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南京工程學(xué)院 畢業(yè)設(shè)計(論文)外文資料翻譯 原 文 題 目：Five-axis milling machine tool kinematic chain design and analysis 原 文 來 源： International Journal of Machine Tools & Manufacture 42 (2002) 505–520 學(xué) 生 姓 名： 石佳倩 學(xué) 號： X00231120404 所在院(系)部： 工業(yè)中心 專 業(yè) 名 稱： 機械設(shè)計制造及其自動化 五軸銑床運動鏈的設(shè)計與分析 作者：E.L.J. Bohez 摘要：現(xiàn)如今五軸數(shù)控加工中心已經(jīng)非常普及。大部分機床的運動學(xué)分析都基于笛卡爾直角坐標(biāo)系。本文羅列了現(xiàn)有的概念設(shè)計與實際應(yīng)用，這些從理論上都基于自由度的綜合。一些有用的參數(shù)都有所規(guī)定，比如工件使用系數(shù)，機床空間效率，方向空間搜索以及方向角等。每一種概念，它的優(yōu)缺點都有所分析。選擇的標(biāo)準(zhǔn)及機器參數(shù)設(shè)置的標(biāo)準(zhǔn)都給出來了。據(jù)于Stewart平臺的新概念最近行業(yè)內(nèi)已有介紹并作簡短討論。 關(guān)鍵詞：五軸；機床；運動鏈；工作區(qū)；CNC；旋轉(zhuǎn)軸 1.緒論 設(shè)計一臺數(shù)控機床主要遵循以下規(guī)則： 1、刀具和工件在空間方向上要有足夠的靈活性。 2、方向和位置的改變要盡可能的快。 3、方向和位置的改變要盡可能的準(zhǔn)確。 4、刀具和工件快速變、換。 5、環(huán)保 6、切削材料速度快 一臺數(shù)控機床的軸的數(shù)目通常取決于其自由度數(shù)目或者獨立控制運動的導(dǎo)軌數(shù)目。國際標(biāo)準(zhǔn)委員會推薦通過右手笛卡兒坐標(biāo)系來命名坐標(biāo)軸，刀具相應(yīng)的為Z軸。一臺三軸銑床上有三條導(dǎo)軌，X,Y,Z向，它們可用來在長度范圍內(nèi)在任意位置上移動。加工過程中刀具軸的位置始終不變。這就限制了刀具相對于工件在方向上變化的靈活性，并且導(dǎo)致多次工裝的出現(xiàn)。為了盡可能的提高刀具相對于工件的靈活性，無需再次裝夾工件，必須要加入多個自由度。對于傳統(tǒng)三軸機床來說這可以通過提供旋轉(zhuǎn)滑臺來實現(xiàn)。圖1給出了一個五軸銑床的例子。 圖1 五軸數(shù)控機床 2.運動鏈圖表 通過制作機器的運動鏈圖表對于機器的分析來說十分有用。通過運動簡圖得知兩組軸可以迅速的區(qū)分開：工件裝夾軸和刀具軸。圖2給出了圖1五軸機床的運動鏈簡圖。由圖上可以看出工件由四根軸承載，刀具僅在一根軸上。這個五軸機床與兩工位操作機器人很相似，一個機器人夾住工件，另一個夾住刀具。為了獲得刀具工件方向上的最大自由，五個自由度已是最低要求，這就意味著工件和刀具可以在任意角度位置相對定位。最低需求的軸數(shù)也可以通過剛體運動學(xué)的方法來分析。兩個剛體在空間確定相對位置，每個剛體需要6個到12個自由度。然而由于任意的移動或轉(zhuǎn)動并不改變相對位置就允許將自由度減少到6。兩個剛體之間的距離通過刀具軌跡來描述，并且允許去掉一個額外的自由度，結(jié)果也就是5個自由度。 圖2 運動鏈圖 3.參考文獻(xiàn) 最早（1970年）到目前并且仍就有參考價值的對五軸數(shù)控銑床的介紹之一是由 Baughman[1]提出的并清楚的闡述了它的應(yīng)用。APT語言隨后成為唯一的五軸輪廓加工的編程語言之一。后處理階段的問題也在數(shù)控發(fā)展的早期由Sim[2]清楚的表述出來，并且大部分問題到現(xiàn)在仍然有效。Boyd [3]也是最早引進(jìn)數(shù)控機床的先驅(qū)之一。Beziers[4]的書也是非常有用的介紹。Held[5]在他的小型銑削加工的書里對多軸機床也有非常簡短但啟發(fā)性的定義。目前一篇適用于解決五軸數(shù)控機床工作空間計算的文章，通過使用Denawit-Hartenberg發(fā)表并由 Abdel-Malek and Othman[6]改進(jìn)的算法應(yīng)用于多弧段切削。許多對機床的類型和概念設(shè)計，可以被應(yīng)用于五軸機床[7]但不是專門為五軸機床。對部分機床設(shè)置的數(shù)量和最優(yōu)取向上進(jìn)行了探討[8]。關(guān)于對刀具路徑生成的技巧和新需求由B.K. Choi et al[9]給出。工件與刀具的圖像模擬也是研究的熱點并且是一個好的入門讀物[10]。 4.五軸機床運動結(jié)構(gòu)的分類 從R軸（旋轉(zhuǎn)軸）和T軸（移動軸）劃分大致可以分為四大部分：（i）3個移動軸和2個轉(zhuǎn)動軸；（ii）2個T軸和3個R軸；（iii）1個移動軸和4個轉(zhuǎn)動軸以及（iv）5個轉(zhuǎn)動軸；幾乎所有五軸機床都是第一組。也有一些焊接機器人，彎折機器以及激光機器也屬于這一類。只有限距五軸機床屬于第二組，用以制造船舶螺旋槳用。第三組和第四組用于制造機器人，常常另加三個自由度。在不同的制品中，五根軸可以在工件或刀具之間分配。第一分類可以由工件和刀具所承載的軸數(shù)以及每根軸在運動鏈中的功能來劃分。另一種分法是據(jù)于旋轉(zhuǎn)軸的位置，在工件一邊還是在刀具一邊。五自由度基于笛卡爾坐標(biāo)系的機床是：3個移動軸X,Y,Z(通常表述為TTT)和2個旋轉(zhuǎn)運動AB,AC,BC（通常稱作RR）。擁有3個旋轉(zhuǎn)軸和2個移動軸的制品并不多見。如果一個軸裝夾工件，習(xí)慣上不另加?xùn)|西在這根軸上。由圖1五軸機床可記為 X 'Y 'A' B 'Z. XYAB軸裝夾工件，Z軸裝刀具。圖3展示的是XYZA'B'型機床，3個移動軸裝夾刀具，2個旋轉(zhuǎn)軸裝工件。 圖3 XYZA 'B '型機床 4.1基于工件和工具承載軸順序的分類 理論上，如果工具和工件承載軸的兩個運動鏈的順序算作一個不同的配置，可能配置的數(shù)目是相當(dāng)大的。也只有兩個線性軸和三個旋轉(zhuǎn)軸的組合包括在內(nèi)。 一個工具承載軸和四個工件承載軸可以在一個五軸機床組合如下：對每一個可能的工具承載軸X，Y，Z，A，B，C其他四個工件承載軸可以從現(xiàn)存的五個軸中選。所以四軸組合的數(shù)量與另一個配置是考慮不同排列5×4！=120為每個可能的工具軸選擇（1出6或6的可能性）。所以理論上有6×120=720可能五軸機床使用一個工具承載軸。同樣的分析可以用于所有其他組合。t數(shù)量的工具承載軸和w數(shù)量的工件承載軸（w+t=5）的組合的總數(shù)量如下。 (1) (2) 該方程的值總是等于6！或720 W + T =5時。這些720的組合將只包含兩個線性軸。如果只有五軸機床被認(rèn)為帶有三個線性軸，那么只有3×5！=360組合是可能的。 5.五軸機床工作空間 在定義五軸機床工作空間之前，有必要說明一下刀具工作空間和工件工作空間。刀具工作空間就是通過刀具參考點沿著刀具軌跡生成軸來獲得。工件空間也是同樣定義的（工作臺中心可以被選擇為工件參考原點）。這些工作空間可以通過計算切削量來定義。 基于上述定義一些參數(shù)量可以定下來，這些參數(shù)對比較，選擇以及設(shè)計不同類型機床都是十分有用的。 圖11 G2/G3’組中的 R 'R機床 6.選擇五軸機床的標(biāo)準(zhǔn) 完全學(xué)習(xí)好如何為專用機床選擇或設(shè)計一個五軸機床是不現(xiàn)實的。只有使用主要標(biāo)準(zhǔn)，來核實五軸機床并加以討論。 6.1 五軸機床的應(yīng)用 應(yīng)用程序可以分為位置和輪廓。圖12和圖13展示了五軸位置機床和五軸輪廓機床。 6.1.1圖12展示了一個多孔以及不同角度有平臺的工件。要用一個三軸磨床加工這個工件，一步也無法完成。如果用五軸機床則可以加工。輪廓更多的參數(shù)等信息可以在參考文獻(xiàn)[13]中去查看。五軸機床用于加工輪廓的有：（i）葉片類產(chǎn)品，例如空氣壓縮機的葉片和渦輪機的葉片；（ii）燃料泵的噴嘴；（iii）輪胎的輪廓；（iv）醫(yī)學(xué)假肢，例如人工心臟瓣膜；（v）復(fù)雜表面的模具。 圖12 五軸加工多孔復(fù)雜方位角零件 圖13 五軸加工復(fù)雜輪廓零件 6.1.2．五軸輪廓 圖13顯示了一個五軸輪廓的例子，機器的表面形狀復(fù)雜，我們需要控制工具相對部分切割過程中的方向。