Lc-f+-（4）The distance, L, measured from the centre of ring circle with radius R2 is computed as follows:L=Lc-f+The two distinct points on the cutter profile with distances of R1 and R2, respectively,from the tool axis divide the cutter profile into three different regions. On the top is a frustum of cone with radius R, R2 and height L, the median part is a torus of ring radius e and corner circle radius r, and the bottom is a circle cone of radius R1 and height R1 tan_1. Generally, a cutter profile needs not contain all the three regions. As shown in figure 4(a),the shape becomes a cylinder; in figure 4(c), the torus becomes a semisphere; in figure 4(d), it is a taper cutter.4. Algorithm for generating cutter locationsA traditional solid modelling approach to NC machining requires the generation of offset surfaces to approximate the CL surfaces. Although it is simple in concept, there are several shortcomings. First, the generation of offset surfaces in itself is not a trivial problem. In solid modelling, a boundary representation (B-Rep) model is the most popular representational form. The offset of trimmed non-uniform rational B-splines (NURBS) surfaces is a complex and computationally expensive operation. Second, the offset of multiple trimmed surfaces can easily create complex selfintersection and global-intersection (with adjacent surfaces) problems. Third, the uniform offset of surfaces is only useful to the generation of CL points for ballend mills. The offset of CL surfaces for fillet-end mills is a more difficult problem, not to mention the more general case of APT cutters. Overall, the traditional solid modelling approach to NC tool-path generation by surface offsetting is complex in calculation and inefficient in computation.Furthermore, in Cartesian machining, given a parametric part surface, the tool path is generated from the intersection of the offsetting part surface and a series of vertical planes parallel to the tool axis. Non-linear equation solving may be involved for finding the intersection curves. For an STL model, however,since the part surface is already triangulated, the tool path generation is to compute the CLs from the polyhedral surface. In most cases, only linear operations are needed. As shown in figure 5, the point of the c
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