機械設(shè)計制造及其自動化專業(yè)外文資料翻譯

1、畢業(yè)設(shè)計(論文)外文資料翻譯系 （院）：機械工程學(xué)院專 業(yè)：機械設(shè)計制造及其自動化姓 名：學(xué) 號：外文出處：computer-aided design & applications, vol. 2, nos. 1-4, 2005, pp95-104附 件：1.外文資料翻譯譯文；2.外文原文。指導(dǎo)教師評語： 年月日簽名： 附件1：外文資料翻譯譯文多軸數(shù)控加工仿真的自適應(yīng)固體香港t. yau1, lee s. tsou2 and yu c. tong31中正大學(xué)，.tw 2中正大學(xué)，.tw3 中正大學(xué)，.tw

2、摘要：如果在一個復(fù)雜的表面的加工中，通常會產(chǎn)生大量的線性nc段來近似精確的表面。如果沒有發(fā)現(xiàn)，直到切割不準(zhǔn)確的nc代碼，則會浪費時間和昂貴的材料。然而，準(zhǔn)確和視圖獨立驗證的多坐標(biāo)數(shù)控加工仍然是一個挑戰(zhàn)。本文著重介紹了利用自適應(yīng)八叉樹建立一個可靠的多軸模擬程序驗證模擬切割期間和之后的路線和工件的外觀。體素模型的自適應(yīng)八叉樹數(shù)據(jù)結(jié)構(gòu)是用來加工工件與指定的分辨率。隱函數(shù)的使用刀具接觸點的速度和準(zhǔn)確性的檢驗，以代表各種刀具的幾何形狀。它允許用戶做切割模型和原始的cad模型的誤差分析和比較。在加工前運行數(shù)控機床，以避免浪費材料，提高加工精度，它也可以驗證nc代碼的正確性。關(guān)鍵詞：數(shù)控仿真加工，固體素模型

3、，自適應(yīng)1.介紹nc加工是一個基本的和重要的用于生產(chǎn)的機械零件的制造過程。在理想的情況下，數(shù)控機床將運行在無人值守模式。使用nc仿真和驗證是必不可少的，如果要運行的程序有信心在無人操作。因此，它是非常重要的，在執(zhí)行之前，以保證nc路徑的正確性。從文學(xué)來說，數(shù)控仿真主要分為三種主要方法，如下所述。第一種方法使用直接布爾十字路口實體模型來計算材料去除量在加工過程。這種方法在理論上能夠提供精確的數(shù)控加工仿真，但使用實體建模方法的問題是，它是計算昂貴。使用構(gòu)造實體幾何仿真的成本刀具運動的o（n 4）的數(shù)量的四次冪成正比。第二種方法使用空間分割表示，代表刀具和工件。在這種方法中，一個堅實的對象被分解成一

4、個集合的基本幾何元素，其中包括體素并，dexels，g-緩沖器，依此類推，從而簡化了過程的正規(guī)化布爾操作。第三種方法使用離散矢量路口。這種方法是基于對一個表面成的一組點的離散化。切割是模擬計算通過與刀具路徑信封的表面點的矢量的交點。在多軸數(shù)控加工，切削刀具頻繁地旋轉(zhuǎn)，以便計算出工件的模型，該模型是依賴于視圖，這是很困難的。因此在本文中，我們使用的體素的數(shù)據(jù)結(jié)構(gòu)來表示的工件模型。不過，根據(jù)過去的文獻，如果精度是必要的，大量的像素，必須設(shè)立執(zhí)行布爾操作。這會消耗內(nèi)存和時間。因此，我們的方法是使用八叉樹的數(shù)據(jù)結(jié)構(gòu)來表示的工件模型。八叉樹可以適于創(chuàng)建與所需決議所需要的體素。我們利用八叉樹的快速搜索與刀

