外文资料翻译---多轴数控加工仿真的自适应固体

毕业设计(论文)外文资料翻译

系（院）：机械工程学院

专业：机械设计制造及其自动化

姓名：

学号：1091101630

外文出处：Computer-Aided Design & Applications,

V ol. 2, Nos. 1-4, 2005, pp95-104

附件： 1.外文资料翻译译文；2.外文原文。

附件1：外文资料翻译译文

多轴数控加工仿真的自适应固体

香港T. Yau1, Lee S. Tsou2 and Y u C. Tong3

1中正大学，imehty@https://www.360docs.net/doc/2f18965511.html,.tw

2中正大学，lstsou@https://www.360docs.net/doc/2f18965511.html,.tw

3 中正大学，pu@https://www.360docs.net/doc/2f18965511.html,.tw

摘要：

如果在一个复杂的表面的加工中，通常会产生大量的线性NC段来近似精确的表面。如果没有发现，直到切割不准确的NC代码，则会浪费时间和昂贵的材料。然而，准确和视图独立验证的多坐标数控加工仍然是一个挑战。本文着重介绍了利用自适应八叉树建立一个可靠的多轴模拟程序验证模拟切割期间和之后的路线和工件的外观。体素模型的自适应八叉树数据结构是用来加工工件与指定的分辨率。隐函数的使用刀具接触点的速度和准确性的检验，以代表各种刀具的几何形状。它允许用户做切割模型和原始的CAD模型的误差分析和比较。在加工前运行数控机床，以避免浪费材料，提高加工精度，它也可以验证NC代码的正确性。

关键词：数控仿真加工，固体素模型，自适应

1.介绍

NC加工是一个基本的和重要的用于生产的机械零件的制造过程。在理想的情况下，数控机床将运行在无人值守模式。使用NC仿真和验证是必不可少的，如果要运行的程序有信心在无人操作。因此，它是非常重要的，在执行之前，以保证NC路径的正确性。从文学来说，数控仿真主要分为三种主要方法，如下所述。

第一种方法使用直接布尔十字路口实体模型来计算材料去除量在加工过程。这种方法在理论上能够提供精确的数控加工仿真，但使用实体建模方法的问题是，它是计算昂贵。使用构造实体几何仿真的成本刀具运动的O（N 4）的数量的四次幂成正比。第二种方法使用空间分割表示，代表刀具和工件。在这种方法中，一个坚实的对象被分解成一个集合的基本几何元素，其中包括体素并，dexels，G-缓冲器，依此类推，从而简化了过程的正规化布尔操作。第三种方法使用离散矢量路口。这

种方法是基于对一个表面成的一组点的离散化。切割是模拟计算通过与刀具路径信封的表面点的矢量的交点。

在多轴数控加工，切削刀具频繁地旋转，以便计算出工件的模型，该模型是依赖于视图，这是很困难的。因此在本文中，我们使用的体素的数据结构来表示的工件模型。不过，根据过去的文献，如果精度是必要的，大量的像素，必须设立执行布尔操作。这会消耗内存和时间。因此，我们的方法是使用八叉树的数据结构来表示的工件模型。八叉树可以适于创建与所需决议所需要的体素。我们利用八叉树的快速搜索与刀具接触的体素。然而，我们的方法使用了一个隐式的函数来表示的切削工具，因为切割器可以容易且准确地表示的隐式代数方程，并判断切割器保持在与工件接触也很容易。因此，我们的方法是可靠和准确的。

论文内容安排如下。第2节讨论的工件表示，使用八叉树体素模式。第3节给出用于表示各种刀具的几何形状的隐函数。第4节概述了该算法的3轴数控加工仿真程序。第5节说明了所提出的方法可以很容易地适应五轴联动数控仿真通过扩展的隐函数来容纳五轴旋转。实例证明所提出的方法的有效性和简单的。第6节说明了NC仿真所需的存储器空间和计算时间的实验结果。最后，结论在第7节。