該工具工件每一步方向的改變。CNC控制器需要同時控制在材料去除過程中所有的五軸。輪廓上更多詳情可參考文獻(xiàn)[13]。五軸機床用于加工輪廓的有：（i）葉片類產(chǎn)品，例如空氣壓縮機的葉片和渦輪機的葉片；（ii）燃料泵的噴嘴；（iii）輪胎的輪廓；（iv）醫(yī)學(xué)假肢，例如人工心臟瓣膜；（v）復(fù)雜表面的模具。 6.2軸配置選擇 軸配置選擇的大小和重量是非常重要的一部分作為第一標(biāo)準(zhǔn)來設(shè)計或選擇一個配置。非常沉重的工件需要短的工件運動鏈。也有一個工件偏愛橫機表內(nèi)，使之更方便修復(fù)和處理。把非常沉重的工件放在一個旋轉(zhuǎn)軸運動鏈上將增加方向的靈活性。提供一個單一的橫向旋轉(zhuǎn)軸的工件，會使機器更加靈活。在大多數(shù)情況下，工具攜帶的運動鏈將盡可能保持簡短，因為刀具主軸驅(qū)動器必須同時進(jìn)行。 6.3例1—軸加工的首飾 圖14中一個典型的工件可以作為花形圖的一部分。此應(yīng)用程序是清楚輪廓。將部分組裝成相對比較小的工具。小直徑工具也需要一個高速主軸。水平旋轉(zhuǎn)表將作為經(jīng)營者一個很好的選擇將有一個良好的視圖部分（360 °范圍）。所有工件承載軸將是一個很好的選擇，因為刀具主軸可以固定，并且非常嚴(yán)格。有20種方法，可以使軸合并起來，組合成工件運動鏈。這里只有兩個運動鏈將被考慮。案例一是圖15 TTTRR運動鏈圖。案例二是圖16 RRTTT運動鏈圖。對于機器模型I X= 300mm，Y = 250mm和Z = 200mm，C = N的360 °，A= 360 °，以及100mm直徑機床表將被考慮。為此運動工作區(qū)的工具鏈?zhǔn)且粋€單點。參考點的設(shè)置也可以選擇小。如果兩軸中心線的相交點在旋轉(zhuǎn)參考，移動工件的工作空間將得到大小XYZ或300 × 250 ×200立方毫米。如果兩個旋轉(zhuǎn)軸的中心線的不相交的工件，工件參考點，有較大的工作空間。這將是一個圓邊棱柱形。這個圓角半徑邊緣的偏心距，工件相對于每一個參考中心線。模型II圖15中RRTTT旋轉(zhuǎn)軸運動鏈開始時的運動。這里還有兩種不同值的旋轉(zhuǎn)偏心距將被考慮。第5條中定義的參數(shù)計算為每個模型的偏心率總結(jié)于表1?？梢钥闯?，隨著旋轉(zhuǎn)軸的運動鏈的結(jié)束（模型I），機床工作空間要小得多。有兩個主要原因。工具和工件的波及體積要小得多，第二個原因是由于很大一部分機床的工作空間無法使用的情況，因為線性干擾軸。然而工作區(qū)利用系數(shù)較大的模型沒有偏心距，因為工具的工作區(qū)與工件的工作區(qū)相比還是相對較小的模型I并且e = 50mm?？臻g索引的定位是相同的這兩種情況，如果該表直徑保持不變。模型II可以處理更大的工件在相同范圍的線性轉(zhuǎn)動軸運動鏈開始時，形成一個更大的機床工作空間，也少了很多干擾機床工作空間的幻燈片。其他18個可能的選擇將在索引值之間。 圖14 珠寶的應(yīng)用程序 模型 I - 偏心距 = 50 mm 模型 I - 偏心距 = 0 mm 圖15 TTTRR型五軸機床 模型 II - 偏心距 = 150 mm 模型 II - 偏心距 = 0 mm 圖16 RRTTT型五軸機床 T'T'TR'R' 模型 I R'R'T'T'T' 模型 II 偏心距 e=0 mm e=50 mm e=0 mm e=150 mm WSmt 25.13 dm3 25.13 dm3 48.0 dm3 32.4 dm3 WSmt 剪除 14.57 dm3 14.57 dm3 48.0 dm3 32.4 dm3 WSTOOlU WSWORK 15 dm3 30.85 dm3 107.7 dm3 39.8 dm3 WR 0.97 0.47 0.44 0.814 最大范圍 中 100 mm 中 100 mm 中300 mm 中250 mm OS： 0.036 0.036 0.29 0.25 OA： 2 2 2 2 空間 32 dm3 45 dm3 100.53 dm3 56.55 dm3 MTs 0.47 0.57 0.48 0.57 表一 比較兩種機床的工作區(qū) 6.4 例二—轉(zhuǎn)盤選擇 兩臺機器使用相同的運動圖并且在相同的范圍將會比較（圖17）。有兩個可供選擇的旋轉(zhuǎn)軸：兩軸垂直表（模型一），兩軸水平表（模型二）。表2和表3提供的功能比較重要?？梢钥闯觯瑴p少軸旋轉(zhuǎn)范圍則增加了機床的工作空間。所以模型I適合大范圍較小工件定位，通常需要輪廓申請。模型II將適合大工件的定位與變化較少的工具或?qū)⑿枰獌蓚€設(shè)置。這種額外的安裝要求可能沒有那么重要那么大的規(guī)模。水平表可以使用托盤的內(nèi)部體制轉(zhuǎn)換的外部設(shè)置。在B軸較大的角度范圍-105到+105。模型I和模型II相比-45到+20，模型I更適合復(fù)雜的雕刻表面，還因為高的角速度范圍。選擇最高的主軸轉(zhuǎn)速，應(yīng)該選擇并且允許使用較小直徑的切削刀具導(dǎo)致更少的削弱和較小的切削力。高主軸轉(zhuǎn)速會更容易降低開模電火花加工機床的銅電極。垂直表也更好的去除芯片。但是，大范圍的角定位降低了工件的最大尺寸為300毫米和100公斤。模型II與模型I具有相同的線性軸范圍，但在旋轉(zhuǎn)范圍要小的多，可以很容易地處理工件雙倍的大小和重量。模型II將有利于定位的應(yīng)用。模型我I不能提供自動交換工件，它不適合大規(guī)模生產(chǎn)。模型II的自動工件交換適用于大規(guī)模生產(chǎn)的應(yīng)用程序。不過模型I可以選擇定位零件如液壓閥外殼，小型的并且需要大角度范圍。 模型I 模型II 圖17模型I和模型II型 TTRRT型機床 表二 圖17中五軸機床的規(guī)格 機型 模型 I 模型 II B軸的范圍 -105 to +105° - 45 to +20° A軸的范圍 nx360° nx360° 角速度 °/s 8500/14,000 3000/3000 表的種類 垂直 水平 直徑表 320 mm 520 mm 最大負(fù)載表 100 kg 200 kg X Y Z 范圍 600x450x500 mm 600x450x500 mm 刀尖 ISO 40 ISO 40 重量 4500 kg 5500kg 交換板 No Yes 表三 圖17中五軸機床工作區(qū)的比較 模型 I 模型 II WSMT 212.0dm3 145.6 dm3 WSMT 剪除 84.0dm3 145.6 dm3 WSTOOlUWSwORK 334.5 dm3 180.5 dm3 Wr 0.25 0.81 最大范圍 中300 mm 中930 mm OSI 0.17 0.05 OAI 1.17 0.36 空間 9000 dm3 9000 dm3 MTs 0.037 0.07 7.基于Stewart 平臺的新型加工概念 傳統(tǒng)機床結(jié)構(gòu)是基于笛卡爾坐標(biāo)系的。很多表面輪廓的應(yīng)用只有通過五軸數(shù)控機床加工，才是最合適的。這種五軸機床結(jié)構(gòu)還需要另外兩個旋轉(zhuǎn)軸。為了加工精確，達(dá)到加工硬度，能夠裝夾大型工件，又大又重的機械裝置是必要的。從經(jīng)典五軸機床的運動鏈圖表可以看出，在運動鏈中設(shè)計第一個軸來承載后續(xù)的軸。因此傳輸動力將受到鏈接處慣性的限制。如果有這么樣一個機構(gòu)，它能夠在移動工件的時候不帶動其它軸，是理想的。一種新的設(shè)計理念就是“HEXAPOD”的使用。Stewart [16]在1965年描述了hexapod的理論。它最早是由Gough和 Whitehall [20] 于1954年建成并擔(dān)任輪胎測試儀。很多可能的使用都被提出來了，但它僅被使用在輕型模擬裝置上。原因就在于控制六個執(zhí)行機構(gòu)十分復(fù)雜。 最近由于在計算速度上驚人的加快，計算費用的降低，Stewart 平臺技術(shù)被美國的兩家公司使用，第一臺機床是來自美國的Giddings and Lewis公司。第二臺機床是來自美國Ingersoll 公司的叫做HEXAPOD。Hexapods系統(tǒng)的設(shè)計以及其他類似系統(tǒng)有所討論 [17]。虛擬軸機床的工件的定義以及定位問題也有所討論 [18]。從機構(gòu)設(shè)計可以看出一旦裝刀平臺由點到矢量的唯一確定之后，裝刀平臺仍就可以繞刀具軸旋轉(zhuǎn)。這就極大可能的擴(kuò)展伸縮致動器對同種編程語言的結(jié)合。 8.總結(jié) 從理論上來講，有很多方法可以用來制造五軸機床。