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畢業(yè)設(shè)計(論文)外文資料翻譯 院 系 ?！　I(yè) 學(xué)生姓名 班級學(xué)號 外文出處 The International Journal of Advanced manufacturing Technology 附件：1外文資料翻譯譯文（約3000漢字）； 2外文資料原文（與課題相關(guān)的1萬印刷符號左右）。 指導(dǎo)教師評語： 指導(dǎo)教師簽名：　　　　　 　　　　　　　　　　　　　　　　　　　　　　 年 月 日 附錄A：外文資料翻譯譯文 對由棱柱體-棱柱-球形組合旋轉(zhuǎn)并聯(lián)而成的機(jī)械手的六個自由度的工作空間的分析 M. Z. A. Majid, Z. Huang and Y. L. Yao Department of Mechanical Engineering, Columbia University, New York, USA 摘要：本文主要討論了一般棱柱體-棱柱-球形類型的并聯(lián)機(jī)械手的6個自由度的工作空間。我們知道，并聯(lián)機(jī)械手一個缺陷就是它有限的工作空間。所以，一個潛在的應(yīng)用就是擴(kuò)大它的有限工作區(qū)。在這里，首先對此類型的機(jī)制與機(jī)制的變化作了簡單的分析，然后對工作空間作了研究，并且把工作空間的形狀和大小對聯(lián)合限制和肢體干擾限制作了大量的研究。結(jié)果表明，此種并聯(lián)機(jī)械手的工作空間后遠(yuǎn)遠(yuǎn)大于同等的Stewart平臺上的工作空間，特別是在豎直方向上。 關(guān)鍵字：并聯(lián)機(jī)械手；Stewart平臺；工作空間 1介紹： 在最近幾十年，許多研究者顯示了對并聯(lián)機(jī)械手的興趣。相比于更加普遍和常用的串聯(lián)機(jī)械手，并聯(lián)機(jī)械手在精度，硬度、容量和裝載對重量比率上有更大的優(yōu)勢。并聯(lián)機(jī)械手由一個移動的平臺，一個基本平臺和幾個通過恰當(dāng)?shù)倪\動關(guān)節(jié)和控制器來連接兩個平臺的支架構(gòu)成。最著名的并聯(lián)機(jī)械手是被廣泛研究的Stewart平臺。在此平臺中，6個杠連接，并能移動，基本平臺能夠控制移動平臺的位置與方向。 許多不同的6自由度并聯(lián)機(jī)械手也被提及。最近Tahmasebi and Tsai發(fā)明了一款新穎的并聯(lián)機(jī)械手（圖1）。它包括上下2個平臺和3個可擴(kuò)展的肢。每個肢的末端通過舵機(jī)的球狀關(guān)節(jié)來連接。舵機(jī)是直線步進(jìn)類型，但能在基本平臺上實現(xiàn)x 和y方向的同時移動。上肢的末端則通過旋轉(zhuǎn)關(guān)節(jié)與移動平臺相連。因此這個機(jī)械手是一個3PPSR機(jī)制，其中，P代表柱體，S代表球體，R代表旋轉(zhuǎn)體。2維步進(jìn)電動機(jī)的輸出動作在基本平臺上和交叉的柱體相似，需要在上平臺上獲得的動作就通過移動基本平臺上的舵機(jī)來實現(xiàn)?；酒脚_還連接著3個下肢的末端。除了前面提到的一般并聯(lián)機(jī)械手相對于串聯(lián)機(jī)械手的特性以外，此種3PPSR機(jī)制還有 些其他的優(yōu)點，比如更簡單的結(jié)構(gòu)和更高的剛度。由于在Stewart 平臺上用3個可擴(kuò)展的肢代替了6個可擴(kuò)展的肢，因此，肢之間發(fā)生干涉的可能性很小。 