游戲機(jī)手柄注塑模具成型部件數(shù)控加工編程帶開題報(bào)告.zip

游戲機(jī)手柄注塑模具成型部件數(shù)控加工編程帶開題報(bào)告.zip,游戲機(jī),手柄,注塑,模具,成型,部件,數(shù)控,加工,編程,開題,報(bào)告 外文出處：Computer-Aided Design & Applications, Vol. 2, Nos. 1-4, 2005, pp95-104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)，imehtyccu.edu.tw 2中正大學(xué)，lstsoucad.me.ccu.edu.tw3 中正大學(xué)，pume.ccu.edu.tw摘要：如果在一個(gè)復(fù)雜的表面的加工中，通常會(huì)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(huì)浪費(fèi)時(shí)間和昂貴的材料。然而，準(zhǔn)確和視圖獨(dú)立驗(yàn)證的多坐標(biāo)數(shù)控加工仍然是一個(gè)挑戰(zhàn)。本文著重介紹了利用自適應(yīng)八叉樹建立一個(gè)可靠的多軸模擬程序驗(yàn)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(diǎn)的速度和準(zhǔn)確性的檢驗(yàn)，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(yùn)行數(shù)控機(jī)床，以避免浪費(fèi)材料，提高加工精度，它也可以驗(yàn)證NC代碼的正確性。關(guān)鍵詞：數(shù)控仿真加工，固體素模型，自適應(yīng)1.介紹NC加工是一個(gè)基本的和重要的用于生產(chǎn)的機(jī)械零件的制造過程。在理想的情況下，數(shù)控機(jī)床將運(yùn)行在無人值守模式。使用NC仿真和驗(yàn)證是必不可少的，如果要運(yùn)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(shí)體模型來計(jì)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(shí)體建模方法的問題是，它是計(jì)算昂貴。使用構(gòu)造實(shí)體幾何仿真的成本刀具運(yùn)動(dòng)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(gè)堅(jiān)實(shí)的對(duì)象被分解成一個(gè)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(duì)一個(gè)表面成的一組點(diǎn)的離散化。切割是模擬計(jì)算通過與刀具路徑信封的表面點(diǎn)的矢量的交點(diǎn)。在多軸數(shù)控加工，切削刀具頻繁地旋轉(zhuǎn)，以便計(jì)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(xiàn)，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(huì)消耗內(nèi)存和時(shí)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具接觸的體素。然而，我們的方法使用了一個(gè)隱式的函數(shù)來表示的切削工具，因?yàn)榍懈钇骺梢匀菀浊覝?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(dòng)數(shù)控仿真通過擴(kuò)展的隱函數(shù)來容納五軸旋轉(zhuǎn)。實(shí)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(chǔ)器空間和計(jì)算時(shí)間的實(shí)驗(yàn)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(gè)像素的數(shù)據(jù)結(jié)構(gòu)來表示研磨工件的自由形式的幾何體素的新型固體，因?yàn)槟Ｐ偷妮S對(duì)準(zhǔn)和視圖獨(dú)立的性質(zhì)。同時(shí)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(dá)到八像素遞歸直到達(dá)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?cái)?shù)據(jù)結(jié)構(gòu)來表示一個(gè)研磨工件，精度提高了體素?cái)?shù)據(jù)時(shí)，將產(chǎn)生大量的工件。這將使加工仿真變慢，這是因?yàn)榇罅坑?jì)算機(jī)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)表示空間均勻的體素分割方法未能解決多維數(shù)控驗(yàn)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(jìn)行布爾集合運(yùn)算。這將消耗大量的內(nèi)存和時(shí)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(gè)研磨的工件，并使用隱式函數(shù)來表示的刀具實(shí)體模型。由于刀具不分解成一個(gè)集合的基本幾何元素，因此可以實(shí)現(xiàn)高的精度。同時(shí)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(gè)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(diǎn)位于原點(diǎn)。因此，一個(gè)平面銑刀的隱函數(shù)：F ( X , Y , Z ) = maxabs ( Z L/2 ) L/2, 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(gè)像素頂點(diǎn)坐標(biāo)為隱函數(shù)。方程式（2）描述了一個(gè)頂點(diǎn)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(diǎn)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(gè)圓柱和一個(gè)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(diǎn)平移到球體的中心，則一個(gè)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 + 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(diǎn)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(gè)氣缸和一個(gè)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(diǎn)，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(diǎn)開始測量距離圖5：圓角銑刀和相關(guān)的坐標(biāo)系統(tǒng)作為一個(gè)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(diǎn)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(xiàng)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(gè)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(diǎn)都在模型所有頂點(diǎn)都在模型外細(xì)分所有頂點(diǎn)都在模型達(dá)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(gè)數(shù)控段的開始和結(jié)束的刀具位置（CL）的刀具運(yùn)動(dòng)點(diǎn)。因此，三軸運(yùn)動(dòng)模型是任何兩個(gè)工具的配置點(diǎn)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(diǎn)將被取代刀隱函數(shù)來決定是否像素頂點(diǎn)位于刀具的內(nèi)部或外部。（3） 如果所有的體素點(diǎn)符合條件的F（x，y，z）0時(shí)，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(diǎn)部分落在里面和其他人以外，這意味著需要進(jìn)一步劃分體素。為了細(xì)分每一個(gè)體素，步驟（2）和步驟（3）將進(jìn)行遞歸，直至達(dá)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(gè)球立銑刀接觸將細(xì)分。