2.立体几何体素表示

在本文中，我们使用一个像素的数据结构来表示研磨工件的自由形式的几何体素的新型固体，因为模型的轴对准和视图独立的性质。同时利用八叉树来避免创建大量的体素。该方法判断刀具保持与工件接触，发现接触的所有体素，然后将这些体素空间分辨率达到八像素递归直到达到所需的精度水平。因此，如果有与刀具接触体素，也没有必要细分模型。图1显示了八叉树数据结构和它所代表的体素模型。

图1. 八叉树数据结构和相关的体素模型

传统上，由于加工仿真采用均匀的体素数据结构来表示一个研磨工件，精度提高了体素数据时，将产生大量的工件。这将使加工仿真变慢，这是因为大量计算机内存的需要。因此，我们使用八叉树数据结构的自适应创造体素，需要仿真。

3. 刀具几何使用隐式函数表示

空间均匀的体素分割方法未能解决多维数控验证相当复杂与准确的工件。如果需要高精度，大量的体素必须成立进行布尔集合运算。这将消耗大量的内存和时间。但我们三轴仿真的新方法采用自适应的体素模型来表示一个研磨的工件，并使用隐式函数来表示的刀具实体模型。由于刀具不分解成一个集合的基本几何元素，因此可以实现高的精度。同时工件模型利用八叉树模型来减少不必要的体素。下面，我们描述了使用隐函数表示的各种刀具。

平立铣刀可以由一个圆柱体代表。图2显示平立铣刀切削方向一致。如果工具是平行于Z轴，且坐标系统转换，中心点位于原点。因此，一个平面铣刀的隐函数：

F ( X , Y , Z ) = max{abs ( Z ?L/2 ) ? L/2, X 2 + Y 2? R 2 } if Z≥ 0

该隐函数被用于确定体素在里面，外面，或交叉的刀不损失任何精度。裁判可以通过插入一个像素顶点坐标为隐函数。方程式（2）描述了一个顶点与刀具间的关系，如图3所示。

< 0 在内部表面

F ( X , Y , Z) = 0 在表面（2）

> 0 在外部表面

表示

R：刀具半径

L：从沿刀轴的中心点开始测量距离

图2：平面铣刀和相关的坐标系统

图3：隐函数用于确定刀具的内部或外部

球立铣刀是由一个圆柱和一个球体组成，如图4所示。如果工具是平行于Z轴，且坐标系统的原点平移到球体的中心，则一个球立铣刀的隐函数可以被描述为：

max{ abs ( Z ) ? L , X 2 + Y 2? R 2 } 若Z ≥ 0

F ( X , Y , Z )= （3）

X 2 + Y 2 + Z 2 ? R 2其他

表示

R：刀轴刀角中心径向距离

r：刀角半径

L：从沿刀轴的中心点开始测量距离

图4：球立铣刀和相关的坐标系统

圆角立铣刀可以由两个气缸和一个圆环表示。如果工具是平行于Z轴，且坐标系统转换到中心点，如图5所示，圆角端铣刀隐函数可推导为：

max{ abs ( Z ) ? L , X 2 + Y 2?（R+r）2 } 若Z ≥ 0

F ( X , Y , Z )= max{ abs ( Z ) ? L , X 2 + Y 2? R 2 } 否则abs（X）≤R

（X 2 + Y 2 + Z 2 +R 2）?4R2(X 2+ Z 2) 其他

表示

R：刀轴刀角中心径向距离

r：刀角半径

L：从沿刀轴的中心点开始测量距离

图5：圆角铣刀和相关的坐标系统

作为一个简单的平面铣刀，或复杂的圆角立铣刀，隐函数可以用来确定一点是

否是内部或外部的刀具直接应用的方程式。隐函数表示刀具的使用不仅是精确的几何形状，简单的概念，而且隐函数编程也很容易和简单。因此，我们可以很容易地知道隐式函数F（x，y，z）的存量和刀具之间的几何关系。