幾乎所有傳統(tǒng)的五軸機床都屬于3根移動軸，2根轉(zhuǎn)動軸或者3根轉(zhuǎn)動軸2根移動軸的類型。這一類還可以再細(xì)分成為6個小群體，720個例子。只有當(dāng)考慮三軸實體時，每組仍然有360個實體。不同例子根據(jù)裝夾在運動鏈上的刀具和工件所在軸的順序來劃分。一旦在工件與刀具的運動鏈上的轉(zhuǎn)動軸的位置被認(rèn)為是區(qū)分3根移動軸和2根轉(zhuǎn)動軸的五軸機床時，3個組可以明顯的區(qū)分開來。在第一組中2根轉(zhuǎn)動軸被應(yīng)用于工件運動鏈中。在第二組中2根轉(zhuǎn)動軸應(yīng)用于刀具鏈中。在第三組中，每個運動鏈有一個轉(zhuǎn)動軸。每組有20個可能的形式。確定選擇哪中形式應(yīng)用到具體領(lǐng)域還是一個復(fù)雜的問題。為了解決這個問題，就定義了一些比較檢索，例如機床工作空間，工作空間使用面，工作方向索引，工作角索引以及機床空間效率。一個方程來計算機床工作空間，最大球形直徑，當(dāng)機床選擇好后它就可以用來加工此球形。這些索引的使用有兩個例子進(jìn)行了討論。第一個就是五軸機床加工珠寶。第二個就是當(dāng)相同移動軸數(shù)范圍確定時，旋轉(zhuǎn)軸的選擇。 應(yīng)用最廣泛的五軸機床在運動鏈的工件端有兩個旋轉(zhuǎn)軸。這種結(jié)構(gòu)為機床制造商提供了模塊化設(shè)計。這種模塊化設(shè)計，并不總是從應(yīng)用的最佳角度看。因為大量理論上可能的配置，很顯然，特定的五軸機床最適合一組特殊的工件。模塊化設(shè)計應(yīng)根據(jù)所有的五軸的模塊化組合。當(dāng)前的模塊化設(shè)計是基于三個線性軸機器。 五軸銑削提供減少裝置。這有助于提高精度和減少批量大小。然而，有一些缺點：（一）五軸機床價格高；（二）附加旋轉(zhuǎn)軸造成額外的位置誤差；（三）為了相同的供給在機床軸上更高的切削速度 購買五軸機床之前必須深刻研究被加工產(chǎn)品的范圍。這部分應(yīng)分類為五軸定位或五軸輪廓這兩種。用轉(zhuǎn)臺機為例很好可以產(chǎn)生旋轉(zhuǎn)的工件，如壓縮機。在工具上側(cè)的一個旋轉(zhuǎn)軸和在工件上側(cè)的一個旋轉(zhuǎn)軸將提供一個較大的工作空間利用率因素。 最近推出的虛擬軸機器有一個主要優(yōu)勢，高動態(tài)響應(yīng)和高剛性的可能性。然而工作空間利用率與經(jīng)典五軸機床相比要低得多。這些機器的高剛性的設(shè)計使它們非常適合于高速主軸[19]需要高速銑削的設(shè)計。 參考 1. 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Chou, Separating and intersecting spherical polygons: computing machinability on three-, four, and five-axis numerically controlled machines, ACM Transactions on Graphics 12 (4) (1993) 305-326. 9. B.K. Choi, D.H. Kim, R.B. Jerard, C-space approach to tool-path generation for die and mold machining, Computer Aided Design 29 (9) (1997) 657-669. 10. R.B. Jerard, R.L. Drysdale, Methods for geometric modeling, simulation and spacial verification of NC machining programs, in: J. Turner, J. Pegna, M. Wozny (Eds.), Product Modeling for Computer-Aided Design and Manufacturing, Elsevier Science Publishers B.V./North Holland, Amsterdam, 1990. 11. R.P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, (3rd ed), Addison-Wesley, Reading, MA, 1994. 12. M.G. Kendal, A Course in the Geometry of n Dimensions, John Wright and Sons Ltd., 1961. 13. L.J. Erik, S.D. Bohez, R. Senadhera, K. Pole, J.R. Duflou, T. Tar, A geometric modeling and five-axis machining algorithm for centrifugal impellers, Journal of Manufacturing Systems 16 (16) (1997). 14. E.L.J. Bohez, T. Mahasan, An expert system for diagnosing CNC machines: a case study, Computers in Industry 32 (1997) 233248. 15. W.W. Luggen, Flexible Manufacturing Cells and Systems, Prentice-Hall, Englewood Cliffs, NJ, 1991. 16. D. Stewart, A platform with six degrees of freedom, The Institution of Mechanical Engineers, Proceedings 1965-1966, Vol. 180 Part 1, No. 15, pp. 371-386. 17. G. Pritschow, K.-H. Wurst, Systematic design of hexapods and other parallel link systems, Annals of the CIRP 46 (1) (1997). 18. T. Huang, J. Wang, D.J. Whitehouse, Closed form solution to workspace of hexapod-based virtual axis machine tools, Transactions of ASME 121 (March) (1999). 19. E.L.J. Bohez, Computer aided dynamic design of rotating shafts, Computers in Industry 13 (1) (1989). 20. V.E. Gough, S.G. Whitehall, Universal tyre test machine, in: Proc. 9th Int. Tech. Congress F.I.S.I.T.A., May 1962, p. 117. 13 Pergamon MACHINE TOOLS & MANUFACTURE DESIGN, RESEARCH AND APPLICATION International Journal of Machine Tools & Manufacture 42 (2002) 505-520 Five-axis milling machine tool kinematic chain design and analysis E.L.J. Bohez * Department of Design and Manufacturing Engineering, Asian Institute of Technology, P.O. Box 4, Klong Luang, 12120 Pathumthani, Thailand Received 23 May 2000; received in revised form 12 September 2001; accepted 13 September 2001 Abstract Five-axis CNC machining centers have become quite common today. The kinematics of most of the machines are based on a rectangular Cartesian coordinate system. This paper classifies the possible conceptual designs and actual existing implementations based on the theoretically possible combinations of the degrees of freedom. Some useful quantitative parameters, such as the workspace utilization factor, machine tool space efficiency, orientation space index and orientation angle index are defined. The advantages and disadvantages of each concept are analyzed. Criteria for selection and design of a machine configuration are given. New concepts based on the Stewart platform have been introduced recently in industry and are also briefly discussed. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Five-axis; Machine tool; Kinematic chain; Workspace; CNC; Rotary axis 0890-6955/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S0890-6955(01)00134-1 1. Introduction The main design specifications of a machine tool can be deduced from the following principles: ? The kinematics should provide sufficient flexibility in orientation and position of tool and part. ? Orientation and positioning with the highest possible speed. ? Orientation and positioning with the highest possible accuracy. ? Fast change of tool and workpiece. ? Save for the environment. ? Highest possible material removal rate. The number of axes of a machine tool normally refers to the number of degrees of freedom or the number of independent controllable motions on the machine slides. The ISO axes nomenclature recommends the use of a right-handed coordinate system, with the tool axis corresponding to the Z-axis. A three-axis milling machine has three linear slides X, Y and Z which can be positioned everywhere within the travel limit of each slide. The tool axis direction stays fixed during machining. This limits * Tel.: +66-2-524-5687; fax: +66-2-524-5697. E-mail address: bohez@ait.ac.th (E.L.J. Bohez). the flexibility of the tool orientation relative to the workpiece and results in a number of different set ups. To increase the flexibility in possible tool workpiece orientations, without need of re-setup, more degrees of freedom must be added. For a conventional three linear axes machine this can be achieved by providing rotational slides. Fig. 1 gives an example of a five-axis milling machine. E.L.J. Bohez /International Journal of Machine Tools & Manufacture 42 (2002) 505-520 511 2. Kinematic chain diagram To analyze the machine it is very useful to make a kinematic diagram of the machine. From this kinematic (chain) diagram two groups of axes can immediately be distinguished: the workpiece carrying axes and the tool carrying axes. Fig. 2 gives the kinematic diagram of the five-axis machine in Fig. 1. As can be seen the workpiece is carried by four axes and the tool only by one axis. The five-axis machine is similar to two cooperating robots, one robot carrying the workpiece and one robot carrying the tool. Five degrees of freedom are the minimum required to obtain maximum flexibility in tool workpiece orientation, this means that the tool and workpiece can be oriented relative to each other under any angle. The minimum required number of axes can also be understood from a rigid body kinematics point of view. To orient two rigid bodies in space relative to each other 6 degrees of freedom are needed for each body (tool and workpiece) or 12 degrees. However any common translation and rotation which does not change the relative orientation is permitted reducing the number of degrees by 6. The distance between the bodies is prescribed by the toolpath and allows elimination of an additional degree of freedom, resulting in a minimum requirement of 5 degrees. 