外文翻译基于细分曲面生成的三轴数控切削轨迹的研究

南京理工大学毕业设计(论文)外文资料翻译系部： 机械工程系 专 业： 机械工程及自动化 姓 名： 学 号： 外文出处：Proceedings of the Geometric Modeling and Processing 0-7695-2078-2/04 2004 IEEE 附 件： 1.外文资料翻译译文；2.外文原文。 指导教师评语： 签名： 年 月 日注：请将该封面与附件装订成册。附件1：外文资料翻译译文 基于细分曲面生成的三轴数控切削轨迹的研究摘要在本文中，我们提出数控切削轨迹生成细分曲面的方法和算法。 我们选择环状曲面的细分曲面。一条轨迹示意图，包括粗糙切削完成切割是发展的基础上逐步的LOD （层次细节）的细分曲面。 产生一个粗糙网状覆盖曲面的界线执行切削。 为了完成切削，使用球头磨坊，压榨机和抵销切削产生位置连络放置的地方。 在这两个步骤中，我们使用 Zmap模型和一个检查碰撞和纠正的方法是为了表示这二个步骤没有冲突。 我们执行我们的方法和目前的加工结果。 两类切削可以实现快速自动计算。 1.引言 机械加工自由形态曲面在生产模具方面占据一个重要的角色。这些自由形态曲面模型被曲现为参数曲面小块。举例说明， B 样条曲面和 NURBS 曲面，哪一个被 CAD 产生（计算机半自动设计）系统 267。另一方面，这是很难获得通过运用这些参数曲面的完美模型数据。例如，构造一个用多片的曲面模型能引起缝隙尤其当模型是复杂的时候。一个在小片中间的装饰品工序易于连通性错误。现在 CAD 系统不能自动的解决这些问题，工程师必须用手改正。从如此差的 CAD 模型中产生的数控资料这个严重的问题经常影响产品的质量。在这研究中，我们采用细分曲面模型工艺来解决这些问题。细分曲面的基本观念是产生一个平滑曲面通过重复地细分一个最初的多面体。例如三角形网状物。它是在只是在单频率激光震荡片只有一个很好的性能，这个性能能曲现一个形状复杂的型钢。如此很少需要有一个多片组织或装饰品甚至是个复杂的形状。另外，那细分曲面有特定的连续性，举例来说， C2 的连续性几乎在曲面的任何地方。因为这些细分曲面在计算机直观显示的领域内是一个严重的曲现 3。然而，在 CAD凸轮的领域中，它不是如此多的被运用。一个原因是我们仍然不知道是否细分曲面适用于凸轮。这是在这项研究后的基本原则。这项研究的目的是为了通过细分曲面发展一个有效的数控通路形成法。而且得到自动化的最好性能和速度。在这研究中，当做我们的目标细分升至水面，我们选择环状细分曲面 1 。它的领域是三角形的网孔。在第 2 节中，我们介绍它的基本原理。在我们的研究中，我们提议包括二个阶段的一个轨迹示意图粗糙的切割和完成切割。这个方法的开发LoD 细分曲面的性能为了这二个阶段。换句话说，使用粗糙的网孔作为粗糙切割和好的网孔来完成切割。产生的轨迹示意图被申请用来三轴加工。这二个阶段使用 Zmap模型。在第 3 段中，我们详细描述一下Z地图模型。在第 4 和 第5 段中我们介绍粗糙切割的轨迹运算法则和完成切割的轨迹。在第 6 段中我们提议发现碰撞和控制加工准确性和质量的修正。在第7段中，切削轨迹工具执行和他们的加工结果被示范。最后一段我们得出最后的结论。2. 环状细分曲面在这一个段中，我们简要地介绍环状细分曲面。更多的详细的在13 中提到。在 1987年环状细分曲面在方形齿轮到不规则的网状物反复被推广19 环状的细分方案以三方向方形齿轮为基础。它生产除了特别顶点以外的各处地方的C2 连续曲面。特别的顶点是当原子价六是规则的时候，谁的原子价不等于六。环细分方案能被适用于任意的三角形的网孔在下列的二个步骤：1.三角形的每边缘被分为两边缘。2.每个三角形被分为四个三角形。在这些操作中旧的顶点 （甚至被称为顶点）被移动到新的位置。同时新的顶点进入边缘之内是被称为奇数顶点。奇数的位置和顶点被环状细分规则所计算。有四种规则为一奇数顶点顶点进入网孔 （内部的顶点）, 和一奇数顶点顶点在边界上。通过无限地细分那最初的三角形的网孔 （控制网孔或控制多角形）, 它能聚合一个平滑的曲面. （界限曲面）用控制网孔来加工顶点界限曲面上找到一个位置。这个界限位置能容易的，没有任何细分地被计算用做类似于一个细分规则公式13。除此之外，切线矢量以及界限曲面的正常矢量在界限位置升至水面顶点时候能被计算。3.Zmap模型Zmap模型是一个分离的非参数的在那一高度的曲面代理的特殊形式。在格子上，点被储存在二维数组中，如图 3.1 所描述. 它已经广泛地用于数控切削轨迹上2。格子是以和 而定， 是参考点，是格网间距。图 3.1 Zmap的模型我们首先将细分曲面转换成 Z地图模型是为了要产生一条数控切削轨迹。在这个 Zmap赋值过程中，我们通过交叉线沿着 z轴曲面方向为每个格点进行z轴方向上赋值。在 Zmap赋值之后，我们通过x（或 y）常数格网线生成一条切削轨迹，以便切削连接Zmap曲面。4.粗糙的切割轨迹基本上为了除去大量的东西，我们通过平端磨细的分层来切削工件层，这叫做切削片以及采用在其他的早先的研究中10。切削片沿着粗糙切削曲面完成。切削层的厚度由工具参数决定。粗糙切削曲面应该带有某一数量缺口的细分曲面围起来，而且应该尽可能简单的计算。简单的洞察力是由于粗切削曲面而使用一中间的粗糙网孔。然而，这一粗糙网孔不总是没有在之上减少的情况而满意的。因此我们将它转换成网孔，是一个近似使这些情况得到满意。我们称它是掩护网孔。4.1.一直覆盖网孔图 4.1 告诉我们的基本思是在两个空间的情况下为一个粗糙的网孔产生一个覆盖的网孔。我们移动那些顶点在曲面以下当较低的部份所显示的曲面在曲面以上是为了克服超过切削的问题。这些新的位置通过尖锐的补助来弥补他们的界限位置。这样，它看起来我们能产生一个总是在曲面的覆盖网孔。图4.1 二维覆盖网孔然而，三维空间的情形并不是如此的容易。部份界限曲面可能在网孔上面，即使所有的顶点在曲面上。如此的一个部份可能被检验控制的局部凸壳的网孔所发现。为了解决这个问题，我们采用Jos Stam 的赋值方法4 去计算界限曲面上的点，如同普通的参数曲面。首先，如图4.2所显示我们估计在二个三角形的面之间的边缘一系列的点。图4.2三维覆盖网孔然后我们判断是否被抽取样品的点在两个邻接三角形下面。当有一些点在这些三角形上面，我们找到最高点，计算从最高的点到估计边缘的距离（dmax )。在图 4.2 ，我们使用公式（4.1)弥补对他们的新位置 V 的顶点，是切削补助和N是顶点最平常的地方。在边界上，我们不能够估算边缘曲线上的点，除了被增加。图 4.3 举例说明一个方法。图4.3(a）用红色展示了最初的网孔 M0 。图4.3(b）用绿色曲示 M1 网孔 （细分 M0 一次）。图 4.3(c）曲示界限曲面。图 4.3（d）用绿色曲示M1网孔和用红色曲示界限曲面。可能 M1网孔切断界限曲面，换句话说，那绿色的部份正在切断红色的部份。图 4.3(e）曲示覆盖网孔用蓝色从M1中生成。在图 4.3(f）中，红色界限曲面不切断蓝色覆盖网孔。意思是覆盖网孔完全覆盖界限曲面。虽然覆盖网孔的方法不是精确的，它仍然是一个适用的方法用作粗糙切削曲面。当然，我们在细分曲面和覆盖网孔之间能应用交叉检查。 (a)最初的网孔 (b)网孔M1 (c)界限曲面 (d) M1和界限曲面 (e)覆盖网孔 (f)覆盖网孔和界限曲面图4.3产生覆盖网孔的例子4.2.Z-map的粗糙切削曲面然后我们为覆盖网孔产生一个Zmap模型当做被Z-map抽样的粗糙切削曲面。对于一个Zmap模型，我们为粗糙切削曲面设定格子间隔的锐利补助的一半？哪一个能被使用者分配。R是工具半径。4.3.CL材料为粗糙切削计算 粗糙切削的最后一个步骤是计算 CL（切削位置）每个部分的数据，为了由加工者输入程序。在产生覆盖网孔的Zmap模型之后，我们在垂直于Z方向上把它一层一层的切成薄片。对于每个x （或y）常数格子线，我们规定在Zmap上沿着这一条线抽取样品。然后我们在薄片飞机和聚合点之间发现交叉点。