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用普通刀具在立體平版印刷格
式中對三角雕塑面的數(shù)控加工
一個(gè)產(chǎn)生數(shù)字控制刀具路徑的統(tǒng)一方法已經(jīng)出現(xiàn)，這個(gè)刀具路徑是用普通刀具在立體平版印刷格式中對三角形雕塑表面的數(shù)字控制加工產(chǎn)生的。這個(gè)方法的產(chǎn)生很重要，這是是因?yàn)橐粋€(gè)立體平板印刷格式的應(yīng)用象征著一個(gè)計(jì)算機(jī)輔助設(shè)計(jì)模型已經(jīng)在很短的一段時(shí)間內(nèi)被工業(yè)界廣泛接受。這不僅是因?yàn)楸热缣貏e需要運(yùn)用這種方法的快速設(shè)計(jì)模型的應(yīng)用，而且還歸結(jié)于現(xiàn)在可以直接用數(shù)字化和反向工程過程來制造復(fù)雜的立體平板印刷模型。雖然有很多支持立體平板印刷文件的計(jì)算機(jī)輔助設(shè)計(jì)和計(jì)算機(jī)輔助制造的軟件系統(tǒng)，但在這些文章中只有幾頁紙直接涉及到從立體平板印刷文件來的數(shù)字控制加工問題。一種用普通自動(dòng)編程切削刀具產(chǎn)生刀具路徑的一般計(jì)算算法已經(jīng)出現(xiàn)。這種算法看起來普遍，它可適用于包括球頭刀、平底刀和內(nèi)圓角精銑刀在內(nèi)的各種各樣的切削刀具。為了減少計(jì)算時(shí)間，一個(gè)用彩色小石塊鑲嵌產(chǎn)生網(wǎng)狀詢問區(qū)域高的效方法也已經(jīng)產(chǎn)生。用模仿加工和真正的機(jī)械加工制造的實(shí)例來說明所提出的方法的效率。
1、介紹
目前，多數(shù)計(jì)算機(jī)輔助設(shè)計(jì)(CAD) 系統(tǒng)利用參數(shù)表面代表CAD 模型的幾何形狀。由于各種各樣的設(shè)計(jì)或制造過程的不同，需要轉(zhuǎn)換不同的計(jì)算機(jī)輔助設(shè)計(jì)系統(tǒng)（CAD）和計(jì)算機(jī)輔助制造系統(tǒng)（CAM）之間的模型。中立數(shù)據(jù)文件譬如最初的圖表交換規(guī)格(IGES) 廣泛地被用于(美國產(chǎn)品數(shù)據(jù)協(xié)會(huì)1996) 。最初的圖表交換規(guī)格(IGES)描述在建立CAD 模型時(shí)可能被使用的信息、定義個(gè)體模型時(shí)的參數(shù)量和各個(gè)個(gè)體之間不同的關(guān)系。但是，使用最初的圖表交換規(guī)格(IGES)翻譯CAD 模型并不是總是容易的，這是因?yàn)榇蠖鄶?shù)計(jì)算機(jī)輔助設(shè)計(jì)系統(tǒng)（CAD）使用不同的內(nèi)部表示方法而且轉(zhuǎn)換并不是總是直接的而錯(cuò)誤又比較多。與最初的圖表交換規(guī)格(IGES) 相比，立體平版印刷(STL) 格式比較簡單，并且它的實(shí)施比較容易 (Jacobs 和Reid 1992 年, Kochan 1993) ?；旧希粋€(gè)立體平版印刷(STL)文件只包含一個(gè)三角形和它們的法線傳播媒介。立體平版印刷(STL)文件不能替換最初的圖表交換規(guī)格(IGES)，最初的圖表交換規(guī)格(IGES)包含更多與設(shè)計(jì)相關(guān)的信息，然而對于許多順流制造業(yè)活動(dòng)譬如迅速設(shè)計(jì)模型、數(shù)字控制(NC) 加工制造甚至有限元素分析來說，這些信息包含在立體平版印刷(STL)之中的信息是充足的。由于它的簡單和在各種各樣的工程學(xué)領(lǐng)域中的不同用途，立體平版印刷(STL)翻譯受到大多數(shù)計(jì)算機(jī)輔助設(shè)計(jì)和計(jì)算機(jī)輔助制造（CAD/CAM）系統(tǒng)的支持。在過去，立體平版印刷(STL)文件是負(fù)擔(dān)對內(nèi)存分配和計(jì)算速度的任務(wù)。但是，隨著中央處理單元的加速，更多力量和存儲(chǔ)芯片逐漸變得更加便宜，這不再是轉(zhuǎn)換和處理立體平版印刷(STL) 文件時(shí)的一個(gè)障礙。此外，最新的三維掃描技術(shù)也促使反向工程的應(yīng)用迅速增長，在反向工程應(yīng)用中，創(chuàng)造很大并且很復(fù)雜的模型后，將它存放在立體平版印刷(STL)文件之中(Chuang 等2002) 。人們現(xiàn)在已經(jīng)普遍接受這樣一個(gè)事實(shí)，分成三角形的表面和立體平版印刷(STL) 文件的應(yīng)用將使設(shè)計(jì)和制造業(yè)應(yīng)用變得越來越普遍。
