四工位全自動(dòng)回轉(zhuǎn)工作臺(tái)設(shè)計(jì)【定位銷式分度工作臺(tái)】【含12張CAD圖紙】

四工位全自動(dòng)回轉(zhuǎn)工作臺(tái)設(shè)計(jì)【定位銷式分度工作臺(tái)】【含12張CAD圖紙】,定位銷式分度工作臺(tái),含12張CAD圖紙,四工位,全自動(dòng),回轉(zhuǎn),工作臺(tái),設(shè)計(jì),定位,分度,12,CAD,圖紙 第 18 頁(yè) 非正交主軸與工作臺(tái)型五軸工具機(jī)后處理程序開發(fā) 黃昭堂.佘振華摘要：后處理程序是將刀具位置數(shù)據(jù)轉(zhuǎn)換成加工操作所需數(shù)據(jù)的重要接口，其對(duì)五軸工具機(jī)來說是非常復(fù)雜的，因?yàn)樵谖遢S工具機(jī)中線性軸和旋轉(zhuǎn)軸是同動(dòng)的。以前大部分的五軸后處理方法研究只局限于正交的工具機(jī)構(gòu)型，本論文針對(duì)主軸型與工作臺(tái)型及工作臺(tái)/主軸型有非正交旋轉(zhuǎn)軸的五軸工具機(jī)開發(fā)其后處理算法，這種構(gòu)型的工具機(jī)具有從立式加工轉(zhuǎn)換為臥式加工的優(yōu)點(diǎn)。本文以齊次坐標(biāo)轉(zhuǎn)換為基礎(chǔ)，利用運(yùn)動(dòng)學(xué)的前向轉(zhuǎn)換，求得五軸工具機(jī)的形狀創(chuàng)成函數(shù)矩陣，再由逆向轉(zhuǎn)換，解出工具機(jī)各軸運(yùn)動(dòng)的解析方程。后處理程序中的線性算法是為了保證加工的精確性而開發(fā)的。五軸后處理程序接口是利用Borland C++、Builder與OpenGL開發(fā)，以產(chǎn)生三種構(gòu)型的NC碼，經(jīng)由商業(yè)實(shí)體切削仿真軟件VERICUT驗(yàn)證及試加工實(shí)驗(yàn)，證實(shí)所提出的后處理方法論的可行性。 關(guān)鍵詞：后處理、五軸加工、形狀創(chuàng)成函數(shù)、非正交旋轉(zhuǎn)軸 1、引言 五軸工具機(jī)被越來越多地的用戶所使用的，特別是用于加工復(fù)雜自由曲面。傳統(tǒng)的五軸工具機(jī)有三個(gè)正交的線性軸和旋轉(zhuǎn)軸。這里所說的旋轉(zhuǎn)軸通常是指與相互正交的中心線平行的線性軸。各國(guó)的機(jī)械工具制造商，如Makino，Ingersol和Deckel Maho，將非正交旋轉(zhuǎn)軸或工作臺(tái)進(jìn)行改進(jìn)使機(jī)器具有更好的多功能性和靈活性?！胺钦弧笔侵篙S旋轉(zhuǎn)體的振蕩運(yùn)動(dòng)，這類似與一張桌子上的硬幣的緩慢旋轉(zhuǎn)。五軸工具機(jī)有一個(gè)旋轉(zhuǎn)軸的傾斜面[1]，而不同于平行的直線軸，它提供的優(yōu)勢(shì)可使切削刀具在一個(gè)半球內(nèi)指向任意角度[2，3]。這種機(jī)器可以在連續(xù)的水平和垂直位置移動(dòng)。非正交旋轉(zhuǎn)軸為生產(chǎn)航空部件及汽車頭部提供了便利。運(yùn)動(dòng)經(jīng)電機(jī)主軸傳遞給空心軸和齒輪[4]。由于線性和旋轉(zhuǎn)運(yùn)動(dòng)同時(shí)作用在五軸數(shù)控機(jī)床上，導(dǎo)致了五軸數(shù)控程序比三軸數(shù)控程序更加的復(fù)雜。后處理程序必須利用刀具位置（CL）將數(shù)據(jù)從凸輪系統(tǒng)轉(zhuǎn)化為機(jī)器控制數(shù)據(jù)。盡管先進(jìn)的控制器可以接受實(shí)時(shí)的數(shù)據(jù)，而不需要后處理，但他們是相當(dāng)昂貴的[5]。該方法主要可以分為三類：圖形[ 6]，[7 ]和坐標(biāo)數(shù)值迭代[8-10]。由坐標(biāo)變換方法解析方程，產(chǎn)生的數(shù)控?cái)?shù)據(jù)最有效，它已被廣泛采用在最近的研究中。然而，幾乎所有的這些方法包括后處理方法均采用正交旋轉(zhuǎn)軸五軸工具機(jī)。研究解決非正交配置的相對(duì)較少。例如，有為主軸傾斜式發(fā)展的非正交旋轉(zhuǎn)軸五軸機(jī)床后處理程式[11]。最近，Sorby [ 12]發(fā)表了一篇關(guān)于封閉形式五軸工具機(jī)的非正交旋轉(zhuǎn)工作臺(tái)論文。然而，該解決方案具有一定的局限性。例如，工件原點(diǎn)的偏移向量和二次主旋轉(zhuǎn)不明確，及角度傾斜45度的非正交軸的固定。 本研究開發(fā)一種后置的雙主軸和工作臺(tái)五軸工具機(jī)?；邶R次坐標(biāo)變換矩陣的解析方程，確定方程的一般形式；偏移向量定義為從工件的起始位置回轉(zhuǎn)至工作臺(tái)，偏移向量在非正交軸中是可變的。此外還包括線性化算法的后處理開發(fā)，保證加工精度。 一個(gè)基于后處理是開發(fā)和圖形界面動(dòng)態(tài)顯示的表面模型議案的提出幫助用戶輸入相關(guān)參數(shù)正確。此外，生成的NC數(shù)據(jù)進(jìn)行驗(yàn)證，使用商業(yè)實(shí)體切削仿真軟件VERICUT [13]進(jìn)行五軸加工實(shí)驗(yàn)工具機(jī)的非正交旋轉(zhuǎn)工作臺(tái)的后處理方法確認(rèn)。 2、五軸工具機(jī)的配置與類型 大多數(shù)五軸工具機(jī)有兩個(gè)旋轉(zhuǎn)軸作為常規(guī)X軸，X軸和Z軸。五軸機(jī)床可分為三種類型：主軸型，工作臺(tái)型和工作臺(tái)/主軸型。商業(yè)方面用正交配置，如圖1所示三種類型。圖1（a）為非正交旋轉(zhuǎn)主軸型。圖1（b）為非正交旋轉(zhuǎn)工作臺(tái)型，如Deckel Maho DMU 70改進(jìn)型[ 15]，其在工作臺(tái)上具有兩個(gè)旋轉(zhuǎn)軸，和一個(gè)平行與Z軸而與非正交旋轉(zhuǎn)軸存在一定的傾斜角度的旋轉(zhuǎn)軸（C軸）。圖1（c）為工作臺(tái)\主軸型，如Deckel Maho 200P [ 15]，其中一個(gè)旋轉(zhuǎn)工作臺(tái)（c）是以在工作臺(tái)上的非正交旋轉(zhuǎn)軸（B軸）為主軸。由于作者已經(jīng)提出過主軸型非正交旋轉(zhuǎn)主軸的后處理程序，本研究著重于發(fā)展與其他兩種配置的后處理。 五軸機(jī)床可以看作是平動(dòng)與旋轉(zhuǎn)運(yùn)動(dòng)組合的機(jī)床。正向運(yùn)動(dòng)學(xué)方程必須建立數(shù)學(xué)模型來描述刀具相對(duì)于工件的切削運(yùn)動(dòng)。基本的坐標(biāo)變換矩陣，包括Trans矩陣和Rot矩陣 [ 16 ]。Trans矩陣式可以表示如下： Trans（a，b，c）表示矢量a i+b j+c k 一般Rot矩陣用來描述旋轉(zhuǎn)的主軸單元。本坐標(biāo)系設(shè)定；則Rot矩陣可以表示為： 其“C”和“S” 分別為余弦和正弦函數(shù)，且 圖1五軸工具類型：a.主軸型 b.工作臺(tái)型 c.工作臺(tái)\主軸型 3、后處理程序 3.1工作臺(tái)傾斜型 圖2描繪了相關(guān)的坐標(biāo)系配置。工件坐標(biāo)系為而為刀具坐標(biāo)系。由于這兩個(gè)旋轉(zhuǎn)軸并不相交，則必存在一條公法線垂直于兩軸。公法線分別與C軸和B軸相交于RC和RB點(diǎn)。偏移向量為從原點(diǎn)至RC，而偏移向量為從RC至RB。 圖2傾斜型坐標(biāo)系 組成機(jī)床的結(jié)構(gòu)有：回轉(zhuǎn)工作臺(tái)C，回轉(zhuǎn)工作臺(tái)B，機(jī)床床身， X軸方向工作臺(tái)，Y軸方向工作臺(tái)，Z軸方向工作臺(tái)，主軸和刀具。根據(jù)刀具與工件的相對(duì)位置和方向，將從工件開始至刀具完成的過程稱為形式塑造功能，[17]。這種機(jī)床的形式塑造過程，用數(shù)學(xué)矩陣形式表示如下： 其中Px，Py和Pz分別表示X，Y和Z軸的相對(duì)距離。和分別為與C軸和B軸的旋轉(zhuǎn)角度。采用右手螺旋定則判定+C和+B。方程（3）表示的函數(shù)矩陣，結(jié)合機(jī)床參數(shù)Px，Py，Px，和。第一步是計(jì)算刀具所需的旋轉(zhuǎn)角度，二是根據(jù)已知的旋轉(zhuǎn)角中心位置的直線計(jì)算所需的位置關(guān)系。 當(dāng)?shù)毒呶恢煤偷毒叩姆较蛳蛄看_定后，CL數(shù)據(jù)可用矩陣形式表示如下： 由于方程（3）和（4）都表示相同的刀具和工件之間的關(guān)系，聯(lián)立這兩個(gè)矩陣，確定所需的參數(shù)。結(jié)合兩個(gè)矩陣得到下列公式： 首先可以確定，的值。