4. 三轴数控加工仿真

配制后的切削刀具的隐函数，需要被执行的一项重要任务是确定哪些需要在球磨过程中细分或删除的体素。图6则表示一个三轴NC路径模拟流程图：

图6：三轴加工仿真流程图

三轴NC路径模拟过程描述如下：

（1）第一步是读NC代码。然后我们可以得到每个数控段的开始和结束的刀具位置（CL）的刀具运动点。因此，三轴运动模型是任何两个工具的配置点位置的联合结构的CL插值。

（2）刀具的边界框是用来初步判断刀具接触体素模型的哪部分。其目的是摆脱不与刀具的体素接触。如果确定体素是与刀具接触，体素的顶点将被取代刀隐函数来决定是否像素顶点位于刀具的内部或外部。

（3）如果所有的体素点符合条件的F（x，y，z）＜0时，则确认它的体素已被完全切断刀；也就是说，体素在落刀应该被消除。如果顶点部分落在里面和其他人以外，这意味着需要进一步划分体素。为了细分每一个体素，步骤（2）和步骤（3）将进行递归，直至达到预定的精度水平。

在NC路径当前段完成之后，步骤1再次上演，读取下一段数控代码。该程序是

遵循直到所有NC代码已读才结束。

在上述三轴加工仿真程序中，它是明确的，像素被细分需要根据刀具与工件之间的几何关系；大量的体素不一次全部在开始创建。图7则表明体素通过八叉树只与一个球立铣刀接触将细分。体素不与刀具接触则不会细分。因此，这种方法大大降低了体素的数量，节省了空间。

图7：三轴数控加工仿真

在Z-map模型的比较，使用体素模型仿真的结果是可以显示多轴加工。DEXEL 模型的视图有相关的限制，但体素模型没有这个限制。因此，三轴、五边加工可以利用本文的三轴仿真方法，如图8所示。

图8：三轴和五边的仿真实例

5. 五轴联动数控加工仿真

在五轴加工中，除了三个平移运动，刀具轴也会旋转。因此，我们对五轴仿真方法只修改刀隐函数。所有其他的步骤与三轴仿真是相同的。因此，刀具的三种可以表示如下：

平立铣刀可以由一个圆柱体代表。图9结果表明刀具轴沿着{n}，中心点位于{ p}。因此，一个平面铣刀的隐函数：

F ( X , Y , Z ) = ? ({ x } ? { p }) T [ n ] 2({ x }?{ p }) ? R 2

若0 ≤ { n } T({ x } ? { p })≤L(5) 表示

R：刀具半径

{ x } ={ X Y Z}T：一个像素点的位置

{ n } ={ n x n y n z}T ：刀具轴的单位矢量

{ p } ={ p x p y p z }T：中心点

0 -n z n y n x2-1 n x n y n x n z

[n]= n z 0 -n x [n]2= n x n y n y2-1 n y n z

-n y n x 0 n x n z n y n z n z2-1

图9：五轴旋转的平面铣刀模式

球立铣刀可以由一个圆柱和一个球体，工会代表。图10则结果表明刀具轴沿{ N}和中心点位于{P }。因此，一个球立铣刀的隐函数：

?({ x } ? { p }) T [ n ]2 ({ x } ? { p }) ? R 2

F ( X , Y , Z ) = 如果0 ≤ { n } T ({ x } ? { p }) ≤ L

({ x } ? { p }) T({ x } ? { p })? R2 其他

表示

R：刀具半径

{n}：刀具轴的单位矢量

图10：五轴模式旋转球立铣刀

圆角立铣刀可以由两个气缸和一个圆环工会代表。图11则结果表明刀具轴沿｛n}，中心点位于{P}。因此，一个圆形立铣刀的隐函数：

( ? { v } T [ n ] 2{ v }) ? ( R + r ) 2若0 ≤ { n } T{ v } ≤ L

F ( X , Y , Z ) = ( ? { v } T [ n ] 2{ v }) ? R 2否则0 ≤ { n } ({ v } ? r { n }) < r and