在過去，人們已經(jīng)學(xué)到了許多三軸機(jī)械加工刀具路徑的計(jì)劃方法 (Dragomatz 和Mann 1997) 。刀具路徑的產(chǎn)生方法可以分成兩種類型： 解析和參數(shù)(Zeid 1991) 。前者產(chǎn)生于橫切機(jī)械加工表面的短剖面飛機(jī)。而后者產(chǎn)生于沿著平面或者曲面走刀的數(shù)字控制刀具路徑，而刀具切削點(diǎn)(CL) 通常是用計(jì)算機(jī)從機(jī)械制造加工表面的設(shè)置來進(jìn)行計(jì)算的(Kishinami等1987 年，Tang 等1995 年，Choi 等1997 年，Lee 2003 年)。參數(shù)設(shè)置方法在應(yīng)用精確表面信息之中有其特定的優(yōu)點(diǎn)，但是它可能不適合應(yīng)用于加工帶有很多凸臺(tái)的復(fù)合表面并且很容易受到凹平面的損壞(Choi 和Jerard 1998) 。在另一方面，解析的方法的優(yōu)點(diǎn)是可以產(chǎn)生沒有凹平面損壞現(xiàn)象的刀具路徑，但是它的缺點(diǎn)的是不能產(chǎn)生直線的走刀路徑和進(jìn)行清角加工的刀具路徑(Dragomatz 和Mann 1997) 。所以，在零件的現(xiàn)實(shí)加工之中，解析和參數(shù)路徑的應(yīng)用策略在互換性之中被運(yùn)用。立體平版印刷(STL)加工主要是應(yīng)用解析的加工策略，這是因?yàn)樗话型暾谋砻嫘畔?。在解析路徑?jì)劃之中，當(dāng)切削刀具接觸到機(jī)械加工表面時(shí)， CLs就被計(jì)算出來。在解析路徑計(jì)劃之中一個(gè)最有效的方法是繪制Z軸 (Choi 等1988 年，Choi 1991 年，Saito 和Takahashi 1991 年，林和劉1998) 。繪制Z軸的方法計(jì)算柵格數(shù)據(jù)庫設(shè)置中的無干涉CLs。機(jī)械加工是精確度取決于庫柵格數(shù)據(jù)的密度。這通常是一個(gè)對柵格數(shù)據(jù)庫大的存儲(chǔ)空間的分配的需要。Hwang 和他的同事提出這樣一個(gè)方法，就是用平底刀、球頭刀和圓角精銑刀從三角形表面產(chǎn)生無干涉機(jī)械機(jī)械刀具路徑（Hwang 1992 年，Hwang 和Chang 1998) 。但是，這種方法是相對于不同的切削刀, 而且它的算法限制可切削刀類型的開發(fā)。盡管在現(xiàn)實(shí)之中使用著更多不同的切削刀具類型。例如，經(jīng)常用一把細(xì)而利的精銑刀具作為淺槽的標(biāo)號。它會(huì)繁瑣而笨拙地產(chǎn)生所有需要的刀具的不同算法和代碼。
本論文介紹一種直接產(chǎn)生刀具路徑的統(tǒng)一方法，這種方法是用通用的自動(dòng)編程加工刀具(APT)在立體平版印刷(STL)的三角形表面上產(chǎn)生的(Kral 1986)。APT切削刀的拓?fù)浣Y(jié)構(gòu)定義通常被用于數(shù)字控制的應(yīng)用實(shí)例，但大多數(shù)刀具路徑的形成方法都是為特殊的刀具類型開發(fā)的，不的通用的 (Chung?等1998 年，Chiou and Lee 1999) 。在這里為所有刀具類型產(chǎn)生的刀具路徑形成是普遍的，在這里以包括球頭刀、平底刀和圓角精銑刀以及其它更多的刀具作為代表(圖3 和4)。從這個(gè)研究結(jié)果來看，只有一種系統(tǒng)的和統(tǒng)一的算法是必要的, 這個(gè)算法對通用APT切削刀的原則是非常兼容的。為了減少在處理一個(gè)大立體平版印刷(STL)文件時(shí)的計(jì)算時(shí)間，一個(gè)在區(qū)域詢問方面的高效方法產(chǎn)生了。
2、在三軸機(jī)械加工中的數(shù)字控制刀具路徑計(jì)劃
在實(shí)際的應(yīng)用之中，數(shù)字控制刀具路徑可以適用于各種不同的機(jī)械加工過程(Choi 等1994) (圖1)：
圖1 數(shù)字控制刀具路徑計(jì)劃中不同的機(jī)械加工過程
