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題目 五軸磨床加工工具運(yùn)動(dòng)鏈的設(shè)計(jì)和分析
專 業(yè) 名 稱 機(jī)械設(shè)計(jì)制造及其自動(dòng)化
班 級(jí) 學(xué) 號(hào) 068105337
學(xué) 生 姓 名 鄭帥棟
指 導(dǎo) 教 師 羅海泉
填 表 日 期 2010 年 5 月 18 日
五軸磨床加工工具運(yùn)動(dòng)鏈的設(shè)計(jì)和分析
E.L.J. Bohez，設(shè)計(jì)與制造工程部門，亞洲技術(shù)研究所
摘要：五軸CNC加工中心現(xiàn)在應(yīng)用得非常廣泛。大多數(shù)機(jī)器的運(yùn)動(dòng)學(xué)原理都是以直角笛卡兒坐標(biāo)系統(tǒng)為基礎(chǔ)的。這篇文章對(duì)有可能的概念上的設(shè)計(jì)和基于理論上有可能的自由度的結(jié)合并且真實(shí)存在的器械進(jìn)行了分類。本文還定義了一些有用的定量參數(shù)，例如：工作空間利用因素、機(jī)器加工工具的空間利用率、方位空間的指標(biāo)和方位角。同時(shí)還分析了不同概念的優(yōu)缺點(diǎn)，給出了選擇的標(biāo)準(zhǔn)和機(jī)器結(jié)構(gòu)的設(shè)計(jì)。最近在工業(yè)中提出的一些基于斯圖爾特平臺(tái)的概念也將在這篇文章中進(jìn)行簡(jiǎn)要的論述。
關(guān)鍵詞：五軸；機(jī)器加工工具；運(yùn)動(dòng)鏈；工作空間；CNC；旋轉(zhuǎn)軸
1.介紹
機(jī)器加工工具的主要設(shè)計(jì)規(guī)范應(yīng)該滿足以下法則：
1） 運(yùn)動(dòng)件在工具和零件的定位和安置上應(yīng)該
有足夠的彈性。
2） 以可能的最快的速度進(jìn)行定位和安置。
3） 以可能的最高的精確度進(jìn)行定位和安置。
4） 加工工具和工件的快速切換。
5） 保護(hù)環(huán)境。
6） 可能的高速材料移動(dòng)率。
一臺(tái)機(jī)器的加工工具的軸的個(gè)數(shù)通常是由機(jī)器自由度數(shù)或者是在機(jī)器滑動(dòng)過程中獨(dú)立可控的運(yùn)動(dòng)數(shù)來決定的。隨著加工工具軸對(duì)應(yīng)Z坐標(biāo)軸的產(chǎn)生，ISO軸命名法推薦使用右手坐標(biāo)法則。一個(gè)三軸磨床有三個(gè)方向的線性滑動(dòng)：X、Y和Z，這使得機(jī)器能放置在相應(yīng)軸向滑動(dòng)范圍內(nèi)的任何一個(gè)位置。加工工具軸的方向在加工的時(shí)候保持不變。這就限制了與工件連接的加工工具的彈性，并最終導(dǎo)致很多不同的結(jié)構(gòu)。為了增加在可能的加工工具、工件定位中的彈性而不用重新設(shè)計(jì)結(jié)構(gòu)，我們將要在增加更多的機(jī)器的自由度。對(duì)于一個(gè)傳統(tǒng)的三線性軸機(jī)器，能通過提供旋轉(zhuǎn)滑動(dòng)來實(shí)現(xiàn)。圖1就展示了一個(gè)五軸磨床的例子。
2.運(yùn)動(dòng)鏈接圖
制作一個(gè)機(jī)器的運(yùn)動(dòng)鏈接圖對(duì)于分析機(jī)器是很有用的。從運(yùn)動(dòng)鏈接圖中我們可以很快區(qū)別兩組軸：圖2展示了在圖1中五軸磨床的運(yùn)動(dòng)鏈接圖。從圖中我們可以看到，工件由四根軸運(yùn)載，而加工工具只由一根軸運(yùn)載。
五軸機(jī)器就像兩個(gè)相互協(xié)作的機(jī)器人，一個(gè)機(jī)器人運(yùn)載工件，另一個(gè)機(jī)器人則運(yùn)載加工工具。
為了得到工件和工具定位上最大的彈性，機(jī)器至少需要5個(gè)自由度，這意味著工具和工件能在任何角度下連接起來。從一個(gè)剛性的物體運(yùn)動(dòng)連接點(diǎn)的觀點(diǎn)來說，我們也可以理解對(duì)軸的個(gè)數(shù)的最低要求。為了定位兩個(gè)在空間上相互連接的剛體，每個(gè)剛體（工具和工件）需要6個(gè)自由度或者12個(gè)自由度。然而，任何不改變兩者之間定位的共同的平移和旋轉(zhuǎn)的存在將會(huì)使自由度數(shù)目減少6個(gè)。兩個(gè)剛體之間的距離是由工具的加工路徑來決定的，這個(gè)距離也將會(huì)允許減少一個(gè)多余自由度，這樣就使得最小的自由度數(shù)為5。
3.文化背景（略）
4.五軸機(jī)器運(yùn)動(dòng)結(jié)構(gòu)的分類
按照機(jī)器的旋轉(zhuǎn)和平移軸分類，我們可以把機(jī)器運(yùn)動(dòng)結(jié)構(gòu)分為四大類：（1）三個(gè)平移軸和兩個(gè)旋轉(zhuǎn)軸；（2）兩個(gè)平移軸和三個(gè)旋轉(zhuǎn)軸；（3）一個(gè)平移軸和四個(gè)旋轉(zhuǎn)軸；（4）五個(gè)旋轉(zhuǎn)軸。近乎所有存在的五軸機(jī)器設(shè)備都屬于（1）類。很多的定位焊接機(jī)器人、圈絲機(jī)器和激光加工中心也屬于這一類。只有有限的一些用來加工輪船推進(jìn)器的五軸機(jī)器屬于（2）類。（3）和（4）類只有在設(shè)計(jì)需要增加更多自由度的機(jī)器人的時(shí)候才會(huì)用到。
五根軸可以分布在工具和工件之間的結(jié)合處。第一種分類是根據(jù)運(yùn)載軸的工具和工件的數(shù)量和在運(yùn)動(dòng)鏈中各個(gè)軸的次序來劃分的。另一種分類是根據(jù)旋轉(zhuǎn)軸放置的位置（是在工具那邊還是在工件那邊）來劃分的?；诘芽▋鹤鴺?biāo)的機(jī)器中的五個(gè)自由度是：三個(gè)平移運(yùn)動(dòng)X、Y、Z（一般表示為TTT）和兩個(gè)旋轉(zhuǎn)運(yùn)動(dòng)AB、AC、BC（一般表示為RR）。三個(gè)旋轉(zhuǎn)軸（RRR）和兩個(gè)直線運(yùn)動(dòng)軸（TT）的結(jié)合是很少見的。如果一根軸承載著工件，習(xí)慣上是用一個(gè)附加的標(biāo)記來注釋它。圖1中的機(jī)器能以X’Y’A’B’Z。XYAB軸運(yùn)載著工件，Z軸運(yùn)載著工具。圖3中展示的是XYZA’B’，三根直線運(yùn)動(dòng)軸運(yùn)載工具,兩根旋轉(zhuǎn)軸運(yùn)載著工件。
4.1基于工件和工具運(yùn)載軸次序的分類
理論上，如果認(rèn)為在工具和工件運(yùn)載軸的兩個(gè)運(yùn)動(dòng)鏈上的軸的次序有不同的結(jié)構(gòu)，可能的結(jié)構(gòu)的數(shù)目會(huì)非常大。同時(shí)只有兩根直線運(yùn)動(dòng)軸和三根旋轉(zhuǎn)軸結(jié)合也包括在內(nèi)。
在一個(gè)五軸機(jī)器中能以以下方式將一根工具運(yùn)載軸和四根工件運(yùn)載軸結(jié)合：對(duì)于X，Y，Z，A，B，C中任意一個(gè)可能作為工具運(yùn)載的軸，其他工件運(yùn)載軸可以在剩下的五根軸中選取。所以，對(duì)于任意可能的工具運(yùn)載軸選擇（六選一或者有六種可能），在剩下的五根軸中選取四根進(jìn)行不同結(jié)構(gòu)的排列個(gè)數(shù)為5*4！=120。所以，理論上只有一根工具運(yùn)載軸的五軸機(jī)器就有6*120=720種可能。其他的結(jié)合方式也可以用這種方法分析。假設(shè)t代表工具運(yùn)載軸的數(shù)目，w表示工件運(yùn)載軸的數(shù)目（w+t=5），那么全部可能的結(jié)合數(shù)如下所示：