代入式（5）得： 值得注意的是，在范圍內(nèi)的表達(dá)方式如下： 如果范圍在–π到0之間，方程應(yīng)修改為式（8）所示。另一方面，如果同時(shí)滿足以上兩種情況，則以最小的旋轉(zhuǎn)角選擇算法。 此外，將式（5）對(duì)應(yīng)的第一值第二值聯(lián)立求解線性方程組得到： 由于方程（9）和（10）分母是相同的，總是正的，C軸轉(zhuǎn)角可以確定如下： 其中arctan2（y，x）是在范圍內(nèi)的函數(shù)返回值，表示y和x的夾角[16]。 此外，結(jié)合矩陣（6）式兩邊的相應(yīng)參數(shù)，產(chǎn)生三個(gè)未知數(shù)Px，Py和Pz。聯(lián)立方程組，設(shè)定程序坐標(biāo)系為工件坐標(biāo)系。因此，可以得到所需的NC數(shù)據(jù)（記為x，y和z），考慮兩個(gè)偏移向量和，并表示為如下： 3.2工作臺(tái)/主軸傾斜型 工作臺(tái)/主軸傾斜型有一個(gè)旋轉(zhuǎn)主軸和一個(gè)旋轉(zhuǎn)軸的工作臺(tái)。圖3分別顯示了C軸和B軸上的兩個(gè)交點(diǎn)RC和RB。交點(diǎn)RC位于C軸上任意點(diǎn)，交點(diǎn)B為非正交旋轉(zhuǎn)B軸和刀具的交點(diǎn)。偏移向量是按從原點(diǎn)到交點(diǎn)RC，有效刀具長(zhǎng)度代表交點(diǎn)RB和刀尖中心之間的距離，。由其造型函數(shù)矩陣可以得到坐標(biāo)變換矩陣如下： 圖 圖3工作臺(tái)/主軸傾斜型坐標(biāo)系統(tǒng) 等值式（14）式（15）聯(lián)立得： 結(jié)合參數(shù)，可以采用同工作臺(tái)傾斜型的計(jì)算過程。但要注意的是， NC參考點(diǎn)，在此假設(shè)為交點(diǎn)RB。這個(gè)定義是根據(jù)主軸傾斜和工作臺(tái)/主軸傾斜型得來，而且使用的是相同的商業(yè)后處理器程序的軟件包。完整的NC數(shù)據(jù)的分析方程可以表示為： 3.3線性問題 從理論上講， CAD / CAM系統(tǒng)生成的CL數(shù)據(jù)是以假設(shè)刀具在連續(xù)兩個(gè)點(diǎn)之間的線性移動(dòng)為基礎(chǔ)。然而，實(shí)際的刀具與工件的運(yùn)動(dòng)軌跡并不是直線和旋轉(zhuǎn)軸移動(dòng)同時(shí)進(jìn)行。彎曲的路徑偏離線性插值的連續(xù)路徑點(diǎn)之間的直線路徑被稱為線性問題。以下算法可以解決這個(gè)問題。 假設(shè)，在圖4中 為三個(gè)相鄰的CL數(shù)據(jù)點(diǎn)。矢量Pn的矩陣形式可表示為，其中和組成刀尖的中心位置，和組成刀具的方向。 相應(yīng)的機(jī)床數(shù)控代碼Pn為。由于五軸同時(shí)從當(dāng)前位置Pn移動(dòng)到隨后的位置的Pn+1，每個(gè)軸之間移動(dòng)假定為線性的[18]。因此，實(shí)際的曲線路徑的每個(gè)點(diǎn)可以表示如下： 其中t是一個(gè)虛擬的時(shí)間坐標(biāo)。其中CL數(shù)據(jù)和為正值。例如，工作臺(tái)傾斜型的式（5）、（6）和工作臺(tái)/主軸傾斜型的式（16）、（17）。此外，在理想的線性刀具路徑下每個(gè)點(diǎn)可以決定如下： 理想的線性刀具路徑 實(shí)際曲面刀具路徑 內(nèi)插刀具路徑 圖4多軸加工線性問題 圖5后處理程式對(duì)話框：a工作臺(tái)傾斜型 b工作臺(tái)/主軸傾斜型 圖6工作臺(tái)傾斜型生成NC數(shù)據(jù)對(duì)話框 之間的距離為偏差。如果最大偏差超過規(guī)定的公差，應(yīng)將插入到原CL數(shù)據(jù)。理論上，必須采取數(shù)值迭代方法計(jì)算。實(shí)際上，中間點(diǎn)，t=0.5，常被選為候選點(diǎn)[10]。將中間點(diǎn)插入后，即可以生成相應(yīng)的數(shù)控代碼。 4、討論 1．非正交旋轉(zhuǎn)軸的主要特征是在同一臺(tái)機(jī)床上水平位置和垂直位置之間的連續(xù)運(yùn)動(dòng)。在當(dāng)前商業(yè)工具機(jī)的配置中，可以由以上方程得出非正交旋轉(zhuǎn)軸傾斜45度。可以拿工作臺(tái)型傾斜是用來作為一個(gè)例子，方程（5）表示刀具相對(duì)于工件的方向。在初始位置，工作臺(tái)水平，可以確定。非正交旋轉(zhuǎn)軸假定繞x軸旋轉(zhuǎn)θ角使矢量及。將以上條件代入式（5）且，，產(chǎn)生了如下方程： 解得，。因此，當(dāng)工作臺(tái)轉(zhuǎn)動(dòng)角度π，非正交旋轉(zhuǎn)軸B軸轉(zhuǎn)動(dòng)π/4時(shí)工作臺(tái)的處于垂直位置。 圖7工作臺(tái)/主軸傾斜型生成NC數(shù)據(jù)對(duì)話框 2、非正交坐標(biāo)系的采用提高了五軸工具機(jī)床靈活性。然而，在CL數(shù)據(jù)方面是有限制的。只有在方程（7）顯示的條件滿足時(shí)方能使用。當(dāng)非正交軸設(shè)置在45度角時(shí)，的取值范圍在。所以，為負(fù)值時(shí)，通過CAD / CAM軟件生成的CL數(shù)據(jù)無法進(jìn)行加工。 3、生成的NC數(shù)據(jù)是一個(gè)普遍的形式，它可以運(yùn)用到正交配置中。工作臺(tái)傾斜型就是一個(gè)例子。如果向量W是在X軸方向，且Wx =1 Wy=Wz= 0就是CA工作臺(tái)傾斜性的配置。分析方程中的NC數(shù)據(jù)，例如Y軸的值，與文獻(xiàn)[8]中的一致，可以表示如下： 注意，在所列舉的例子中，假設(shè)兩個(gè)旋轉(zhuǎn)軸相交且偏移向量用于推導(dǎo)上述方程。 4、基于和，刀具解可能通過，，且[12, 18]未知的點(diǎn)。該點(diǎn)發(fā)生在且C軸平行于刀具軸時(shí)。正如在圖4所示，如果當(dāng)Pn+1是該點(diǎn)時(shí)， 在理論上可以是任意的值，因?yàn)镻n+2是未知的。Pn+2應(yīng)進(jìn)一步確定，以確保的值是在該連續(xù)兩個(gè)點(diǎn)之間的線性變化。 的值可定義為Pn到Pn+2之間的距離。 5、在實(shí)際的多軸加工中進(jìn)給速度控制是一個(gè)重要的問題。大多數(shù)控制器，如FANUC公司和Cincinnati Milacron公司采用字符（FRN）和G93代碼來控制進(jìn)給速度。FRN由工件的進(jìn)給率的所決定。當(dāng)兩個(gè)或兩個(gè)以上線性軸旋轉(zhuǎn)運(yùn)動(dòng)時(shí)，路徑長(zhǎng)度的確定變得非常復(fù)雜。在大多數(shù)情況下，實(shí)際的路徑長(zhǎng)度可以充分接理論的線性位移[19]。 5執(zhí)行和核查 5.1軟件實(shí)現(xiàn) 在Windows XP環(huán)境中使用BorlandC ++ 、Builder編程語(yǔ)言和OpenGL圖形庫(kù)。采用一個(gè)半徑為35mm、的半球進(jìn)行加工說明。 CL數(shù)據(jù)通過商業(yè)CAD / CAM軟件與PowerMILL[20]產(chǎn)生。機(jī)床采用工作臺(tái)傾斜型與工作臺(tái)/主軸傾斜型的二種形式的工具機(jī)，進(jìn)行了測(cè)試。圖5（a）所示工作臺(tái)傾斜型配置后處理器開發(fā)軟件對(duì)話框。用戶可以用鼠標(biāo)的旋轉(zhuǎn)放大機(jī)床表面模型。當(dāng)用戶輸入相關(guān)參數(shù)，如偏移向量從C軸中心點(diǎn)開始時(shí)，系統(tǒng)會(huì)顯示數(shù)字，以幫助用戶輸入正確的參數(shù)，如圖6所示。最后，點(diǎn)擊“文件”菜單打開CL數(shù)據(jù)，生成NC代碼。圖5（b）和圖7顯示的是工作臺(tái)/主軸傾斜型啟動(dòng)和實(shí)施環(huán)節(jié)的對(duì)話框，。值得注意的是，設(shè)值長(zhǎng)度是從壓刀尖中心到工作臺(tái)表面。 5.2實(shí)體切削仿真 實(shí)體切削仿真軟件VERICUT是用來生成數(shù)控加工數(shù)據(jù)。軟件中有可供選擇的原材料，刀具的規(guī)格尺寸，數(shù)控?cái)?shù)據(jù)，控制器的類型，及物理性能不同的數(shù)控加工工具，它可以用數(shù)控?cái)?shù)據(jù)來模擬材料去除過程。工作臺(tái)傾斜型工具機(jī)用產(chǎn)品仿真和成品加工進(jìn)行驗(yàn)證，如圖8所示。相關(guān)參數(shù)如圖6所示。 圖8 工作臺(tái)傾斜型的VERICUT軟件模擬 圖9工作臺(tái)/主軸傾斜類型的VERICUT軟件模擬 圖9所示工作臺(tái)/主軸傾斜類型的VERICUT軟件模擬。