主要規(guī)程包括粗加工、半精加工和去除咬邊加工(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進(jìn)給量進(jìn)行粗加工（通常用平底精銑刀）可以高效率地去除龐大的重復(fù)材料。為得到一個(gè)更好的加工表面，在精加工之前通常通常要進(jìn)行幾次半精加工(通常是用圓精銑刀和球頭精銑刀)。完成了半精加工之后，工件表面就留下均勻厚度的材料，這個(gè)厚度是作為精加工（通常是用小的球頭精銑刀）要去除的工序余量薄層。有時(shí)完成精加工后還需要進(jìn)行平行加工和清角加工，因?yàn)檠乇诮沁吘売幸粋€(gè)咬邊的區(qū)域 (圖2)。一種更小的切削刀具是用在精加工之后的加工過程，以塑造局部角度外形或邊緣并且去除未切削的材料。根據(jù)上述討論，產(chǎn)生CLs通用切削刀具的一個(gè)統(tǒng)一方法的形成是不僅實(shí)用的, 而且更加容易實(shí)施和維護(hù)。因?yàn)橛杏行вX得方法來將三角形參數(shù)或所包含的表面的公差控制在允許的范圍內(nèi)，在這里開發(fā)的算法能夠?qū)ζ胀ǖ挠?jì)算機(jī)輔助制造（CAM）產(chǎn)生一個(gè)‘核心引擎作用’。
圖2 咬邊區(qū)域
3、普通幾何形狀的APT 切削刀
根據(jù)APT 的定義，可以用如圖3所示的參數(shù)來將普通幾何形狀的切削刀具完整地描述出來：
圖3 普通幾何形狀刀具的參量
d、切削刀具直徑，刀具直徑是輻形距離兩倍，這個(gè)輻形距離的切削刀具軸到上部和下部直線段交叉點(diǎn)的距離計(jì)量的；
r、壁角半徑；
e 輻形距離，它是從切削軸到壁角圈子中心的距離，如果它的壁角和中心是在工具軸的同一邊，那么它是正面的，否則它就是反面的；
f、從終點(diǎn)到壁角圈子中心的距離，這個(gè)距離是通過平行于工具軸來測量的。
切削刀具參數(shù)的值的必須與其內(nèi)部的各種參數(shù)相一致，而且不能違背某些規(guī)定的約束，以便適當(dāng)?shù)孛枋鲈试S范圍之內(nèi)加工刀具的幾何形狀(Kral 1986)如圖4所示是切削刀具集合形狀的幾種選擇:
圖4 根據(jù)APT 定義的機(jī)械加工刀具形狀的幾種選擇
一些附屬參量的方法如下，被使用幫助描述CL 點(diǎn)的計(jì)算。這些附屬參數(shù)可用于幫助描述刀具切削點(diǎn)的計(jì)算。
R=+（Lc-tan1）tan2--------（1）
那里的半徑R，是切削刀具在加工零件表面上的最大伸出界限。這個(gè)界限將用于幫助尋找在伸出區(qū)域內(nèi)的相交的三角形。從機(jī)械加工的切削刀具的幾何學(xué)外形來看，圓環(huán)圈子的半徑R1和R2 的計(jì)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）
距離L，是用半徑R2從圓環(huán)圈子的中心開始計(jì)量來進(jìn)行計(jì)算的，計(jì)算的方法如下：
L=Lc-f+
在刀具參側(cè)面上距離分別為R1和R2的兩個(gè)不同的點(diǎn)從工具軸開始將機(jī)械加工切削刀具分成三個(gè)不同的區(qū)域。在上面的部分是錐體截面體其半徑是R，R2高度是L，中間部分是圓環(huán)半徑e和壁角圓半徑r，底部是半徑R1 和高度R1 圈子錐體tan_1。
通常,切削刀具側(cè)面不需要包含所有三個(gè)區(qū)域。比如在上圖4中是（a）圖，它的形狀是一個(gè)圓筒；在圖4中的(c)圖，花托成為一個(gè)半圓球；在圖4中的中(d)，它是一把逐漸變得尖細(xì)切削刀。
4、形成刀具切削區(qū)域的算法
數(shù)字控制加工中形成塑造傳統(tǒng)的實(shí)體模型的方法需要垂直于刀具切削表面（CL）的平面。雖然它在概念是簡單的，但是還是有幾的缺點(diǎn)。首先,垂直表面的形成不是一個(gè)瑣細(xì)的問題。在固體模型中，界限表示法(B Rep)模型是最普遍的代表形式。被整理的不均勻的B多槽軸(NURBS)表面的垂距是復(fù)雜的而且計(jì)算費(fèi)用昂貴的操作。其次,被整理的表面的垂距可以容易地制造復(fù)雜的自身內(nèi)部的交叉點(diǎn)和外部的交叉點(diǎn)(以毗鄰表面)問題。此外，這個(gè)統(tǒng)一的垂距表面只實(shí)用于球頭銑刀刀具切削區(qū)域的產(chǎn)生。用圓角銑刀加工形成的刀具切削區(qū)域垂距表面是一個(gè)更加困難的問題，更不用說更加通用的APT刀具?？偟膩碚f，用垂距表面在數(shù)字控制刀具路徑中產(chǎn)生傳統(tǒng)的實(shí)體模型在計(jì)算過程中是復(fù)雜的，而且計(jì)算的效率也很底。