這個(gè)方程式的值恒等于6！或者當(dāng)w+t=5時(shí)這個(gè)值等于720。在這些720的結(jié)合中，有一些只包含兩根直線運(yùn)動(dòng)軸。如果只考慮有三根直線運(yùn)動(dòng)軸的五軸機(jī)器，只有3*5！=360的結(jié)合也是仍是有可能的。這些結(jié)合的預(yù)設(shè)值Gt主要是由t的預(yù)設(shè)值決定的。這個(gè)預(yù)設(shè)值和由w的預(yù)設(shè)值所決定的G`w 的預(yù)設(shè)值是一致的，其中w=5-t。運(yùn)用以上的定義，我們可以把五軸機(jī)器分為以下小群：（1）G0/G`5組；（2）G1/G`4組；（3）G2/G`3組；（4）G3/G`2組；（5）G4/G`1組；（6）G5/G`0組。
4.2基于旋轉(zhuǎn)軸的位置的分類
我們能根據(jù)旋轉(zhuǎn)軸裝配的位置對(duì)機(jī)器進(jìn)行分類。
只有那些有兩根旋轉(zhuǎn)軸和三根線性軸的機(jī)器我們才會(huì)進(jìn)一步考慮?？赡艿慕Y(jié)構(gòu)如下：
（a）旋轉(zhuǎn)軸裝配在工具桿上；
（b）旋轉(zhuǎn)軸裝配在機(jī)器平臺(tái)；
（c）兩者的結(jié)合。
如果機(jī)器的軸的R或者T的類型一樣，那么在工具或者工件運(yùn)載運(yùn)動(dòng)鏈中軸的次序就不重要了。一般來說，如果在工件運(yùn)載運(yùn)動(dòng)鏈中有根平移軸和根旋轉(zhuǎn)軸，在工具運(yùn)動(dòng)鏈中有根平移軸和根旋轉(zhuǎn)軸，那么結(jié)合的個(gè)數(shù)為[11]：
其中
每一組的結(jié)合的個(gè)數(shù)在下面中將會(huì)一個(gè)個(gè)給出。所有組的結(jié)合總數(shù)為60。從設(shè)計(jì)的觀點(diǎn)來說，這是我們所考慮的選擇的數(shù)量中較為容易處理的一個(gè)。
5.五軸機(jī)器的工作空間
在定義五軸機(jī)器設(shè)備的工作空間之前，要適當(dāng)?shù)亩x加工工具的工作空間和工件的工作空間。加工工具的工作空間就是通過沿著工具運(yùn)載軸路線描繪出工具掃過參考點(diǎn)（例如工具尖端）而得到的。工件運(yùn)載軸的工作空間也是用同樣方法定義的（把機(jī)器平臺(tái)的中心選擇作為參考點(diǎn)）。這些工作空間能由計(jì)算掃過過的體積決定[6]。
基于以上的定義，我們能定義一些對(duì)于不同類型機(jī)器比較、選擇和設(shè)計(jì)有用的定量參數(shù)。
5.1.工作空間利用因素
這個(gè)因素可以定義為，工件空間和工具空間的交集與工具空間和工件空間的并集的比。公式為
5.2.可加工的體積大小
一旦工件相對(duì)于工件參考點(diǎn)是固定的，并且一個(gè)特殊的工具相對(duì)于工具參考點(diǎn)也是固定的，那么我們就有可能確定可加工的體積的大小?？杉庸さ捏w積就是能夠在工件上切除的全部體積。機(jī)器工具空間和工件的交集給出了可以切除的材料的總量，或者說是可加工的體積（這是對(duì)于特殊的工件和工具機(jī)構(gòu)來說的）。
5.3.機(jī)器工具空間效率
機(jī)器工具空間效率的定義為：機(jī)器工具空間（省略了一部分）和包含著機(jī)器的最小凸起體積。
5.4.五軸機(jī)器的定位空間指數(shù)
一個(gè)我們用來估計(jì)定位的最大范圍的方法是為了決定能在機(jī)器上用兩根旋轉(zhuǎn)軸加工的球的最大部分?？臻g定位指數(shù)定義為能夠由用所有旋轉(zhuǎn)軸加工的機(jī)器來加工的最大的球頂體積除以機(jī)器工具空間。
如果這個(gè)指數(shù)趨近于1，這就意味著所有的旋轉(zhuǎn)軸能夠在整個(gè)機(jī)器工具空間中運(yùn)用。如果這個(gè)指數(shù)比1小，這就意味著大概百分之的工作空間能運(yùn)用所有的旋轉(zhuǎn)軸。
以上的定義都是理論上的定義。實(shí)際上的定位空間指數(shù)會(huì)因?yàn)楸苊饬慵蜋C(jī)器、工具和工件之間的碰撞而進(jìn)一步受到限制。能夠加工的球頂變小就說明了這一點(diǎn)。
6.五軸機(jī)器的選擇標(biāo)準(zhǔn)
我們的目的不是對(duì)五軸機(jī)器對(duì)于某一項(xiàng)特定的運(yùn)用的選擇或者設(shè)計(jì)進(jìn)行一個(gè)徹底的研究。我們只是論述能用來判斷五軸機(jī)器的選擇的主要的標(biāo)準(zhǔn)。
6.1.五軸機(jī)器設(shè)備的應(yīng)用
應(yīng)用能在布置和造型上進(jìn)行區(qū)分。圖12和圖13說明了五軸的布置和五軸造型上的區(qū)別。
6.1.1.五軸布置
如圖12所示，一個(gè)在不同角度有著很多孔和平面板的零件，僅僅用一臺(tái)三軸磨床來加工這個(gè)零件是不可能的。如果我們?cè)谟靡慌_(tái)五軸機(jī)器，那么工具能在任何方向和工件定位連接起來。一旦達(dá)到了正確的位置，在大多數(shù)軸固定的情況下，我們就可以對(duì)孔和平面板進(jìn)行加工了。平面板中能包括獨(dú)立結(jié)構(gòu)的2D平面。如果我們僅僅是要鉆孔，那么理論上一軸CNC同步控制就足夠了，而加工2D平面時(shí)兩軸同步控制就夠了。然而，三軸同步現(xiàn)在也很普遍了。當(dāng)我們把工件和工具放置在連接在一起的時(shí)候，這就增加了在開始切削前的快進(jìn)的速度。
6.1.2.五軸造型
圖13所示為一個(gè)五軸造型的例子，為了加工這個(gè)形狀復(fù)雜的表面，我們需要在切削時(shí)控制好與零件接觸的刀具的位置。刀具工件的位置在每一步工序中都會(huì)改變。CNC控制器需要在材料切除過程中同步控制五軸。更多關(guān)于造型的細(xì)節(jié)能在參考文獻(xiàn)[13]中找到。五軸的機(jī)器有如下的應(yīng)用：（1）生產(chǎn)刀刃，如： 壓縮機(jī)和渦輪的漿；（2）燃料泵的注射器；（3）頭飾的外形；（4）醫(yī)學(xué)器官例如人造心臟閥；（5）復(fù)雜表面的鑄型。
6.2.軸結(jié)構(gòu)的選擇
在設(shè)計(jì)和選擇一個(gè)結(jié)構(gòu)時(shí)，零件的尺寸和重量是首要的標(biāo)準(zhǔn)。重型工件要求工件運(yùn)動(dòng)鏈短。同時(shí)，水平加工平面又是較好的一種設(shè)計(jì)，這種設(shè)計(jì)會(huì)使定位和處理工件變得很便利。把一個(gè)重型工件放在一個(gè)單獨(dú)旋轉(zhuǎn)軸運(yùn)動(dòng)鏈上將會(huì)很大程度上增加定位的彈性。從圖4中我們可以看出，用一個(gè)單獨(dú)水平旋轉(zhuǎn)軸來運(yùn)載工件會(huì)使得機(jī)器更加具有彈性。
在很多情況下，我們應(yīng)該把工具運(yùn)動(dòng)鏈保持得盡量短，因?yàn)槲覀冞€必須運(yùn)載工具軸驅(qū)動(dòng)裝置。
7.基于斯圖爾特平臺(tái)的新的機(jī)器概念（略）
8.結(jié)論
理論上，五軸機(jī)器有很多構(gòu)成方式。近乎所有經(jīng)典的笛卡兒坐標(biāo)五軸機(jī)器都屬于由三根線性軸和兩根旋轉(zhuǎn)軸，或者三根旋轉(zhuǎn)軸和兩根線性軸組成的系列。這個(gè)系列又可以細(xì)分為有著720種情況的六組。就算只考慮三根線性軸的情況，在每個(gè)系列中仍然有360種組合。這些不同的組合是根據(jù)在工具和工件運(yùn)載運(yùn)動(dòng)鏈中軸的次序來區(qū)分的。