Tahmasebi和Tsai察覺到，此種機(jī)制能夠安裝在串聯(lián)機(jī)械手的手腕與結(jié)束效應(yīng)之間，用于誤差補(bǔ)償和微小的位置變化以及力的控制。所以當(dāng)工作空間被考慮的時候，由于只需要很小的工作空間，每臺舵機(jī)的動作也只能限定在一塊很小的圓面積之內(nèi)。盡管如此，載著肢的這些舵機(jī)不用如此的嚴(yán)格受限，它們能在整個基本平臺上運動從而獲得較大的工作區(qū)域。所以，這種3PPSR機(jī)制能夠作為獨立的機(jī)械手來用。另外，它特殊的裝配運動學(xué)配對讓它能在同等的Stewart平臺上獲得截然不同的和更大的工作空間。對機(jī)械手的工作空間的研究是設(shè)計機(jī)器人手臂的基本問題。很多學(xué)者指出，并聯(lián)體制的主要缺陷就是它有限的工作空間。而3PPSR這種并聯(lián)機(jī)制克服了傳統(tǒng)機(jī)械手的這個缺陷，并拓展了并聯(lián)機(jī)制的應(yīng)用。本文分析了這種3PPSR并聯(lián)機(jī)械手的工作空間的大小，形狀，構(gòu)成和限制因素。 并聯(lián)機(jī)械手的工作區(qū)域在過去十年中吸引了很大學(xué)者的注意。 有不少報道說2個或3個自由度的平面或球面機(jī)器人用到了并聯(lián)機(jī)制工作空間。Asada and Ro [7] and Bajpai and Roth [8]分析了閉合回路平面2個自由度5桿的并聯(lián)機(jī)制。Gosselin and Angeles [9,10]分析了平面和球面3自由度機(jī)制的工作空間。Lee and Shah [11] and Waldron et al.[12]則闡述了空間3自由度并聯(lián)機(jī)制的工作空間。 關(guān)于6自由度并聯(lián)機(jī)制的工作空間的研究較少，Yang and Lee [13], Fichter [14],and Merlet [15]用一種基于笛卡爾空間的方法描述了6自由度并聯(lián)機(jī)制。Gosselin [16]從幾何角度介紹了一種決定6自由度 平臺工作區(qū)域的算法。他的結(jié)果表明，工作空間就是6個環(huán)形區(qū)域的交匯點。Masory and Wang[17]則更系統(tǒng)的研究了6自由度的Stewart平臺。他們討論了幾個用于計算Stewart平臺工作區(qū)域的限定條件，包括運動學(xué)配對的轉(zhuǎn)角度的區(qū)域和機(jī)制里任意兩個肢之間的干涉。另外，他們還分析了工作空間的形狀和工作空間與機(jī)制幾何參數(shù)之間的關(guān)系。Tahmasebi and Tsai [6]還研究了一種新的3PPSR并聯(lián)機(jī)械手，其中，由舵機(jī)連接的每個下肢的末端的動作都被限制在很小的圓面積范圍內(nèi)。在本文對工作空間的分析中，肢之間的干涉和聯(lián)動限制都是考慮過的，而且舵機(jī)能在一個較大的直徑范圍內(nèi)移動。通過構(gòu)成機(jī)械手的不同類型來確定制憲地區(qū)，從而確定了工作空間的組成。 2 機(jī)制分析 如上所述，3PPSR機(jī)制的上下平臺通過三個確定的肢用如下的運動學(xué)配對來連接：兩個棱柱，球形一雙和一個 旋轉(zhuǎn)體（圖2a）。一個球形關(guān)節(jié)等同于三個旋轉(zhuǎn)關(guān)節(jié)和一個共通點，用RRR來表達(dá)。因此，如圖2b所示的這個PPRRRR系統(tǒng)同PPSR系統(tǒng)在運動學(xué)上是等價的。