( ? { v }T [ n ]2{ v }) ≤ R 2

(( ? { v } T [ n ] 2 { v })+({n}T{v})2+R2-r2 )+4R2({ v } T [ n ] 2 { v })

表示

R：从刀轴刀角中心径向距离

r：刀角半径

{n}：刀具轴的单位矢量

{ v } = { x } ? { p}

图11：五轴式旋转圆角立铣刀

图12显示了一个简单的例子，从70度到50度左右的X轴和沿X轴移动的刀具轴的旋转。图13显示用于五轴加工叶轮的刀具路径和仿真过程。图14显示叶片五轴数控加工仿真的另外一个例子。

图12：五轴数控加工仿真。

（a）(b)

(c) (d)

图13：五轴模拟叶轮的例子。（a）刀具路径，（b）（C）与刀具加工工件，（d）完成的部分

(a) (b)

（c）(d)

图14：五轴模拟叶片的例子。（a）刀具路径，（b）（C）与刀具加工工件，（d）完成的部分

6. 实验结果

该方法已运行在2.4 GHz的奔腾4电脑，实现了在C + +和一些测试用例。标签1给出了自适应数控仿真所需要的内存空间和计算时间的比较。第一行显示的图片的数控轨迹的四个不同的模型。第二和第三行显示NC代码的信息。第四行是工件模型的分辨率。刀具模型由隐式函数正确表达，所以没有精度问题，第五行显示刀具使用的类型，最后三行是所需要的内存空间，计算时间和所提供的仿真结果。

推动

标签1：需要的内存空间和计算时间的自适应数控仿真。

标签2给出了所需要的内存空间，使用均匀的体素模型的数控加工仿真的计算时间的比较。为适应数控加工仿真的参数保持不变。在这种情况下，我们不能够模拟案例1（叶轮）和成功案例3（鞋），因为这样的案例超过内存限制。可以观察到的结果是壳体2（刀片）和壳体4（瓶）。相比较而言，该自适应数控仿真的优势是显而易见的，通过实现使用自适应数控仿真，时间和空间可以大大减少

标签2：所需的存储空间和均匀的体素模型的数控仿真计算时间。

7.结论

在本文中，我们提出了一个新的多轴模拟方法。本文的目的是使用自适应的体素模型来建立一个可靠的多轴模拟程序，可以模拟的切割路线和模拟之后的工件的外观。它允许用户进行切削模型与原始CAD模型比较及误差分析。它可以在加工数控机床之前验证其NC代码的准确性，以避免浪费材料和提高加工精度。

总之，对多轴模拟方法的优点如下所述。（1）模拟的方法比其他基于体素模拟的方法使用更少的内存。（2）仿真是独立的。DEXEL模型的视图有相关的限制，

而体素模型没有这个限制。（3）模拟是可靠和准确的。无论刀具的五轴还是三轴仿真都是采用隐函数来表示，整个方法简单可靠。

8.参考文献

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Kluwer Academic Publishers, 1998.

[2]. Wang, W. P., Wang, K. K., Geometric Modeling for Swept V olume of Moving Solids,

IEEE Computer Graphics & Applications, V ol. 6, No.12, 1986, pp 8-17

[3]. Atherton, P. R., A Scan-Line Hidden Surface Removal Procedure for Constructive

Solid Geometry, Computer Graphics, V ol. 17, No. 3, 1983, pp 73-82.

[4]. Kawashima, Y., Itoh, K., Ishida, T., Nonaka, S., Ejiri, K., A Flexible Quantitative

Method for NC Machining Verification Using a Space-Division Based Solid Model, The Visual Computer, V ol. 7, 1991, pp 149-157.