如果在對(duì)由三根線性軸和兩根旋轉(zhuǎn)軸組成的五軸機(jī)械進(jìn)行分組時(shí)，只考慮在工具和工件運(yùn)動(dòng)鏈中旋轉(zhuǎn)軸的位置，那么我們能五軸機(jī)器分為三組。在第一組中，兩根旋轉(zhuǎn)軸安置在工件運(yùn)動(dòng)鏈。在第二組中，兩根旋轉(zhuǎn)軸安置在工具運(yùn)動(dòng)鏈。在第三組中，每個(gè)運(yùn)動(dòng)鏈都安置一根旋轉(zhuǎn)軸。每一組仍然有20種可能的情況。對(duì)于一個(gè)特定的應(yīng)用領(lǐng)域，要從這些組合中選出一組最好的是一項(xiàng)很復(fù)雜的工作。為了使這項(xiàng)工作變得容易些，我們定義了一些用于比較的指數(shù)，例如：機(jī)器刀具空間、空間利用因素、定位空間指數(shù)、定位角度指數(shù)和機(jī)器刀具空間效率。列出了用來計(jì)算機(jī)器刀具空間和在機(jī)器上能加工的最大球頂?shù)闹睆降乃惴?。詳?xì)論述了兩個(gè)運(yùn)用這些指數(shù)的例子。第一個(gè)例子論述的是加工珠寶的五軸機(jī)器的設(shè)計(jì)。第二個(gè)例子則闡明了一臺(tái)機(jī)器在線性軸中有著相同范圍，在這種情況下，旋轉(zhuǎn)軸選項(xiàng)的選擇（略）。
運(yùn)用得最廣泛的五軸機(jī)器的兩根旋轉(zhuǎn)軸安置在運(yùn)動(dòng)鏈末端處的工件一側(cè)。這種結(jié)構(gòu)給出了一種對(duì)于機(jī)器刀具結(jié)構(gòu)的模塊設(shè)計(jì)。然而，從應(yīng)用的觀點(diǎn)來說，這種模塊設(shè)計(jì)并不總是最理想的。因?yàn)槔碚撋洗嬖诤芏嗫赡艿慕Y(jié)構(gòu)，很明顯的是，對(duì)于一個(gè)特殊的工件裝置需要一個(gè)合適的特定的五軸機(jī)器。模塊設(shè)計(jì)應(yīng)該以在所有的五軸結(jié)合中的模塊性為基礎(chǔ)。當(dāng)前在設(shè)計(jì)中的模塊性是以三線性軸機(jī)器為基礎(chǔ)的。
五軸磨床使得機(jī)器結(jié)構(gòu)的數(shù)量變小。這對(duì)增加精確度和減小大部分尺寸是有幫助的。然而，它也有一些缺點(diǎn)：（1）五軸機(jī)器的高價(jià)；（2）增加的旋轉(zhuǎn)軸的同時(shí)也增加了定位誤差；（3）在同等的進(jìn)給下，在機(jī)器軸上的切削速度更高。
在購(gòu)買五軸機(jī)器之前必須要對(duì)需要加工的產(chǎn)品的范圍進(jìn)行深入的研究。那些零件也應(yīng)該分為五軸定位或者是五軸造型，或者兩者都是。例如，有著旋轉(zhuǎn)平臺(tái)的機(jī)器對(duì)于生產(chǎn)諸如壓縮機(jī)的旋轉(zhuǎn)工件是很好的。一根旋轉(zhuǎn)軸在刀具側(cè)，一根旋轉(zhuǎn)軸在工件側(cè)，這樣的布置將會(huì)提供更大的工作空間利用因素。
最近所介紹的虛擬軸機(jī)器有著一個(gè)主要的優(yōu)點(diǎn)：潛在的更高的動(dòng)力響應(yīng)和更高的硬度。然而，它的工作空間利用因素比經(jīng)典的五軸機(jī)器要低。這些機(jī)器的更高強(qiáng)度使得他們非常適于高速磨所需要的高速桿[19]的設(shè)計(jì)。
9
International Journal of Machine Tools & Manufacture 42 (2002) 505520Five-axis milling machine tool kinematic chain design and analysisE.L.J. Bohez*Department of Design and Manufacturing Engineering, Asian Institute of Technology, P.O. Box 4, Klong Luang, 12120 Pathumthani, ThailandReceived 23 May 2000; received in revised form 12 September 2001; accepted 13 September 2001AbstractFive-axis CNC machining centers have become quite common today. The kinematics of most of the machines are based on arectangular Cartesian coordinate system. This paper classifies the possible conceptual designs and actual existing implementationsbased on the theoretically possible combinations of the degrees of freedom. Some useful quantitative parameters, such as theworkspace utilization factor, machine tool space efficiency, orientation space index and orientation angle index are defined. Theadvantages 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. 2002Elsevier Science Ltd. All rights reserved.Keywords: Five-axis; Machine tool; Kinematic chain; Workspace; CNC; Rotary axis1. IntroductionThe main design specifications of a machine tool canbe deduced from the following principles:? The kinematics should provide sufficient flexibility inorientation and position of tool and part.? Orientation and positioning with the highest poss-ible speed.? Orientation and positioning with the highest poss-ible 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 refersto the number of degrees of freedom or the number ofindependent controllable motions on the machine slides.The ISO axes nomenclature recommends the use of aright-handed coordinate system, with the tool axis corre-sponding to the Z-axis. A three-axis milling machine hasthree linear slides X, Y and Z which can be positionedeverywhere within the travel limit of each slide. The toolaxis direction stays fixed during machining. This limits* Tel.: +66-2-524-5687; fax: +66-2-524-5697.E-mail address: bohezait.ac.th (E.L.J. Bohez).0890-6955/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0890-6955(01)00134-1the flexibility of the tool orientation relative to the work-piece and results in a number of different set ups. Toincrease the flexibility in possible tool workpiece orien-tations, without need of re-setup, more degrees of free-dom must be added. For a conventional three linear axesmachine this can be achieved by providing rotationalslides. Fig. 1 gives an example of a five-axis millingmachine.Fig. 1.Five-axis machine tool.506E.L.J. Bohez / International Journal of Machine Tools & Manufacture 42 (2002) 5055202. Kinematic chain diagramTo analyze the machine it is very useful to make akinematic diagram of the machine. From this kinematic(chain) diagram two groups of axes can immediately bedistinguished: the workpiece carrying axes and the toolcarrying axes. Fig. 2 gives the kinematic diagram of thefive-axis machine in Fig. 1. As can be seen the work-piece is carried by four axes and the tool only by oneaxis.The five-axis machine is similar to two cooperatingrobots, one robot carrying the workpiece and one robotcarrying the tool.Five degrees of freedom are the minimum required toobtain maximum flexibility in tool workpiece orien-tation, this means that the tool and workpiece can beoriented relative to each other under any angle. Theminimum required number of axes can also be under-stood from a rigid body kinematics point of view. Toorient two rigid bodies in space relative to each other 6degrees of freedom are needed for each body (tool andworkpiece) or 12 degrees. However any common trans-lation and rotation which does not change the relativeorientation is permitted reducing the number of degreesby 6. The distance between the bodies is prescribed bythe toolpath and allows elimination of an additionaldegree of freedom, resulting in a minimum requirementof 5 degrees.3. Literature reviewOne of the earliest (1970) and still very useful intro-ductions to five-axis milling was given by Baughman 1clearly stating the applications. The APT language wasthen the only tool to program five-axis contouring appli-cations. The problems in postprocessing were alsoFig. 2.Kinematic chain diagram.clearly stated by Sim 2 in those earlier days of numeri-cal control and most issues are still valid. Boyd in Ref.3 was also one of the early introductions. Beziers book4 is also still a very useful introduction. Held 5 givesa very brief but enlightening definition of multi-axismachining in his book on pocket milling. A recent paperapplicable to the problem of five-axis machine work-space computation is the multiple sweeping using theDenawit-Hartenberg representation method developedby Abdel-Malek and Othman 6.Many types and design concepts of machine toolswhich can be applied to five-axis machines are discussedin Ref. 7 but not specifically for the five-axis machine.The number of setups and the optimal orientation ofthe part on the machine table is discussed in Ref. 8. Areview about the state of the art and new requirementsfor tool path generation is given by B.K. Choi et al. 9.Graphic simulation of the interaction of the tool andworkpiece is also a very active area of research and agood introduction can be found in Ref. 10.4. Classification of five-axis machines kinematicstructureStarting from Rotary (R) and Translatory (T) axes fourmain groups can be distinguished: (i) three T axes andtwo R axes; (ii) two T axes and three R