此外，解析機(jī)械加工中，在給定部分參數(shù)表面后，機(jī)械加工刀具路徑是從垂距部分表面和平行于工具軸的一系列垂直平面的交叉點(diǎn)產(chǎn)生的。非線性等式解決的方法也許包含在要尋找的交叉點(diǎn)曲線中。對于一個(gè)立體平版印刷(STL)模型，因?yàn)榱慵砻嬉呀?jīng)被分成三角形，刀具路徑的形成是從多面體表面開始計(jì)算CLs的。在許多情況下,唯一線性運(yùn)算是很有必要的。如圖5所示，切削刀具與零件表面接觸的地方叫做刀具接觸點(diǎn)（CC），而刀具的端點(diǎn)被定義為刀具切削點(diǎn)。在機(jī)械加工中，刀具接觸點(diǎn)（CC）并不是固定的，而刀具切削點(diǎn)（CL）的x-y 坐標(biāo)值可以任意確定(多數(shù)情況下是落在固定的網(wǎng)格點(diǎn))。唯一的未知數(shù)就是刀具切削點(diǎn)的Z軸坐標(biāo)值。因此，刀具路徑通常的由很多連續(xù)的刀具切削點(diǎn)組成。當(dāng)工具軸向二維點(diǎn)(xc,yc)移動(dòng)時(shí)，零件表面將形成一個(gè)區(qū)域，這個(gè)區(qū)域是半徑為R的二維圓組成的，這個(gè)二維圓的圓心在(xc,yc)點(diǎn)上。這個(gè)區(qū)域叫做刀具接觸點(diǎn)（CC）（圖5）。
圖5 CC點(diǎn)、CL點(diǎn)和CC區(qū)域
本文提出一種計(jì)算無推斷從零件表面的那些小的三角形來的刀具切削點(diǎn)（CL）的算法，這些零件表面與刀具接觸點(diǎn)（CC）區(qū)域是重疊的。當(dāng)切削刀具與一個(gè)三角形多面體接觸時(shí)，刀具接觸點(diǎn)（CC）可能位于端點(diǎn)、小平面或者是邊沿上。對于切削刀具本身，刀具接觸點(diǎn)（CC）可能與包絡(luò)線、花托區(qū)域或更低的錐體連在一起。對于這些各種各樣的聯(lián)系情況，刀具切削點(diǎn)（CL）的計(jì)算方法是不相同的。先確定接觸區(qū)域和刀具接觸點(diǎn)（CC）是非常必要的，然后從刀具接觸點(diǎn)（CC）區(qū)域可以計(jì)算出刀具路徑的刀具切削點(diǎn)（CL）。對于一把普通的APT 切削刀具，有九類型計(jì)算模型。實(shí)際上，
并不是每把切削刀具都包含有三個(gè)區(qū)域。通常，一把切削刀具只包含有一個(gè)或者兩個(gè)切削區(qū)域（圖4）。刀具切削點(diǎn)（CL）的計(jì)算過程首先是從分成三角形小平面的刀具接觸點(diǎn)（CC）區(qū)域開始的。這是一個(gè)節(jié)約時(shí)間的策略，因?yàn)槿绻毒呓佑|點(diǎn)（CC）小平面里面， 切削刀具就不接觸到端點(diǎn)或者三角形的邊沿，因此，后面二者更加費(fèi)時(shí)的步驟就可能得到避免。
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Numerical control machining of triangulated
sculptured surfaces in a stereo lithography
format with a generalized cutter
A 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 automatically
programmed 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. Introduction
Currently, 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 always
straightforward 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 paths
from 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 machining
In 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 cutter
According 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）
Where
R2=e+（vsin(22)+）/2--------（3）
And
V=（（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 locations