3. Literature review One of the earliest (1970) and still very useful introductions to five-axis milling was given by Baughman [1] clearly stating the applications. The APT language was then the only tool to program five-axis contouring applications. The problems in postprocessing were also Fig. 2. Kinematic chain diagram. clearly stated by Sim [2] in those earlier days of numerical control and most issues are still valid. Boyd in Ref. [3] was also one of the early introductions. Beziers’ book [4] is also still a very useful introduction. Held [5] gives a very brief but enlightening definition of multi-axis machining in his book on pocket milling. A recent paper applicable to the problem of five-axis machine workspace computation is the multiple sweeping using the Denawit-Hartenberg representation method developed by Abdel-Malek and Othman [6]. Many types and design concepts of machine tools which can be applied to five-axis machines are discussed in Ref. [7] but not specifically for the five-axis machine. The number of setups and the optimal orientation of the part on the machine table is discussed in Ref. [8]. A review about the state of the art and new requirements for tool path generation is given by B.K. Choi et al. [9]. Graphic simulation of the interaction of the tool and workpiece is also a very active area of research and a good introduction can be found in Ref. [10]. 4. Classification of five-axis machines9 kinematic structure Starting from Rotary (R) and Translatory (T) axes four main groups can be distinguished: (i) three T axes and two R axes; (ii) two T axes and three R axes; (iii) one T axis and four R axes and (iv) five R axes. Nearly all existing five-axis machine tools are in group (i). Also a number of welding robots, filament winding machines and laser machining centers fall in this group. Only limited instances of five-axis machine tools in group (ii) exist for the machining of ship propellers. Groups (iii) and (iv) are used in the design of robots usually with more degrees of freedom added. The five axes can be distributed between the workpiece or tool in several combinations. A first classification can be made based on the number of workpiece and tool carrying axes and the sequence of each axis in the kinematic chain. Another classification can be based on where the rotary axes are located, on the workpiece side or tool side. The five degrees of freedom in a Cartesian coordinates based machine are: three translatory movements X,Y,Z (in general represented as TTT) and two rotational movements AB, AC or BC (in general represented as RR).Combinations of three rotary axes (RRR) and two linear axes (TT) are rare. If an axis is bearing the workpiece it is the habit of noting it with an additional accent. The five-axis machine in Fig. 1 can be characterized by XfYfAfBfZ. The XYAB axes carry the workpiece and the Z-axis carries the tool. Fig. 3 shows a machine of the type XYZArBr, the three linear axes carry the tool and the two rotary axes carry the workpiece. 4.1. Classification based on the sequence of workpiece and tool carrying axes Theoretically the number of possible configurations is quite large if the order of the axes in the two kinematic chains of the tool and workpiece carrying axes is counted as a different configuration. Also the combinations with only two linear axes and three rotary axes are included. One tool carrying axis and four workpiece carrying axes can be combined in a five-axis machine as follows: for each possible tool carrying axis X,Y,Z,A,B,C the other four workpiece carrying axes can be selected from the five remaining axes. So the number of combinations of four axes out of five with considering different permutation as another configuration is 5x4!=120 for each possible tool axis selection (1 out of 6 or 6 possibilities). So theoretically there are 6x120=720 possible five-axis machines with one tool carrying axis. The same analysis can be done for all other combinations. With t the number of tool carrying axes and w the number of workpiece carrying axes (w+t=5) the total number of combinations is as follows. Ncomb=(t)t!