[5]. Jang, D., Kim, K., Jung, J., V oxel-Based Virtual Multi-Axis Machining, Advanced

Manufacturing Technology, V ol. 16, No. 10, 2000, pp 709-713.

[6]. Van Hook, T., Real Time Shaded NC Milling Display, Computer Graphics, V ol. 20,

No. 4, 1986, pp 15-20.

[7]. Huang, Y., Oliver, J. H., Integrated Simulation, Error Assessment, and Tool Path

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[8]. Saito, T., Takahashi, T., NC Machining with G-buffer Method, Computer Graphics,

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for A Multi-Axis Milling System, ASME Control of Manufacturing Process, DSC-V ol.28/PED-V ol.52, 1991, pp 23-30.

附件2：外文原文

Adaptive NC Simulation for Multi-axis Solid Machining

Hong T. Yau 1 , Lee S. Tsou 2 and Yu C. Tong 3

1National Chung Cheng University，imehty@https://www.360docs.net/doc/2f18965511.html,.tw

2National Chung Cheng University，lstsou@https://www.360docs.net/doc/2f18965511.html,.tw

3National Chung Cheng University，pu@https://www.360docs.net/doc/2f18965511.html,.tw

ABSTRACT

For the machining of a complicated surface, a large amount of linear NC segments are usually generated to approximate the surface with precision. If inaccurate NC codes are not discovered until the end of cutting, time and expensive material would be wasted. However, accurate and view-independent verification of multi-axis NC machining is still a challenge. This paper emphasizes the use of adaptive octree to develop a reliable multi-axis simulation procedure which verifies the cutting route and the workpiece appearance during and after simulation. V oxel models with adaptive octree data structure are used to approximate the machined workpiece with specified resolution. Implicit functions are used to represent various cutter geometries for the examination of cutter contact points with speed and accuracy. It allows a user to do error analysis and comparison between the cutting model and the original CAD model. It can also verify the exactness of NC codes before machining is carried out by a CNC machine in order to avoid wasting material and to improve machining accuracy.

Keywords: NC Simulation, Solid Machining, Adaptive V oxel Model

1.INTRODUCTION

NC machining is a fundamental and important manufacturing process for the production of mechanical parts. Ideally, an NC machine would be running in unmanned mode. The use of NC simulation and verification is essential if programs are to be run with confidence during an unmanned operation . Therefore, it is of vital importance to guarantee the exactness of NC paths before execution. From literature, NC simulation can mainly be divided into three major approaches , described as follows.

The first kind of approach uses direct Boolean intersections of solid models to calculate the material removal volumes during machining . This approach is theoretically capable of providing accurate NC simulation, but the problem with using the solid modeling approach is that it is computationally expensive. The cost of simulation using constructive solid geometry is proportional to the fourth power of the number of tool movements O(N 4 ) . The second kind of approach uses spatial partitioning representation to represent the cutter and the workpiece. In this approach, a solid object is decomposed into a collection of basic geometric elements, which include voxels , dexels，G-buffers , and so on, thus simplifying the processes of regularized Boolean set operations. The third

kind of approach uses discrete vector intersection . This method is based on a discretization of a surface into a set of points. Cutting is simulated by calculating the intersection of vectors which pass through the surface points with tool path envelopes. During multi-axis NC machining, the cutting tool frequently rotates so that it is very difficult to calculate a workpiece model that is view-dependent. Thus, in this paper, we use the voxel data structure to represent the workpiece model. But according to past literature, if precision is needed, a large number of voxels must be set up to carry out Boolean set operations. This consumes memory and time. Thus, our approach uses the octree data structure to represent the workpiece model. The octree can be adapted to create voxels with the desired resolutions that are needed. We utilize the octree to quickly search for voxels which have contact with the cutter. However, our approach uses an implicit function to represent the cutting tool because a cutter can be easily and exactly represented by implicit algebraic equations, and judging whether the cutter keeps in contact with the workpiece is also easy. Thus, our approach is reliable and precise.