5、具接觸的體素。然而，我們的方法使用了一個隱式的函數(shù)來表示的切削工具，因為切割器可以容易且準(zhǔn)確地表示的隱式代數(shù)方程，并判斷切割器保持在與工件接觸也很容易。因此，我們的方法是可靠和準(zhǔn)確的。論文內(nèi)容安排如下。第2節(jié)討論的工件表示，使用八叉樹體素模式。第3節(jié)給出用于表示各種刀具的幾何形狀的隱函數(shù)。第4節(jié)概述了該算法的3軸數(shù)控加工仿真程序。第5節(jié)說明了所提出的方法可以很容易地適應(yīng)五軸聯(lián)動數(shù)控仿真通過擴展的隱函數(shù)來容納五軸旋轉(zhuǎn)。實例證明所提出的方法的有效性和簡單的。第6節(jié)說明了nc仿真所需的存儲器空間和計算時間的實驗結(jié)果。最后，結(jié)論在第7節(jié)。2.立體幾何體素表示在本文中，我們使用一個像素的數(shù)據(jù)結(jié)構(gòu)來表示研

6、磨工件的自由形式的幾何體素的新型固體，因為模型的軸對準(zhǔn)和視圖獨立的性質(zhì)。同時利用八叉樹來避免創(chuàng)建大量的體素。該方法判斷刀具保持與工件接觸，發(fā)現(xiàn)接觸的所有體素，然后將這些體素空間分辨率達到八像素遞歸直到達到所需的精度水平。因此，如果有與刀具接觸體素，也沒有必要細(xì)分模型。圖1顯示了八叉樹數(shù)據(jù)結(jié)構(gòu)和它所代表的體素模型。圖1. 八叉樹數(shù)據(jù)結(jié)構(gòu)和相關(guān)的體素模型傳統(tǒng)上，由于加工仿真采用均勻的體素數(shù)據(jù)結(jié)構(gòu)來表示一個研磨工件，精度提高了體素數(shù)據(jù)時，將產(chǎn)生大量的工件。這將使加工仿真變慢，這是因為大量計算機內(nèi)存的需要。因此，我們使用八叉樹數(shù)據(jù)結(jié)構(gòu)的自適應(yīng)創(chuàng)造體素，需要仿真。3. 刀具幾何使用隱式函數(shù)表示空間均勻的

7、體素分割方法未能解決多維數(shù)控驗證相當(dāng)復(fù)雜與準(zhǔn)確的工件。如果需要高精度，大量的體素必須成立進行布爾集合運算。這將消耗大量的內(nèi)存和時間。但我們?nèi)S仿真的新方法采用自適應(yīng)的體素模型來表示一個研磨的工件，并使用隱式函數(shù)來表示的刀具實體模型。由于刀具不分解成一個集合的基本幾何元素，因此可以實現(xiàn)高的精度。同時工件模型利用八叉樹模型來減少不必要的體素。下面，我們描述了使用隱函數(shù)表示的各種刀具。平立銑刀可以由一個圓柱體代表。圖2顯示平立銑刀切削方向一致。如果工具是平行于z軸，且坐標(biāo)系統(tǒng)轉(zhuǎn)換，中心點位于原點。因此，一個平面銑刀的隱函數(shù)：f ( x , y , z ) = maxabs ( z l/2 ) l/2

8、, x 2 + y 2 r 2 if z 0 該隱函數(shù)被用于確定體素在里面，外面，或交叉的刀不損失任何精度。裁判可以通過插入一個像素頂點坐標(biāo)為隱函數(shù)。方程式（2）描述了一個頂點與刀具間的關(guān)系，如圖3所示。 0 在外部表面表示r：刀具半徑l：從沿刀軸的中心點開始測量距離圖2：平面銑刀和相關(guān)的坐標(biāo)系統(tǒng)圖3：隱函數(shù)用于確定刀具的內(nèi)部或外部球立銑刀是由一個圓柱和一個球體組成，如圖4所示。如果工具是平行于z軸，且坐標(biāo)系統(tǒng)的原點平移到球體的中心，則一個球立銑刀的隱函數(shù)可以被描述為：max abs ( z ) l , x 2 + y 2 r 2 若z 0 f ( x , y , z )= （3） x 2 +