所有的交叉规定为 CC （切削接触）点，然后被抵消计算 CL 点的 R 。通过连接那些 CL 点，计算这个格子线的数控切削轨迹。这个程序为所有的格子线所重复利用，以及为所有的飞机零件生成完整的数控切削轨迹。附件2：外文原文Three-axis NC Cutter Path Generation for Subdivision SurfaceAbstractIn this paper we propose methodologies and algorithms of NC cutter path generation for subdivision surfaces. We select Loop surface as the subdivision surface. A path plan including rough cut and finish-cut is developed based on LoD (Level of Detail) property of the subdivision surface. We generate a coarse mesh that covers the limit surface to implement rough cut. For finish-cut we use ball-end mills and offset cutter contact positions to generate cutter location. In these two steps we use a Z-map model and a collision detection and correction method is presented for the interference-free of these two steps. We implement our methods and present machining results. All of these two kinds of cutter paths are computed rapidly and automatically.1. Introduction Machining free form surfaces plays an important role in producing dies and moulds. These free-form surface models are represented with parametric surface patches, e.g. B-spline surfaces and NURBS surfaces, which are generated by CAD (Computer Aided Design) systems 267. On the other hand it is difficult to obtain perfect modeling data using these parametric surfaces. For example, constructing a surface model with multi-patches can cause gaps especially when the model is complex. A trimming operation among the patches is prone to connectivity errors. Current CAD systems cannot solve these problems automatically, and engineers have to correct them manually. It is a serious problem that NC data generated from such bad CAD models often affects the quality of products. In this research we adopt subdivision surface modeling technologies to solve these problems. The basic concept of the subdivision surface is to generate a smooth surface by repeatedly subdividing an initial polyhedron such as a triangular mesh. It has such a nice property that it can represent a complex shape in only “single” patch. Thus it is seldom required to have a multi-patch structure or trimming even for a complex shape. In addition, the subdivision surface has certain continuity, for instance C2-continuity almost everywhere on the surface. For these characteristics the subdivision surface has been a major representation in the field of computer animation 3. However, in the field of CAD/CAM, it has not been used so much. One reason is that we have not yet known if subdivision surface can be applied to CAM. It is our fundamental motivation behind this research. The objective of this research is to develop an effective NC path generation method for subdivision surfaces, and get the best performance of automation and rapidity. In this research we select Loop subdivision surface 1 as our target subdivision surface. Its domain is a triangular mesh. In section 2 we introduce its basics. In our research we propose a path plan including two stages: rough-cut and finish-cut. The approach exploits LoD property of subdivision surface for these two stages. In other words, use a rough mesh for rough cut and a fine mesh for finish cut. The generated path plan is applied for three-axis machining. These two stages utilize a Z-map model. In section 3 we describe the Z-map model in detail. In sections 4 and 5 we present algorithms of the rough-cut path generation and