axes; (iii) oneT axis and four R axes and (iv) five R axes. Nearly allexisting five-axis machine tools are in group (i). Also anumber of welding robots, filament winding machinesand laser machining centers fall in this group. Only lim-ited 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 withmore degrees of freedom added.The five axes can be distributed between the work-piece or tool in several combinations. A first classi-fication can be made based on the number of workpieceand tool carrying axes and the sequence of each axis inthe kinematic chain. Another classification can be basedon where the rotary axes are located, on the workpieceside or tool side. The five degrees of freedom in a Car-tesian coordinates based machine are: three translatorymovements X,Y,Z (in general represented as TTT) andtwo rotational movements AB, AC or BC (in general rep-resented as RR).Combinations of three rotary axes (RRR)and two linear axes (TT) are rare. If an axis is bearingthe workpiece it is the habit of noting it with anadditional accent. The five-axis machine in Fig. 1 canbe characterized by X?Y?A?B?Z. The XYAB axes carry theworkpiece and the Z-axis carries the tool. Fig. 3 showsa machine of the type XYZA?B?, the three linear axescarry the tool and the two rotary axes carry the work-piece.507E.L.J. Bohez / International Journal of Machine Tools & Manufacture 42 (2002) 505520Fig. 3.XYZA?B? machinery.4.1. Classification based on the sequence of workpieceand tool carrying axesTheoretically the number of possible configurations isquite large if the order of the axes in the two kinematicchains of the tool and workpiece carrying axes is countedas a different configuration. Also the combinations withonly two linear axes and three rotary axes are included.One tool carrying axis and four workpiece carryingaxes can be combined in a five-axis machine as follows:for each possible tool carrying axis X,Y,Z,A,B,C the otherfour workpiece carrying axes can be selected from thefive remaining axes. So the number of combinations offour axes out of five with considering different permu-tation as another configuration is 54!=120 for eachpossible tool axis selection (1 out of 6 or 6 possibilities).So theoretically there are 6120=720 possible five-axismachines with one tool carrying axis. The same analysiscan be done for all other combinations. With t the num-ber of tool carrying axes and w the number of workpiececarrying axes (w+t=5) the total number of combinationsis as follows.Ncomb=?6t?t!?6tw?w! t?3, t+w=5(1)Ncomb=?6w?w!?6wt?t! t?3, t+w=5(2)The value of this equation is always equal to 6! or720 when w+t=5. Some of these 720 combinations willbe containing only two linear axis. If only five-axismachines with three linear axes are considered, only35!=360 combinations are still possible.The set Gt of combinations is characterized by a fixedvalue of t. This set is identical to the set G?w charac-terized by a fixed value of w, w=5?t. Using above defi-nitions following subgroups of five-axis machines exist:(i) Group G0/G?5; (ii) Group G1/G?4; (iii) GroupG2/G?3; (iv) Group G3/G?2; (v) Group G4/G?1; (vi)Group G5/G?0.4.1.1. G5/G0? machineAll axes carry the tool and the workpiece is fixed ona fixed table. Fig. 4 shows a machine with all the fiveaxes carrying the tool. The kinematic chain is XBYAZ(TRTRT). This machine was one of the earliest modelsof five-axis machines to handle very heavy workpieces.As there are many links in the tool