如前所述，根據(jù)圖7，應(yīng)設(shè)置相關(guān)參數(shù)。B軸的向量為。偏移向量從程序原點(diǎn)到旋轉(zhuǎn)刀具軸。 5.3實(shí)驗(yàn)驗(yàn)證 生成的五軸聯(lián)動(dòng)數(shù)控?cái)?shù)據(jù)要進(jìn)一步驗(yàn)證。工作臺(tái)傾斜型五軸加工中心（DECKEL MAHO DMU70改進(jìn)型）配備Heidenhain iTNC530用于半球形工件加工。這項(xiàng)實(shí)驗(yàn)是在下列條件下進(jìn)行： （1）兩個(gè)球頭直徑為10毫米和4毫米的刀具分別用于粗加工和精加工 （2）主軸轉(zhuǎn)速5000r\min,進(jìn)給速度為1000mm/min （3）工作臺(tái)采用7075鋁合金材料制造。 應(yīng)該注意的是，本機(jī)床C軸的正方向是刀具沿著Z軸的負(fù)方向。C軸的實(shí)際數(shù)控?cái)?shù)值再式（11）中為負(fù)值。圖10顯示了實(shí)際的加工過程，揭示正確的后處理程式，能成功生成NC數(shù)據(jù)。 圖10 DECKEL MAHO DMU1070改進(jìn)型機(jī)床的實(shí)際加工實(shí)驗(yàn) a.粗加工 b.精加工 六、結(jié)論 非正交工作臺(tái)和主軸型五軸工具機(jī)床的后處理程序有了一定的發(fā)展。一般的NC數(shù)據(jù)是由齊次坐標(biāo)變換矩陣，正向和逆向運(yùn)動(dòng)學(xué)的分析來確定的。生成的NC數(shù)據(jù)對(duì)那些旋轉(zhuǎn)軸需要相互交叉和非正交軸的傾斜角度為變量的這類機(jī)床是有用的。產(chǎn)生的可變傾斜角能增加派生方程的有效性，從而NC數(shù)據(jù)可降低正交型的配置。該種算法也可以應(yīng)用到線性軸和旋轉(zhuǎn)軸非正交的多功能磨/轉(zhuǎn)機(jī)床中[21]，目前這項(xiàng)工作正在進(jìn)行。 致謝 對(duì)中華人民共和國(guó)理事會(huì)NSC95-2221-E-150-101的財(cái)政資助深表感謝。同時(shí)也對(duì)金屬工業(yè)研究發(fā)展中心提供五軸設(shè)備，及對(duì)在臺(tái)灣Delcam公司的Bacchus Yu先生提出的有效建議意見表示感謝。 參考文獻(xiàn) 1.Goode KF, Rockford IL (1983) Tool head having nutating spindle,US Patent No. 4370080 2.Makino (2003) High-Productivity Aerospace Machining Center.MMS Online?, URL: http://www.mmsonline.com/equipment/mcen389.html 3. Ingersol Mastercenter? (2007), URL: http://www.ingersoll.com/ind/mastercenterH.htm 4. Hagiz G (2006) 5-axis machining, URL: http://numeryx.com/cnc/5axes.htm 5.Affouard A, Duc E, Lartigue C, Langeron J-M, Bourdet P (2004)Avoiding 5-axis singularities using tool path deformation. InternationalJournal of Machine Tools and Manufacture 44:415–425 6. Fauvel OR, Vaidyanathan J, Norrie DH (1990) An analysis oflinearization errors in multi-axis APT-based programming systems.Journal of Manufacturing Systems 9(4):353–362 7. Nagasaka M, Takeuchi Y (1996) Generalized post-processor for5-axis control machining based on form shape function. Journalof the Japan Society for Precision Engineering 62(11):1607–1611 8. Lee RS, She CH (1997) Developing a postprocessor for threetypes of five-axis machine tools. International Journal of AdvancedManufacturing Technology 13(9):658–665 9. She CH, Lee RS (2000) A postprocessor based on the kinematicsmodel for general five-axis machine tools. SME Journal ofManufacturing Processes 2(2):131–141 10. Chou HL (1989) Development of an APT universal postprocessorfor multi-axis CNC milling machine tools. Master’s thesis, NorthCarolina State University, USA 11. She CH, Chang CC (2007) Development of a five-axis postprocessorsystem with a nutating head. Journal of MaterialsProcessing Technology 187–188:60–64 12. Sorby K (2007) Inverse kinematics of five-axis machines nearsingular configurations. International Journal of Machine Toolsand Manufacture 47(2):299–306 13. VERICUT? V5.3 User Manual, URL: http://www.cgtech.com 14. Sakamoto S, Inasaki I (1993) Analysis of generating motion forfive-axis machining centers. Transactions of the Japan Society ofMechanical Engineers, Series C. 59(561):1553–1559 15. Deckel Maho, URL: http://www.dmgtaiwan.com 16. Paul RP (1981) Robot Manipulators: Mathematics, Programmingand Control. MIT press, Cambridge, MA 17. Reshetov DN, Portman VT (1988) Accuracy of Machine Tools.ASME press, New York 18. Bohez E, Makhanov SS, Sonthipermpoon K (2000) Adaptivenonlinear tool path optimization for five-axis machining. InternationalJournal of Production Research 38(17):4329–4343 19. Cincinnati Milacron (1994) Programming Manual for CincinnatiMilacron Acramatic 950MC Rel 3.0 Computer NumericalControl. Ohio 20. Delcam, URL: http://www.delcom.com 21. OKUMA MULTUS, URL: http://www.okuma.co.jp/english/ ORIGINAL ARTICLEPostprocessor development of a five-axis machine toolwith nutating head and table configurationChen-Hua She&Zhao-Tang HuangReceived: 21 August 2006 /Accepted: 7 June 2007#Springer-Verlag London Limited 2007Abstract The postprocessor is an