A 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 cutter contacting the part surface is called a cutter-contact (CC) point, and the endpoint of a cutter is defined as a CL point. During machining,the CC point is not fixed, but the x–y locations of the CL point can be arbitrarily determined (most times falling on fixed grid points). The only unknown to be found is the z component of the CL point. Therefore, the tool path usually consists of a sequenced series of CL points. When the tool axis moves to a two-dimensional point (xc, yc), there exists a region on the part surface, enclosed by a two-dimensional circle with radius R, and centred at (xc, yc). It is called a CC region (figure 5).
Figure 5. CC point, CL point and CC region.
This paper presents an algorithm to compute the inference-free CL point from those triangular facets of the part surface that are overlapped with the CC region. When a cutter makes contact with a triangular polyhedron, the CC point may be located at a vertex, a facet or an edge. For the cutter itself, the CC point may be contacted on the taper envelope, the torus region or the lower cone. For these various contact conditions, the computations of CL points are not the same. It is necessary to determine the contact regions and CC points first, and then from the locations of CC points, the CL points of the tool path can be calculated. For a generalized APT cutter, there are nine types of computation model. In reality, not every cutter profile contains all the three regions. Very often, a cutter usually contains only one or two cutting regions (figure 4).The CLpoint_computation procedure begins the computation of the CC point with the triangular facet first. This is a time-saving strategy because if the CC point is located inside the facet, then the cutter does not touch the vertices or edges of the triangle and, therefore, the latter two more time-consuming steps can be avoided.
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