the finish-cut path generation. In section 6 we propose collision detection and correction for controlling machining accuracy and quality. In section 7 the implementation of cutter path generation and their machining result are demonstrated. Finally in the last section we discussed conclusions.2. Loop Subdivision SurfaceIn this section we briefly introduce Loop subdivision surface. More details are available in 13. In 1987 Loop generalized the recurrence relations for box-spline to irregular meshes 19. The Loops subdivision scheme is based on the three-directional box-spline. It produces surfaces that are C2-continuous everywhere except extraordinary vertices. The extraordinary vertices are those whose number of adjacent vertices (valence) is not six, while ones with valence six are regular. The Loop subdivision scheme can be applied to arbitrary triangular meshes at the following two steps:1. Each edge of a triangle is divided to two edges.2. Each triangle is divided to four triangles.In these operations old vertices (called even vertices) are moved to new positions. At the same time new vertices are inserted into the edges, which are called odd vertices. The positions of the odd and even vertices are computed by Loop subdivision rules. There are four kinds of rules for an odd/even vertex lying inside a mesh (inner vertex), and an odd/even vertex on the boundary. By infinitely subdividing the initial triangular mesh (control mesh or control polygon), it converges to a smooth surface (limit surface). In this process a vertex on a control mesh approaches to a position on the limit surface. This limit position can be easily calculated without any subdivision using a formula similar to subdivision rules 13. In addition, the tangent vectors and also the normal vector of the limit surface at the limit position of a vertex can be computed.3. Z-map modelA Z-map model is a special form of discrete nonparametric surface representation in which the heights at the grid points are stored in a two-dimensional array, as depicted in Fig 3.1. It has been widely used in NC cutter path generation 2. The grid is determined by and , where is the reference point and is the grid interval.We first convert the subdivision surface to Z-map model in order to generate an NC cutter path. In this Zmap sampling process, for each grid point we sample its z-value by intersecting the line from the grid point along z-direction with the surface. After the Z-map sampling, we generate a cutter path by traversing each x (or y)-constant grid line so that the cutter contacts with the Z-map surface.4. Rough cut path generationBasically in order to remove large amount of volume we cut a workpiece layer by layer with flat-end mill, which was called slice-cut and also adopted in other previous research 10. The slice-cut is done along a rough-cut surface. Its thickness of a cuttinglayer can be decided by the tool parameters. The rough-cut surface should enclose the subdivision surface with a certain amount of gap (cutting allowance) and should be as simple as possible for fast computation. A straightforward insight is to use an intermediate coarse mesh as the rough-cut surface. However, this rough mesh does not always satisfy condition for no over-cut. So we convert it to a mesh, which approximately satisfies these conditions. We call it a cover mesh.4.1. Cover mesh generationFig 