9、 y 2 + z 2 r 2 其他表示r：刀軸刀角中心徑向距離r： 刀角半徑l：從沿刀軸的中心點開始測量距離圖4：球立銑刀和相關(guān)的坐標(biāo)系統(tǒng)圓角立銑刀可以由兩個氣缸和一個圓環(huán)表示。如果工具是平行于z軸，且坐標(biāo)系統(tǒng)轉(zhuǎn)換到中心點，如圖5所示，圓角端銑刀隱函數(shù)可推導(dǎo)為： max abs ( z ) l , x 2 + y 2 （r+r）2 若z 0f ( x , y , z )= max abs ( z ) l , x 2 + y 2 r 2 否則abs（x）r （x 2 + y 2 + z 2 +r 2）4r2(x 2+ z 2) 其他表示r：刀軸刀角中心徑向距離r：刀角半徑l：從沿刀軸的中心點開始

10、測量距離圖5：圓角銑刀和相關(guān)的坐標(biāo)系統(tǒng)作為一個簡單的平面銑刀，或復(fù)雜的圓角立銑刀，隱函數(shù)可以用來確定一點是否是內(nèi)部或外部的刀具直接應(yīng)用的方程式。隱函數(shù)表示刀具的使用不僅是精確的幾何形狀，簡單的概念，而且隱函數(shù)編程也很容易和簡單。因此，我們可以很容易地知道隱式函數(shù)f（x，y，z）的存量和刀具之間的幾何關(guān)系。4. 三軸數(shù)控加工仿真配制后的切削刀具的隱函數(shù)，需要被執(zhí)行的一項重要任務(wù)是確定哪些需要在球磨過程中細(xì)分或刪除的體素。圖6則表示一個三軸nc路徑模擬流程圖：建立刀具和工件模型讀取nc代碼結(jié)束nc代碼？完成仿真是刪除停止檢查邊界盒檢查刀隱函數(shù)聯(lián)系所有頂點都在模型所有頂點都在模型外細(xì)分所有頂點都在模

11、型達到要求的精度否是否圖6：三軸加工仿真流程圖三軸nc路徑模擬過程描述如下：（1） 第一步是讀nc代碼。然后我們可以得到每個數(shù)控段的開始和結(jié)束的刀具位置（cl）的刀具運動點。因此，三軸運動模型是任何兩個工具的配置點位置的聯(lián)合結(jié)構(gòu)的cl插值。（2） 刀具的邊界框是用來初步判斷刀具接觸體素模型的哪部分。其目的是擺脫不與刀具的體素接觸。如果確定體素是與刀具接觸，體素的頂點將被取代刀隱函數(shù)來決定是否像素頂點位于刀具的內(nèi)部或外部。（3） 如果所有的體素點符合條件的f（x，y，z）0時，則確認(rèn)它的體素已被完全切斷刀；也就是說，體素在落刀應(yīng)該被消除。如果頂點部分落在里面和其他人以外，這意味著需要進一步劃分體

12、素。為了細(xì)分每一個體素，步驟（2）和步驟（3）將進行遞歸，直至達到預(yù)定的精度水平。在nc路徑當(dāng)前段完成之后，步驟1再次上演，讀取下一段數(shù)控代碼。該程序是遵循直到所有nc代碼已讀才結(jié)束。在上述三軸加工仿真程序中，它是明確的，像素被細(xì)分需要根據(jù)刀具與工件之間的幾何關(guān)系；大量的體素不一次全部在開始創(chuàng)建。圖7則表明體素通過八叉樹只與一個球立銑刀接觸將細(xì)分。體素不與刀具接觸則不會細(xì)分。因此，這種方法大大降低了體素的數(shù)量，節(jié)省了空間。圖7：三軸數(shù)控加工仿真在z-map模型的比較，使用體素模型仿真的結(jié)果是可以顯示多軸加工。dexel模型的視圖有相關(guān)的限制，但體素模型沒有這個限制。因此，三軸、五邊加工可以利用

13、本文的三軸仿真方法，如圖8所示。圖8：三軸和五邊的仿真實例5. 五軸聯(lián)動數(shù)控加工仿真在五軸加工中，除了三個平移運動，刀具軸也會旋轉(zhuǎn)。因此，我們對五軸仿真方法只修改刀隱函數(shù)。所有其他的步驟與三軸仿真是相同的。因此，刀具的三種可以表示如下：平立銑刀可以由一個圓柱體代表。圖9結(jié)果表明刀具軸沿著n，中心點位于 p。因此，一個平面銑刀的隱函數(shù)：f ( x , y , z ) = ( x p ) t n 2 ( x p ) r 2 若 0 n t ( x p )l (5)表示 r：刀具半徑 x = x y zt ：一個像素點的位置 n = n x n y nzt ：刀具軸的單位矢量 p = p x py