important interface thattransforms cutter location data into machine control data, andin a five-axis machine tool is highly complex because thesimultaneous linear and rotary motions occur. Since mostworks of the five-axis postprocessor method have dealt onlywith the orthogonal machine tools configuration, this studypresents a postprocessor scheme for two types of five-axismachine tools, each with a nutating head and a table whoserotational axes are in an inclined plane. The benefit of such aconfiguration is that it allows switching from vertical tohorizontal machining by a single machine. The generalanalytical equations of NC data are obtained from the forwardand inverse kinematics and the homogeneous coordinatetransformation matrix. The linearization algorithm for thepostprocessor is developed to ensure the machining accuracy.Thepresentedalgorithmisimplementedusingawindow-basedfive-axis postprocessor with nutating axes, and programmed inBorland C+ Builder and OpenGL. A simulation is performedusing solid cutting software and a trial-cut experiment wasconducted on a five-axis machine tool with a nutating table toelucidate the accuracy of the proposed scheme.Keywords Postprocessor.Five-axismachining.Form-shapingfunction.Nutatingaxis1 IntroductionFive-axis machining is becoming increasingly used bymachine tool users, especially in machining complexfreeform surfaces. The conventional five-axis machine toolhas three orthogonal linear axes and two rotary axes. Therotary axes are typically orthogonal to each other and thecentre line of the rotary axis is parallel to the direction ofthe linear axis. Various machine tool builders, such asMakino, Ingersol and Deckel Maho, incorporate a nutatinghead or a nutating table in the machine tools to improvetheir versatility and flexibility. The word of “nutating”means oscillatory motion about the axis of a rotating body,which is similar to the slow spinning of a coin on a table. Afive-axis machine tool with a nutating unit has a rotationalaxis in an inclined plane 1, and not parallel to the linearaxis, providing the advantage that allows the cutting tool toorient itself toward any angle within a hemisphere 2, 3.Such machines can move continuously between thehorizontal and vertical positions in a single setup on thesame machine. The nutating head provides very useful inmanufacturing aerospace parts because it has no motors onthe head, and is more rigid. The motors for the spindle areon the machine and the motion is transferred to them byhollow shafts and gears 4.Because the linear and rotary axes move simultaneouslyon a five-axis machine, the derivation of the five-axisprogram is more complex than that of the three-axisprogram. A postprocessor must be utilized to translate thecutter location (CL) data from the CAM system into themachine control data. Although the advanced controllerscan accept the CL data to machine the workpiece in real-time without the need of postprocessor 5, they arerelatively expensive and not commonly used in mostindustries. The methods of developing multi-axis postpro-Int J Adv Manuf TechnolDOI 10.1007/s00170-007-1126-5C.-H. She (*)Department of Mechanical and Computer Aided Engineering,National Formosa University,64 Wen-Hua Road, Huwei,Yunlin 632, Taiwan, Republic of Chinae-mail: chshenfu.edu.twZ.-T. HuangDepartment of Mechanical and Automation Engineering,Da-Yeh University,112 Shan-Jiau Road, Da-Tsuen,Chang-Hua 515, Taiwan, Republic of Chinacessors can be mainly divided into three categories -graphical 6, numerical iterative 7 and coordinatetransformational 810. Since the coordinate transforma-tion method yields the analytical equation of NC data mostefficiently, it has been adopted extensively in recent work.However, almost all of these approaches involve postpro-cessor methods for five-axis machine tools with orthogonalrotary axes. Relatively few studies have addressed non-orthogonal configuration. For example, the authors