The content of the paper is organized as follows. Section 2 discusses the workpiece representation using octree based voxel modes. Section 3 presents the formulation of implicit functions used to represent the geometry of various cutters.Section 4 outlines the procedure of the proposed algorithm for 3-axis NC simulation. Section 5 shows how the proposed approach can be easily adapted to 5-axis simulation by extending the implicit functions to accommodate for the 5-axis rotation. Examples are given to demonstrate the effectiveness and simplicity of the proposed approach. Section 6 shows the experimental results of the required memory space and computation time for NC simulation. At the end, conclusions are made in Section 7.

2.VOXEL REPRESENTATION OF SOLID GEOMETRY

In this paper, we use a voxel data structure to represent the free form solid geometry of a milled workpiece because a voxel model has axis alignment and view-independent properties. At the same time we use the octree to avoid creating a large number of voxels. The method judges whether the cutter keeps in contact with the workpiece, finds all voxels which have such contact, and then subdivides these voxels into eight voxels in space recursively until the resolution reaches the desired precision level. Thus, if there is no voxel in contact with the cutter, there is no need to subdivide the voxel. Fig. 1. shows the Octree data structure and the voxel model it represents.

Fig. 1. Octree data structure and the associated voxel model

Traditionally, since machining simulation uses the uniform voxel data structure to represent a milled workpiece, when precision increases, great quantity of voxel data will be produced to represent the workpiece. This will make the machining simulation slow because a lot of computer memory is needed. Thus, we use the octree data structure to adaptively create voxels as needed during simulation

3.REPRESENTATION OF CUTTER GEOMETRY USING IMPLICIT FUNCTIONS The spatial partitioning approach with uniform voxels has failed to address multi-dimensional NC verification for workpieces of comparable complexity and accuracy. If high precision is needed, a large number of voxels must be set up to carry out Boolean set operations. This will consume considerable amount of memory and time. But our new approach for three-axis simulation uses the adaptive voxel model to represent a milled workpiece, and uses implicit functions to represent a solid model of the cutter. Since the cutter is not decomposed into a collection of basic geometric elements, high accuracy can be achieved. At the same time the workpiece model makes use of the octreemodel to reduce the number of unnecessary voxels. In the following, we describe the representation of various cutters using implicit functions.

Flat endmills can be represented by a cylinder. Fig. 2. shows the flat endmill aligned with the direction of cutting. Assuming the tool is parallel to the z-axis, and the coordinate system is translated such that the center point is located at the origin. Thus, the implicit function of a flat endmill is：

F ( X , Y , Z ) = max{abs ( Z ?L/2 ) ? L/2, X 2 + Y 2? R 2 } if Z≥ 0

This implicit function is used to determine whether a voxel is inside, outside, or intersected with the cutter without losing any accuracy. The judgment can be made by inserting the coordinates of the vertices of a voxel into the implicit function. Eqn. (2) describes the relationship between a vertex and the cutter, which is also illustrated in Fig.

3.

< 0 lie inside the surface

F ( X , Y , Z) = 0 lie on the surface （2） > 0 lie outsid the surface

where

R : the cutter radius

L : the distance measured from the center point along the cutter axis

Fig. 2. Flat endmill and the associated coordinate system

Fig. 3. Implicit function used to determine the interior or exterior of a cutter

Ball endmills can be represented by the union of a cylinder and a sphere, as shown in Fig.