(w )w! t^3,什w=5 ⑴ (6\ (6-w\ Nco^b=\ )w!\ t It! t>3, t+w=5 (2) The value of this equation is always equal to 6! or 720 when w+t=5. Some of these 720 combinations will be containing only two linear axis. If only five-axis machines with three linear axes are considered, only 3x5!=360 combinations are still possible. The set Gt of combinations is characterized by a fixed value of t. This set is identical to the set Gw characterized by a fixed value of w, w=5 —t. Using above definitions following subgroups of five-axis machines exist: (i) Group G0/G'5; (ii) Group G1/G'4; (iii) Group G2/G'3; (iv) Group G3/G'2; (v) Group G4/G'1; (vi) Group G5/G0. 4.1.1. G5/G0f machine All axes carry the tool and the workpiece is fixed on a fixed table. Fig. 4 shows a machine with all the five axes carrying the tool. The kinematic chain is XBYAZ (TRTRT). This machine was one of the earliest models of five-axis machines to handle very heavy workpieces. As there are many links in the tool carrying kinematic chain, there can be a considerable error due to elastic deformations and backlash in the slides. 4.1.2. G0/G5f machine All axes carry the workpiece and the tool is fixed in space. This construction is best used for very small workpieces (see Section 6.3).  4.1.3. G4/G1 machine Four axes carry the tool and one axis carries the workpiece. There are basically two possibilities, the workpiece carrying axis can be R or T. 4.1.4. G1/G4’ machine One axis carries the tool and the other four axes carry the workpiece. There are basically two possibilities, the single axis kinematic chain can be R or T. Fig. 1 is an example of such a machine, with the single tool carrying axis T. 4.1.5. G3/G2f machine Three axes carry the tool and two axes carry the workpiece. There are basically three possibilities, the workpiece carrying axes can be both linear (W) both rotational (R'R') or mixed (T'R'). Fig. 5 gives an example of a machine with the tool carried by two rotary axes and one linear axis. This machine allows processing of large workpieces but the construction of the toolside is complicated. The most common configuration is the workpiece carried by the two rotary axes such as the one given in Figs. 3, 6 and 8. 4.1.6. G2/G3f machine Two axes carry the tool and three axes carry the workpiece. There are basically three possibilities, the tool carrying axis can be both linear (TT) both rotational (RR) or mixed (TR). Fig. 7 shows the mixed construction. Fig. 8 shows two linear axes carrying the tool. 4.2. Classification based on the location of rotary axes Fig. 7. ZrCBY machine. The machines can be classified depending on the place where the rotation axes are implemented. Fig. 5. XZ'CAY machine.  Only machines with two rotary axes and three linear axes will be considered further. The possible configurations are: The number of possible designs is the sum of the following combinations: (i) For the group G0/G5' the tool is fixed in space all the five axes will carry the workpiece. The number of different designs is 10 (NTf=3 and NR =2), (Figs. 15 and 16). (ii) For the group G1/G4r, NT+NR=1, so NT=1 and NR=0, is the only possible choice for the tool kinematic chain. Equation (3) gives NCOMB=6. The combinations are: RfRfTfTfT; TfTfRfRfT; RfTfRfTfT; TRTRT; RTTRT; TRRTT. Fig. 9 shows these six designs. (iii) For the group G2/G3' the tool axes are TT so Nt'=1, Nr =2, Nj=2, Nr=0 and Equation (2) gives Ncomb=3. The three design combinations are: RRTTT; RfTfRfTTand TfRfRfTT. The group G2/G3f contains three instances of the RfRf machine. These instances are represented in Fig. 10. (vi) If the tool axes are TTT the workpiece carrying axes can only be RrRr. So only one design