14、pz t ：中心點 0 -nz n y nx2-1 nxny nxnzn= nz 0 -n x n2= nxny ny2-1 nynz -n y n x 0 nxnz nynz nz2-1圖9：五軸旋轉(zhuǎn)的平面銑刀模式球立銑刀可以由一個圓柱和一個球體，工會代表。圖10則結(jié)果表明刀具軸沿 n和中心點位于p 。因此，一個球立銑刀的隱函數(shù)： ( 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：刀具半徑n：刀具軸的單位矢量圖10：五軸模式旋轉(zhuǎn)球立銑刀圓角立銑刀可以由兩個氣缸和一個

15、圓環(huán)工會代表。圖11則結(jié)果表明刀具軸沿n，中心點位于p。因此，一個圓形立銑刀的隱函數(shù)： ( v t n 2 v ) ( r + r ) 2 若0 n t v lf ( x , y , z ) = ( v t n 2 v ) r 2否則0 n ( v r n ) r and ( v t n 2 v ) r 2 ( v t n 2 v )+(ntv)2+r2-r2 )+4r2( v t n 2 v )表示r：從刀軸刀角中心徑向距離r：刀角半徑n：刀具軸的單位矢量 v = x p圖11：五軸式旋轉(zhuǎn)圓角立銑刀圖12顯示了一個簡單的例子，從70度到50度左右的x軸和沿x軸移動的刀具軸的旋轉(zhuǎn)。圖13顯示用

16、于五軸加工葉輪的刀具路徑和仿真過程。圖14顯示葉片五軸數(shù)控加工仿真的另外一個例子。圖12：五軸數(shù)控加工仿真。 （a） (b) (c) (d)圖13：五軸模擬葉輪的例子。（a）刀具路徑，（b）（c）與刀具加工工件，（d）完成的部分(a) (b)（c） (d) 圖14：五軸模擬葉片的例子。（a）刀具路徑，（b）（c）與刀具加工工件，（d）完成的部分6. 實驗結(jié)果該方法已運行在2.4 ghz的奔騰4電腦，實現(xiàn)了在c + +和一些測試用例。標(biāo)簽1給出了自適應(yīng)數(shù)控仿真所需要的內(nèi)存空間和計算時間的比較。第一行顯示的圖片的數(shù)控軌跡的四個不同的模型。第二和第三行顯示nc代碼的信息。第四行是工件模型的分辨率。刀

17、具模型由隱式函數(shù)正確表達，所以沒有精度問題，第五行顯示刀具使用的類型，最后三行是所需要的內(nèi)存空間，計算時間和所提供的仿真結(jié)果。部分推動布萊德休博特爾nc路徑數(shù)控代碼（線）26542536748033340數(shù)控代碼長度（mm）249508642272873060分辨率（mm）0.1切削刀具flat r3ball r10flat r1.5ball r1計算時間（sec）562387296209內(nèi)存空間（mb）19210167132仿真結(jié)果表明標(biāo)簽1：需要的內(nèi)存空間和計算時間的自適應(yīng)數(shù)控仿真。標(biāo)簽2給出了所需要的內(nèi)存空間，使用均勻的體素模型的數(shù)控加工仿真的計算時間的比較。為適應(yīng)數(shù)控

18、加工仿真的參數(shù)保持不變。在這種情況下，我們不能夠模擬案例1（葉輪）和成功案例3（鞋），因為這樣的案例超過內(nèi)存限制?？梢杂^察到的結(jié)果是殼體2（刀片）和殼體4（瓶）。相比較而言，該自適應(yīng)數(shù)控仿真的優(yōu)勢是顯而易見的，通過實現(xiàn)使用自適應(yīng)數(shù)控仿真，時間和空間可以大大減少部分推動布萊德休博特爾計算時間（sec）x1491x721內(nèi)存空間（mb）x414x280標(biāo)簽2：所需的存儲空間和均勻的體素模型的數(shù)控仿真計算時間。7.結(jié)論在本文中，我們提出了一個新的多軸模擬方法。本文的目的是使用自適應(yīng)的體素模型來建立一個可靠的多軸模擬程序，可以模擬的切割路線和模擬之后的工件的外觀。它允許用戶進行切削模型與原始cad模型