4. Assuming the tool is parallel to the z-axis, and the origin of the coordinate system is translated to the center of the sphere, the implicit function of a ball endmill can be described as:

max{ abs ( Z ) ? L , X 2 + Y 2? R 2 } if Z ≥ 0

F ( X , Y , Z )= （3）

X 2 + Y 2 + Z 2 ? R 2otherwise

where

R : the cutter radius

L : the distance measured from the center point along the cutter axis

Fig. 4. Ball endmill and the associated coordinate system

Fillet endmills can be represented by the union of two cylinders and a torus. Assuming the tool is parallel to the z-axis, and the coordinate system is translated to the center point as shown in Fig. 5., the implicit function of a fillet endmill can be derived as:

max{ abs ( Z ) ? L , X 2 + Y 2?（R+r）2 } if Z ≥ 0

F ( X , Y , Z )= max{ abs ( Z ) ? L , X 2 + Y 2? R 2 } else if abs（X）≤R

（X 2 + Y 2 + Z 2 +R 2）?4R2(X 2+ Z 2) otherwise

Where

R : the radial distance from the cutter axis to the cutter corner center

r : the cutter corner radius

L : the distance measured from the center point along the cutter axis

Fig. 5. Fillet endmill and the associated coordinate system

As simple as a flat endmill, or as complex as a fillet endmill, the implicit functions can be used to decide whether a point is inside or outside a cutter by a direct application of Eqn.

(2). The use of implicit functions to represent a cutter is not only precise in geometry, simple in concept, and the programming of implicit functions is also very easy and straightforward. Thus, we can easily know the geometric relationship between the stock and the cutter by using the implicit functionF ( X , Y , Z).

4.SIMULATION OF THREE-AXIS NC MACHINING

After the implicit function of the cutting tool is formulated, an important task that needs to be performed is to determine which voxels need to subdivided or deleted during the milling process. Fig. 6. shows the flow chart of a three-axis NC path simulation.

Fig. 6. Flow chart of three-axis machining simulation

The process of a three-axis NC path simulation is described as follows.

(1)The first step is to read the NC codes. Then for each NC segment, we can get the start

and end cutter location (CL) points of a tool motion. Thus, the three-axis motion is modeled by a joint interpolation of CL points of the configuration of any two tool positions.

(2)The bounding box of the cutter is used preliminarily to judge which part of a voxel

model the cutter is in contact with. The purpose is to get rid of voxels not in contact with the cutter. If it is determined that a voxel is in contact with the cutter, the voxel vertices will be substituted into the implicit function of the cutter to decide if the voxel vertices lie inside or outside the cutter.

(3)If all of the voxel vertices meet the conditions F ( X , Y , Z) <0, it is confirmed the

voxels have been totally cut by the cutter; that is to say that the voxel falling in the cutter should be eliminated. If part of the vertices fall inside and the others outside, it means the voxel needs to be further divided. For each subdivided voxel, step (2) and step (3) will be carried out recursively until a predetermined precision level is reached.

After the current segment of the NC path is finished, step 1 is performed again, and the next segment of the NC code is read. The procedure is followed to the end until all NC codes have been read.

In the procedure for three-axis machining simulation mentioned above, it is clear that

voxels are subdivided as needed according to the geometric relationship between the cutter and the workpiece; a large number of voxels are not created all at once in the beginning. Fig. 7. shows that voxels in contact only with a ball endmill will be subdivided by the octree. V oxels not in contact with the cutter will not be subdivided. Thus, this approach greatly reduces the number of voxels and saves memory during simulation.

Fig. 7. Three-axis NC machining simulation.

In comparison with z-map model, NC simulation using voxel model can display multi-axis machining result. The dexel model has the restriction of being view-dependent, but the voxel model does not have this restriction. Thus, three-axis, five-side machining can utilize the three-axis simulation method of this paper, as Fig. 8. shows.