havedeveloped the postprocessor for the spindle-tilting typefive-axis machine tool with a nutating head 11. Recently,Sorby 12 has presented a closed-form solution for a table-tilting type five-axis machine tool with a nutating table.However, this solution exhibits some limitations. Forexample, the offset vectors such as from the workpieceorigin to the rotary table and from the secondary rotary tothe primary rotary are not defined, and the angle ofinclination of the nonorthogonal axis is fixed at 45 degrees.This study develops a postprocessor for two five-axismachine tools each with a nutating head and table config-uration. Based on the homogeneous coordinate transforma-tion matrix, the general analytical equations of NC data areobtained from the forward/inverse kinematics and themachine tools form-shaping function matrix. The deter-mined equation is in general form because the rotary axes areassumed not to intersect each other; the angle of inclinationof the nonorthogonal axis is variable, and the offset vectorfrom the origin of the workpiece to the rotary table isdefined. Moreover, the linearization algorithm of the post-processor is developed to ensure the machining accuracy.A window-based postprocessor is developed and a graph-ical interface that dynamically displays the surface model andthemotions ofallofthe axesoftheconfiguredmachinetoolispresented to help the user to input relevant parameterscorrectly. Additionally, the generated NC data are verifiedusingthecommercialsolidcuttingsoftwareVERICUT 13and a machining experiment is conducted on a five-axismachine tool with a nutating table to confirm the effective-ness of the proposed postprocessor methodology.2 Configuration and modeling of five-axis machine toolMost five-axis machine tools have two rotary axes as wellas the conventional X, Y and Z axes. Following Sakamotoand Inasaki 14, the configurations of five-axis machinetools can be categorized into three types: spindle-tilting,table-tilting and table/spindle-tilting. Commercial machinetools with the nonorthogonal configuration, as shown inFig. 1, are also of three types. Figure 1 (a) shows thespindle-tilting type with a nutating head, such as theMakino MAG3 2, which is designed with a rotary axis(C-axis) behind a nutating head that rotates about the B-axis. Figure 1 (b) displays the table-tilting type with anutating table, such as the Deckel Maho DMU 70 eVolution15, which has two rotary axes on the table, and one rotaryaxis (C-axis) is parallel to the Z-axis while the non-orthogonal rotary axis is inclined at an angle to the C-axis.Figure 1 (c) presents the table/spindle-tilting type with anutating head, such as the Deckel Maho 200P 15,inwhichone rotary table (C-axis) is on the table and the nutatingrotary head (B-axis) is on the spindle. Since the authors havealready presented the spindle-tilting postprocessor with anutating head 11, this study focuses on developing thepostprocessors with the other two configurations.A five-axis machine tools can be regarded as amechanism with serially connected links with revolute orprismatic joints. Forward kinematic equations must beestablished to describe mathematically the motion of thecutting tool in relation to the workpiece. The fundamentalcoordinate transformation matrices, including the transla-tion matrix Trans and the rotation matrix Rot 16, areintroduced. The translation matrix Trans can be expressedas follows:Transa;b;c 100a010b001c0001266437751where Trans(a, b, c) implies a translation given by thevector a i+b j+c k.The general rotation transformation matrix should beused to describe the rotation of the nutating unit. Thecoordinate system is assumed to rotate through an angle offwaround any arbitrary vector W=Wxi+Wyj+Wzk; therotational transformation matrix can be expressed as:Rot W;w W2xVw CwWxWyVw? WzSwWxWzVw WySw0WxWyVw WzSwW2yVw CwWyWzVw? WxSw0WxWzVw? WySwWyWzVw WxSwW2zVw Cw00001266437752Int J Adv Manuf Technolwhere “C” and “S” are cosine and sine functions,respectively, and Vfw=1Cfw.3 Postprocessor3.1 Table-tilting type with a nutating tableFigure 2 depicts relevant coordinate systems for thisconfiguration. The coordinate system for the workpiece isOwXwYwZwwhile the system OtXtYtZtis attached to thecutting tool. Since the two rotary axes are assumed not tointersect each other, a common normal line is mutuallyperpendicular to both axes. The common normal line inter-sects with the C-axis and B-axis at two points, RC and RB.The offset vector Lxi+Lyj + Lzk is determined from theorigin Owto the pivot point RC, whereas the offset vectorMxi+Myj+Mzk is calculated from the pivot point RC tothe pivot point RB.Since the structural elements of the machine toolcomprise the C rotary table, the B nutating rotary table,the machine bed, the X linear table, the Y linear table, the Zlinear table, the spindle head and the cutting tool. Therelative position and orientation of the cutting tool withrespect to the workpiece can be determined sequentiallystarting from the workpiece and ending at the cutting tooland is referred to as the form-shaping function 17. TheCBXYZaXYZCBcBCYXZbFig. 1 Configuration for five-axis machine tool with nutating head andtable. a spindle-tilting type with a nutating head. b table-tilting typewith a nutating table. c table/spindle-tilting type with a nutating headtOtXtYtZBRBwOwXwYwZC xyzLLL+ijkRCOffset vectorxyzMMM+ijkOffset vectorFig. 2 Coordinate systems of table-tilting type configurationInt J Adv Manuf Technolform-shaping function of this machine tool can bemathematically expressed in matrix form as follows:Trans Lx;Ly;Lz?Rot z;?zTrans Mx;My;Mz?Rot W;?wTrans Px;Py;Pz?0 00 01 00 1266437753wherePx, Pyand Pzdenote the relative translation distances ofthe X, Y and Z linear tables, respectively. The termsfzandfwrepresent the angles of rotation for the C-axis and the B-axis, respectively. The positive rotation is in the direction of anadvancing right-hand screw about the +C and +B axes.Equation (3) specifies the form-shaping function matrix of thismachine tool and the joint parameters Px, Py, Pz,fzandfwshould be determined by the inverse kinematics. The firststep is to calculate the required rotary angles to yield the toolorientation, and the second is to calculate the required positionin relation to the linear axis to determine the position of thecentre of the tool tip using the known rotary angles.Whenthe CLdataincludingthe positionofthecentreof thetool tip Qxi Qyj Qzk and the tool orientation Kxi Kyj Kzk are known, the CL data can be expressed in thematrix form as follows:KQ01?KxKyKz0QxQyQz1266437754Since both Eq. (3) and Eq. (4) represent the samerelationship between the cutting tool and the workpiece, thedesired joint parameters can be determined by equatingthese two matrices. Equating the CL data matrix and theform-shaping function matrix, and taking the correspondingelements of the two matrices yield the following equations:KxKyKz026643775CzWxWz1 ? Cw ? WySw? SzWyWz1 ? Cw WxSw?SzWxWz1 ? Cw ? WySw? CzWyWz1 ? Cw WxSw?W2z1 ? Cw Cw0266437755QxQyQz126643775PxCzW2z1 ? Cw Cw? SzWxWy1 ? Cw ? WzSw?PyCzWxWy1 ? Cw WzSw? SzW2y1 ? Cw CwhinoPzCzWxWz1 ? Cw ? WySw? SzWyWz1 ? Cw WxSw?Lx CzMx SzMyPx?SzW2x1 ? Cw Cw? CzWxWy1 ? Cw ? WzSw?Py?SzWxWy1 ? Cw WzSw? CzW2y1 ? Cw CwhinoPz?SzWxWz1 ? Cw ? WySw? CzWyWz1 ? Cw WxSw?Ly? SzMx CzMyPxWxWz1 ? Cw WySw? PyWyWz1 ? Cw ? WxSw?PzW2z1 ? Cw Cw? Lz Mz126666666666666666666666664377777777777777777777777756Int J Adv Manuf TechnolThe joint anglesfzandfwshould be determined first.Equating the corresponding third element in Eq. (5) yieldsthe following B-axis angle:B w arccosKz? W2z1 ? W2z?0 ? w? 7Notably, there is another possible solution for B-axisangle in the range of ? ? w? 0, which can be obtainedas follows:B w ?arccosKz? W2z1 ? W2z?8If the operating range of the nutating table is in the rangebetween and 0, the solution should be modified asshown in