體素不與刀具接觸則不會(huì)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(gè)限制。因此，三軸、五邊加工可以利用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(shí)例5. 五軸聯(lián)動(dòng)數(shù)控加工仿真在五軸加工中，除了三個(gè)平移運(yùn)動(dòng)，刀具軸也會(huì)旋轉(zhuǎn)。因此，我們對(duì)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(gè)圓柱體代表。圖9結(jié)果表明刀具軸沿著n，中心點(diǎn)位于 p。因此，一個(gè)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(gè)像素點(diǎn)的位置 n = n x n y nzT ：刀具軸的單位矢量 p = p x py pz T ：中心點(diǎn) 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(gè)圓柱和一個(gè)球體，工會(huì)代表。圖10則結(jié)果表明刀具軸沿 N和中心點(diǎn)位于P 。因此，一個(gè)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(gè)氣缸和一個(gè)圓環(huán)工會(huì)代表。圖11則結(jié)果表明刀具軸沿n，中心點(diǎn)位于P。因此，一個(gè)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(gè)簡單的例子，從70度到50度左右的X軸和沿X軸移動(dòng)的刀具軸的旋轉(zhuǎn)。圖13顯示用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(gè)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(shí)驗(yàn)結(jié)果該方法已運(yùn)行在2.4 GHz的奔騰4電腦，實(shí)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(jì)算時(shí)間的比較。第一行顯示的圖片的數(shù)控軌跡的四個(gè)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。刀具模型由隱式函數(shù)正確表達(dá)，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(jì)算時(shí)間和所提供的仿真結(jié)果。部分推動(dòng)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）0.10.10.10.1切削刀具Flat R3Ball R10Flat R1.5Ball R1計(jì)算時(shí)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(jì)算時(shí)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(jì)算時(shí)間的比較。為適應(yīng)數(shù)控加工仿真的參數(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?yàn)檫@樣的案例超過內(nèi)存限制。可以觀察到的結(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(shí)現(xiàn)使用自適應(yīng)數(shù)控仿真，時(shí)間和空間可以大大減少部分推動(dòng)布萊德休博特爾計(jì)算時(shí)間（sec）X1491X721內(nèi)存空間（MB）X414X280標(biāo)簽2：所需的存儲(chǔ)空間和均勻的體素模型的數(shù)控仿真計(jì)算時(shí)間。7.結(jié)論在本文中，我們提出了一個(gè)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(gè)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(jìn)行切削模型與原始CAD模型比較及誤差分析。它可以在加工數(shù)控機(jī)床之前驗(yàn)證其NC代碼的準(zhǔn)確性，以避免浪費(fèi)材料和提高加工精度?？傊?，對(duì)多軸模擬方法的優(yōu)點(diǎn)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(dú)立的。DEXEL模型的視圖有相關(guān)的限制，而體素模型沒有這個(gè)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(gè)方法簡單可靠。8.參考文獻(xiàn)1.Choi, B. K., Jerard, R. B., Sculptured Surface Machining: Theory and Applications, Kluwer Academic Publishers, 1998. 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 73-82.4. 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 Manufacturing 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 Manufacturing 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 Process, DSC-Vol.28/PED-Vol.52, 1991, pp 23-30.10 Oliver, J. H., Goodman, E. D., Direct Dimensional NC Verification, Computer-Aided Design, Vol. 22, No. 1, 1990, pp 3-10.11 Jerard, R. B., Drysdale, R. L., Hauck, K., Schaudt, B., Magewick, J., Methods for Detecting Errors in Numerically Controlled Machining of Sculptured surface, IEEE Computer Graphics & Applications, Vol. 9, No.1, 1989, pp 26-39. 12 Yamaguchi, K., Kunii, T., Fujimura, K., Octree-Related Data Structures and Algorithms, IEEE Computer Graphics Appl, Vol. 8, No. 3 , 1984, pp 8-68.13 Glaeser, G., Grller, E., Efficient Volume Generation During the Simulation of NC-Milling, Mathematical Visualization, 1997, pp 315-328. 14 Chung, Y. C., Park, J. W., Shin, H., Choi, B. K., Modeling the Surface Swept by a Generalized Cutter for NC Verification, Computer-Aided Design , Vol. 30, No. 8, 1996, pp 587-94. Computer Aided Design 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 Voxels 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 Computer Aided Design Applications Vol 2 Nos 1 4 2005 pp 95 104 100 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 is located at p Thus the implicit function of a flat endmill is 2 2 0 T TF X Y Z x p n x p R if n x p L 5 where R the cutter radius Tx X Y Z the position of a voxel vertex Tx y zn n n n the unit vector of the tool axis Tx y zp p p p the center point 2 2 2 2 10 0 1 0 1 x x y x zz y z x x y y y z y