19、比較及誤差分析。它可以在加工數(shù)控機床之前驗證其nc代碼的準(zhǔn)確性，以避免浪費材料和提高加工精度。總之，對多軸模擬方法的優(yōu)點如下所述。（1）模擬的方法比其他基于體素模擬的方法使用更少的內(nèi)存。（2）仿真是獨立的。dexel模型的視圖有相關(guān)的限制，而體素模型沒有這個限制。（3）模擬是可靠和準(zhǔn)確的。無論刀具的五軸還是三軸仿真都是采用隱函數(shù)來表示，整個方法簡單可靠。8.參考文獻1.choi, b. k., jerard, r. b., sculptured surface machining: theory and applications, kluwer academic publishers, 199

20、8. 2. wang, w. p., wang, k. k., geometric modeling for swept volume of moving solids, ieee computer graphics & applications, vol. 6, no.12, 1986, pp 8-173. atherton, p. r., a scan-line hidden surface removal procedure for constructive solid geometry, computer graphics, vol. 17, no. 3, 1983, pp 73-82

21、.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, vol. 7, 1991, pp 149-157.5. jang, d., kim, k., jung, j., voxel-based virtual multi-axis machining, advanced manu

22、facturing technology, vol. 16, no. 10, 2000, pp 709-713.6. van hook, t., real time shaded nc milling display, computer graphics, vol. 20, no. 4, 1986, pp 15-20.7. huang, y., oliver, j. h., integrated simulation, error assessment, and tool path correction for five-axis nc milling, journal of manufact

23、uring systems, vol. 14, no. 5, 1995, pp 331-334.8. saito, t., takahashi, t., nc machining with g-buffer method, computer graphics, vol. 25, 1991, pp 207-2169. chang, k. y., goodman, e. d., a method for nc tool path interference detection for a multi-axis milling system, asme control of manufacturing

24、 process, dsc-vol.28/ped-vol.52, 1991, pp 23-30.附件2：外文原文adaptive nc simulation for multi-axis solid machininghong t. yau 1 , lee s. tsou 2 and yu c. tong 31national chung cheng university，.tw2national chung cheng university，.tw3national chung cheng university，pume.cc

25、.twabstractfor 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 vie

26、w-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. voxel models with adaptive oct

27、ree 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 cutti

28、ng 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 voxel model1. introductionnc machining is a funda

29、mental 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 impor

30、tance 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 machini

31、ng . 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 movement

32、s 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 processe

33、s 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 env

34、elopes. 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 nee

35、ded, 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

36、 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 work

37、piece 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

38、. 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

39、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 geometryin this paper, we use a voxel data structure to represent the

40、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 hav

41、e 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 mode

42、l it represents.fig. 1. octree data structure and the associated voxel modeltraditionally, 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 m

43、ake 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 simulation3. representation of cutter geometry using implicit functionsthe spatial partitioning approach with uniform voxels has failed to

44、 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 sim

45、ulation 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

46、 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 para

47、llel 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 ) = maxabs ( z l/2 ) l/2, x 2 + y 2 r 2 if z 0this implicit function is used to determine whether a voxel is inside, outsid

48、e, 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 outsid the surfa

49、cewherer : the cutter radius l : the distance measured from the center point along the cutter axis fig. 2. flat endmill and the associated coordinate systemfig. 3. implicit function used to determine the interior or exterior of a cutterball endmills can be represented by the union of a cylinder and

50、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 2 o

51、therwisewherer : the cutter radius l : the distance measured from the center point along the cutter axisfig. 4. ball endmill and the associated coordinate systemfillet endmills can be represented by the union of two cylinders and a torus. assuming the tool is parallel to the z-axis, and the coordina

52、te 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 0f ( 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) otherwisewherer : the radial distance from the cutter axis to the cutter corner centerr : the cutter corner radiusl : the distance measured from the center point along the cutter axisfig. 5. fillet endmill and the associated coordinate systemas simple as a flat endmill, or as complex as a fillet endmill, the implic