combination is possible. From the above-mentioned findings it can also be concluded that the total number of RfRf five-axis machine configurations is 20. Machines with two axes on the clamping table can be seen in Figs. 1, 3, 6 and 8. The advantages are: (a) rotation axes are implemented on tool spindle; (b) rotation axes are implemented on machine table; (c) combination of both. The sequence of the axes in the tool or workpiece carrying kinematic chain is not important if the axes are of the same type R or T. In general, if there are NT transitory axes and Nr rotary axes in the workpiece carrying kinematic chain and NT translatory axes and NR rotary axes in the tool kinematic chain, then the numbers of combinations is [11]: N JNt+Nr)\(Nt+Nr)\ 以一b- Nt!NR Nt!Nr {) with Nt+Nt=3, Nr+Nr=2 The number of combinations of each group will be given below case by case. The total number of combinations over all groups is 60. From the design point of view this is a more tractable number of alternatives to be considered. 4.2.1. RR machine The two rotary axes carry the workpiece. The tool axis can be fixed or carried by one (T), two (TT) or three (TTT) linear axes. ? In case the spindle is horizontal, optimal chip removal is obtained through the gravitational effect of the chips just dropping. ? The tool axis during machining is always parallel to the Z axis of the machine. So the drilling cycles can be executed along the Z-axis of the machine. Circles under a certain orientation of the workpiece are always executed in the XY plane of the machine. The above-mentioned functions can be executed in the simple three-axis numerical control mode. ? The compensation of the tool length happens all the time in the NC control of the machine, as with three- axis machines. Disadvantages: ? Machines with a rotating table are only for workpieces with limited dimensions. ? The useful workspace is usually much smaller than the product of the travel in X,Y and Z axis. ? The transformation of the Cartesian CAD/CAM coordinate (XYZIJK) of the tool position to the machine axes positions (XYZAB or C) is dependent on the position of the workpiece on the machine table. This means that in case the position of the workpiece on the table is changed this cannot be modified by a translation of the axes system in the NC program. They must be recalculated. In case the control of the NC machine cannot transform Cartesian coordinates to machine coordinates, then a new CNC program must be generated with the postprocessor of the CAD/CAM system every time the position of the workpiece changes. Important applications for this type: ? Five-sided cutting of electrodes for EDM and other workpiece. ? Machining of precision workpieces. ? Turbines and tire profiles with a certain workpiece geometry rotated over a certain angle. The same NC program can be repeated after the zero of the rotation axis has been inclined over a certain angle. 4.2.2. RR-machine The number of possible design combinations (Ncomp=20) is the same as in the case of the R'R' machine because of the symmetry. Five-axis machines with the rotation axes implemented on the tool axis spindle can be seen in Figs. 4 and 5. Advantages: ? These machines can machine very large workpieces. ? The machine axis values of the NC program XYZ, depend on the tool length only. A new clamping position of the workpiece is corrected with a simple translation. This happens with a zero translation in the CNC control of the machine. Disadvantages: ? The drive of the main spindle is very complex. Simple design and construction is only obtained when the whole spindle with the motor itself is rotating. ? There is a lower stiffness because the rotation axis of the spindle is limiting the force transmission. At high revolutions per minute (higher than 5000 rpm) there is also a counter acting moment because of the gyroscopic effect which could be a disadvantage in case the tool spindle is turning very fast. ? Circular interpolation in a random plane and drilling cycles under random orientation are often not implemented. ? A change in the tool length cannot be adjusted by a zero translation in the control unit, often a complete recalculation of the program (or postprocessing) is required. Important applications of this type of machine tool are: ? All types of very large workpieces such as air plane wings. 