Fig. 8. Example of three-axis and five-side simulation

5.SIMULATION OF FIVE-AXIS NC MACHINING

In five-axis NC machining, in addition to the three translation movements, the tool axis can also be rotated. Therefore, our approach to five-axis simulation only revises the implicit function of the cutter. All other procedures are the same as the three-axis simulation. Thus, the three kinds of cutters can be expressed as follows:

Flat endmills can be represented by a cylinder. Fig. 9. shows that the tool axis is along{n?} and the center point islocated at{p} . Thus, the implicit function of a flat endmill is:

F ( X , Y , Z ) = ? ({ x } ? { p }) T [ n ] 2({ x }?{ p }) ? R 2

If 0 ≤ { n } T({ x } ? { p })≤L(5) Where

R: the cutter radius

{ x } ={ X Y Z}T：the position of a voxel vertex

{ n } ={ n x n y n z}T ：the unit vector of the tool axis

{ p } ={ p x p y p z }T：the center point

0 -n z n y n x2-1 n x n y n x n z

[n]= n z 0 -n x [n]2= n x n y n y2-1 n y n z

-n y n x 0 n x n z n y n z n z2-1

Fig. 9:Flat endmill rotated in five-axis mode

Ball endmills can be represented by the union of a cylinder and a sphere. Fig. 10. shows that the tool axis is along{n and the center point is located at{p} . Thus, the implicit function of a ball endmill is:

?({ x } ? { p }) T [ n ]2 ({ x } ? { p }) ? R 2

F ( X , Y , Z ) = if 0 ≤ { n } T ({ x } ? { p }) ≤ L

({ x } ? { p }) T({ x } ? { p })? R2 otherwise

Where

R : the cutter radius

{ n }: the unit vector of the tool axis

Fig. 10. Ball endmill rotated in five-axis mode

Fillet endmills can be represented by the union of two cylinders and a torus. Fig. 11. shows that the tool axis is along{n}and that the center point is located at{p} . Thus, the implicit function of a round endmill is:

( ? { v } T [ n ] 2{ v }) ? ( R + r ) 2若0 ≤ { n } T{ v } ≤ L

F ( X , Y , Z ) = ( ? { v } T [ n ] 2{ v }) ? R 2否则0 ≤ { n } ({ v } ? r { n }) < r and

( ? { v }T [ n ]2{ v }) ≤ R 2

(( ? { v } T [ n ] 2 { v })+({n}T{v})2+R2-r2 )+4R2({ v } T [ n ] 2 { v }) Where

R : the radial distance from the cutter axis to the cutter corner center

r : the cutter corner radius

{ n } : the unit vector of the tool axis

{ v } = { x } ? { p}

Fig. 11. Fillet endmill rotated in five-axis mode

Fig. 12. shows a simple example of a cutter axis rotated from 70 degrees to 50 degrees about the x-axis and moved along the x-axis. Fig. 13. shows the tool paths and simulation process used for five-axis machining of an impeller. Fig. 14. shows another example of five-axis machining simulation of a blade.

.

Fig. 12. Five-axis machining simulation

(a) (b)

(c) (d)

Fig. 13. Example of impeller in five-axis simulation. (a) Tool paths. (b)(c) In-process workpiece with a cutter. (d)

Finished part.

(a) (b)

(c) (d)

Fig. 14. Example of blade in five-axis simulation. (a) Tool paths. (b)(c) In-process workpiece with a cutter. (d) Finished

part.

6.EXPERIMENTAL RESULTS

The proposed method has been implemented in C++ and some test cases were run on a 2.4 GHZ Pentium 4 computer. Tab. 1. gives a comparison of the required memory space and computation time for adaptive NC simulation. The first row shows the pictures of NC path for four different models. The second and third rows show the information of NC code. The fourth row is the resolution of the workpiece model. The cutter models are presented by implicit functions exactly, so there is no accuracy issue here. The fifth row shows the types of cutter being used. The last three rows are the required memory space,

Impeller

Tab. 1. Required memory space and computation time for adaptive NC simulation.

Tab. 2. gives a comparison of the required memory space and computation time for NC simulation using uniform voxel models. The parameters remain the same as adaptive NC simulation. Under this condition, we are not able to simulate case1 (Impeller) and case3 (shoe) because such cases exceed our memory limitation. The results that can be observed are case 2 (blade) and case 4 (bottle). By comparison, the advantage of the adaptive NC simulation is clear. A great reduction of time and space can be achieved by using the adaptive NC simulation.