Eq. (8). On the other hand, if the operating rangeof the nutating table meets the two possible solutions, theshortest rotational angle movement of the nutating table isusually chosen in the algorithm.Furthermore, equating the corresponding first andsecond elements in Eq. (5) and solving the simultaneouslinear equations for Sfzand Cfz, yield:Sz?KyWxWz1 ? Cw ? WySw? KxWyWz1 ? Cw WxSw?WxWz1 ? Cw ? WySw?2 WyWz1 ? Cw WxSw?29CzKxWxWz1 ? Cw ? WySw? KyWyWz1 ? Cw WxSw?WxWz1 ? Cw ? WySw?2 WyWz1 ? Cw WxSw?210Since the denominators in Eqs. (9) and (10) are the sameand always positive, the C- axis angle can be determined asfollows:C z arctan2?KyWxWz1 ? Cw ? WySw? KxWyWz1 ? Cw WxSw?;KxWxWz1 ? Cw ? WySw?KyWyWz1 ? Cw WxSw? ? z? 11where arctan2(y,x) is the function that returns angles in therange ?p ? q ? p by examining the signs of both y and x16.In addition, comparing the corresponding elements ofthe matrix on both sides of Eq. (6) yields three simulta-neous equations in three unknowns Px, Pyand Pz. Theprogram coordinate system is assumed to coincide with theworkpiece coordinate system. Accordingly, the expressionsof the desired NC data (denoted as X, Y and Z) can beobtained by considering the two offset vectors Lxi+Lyj+Lzk and Mxi+Myj+Mzk, and are expressed as follows:X Px Lx Mx Qx? CzMx SzMy Lx?CzW2x1 ? Cw Cw? SzWxWy1 ? Cw ? WzSw? Qy? ?SzMx CzMy Ly?SzW2x1 ? Cw Cw? CzWxWy1 ? Cw ? WzSw? Qz? Mz Lz? WxWz1 ? Cw WySw? Lx Mx12Int J Adv Manuf TechnolY Py Ly My Qx? CzMx SzMy Lx?CzWxWy1 ? Cw WzSw? SzW2y1 ? Cw Cwhino Qy? ?SzMx CzMy Ly?SzWxWy1 ? Cw WzSw? CzW2y1 ? Cw Cwhino Qz? Mz Lz? WyWz1 ? Cw ? WxSw? Ly My13Z Pz Lz Mz Qx? CzMx SzMy Lx?CzWxWz1 ? Cw ? WySw? SzWyWz1 ? Cw WxSw? Qy? ?SzMx CzMy Ly?SzWxWz1 ? Cw ? WySw? CzWyWz1 ? Cw WxSw? Qz? Mz Lz? W2z1 ? Cw Cw? Lz Mz143.2 Table/spindle-tilting type with a nutating headThe table/spindle-tilting type configuration has one rotaryaxis on the table and one nutating rotary axis on the spindle.Figure 3 illustrates two pivot points RC and RB on the C andB axes, respectively. The pivot point RC is located arbitrarilyon the C-axis and the pivot point RB is chosen to be theintersection of the nutating rotary B-axis and the axis of thecutting tool. The offset vector Lxi +Lyj +Lzk is calculatedfrom the origin Owto the pivot point RC and the effectivetool length, Lt, represents the distance between the pivotpoint RB and the cutter tip centre. The form-shapingfunction matrix of this configuration can be obtained bythe coordinate transformation matrices:Trans Lx;Ly;Lz?Rot z;?zTrans Px;Py;Pz?Rot W;w00001 ? Lt012664377515Equating Eq. (4) and Eq. (15) leads to the followingequations:KxKyKz026643775CzWxWz1 ? Cw WySw? SzWyWz1 ? Cw ? WxSw?SzWxWz1 ? Cw WySw? CzWyWz1 ? Cw ? WxSw?W2z1 ? Cw Cw02664377516QxQyQz126643775?CzWxWz1 ? Cw WySw? SzWyWz1 ? Cw ? WxSw?LtCzPx SzPy Lx? ?SzWxWz1 ? Cw WySw? CzWyWz1 ? Cw ? WxSw?Lt?SzPx CzPy Ly? W2z1 ? Cw Cw?Lt Pz Lz1266666666666643777777777777517Int J Adv Manuf TechnolThe joint parameters can be evaluated using the sameprocedure similar to the table-tilting configuration. Notably,the reference driving point of NC data in this configuration isassumedtobethepivotpointRB.Thisdefinitionisadoptedtoboth the spindle-tilting and table/spindle-tilting type config-urations, and is consistent with those used in most of thecommercial post-processor software packages. The completeanalytical equations for NC data can be expressed as:B w arccosKz? W2z1 ? W2z?0 ? w? 18C z arctan2 ?KyWxWz1 ? Cw WySw?KxWyWz1 ? Cw ? WxSw?;? KyWxSw? WyWz1 ? Cw?KxWySw WxWz1 ? Cw? ? z? 19X Lx Px LtWxWz1 ? Cw LtWySw SzLy? Qy? CzLx? Qx Lx20Y Ly Py LtWyWz1 ? Cw ? LtWxSw? CzLy? Qy? SzLx? Qx Ly21Z Lz Pz LtW2z1 ? Cw LtCw Qz223.3 Linearization problemTheoretically, the CAD/CAM system generates the CL databased on the assumption that the cutting tool moves linearlybetween two successive points. However, the actual toolmotion trajectory with respect to the workpiece is not linearand becomes curved