carrying kinematicchain, there can be a considerable error due to elasticdeformations and backlash in the slides.4.1.2. G0/G5? machineAll axes carry the workpiece and the tool is fixed inspace. This construction is best used for very smallworkpieces (see Section 6.3).Fig. 4.XBYAZ machine.508E.L.J. Bohez / International Journal of Machine Tools & Manufacture 42 (2002) 5055204.1.3. G4/G1? machineFour axes carry the tool and one axis carries the work-piece. There are basically two possibilities, the work-piece carrying axis can be R? or T?.4.1.4. G1/G4? machineOne axis carries the tool and the other four axes carrythe workpiece. There are basically two possibilities, thesingle axis kinematic chain can be R or T. Fig. 1 is anexample of such a machine, with the single tool carryingaxis T.4.1.5. G3/G2? machineThree axes carry the tool and two axes carry the work-piece. There are basically three possibilities, the work-piece carrying axes can be both linear (T?T?) bothrotational (R?R?) or mixed (T?R?). Fig. 5 gives anexample of a machine with the tool carried by two rotaryaxes and one linear axis. This machine allows processingof large workpieces but the construction of the toolsideis complicated. The most common configuration is theworkpiece carried by the two rotary axes such as the onegiven in Figs. 3, 6 and 8.4.1.6. G2/G3? machineTwo axes carry the tool and three axes carry the work-piece. There are basically three possibilities, the tool car-rying 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 rotaryaxesThe machines can be classified depending on the placewhere the rotation axes are implemented.Fig. 5.X?Z?CAY machine.Fig. 6.B?C?ZYX machine.Fig. 7.Z?X?C?BY machine.509E.L.J. Bohez / International Journal of Machine Tools & Manufacture 42 (2002) 505520Fig. 8.Z?A?B?YX machine.Only machines with two rotary axes and three linearaxes will be considered further. The possible configur-ations 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 workpiececarrying kinematic chain is not important if the axes areof the same type R or T. In general, if there are N?Ttrans-latory axes and N?Rrotary axes in the workpiece carryingkinematic chain and NTtranslatory axes and NRrotaryaxes in the tool kinematic chain, then the numbers ofcombinations is 11:Ncomb?(N?T?N?R)!N?T!N?R!(NT?NR)!NT!NR(3)with N?T?NT?3, N?R?NR?2The number of combinations of each group will be givenbelow case by case. The total number of combinationsover all groups is 60. From the design point of view thisis a more tractable number of alternatives to be con-sidered.4.2.1. R?R? machineThe two rotary axes carry the workpiece. The tool axiscan be fixed or carried by one (T), two (TT) or three(TTT) linear axes.The number of possible designs is the sum of the fol-lowing combinations:(i) For the group G0/G5? the tool is fixed in space allthe five axes will carry the workpiece. The numberof different designs is 10 (NT?=3 and NR? =2), (Figs.15 and 16).(ii) For the group G1/G4?, NT+NR=1, so NT=1 and NR=0,is the only possible choice for the tool kinematicchain. Equation (3) gives NCOMB=6. The combi-nationsare:R?R?T?T?T;T?T?R?R?T;R?T?R?T?T;T?R?T?R?T; R?T?T?R?T; T?R?R?T?T. Fig. 9 showsthese six designs.(iii)For the group G2/G3? the tool axes are TT so NT?=1,NR?=2, NT=2, NR=0 and Equation (2) gives NCOMB=3.The three design combinationsare: R?R?T?TT;R?T?R?TT and T?R?R?TT. The group G2/G3? containsthree instances of the R?R? machine. These instancesare represented in Fig. 10.