4.2.3. R’R machine One rotary axis is implemented in the workpiece kinematic chain and the other rotary axes in the tool kinematic chain (e.g. Fig. 7). The groups G4/G1’，G4’/G1, G3’/G2, G3/G2’ cover this design. Nowadays there are many machines on the market with one rotation axis on the tool spindle and (4) Wr = one rotation axis on the table. They are, however, combining most of the disadvantages of both previous types of machines and are often used for the production of smaller workpieces. The application range of this machine is about the same as with machines with two rotation axes implemented on the table. In all possible designs of this machine the NRf=NR=1 and Nt+Nt=3. The total number of possible designs is: ncomb[nt=0,Nt=3]+ncomb[nt= 1,Nt=2]+ncomb[nt =2,Nt=1]+Ncomb[NT=3, NT' = 0] or 4+6+6+4=20 possible designs. (i) For NTr=0 and Nt=3 the four combinations are: RRTTT; RTRTT; RTTRT; RTTTR. (ii) For NT=1 and Nt=2 the six combinations are: TRRTT; TRTRT; TRTTR; R’T’RTT.， R’T’TRT; R’T’TTR. (iii) For NT'=2 and Nt=1 the six combinations are (see Fig. 11): R'T'T'TR; TR'T'TR; T'TR'TR; RT'TRT; TRTRT; T'TRRT. (iv) For NT=3 and Nt=0 the four combinations are: RT'T'TR; TRT'TR; T'TRTR; T'T'TRR. 5. Workspace of a five-axis machine Before defining the workspace of the five-axis machine tool, it is appropriate to define the workspace of the tool and the workspace of the workpiece. The workspace of the tool is the space obtained by sweeping the tool reference point (e.g. tool tip) along the path of the tool carrying axes. The workspace of the workpiece carrying axes is defined in the same way (the center of the machine table can be chosen as reference point). These workspaces can be determined by computing the swept volume [6]. Based on the above-definitions some quantitative parameters can be defined which are useful for comparison, selection and design of different types of machines. 5.1. Workspace utilization factor WR A possible definition for this is the ratio of the Boolean intersection of the workpiece workspace and tool workspace and the union of the tool workspace and workpiece workspace. WSTOOL^ WSWORKPIECE WSTOOL ^ WSWORKPIECE A large value for WR means that the workspace of the tool and the workspace of the workpiece are about equal in size and overlap almost completely. A small value of WR means that the overlap of tool workspace and workpiece workspace is small and that a large part of the workpiece workspace cannot be reached by the tool. The analogy with two cooperating robots can be clearly seen. It is only in the intersection of the two workspaces of each robot that they can ‘shake hands’. For the five-axis machine tool this corresponds to the volume in which the tool and workpiece reference point can meet. However, in the case where all the five axes carry the workpiece and the tool is fixed in space the above definition would give a zero value for the workspace utilization. In the case of cooperating robots it would mean that there is only one point were they can shake hands. In the case of a five-axis machine, the workpiece can still be moved in front of the tool and remove metal. The reason is that many points from the workpiece can serve as reference point on the workpiece. All