since the linear and rotary axes movesimultaneously. The curved path deviates from the linearlyinterpolated straight line path between successive pathpoints, and this problem is known as the linearizationproblem. An algorithm must be developed to solve thisproblem.Assume that Pn, Pn+1and Pn+2are three continuousadjacent points in CL data, plotted in Fig. 4. The vector ofPnin matrix form can be expressed as Qn,xQn,yQn,zKn,xKn,yKn,z, where Qn,x, Qn,yand Qn,zare the components ofthe position of the center of the tool tip, and Kn,x, Kn,yandKn,zare the components of the tool orientation. Thecorresponding machine NC code of Pnis Mn=XnYnZnBnCn. As the five axes move simultaneously from thecurrent position Pnto the subsequent position Pn+1, eachaxis is assumed to move linearly between the specifiedpoints 18. Therefore, each point in the actual curved pathcan be expressed as follows:Mm;t Mn t Mn1? Mn23where t is a dummy time coordinate0 ? t ? 1. Thecorresponding CL data Pm,tfor Mm,tcan be determined bythe forward kinematics equations, e.g. Eqs. (5) and (6) forthe table-tilting type and Eqs. (16) and (17) for the table/spindle-tilting type. Moreover, each point in the ideal lineartool path can be determined as follows:Pn;t Pn t Pn1? Pn24wOwXwYwZtOtXtYtZCB xyzLLL+ijkRCRBOffset vectortLFig. 3 Coordinate systems of table/spindle-tilting type configurationInt J Adv Manuf Technol,nnP M,n tn tPM,m tm tMP11,nn+PM1,1,ntnt+PM1,1,mtmt+MP,n+2n+2PMInterpolated tool pathActual curved tool pathIdeal linear tool path,n tdFig. 4 Linearization problem inmulti-axis machining abFig. 5 Initiating dialog for the developed postprocessor. a table-tilting type. b table/spindle-tilting typeInt J Adv Manuf TechnolThe distance between Pm,tand Pn,tdenoted as dn,tformsa chordal deviation. If the maximum deviation (dn,t)maxexceeds the prescribed tolerance, then the additionalinterpolated CL data Pn,tshould be inserted into the originalCL data. Theoretically, the numerical iterative method forcalculating (dn,t)maxmust be adopted. Practically, the middlepoint, t = 0.5, is often selected as the candidate point 10.After the intermediate point Pn,thas been inserted, thecorresponding machine NC code can be generated.4 Discussion1.The main characteristic of the nutating rotary axisconfiguration is the continuous motion between thehorizontal and vertical positions in a single setup onthe same machine. In the current configuration of thecommercial machine tool, the angle of inclination ofthe nutating rotary axis is 45 degrees. This fact can beexplained by the equations derived above. The table-tilting type is used as an example. Equation (5)represents the tool orientation in relation to theworkpiece. The tool orientation relative to the work-piece in the initial position, where the table surface ishorizontal, can be determined by substitutingfz=fw=0 into Eq. (5), and is given by 0 0 1 0T. The nutatingrotary axis is assumed to rotate around X-axis by anangle so that the components of the vector W are Wx=0, Wy=S and Wz=C. Substituting the aboveconditions into Eq. (5), and settingfz=0 and KxKyKz0T=0 1 0 0Tfor the table surface in the verticalposition yields the following equation:0?10026643775SSw?SC 1 ? CwC2 1 ? CwCw02664377525The solutions to Eq. (25) for andfware =/4 andfw=. Therefore, the table surface can be set in the verticalposition when the table is rotated through an angle aboutthe nutating B rotary axis at an angle of inclination of /4.2.The nutating units on the five-axis machine tools canenhance the flexibility of the machining strategy. How-ever, the CL data considered are limited. Equation (7)Fig. 6 Implementation dialog for generating NC data for table-tilting type configurationInt J Adv Manuf Technolshows that the conditionKz?W2z1?W2z? ? 1 should be satisfied.When the nutating axis is set at an angle of 45 degrees,i.e. =/4 and Wz=C45, Kzis in the range 0 Kz 1.Consequently, the CL data generated by the CAD/CAMs