(vi) If the tool axes are TTT the workpiece carrying axescan only be R?R?. So only one design combinationis possible.From the above-mentioned findings it can also be con-cluded that the total number of R?R? five-axis machineconfigurations is 20.Machines with two axes on the clamping table can beseen in Figs. 1, 3, 6 and 8. The advantages are:Fig. 9.Members of group G1/G4?.510E.L.J. Bohez / International Journal of Machine Tools & Manufacture 42 (2002) 505520Fig. 10.R?R? machines in group G2/G3?.? In case the spindle is horizontal, optimal chip removalis obtained through the gravitational effect of thechips just dropping.? The tool axis during machining is always parallel tothe Z axis of the machine. So the drilling cycles canbe executed along the Z-axis of the machine. Circlesunder a certain orientation of the workpiece arealways executed in the XY plane of the machine. Theabove-mentioned functions can be executed in thesimple three-axis numerical control mode.? The compensation of the tool length happens all thetime in the NC control of the machine, as with three-axis machines.Disadvantages:? Machines with a rotating table are only for work-pieces with limited dimensions.? The useful workspace is usually much smaller thanthe product of the travel in X,Y and Z axis.? The transformation of the Cartesian CAD/CAM coor-dinate (XYZIJK) of the tool position to the machineaxes positions (XYZAB or C) is dependent on the pos-ition of the workpiece on the machine table. Thismeans that in case the position of the workpiece onthe table is changed this cannot be modified by atranslation of the axes system in the NC program.They must be recalculated. In case the control of theNC machine cannot transform Cartesian coordinatesto machine coordinates, then a new CNC programmust be generated with the postprocessor of theCAD/CAM system every time the position of theworkpiece changes.Important applications for this type:? Five-sided cutting of electrodes for EDM and otherworkpiece.? Machining of precision workpieces.? Turbines and tire profiles with a certain workpiecegeometry rotated over a certain angle. The same NCprogram can be repeated after the zero of the rotationaxis has been inclined over a certain angle.4.2.2. RR-machineThenumberofpossibledesigncombinations(NCOMP=20) is the same as in the case of the R?R?machine because of the symmetry. Five-axis machineswith the rotation axes implemented on the tool axisspindle 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 pos-ition of the workpiece is corrected with a simpletranslation. This happens with a zero translation in theCNC control of the machine.Disadvantages:? The drive of the main spindle is very complex. Simpledesign and construction is only obtained when thewhole spindle with the motor itself is rotating.? There is a lower stiffness because the rotation axis ofthe spindle is limiting the force transmission. At highrevolutions per minute (higher than 5000 rpm) thereis also a counter acting moment because of the gyro-scopic effect which could be a disadvantage in casethe tool spindle is turning very fast.? Circular interpolation in a random plane and drillingcyclesunderrandomorientationareoftennotimplemented.? A change in the tool length cannot be adjusted by azero translation in the control unit, often a completerecalculation of the program (or postprocessing) isrequired.Important applications of this type of machine tool are:? All types of very large workpieces such as air planewings.4.2.3. R?R machineOne rotary axis is implemented in the workpiece kine-matic chain and the other rotary axes in the tool kinem-atic chain (e.g. Fig. 7).The groups G4/G1?, G4?