4.1 shows our basic idea for generating a cover mesh for a rough mesh illustrated in the two dimensional case. We move those vertices under thesurface above the surface as shown in the lower part of the figure in order to overcome the over-cut problem. These new positions are given by offsetting their limit positions by the cutting allowance. In this way it seems that we can generate a cover mesh that always lies above the surface.However the situation in the three dimensional space is not so easy. Some part of the limit surface may come above the mesh even though all the vertices are above the surface. Such a part can be detected by checking the local convex hull of the control mesh. In order to solve this problem we used Jos Stams evaluation method 4 to parametrically compute the points on the limit surface, as is the case with ordinary parametric surfaces. Firstly we evaluate a series of points along edges between two triangular faces as shown in Fig.4.2.Then we judge if a sampled point is below the two neighbor triangles. When there are some points located above these triangles, we find the highest point, and calculate the distance (dmax ) from the highest point to the evaluated edge. In Fig. 4.2, we offset vertices to their new positions V using Eq. (4.1),where is the cutting allowance and N is the normal at the vertex. On the boundary, we cannot evaluate the points on the edge curve, thus only is added.Fig. 4.3 illustrates an example of this method. Fig.4.3 (a) shows the original mesh M0 in red color. Fig.4.3 (b) shows the M1 mesh (subdivide M0 once) in green color. Fig 4.3(c) shows the limit surface. Fig 4.3(d) shows both M1 mesh in green color and limit surface in red color. It can be seen that M1 mesh intersects with the limit surface, in other words, the green part is intersecting with the red part. Fig 4.3 (e) shows the cover mesh generated from M1 with our method in blue color. In Fig 4.3 (f) the limit surface in red does not intersect with the cover mesh in blue. It means the cover mesh completely covers the limit surface. Although this method for the cover mesh is not exact, it is still an applicable way to make a rough-cut surface. Of course, we can apply intersection check between the subdivision surface and the cover mesh.Fig. 4.3 An example of generating cover mesh4.2. Z-map for rough-cut surfaceWe then generate a Z-map model for a cover mesh as a rough-cut surface by Z-map sampling. For a Z-map model, we set the grid interval for rough-cut surface to be half of the cutting allowance , which can be assigned by a user.where R is the radius of a tool.4.3. CL data computation for rough cutThe final step for rough-cut is to compute the CL (cutter location) data of each slice in order to be imported by a machining controller. After generating the Z-map model for a cover mesh, we slice it layerby-layer perpendicular to the z direction. For each x(or y)- constant grid line we define a polyline by sampling the Z-map along this line. Then we find intersection points between the slice plane and the polyline. All these intersection points define CC (cutter contact) points, and then offset by R to compute CL points. By connecting those CL points, an NC cutter path for this grid line is computed. This process is repeated for all the grid lines and also for all slice planes to generate the total rough-cut NC-cutter path.