x x z y z z n n n n nn n n n n n n n n n n n n n n n n n Fig 9 Flat endmill rotated in five axis mode Computer Aided Design Applications Vol 2 Nos 1 4 2005 pp 95 104 101 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 2 2 2 0 T T T x p n x p R if n x p LF X Y Z x p x p R otherwise 6 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 2 2 2 2 2 2 22 2 2 2 2 2 0 0 and 4 T T T T T T T T v n v R r if n v L else if n v r n rF X Y Z v n v R v n v R v n v n v R r R v n v otherwise 7 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 Computer Aided Design Applications Vol 2 Nos 1 4 2005 pp 95 104 102 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 Computer Aided Design Applications Vol 2 Nos 1 4 2005 pp 95 104 103 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 computation time and the rendered simulation result Part Impeller Blade Shoe Bottle NC path Number of NC code line 2654 2536 74803 3340 Length of NC code mm 24950 8642 27287 3060 Resolution mm 0 1 0 1 0 1 0 1 Cutting tool Flat R3 Ball R10 Flat R1 5 Ball R1 Computation time Sec 562 387 296 209 Memory space MB 192 101 67 132 Simulation result Tab 1 Required memory space and computation time for adaptive NC simulation Computer Aided Design Applications Vol 2 Nos 1 4 2005 pp 95 104 104 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 Part Impeller Blade Shoe Bottle Computation time Sec X 1491 X 721 Memory space MB X 414 X 380 Tab 2 Required memory space and computation time for NC simulation with uniform voxel model 7 CONCLUSION In this paper we proposed a novel multi axis simulation method The objective of this paper was to use the adaptive voxel model to develop a reliable multi axis simulation procedure which can simulate the cutting route and the workpiece appearance during and after the simulation It allows the user to do error analysis and comparison between the cutting model and the original CAD model It can verify the accuracy of NC codes before machining on a CNC machine in order to avoid wasting material and to improve machining accuracy In summary the advantages of the multi axis simulation method presented in this paper are as follows 1 The simulation method uses less memory than other voxel based simulation methods 2 The simulation is view independent The dexel model has the restriction of being view dependent but the voxel model does not have this restriction 3 The simulation is reliable and accurate Regardless of whether it is the five axis or three axis simulation which uses the implicit function to represent a cutter the whole method is simple and reliable 8 REFERENCES 1 Choi B K Jerard R B Sculptured Surface Machining Theory and Applications Kluwer Academic Publishers 1998 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 17 3 Atherton P R A Scan Line Hidden Surface Removal Procedure for Constructive Solid Geometry Computer Graphics Vol 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 Vol 7 1991 pp 149 157 5 Jang D Kim K Jung J Voxel Based Virtual Multi Axis Machining Advanced Manufacturing 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 Manufacturing 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 207 216 9 Chang K Y Goodman E D A Method for NC Tool Path Interference Detection for A Multi Axis Milling System ASME Control of Manufacturing Process DSC Vol 28 PED Vol 52 1991 pp 23 30 10 Oliver J H Goodman E D Direct Dimensional NC Verification Computer Aided Design Vol 22 No 1 1990 pp 3 10 11 Jerard R B Drysdale R L Hauck K Schaudt B Magewick J Methods for Detecting Errors in Numerically Controlled Machining of Sculptured surface IEEE Computer Graphics Applications Vol 9 No 1 1989 pp 26 39 12 Yamaguchi K Kunii T Fujimura K Octree Related Data Structures and Algorithms IEEE Computer Graphics Appl Vol 8 No 3 1984 pp 8 68 13 Glaeser G Gr ller E Efficient Volume Generation During the Simulation of NC Milling Mathematical Visualization 1997 pp 315 328 14 Chung Y C Park J W Shin H Choi B K Modeling the Surface Swept by a Generalized Cutter for NC Verification Computer Aided Design Vol 30 No 8 1996 pp 587 94