/G1, G3?/G2, G3/G2? coverthis design. Nowadays there are many machines on themarket with one rotation axis on the tool spindle and511E.L.J. Bohez / International Journal of Machine Tools & Manufacture 42 (2002) 505520one rotation axis on the table. They are, however, com-bining most of the disadvantages of both previous typesof machines and are often used for the production ofsmaller workpieces. The application range of thismachine is about the same as with machines with tworotation axes implemented on the table.In all possible designs of this machine the NR?=NR=1and NT?+NT=3. The total number of possible designs is:NCOMBNT?0,N?T?3?NCOMBNT?1,N?T?2?NCOMBNT?2,N?T?1?NCOMBNT?3, NT?0or 4+6+6+4=20 possible designs.(i) ForNT?=0andNT=3thefourcombinationsare:R?RTTT; R?TRTT; R?TTRT; R?TTTR.(ii) ForNT?=1andNT=2thesixcombinationsare:T?R?RTT;T?R?TRT;T?R?TTR;R?T?RTT;R?T?TRT; R?T?TTR.(iii)For NT?=2 and NT=1 the six combinations are (seeFig.11):R?T?T?TR;T?R?T?TR;T?T?R?TR;R?T?T?RT; T?R?T?RT; T?T?R?RT.(iv)ForNT?=3andNT=0thefourcombinationsare:R?T?T?T?R; T?R?T?T?R; T?T?R?T?R; T?T?T?R?R.5. Workspace of a five-axis machineBeforedefiningtheworkspaceofthefive-axismachine tool, it is appropriate to define the workspaceof the tool and the workspace of the workpiece. TheFig. 11.R?R machines in the group G2/G3?.workspace of the tool is the space obtained by sweepingthe tool reference point (e.g. tool tip) along the path ofthe tool carrying axes. The workspace of the workpiececarrying axes is defined in the same way (the center ofthe machine table can be chosen as reference point).These workspaces can be determined by computing theswept volume 6.Based on the above-definitions some quantitativeparameters can be defined which are useful for compari-son, selection and design of different types of machines.5.1. Workspace utilization factor WRA possible definition for this is the ratio of theBoolean intersection of the workpiece workspace andtool workspace and the union of the tool workspace andworkpiece workspace.WR?WSTOOL?WSWORKPIECEWSTOOL?WSWORKPIECE(4)A large value for WRmeans that the workspace of thetool and the workspace of the workpiece are about equalin size and overlap almost completely. A small value ofWRmeans that the overlap of tool workspace and work-piece workspace is small and that a large part of theworkpiece workspace cannot be reached by the tool. Theanalogy with two cooperating robots can be clearly seen.It is only in the intersection of the two workspaces ofeach robot that they can shake hands. For the five-axismachine tool this corresponds to the volume in whichthe tool and workpiece reference point can meet.However, in the case where all the five axes carry theworkpiece and the tool is fixed in space the above defi-nition would give a zero value for the workspace utiliz-ation. In the case of cooperating robots it would meanthat there is only one point were they can shake hands.In the case of a five-axis machine, the workpiece canstill be moved in front of the tool and remove metal.The reason is that many points from the workpiece canserve as reference point on the workpiece. All pointswhich can cut on the toolsurface c
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