PPSR系統(tǒng)同圖2c所示的肢也等價。這種肢在每個上肢和下肢的末端都有兩個萬向節(jié)，其中，上萬向節(jié)的一根軸線與肢共線，而另一根軸線和下萬向節(jié)的一根軸線則和肢相垂直?？傊?，圖2a,b,c所示的PPSR, PPRRRR, 和 PPUU這3種結(jié)構(gòu)在運動學(xué)上是等價的，其中，u代表萬向節(jié)。如圖2d所示的PPRS結(jié)構(gòu)和先前的類似，通過改變PPSR的球形和旋轉(zhuǎn)體而獲得。盡管如此，它同PPSR系統(tǒng)在運動學(xué)上不能等同，與圖2中的e,f在運動學(xué)上是等價的。因此a,b,c這三張圖同d,e,f這三張圖看起來像，但其實不一樣。 螺旋理論[18,19]用于辨別這兩個系列的結(jié)構(gòu)。旋轉(zhuǎn)關(guān)節(jié)的每根軸線都可以用零間距的螺絲來表達(dá)。我們用分別屬于第一和第二系列的PPRRRR系統(tǒng)和PPRRRR作為例子。兩個系統(tǒng)都定義了oxyz參照系，原點是由同2維舵機(jī)對應(yīng)的兩個方向的兩條直線相交所得。x,y軸與兩個棱柱的移動方向共線。Z軸則由右手法則確定，并與基本平臺相垂直。假設(shè)肢同z軸不共線，兩個系列的螺絲系統(tǒng)可能已如下形式出現(xiàn)： 擰緊系統(tǒng)1 (為PPRRRR)螺絲系統(tǒng)2 (為PPRRRR) 從這兩個螺絲系統(tǒng)我們能看出：每個螺絲系統(tǒng)的六個螺絲是線性獨立的，因為雅克比矩陣并沒有去除 這里，我們就能明白為什么上或下萬向節(jié)的一根軸線必須與肢共線，如圖2c,f所示。如果條件不滿足，雅克比矩陣就變成奇異矩陣。例如，如果$R15與肢不共線（圖2c），而與y軸平行，它的螺絲變成$R15 = (0 1 0; 21 0 0)。因此，由于線性獨立，包含$9的新雅克比矩陣將會成為奇異矩陣。很容易看出，$9與$P11 和$R14.線性相關(guān)。如果所有并聯(lián)機(jī)械手的三只腿都不滿足這個條件，那么它至少會失去三個自由度。 注意，如果最后一個旋轉(zhuǎn)體軸線，在PPRRRR (或PPSR)系統(tǒng)中的$R16與z軸相交，則$R16的最后一部分就為零。也就是說，a,b,c,d為任何實數(shù)，而且a,b,c和d,e不能同時為零。這里有一組數(shù)據(jù)：(0 0 1; 0 0 0)同螺絲系統(tǒng)1的6個螺絲是對等的，也就是說螺絲系統(tǒng)1是線性獨立的，等價于5個螺絲的系統(tǒng)。這個系統(tǒng)失去了一個自由度。整個3PPSR系統(tǒng)都將是奇異矩陣，即使3個肢中只有一個滿足此條件。另一方面，由于最后3個不共面的旋轉(zhuǎn)體與球形等價，因此，這種類型的奇異矩陣不會在螺絲系統(tǒng)2中出現(xiàn)。只可能3根軸線中的2根與z軸同時相交，但球形的第三根軸線就絕不可能與z軸相交。 雖然他們線性獨立，并且都有六個自由度，這兩個系統(tǒng)是不一樣的。結(jié)果，包含這兩個螺絲系統(tǒng)的并聯(lián)機(jī)制，如圖3所證。這兩個機(jī)制有相同的幾何屬性，包括腿長和移動三角形，唯一的區(qū)別是一個用PPSR，另一個用PPRS。假上下兩個平臺最初是平行的，讓兩個上平臺從最初位置旋轉(zhuǎn)到同樣的量，分別用B2,B4和b2,b4表示。容易看出，由這兩個機(jī)制所產(chǎn)生的位置是不同的 3 工作空間分析 如圖1所示，固定的參考系OXYZ處于基本平臺上，原點就是以直徑d為大圓的圓心。X,Y軸都在基本平臺上，Z軸則與基本平臺垂直。動參照系處于移動平臺上。等邊三角形的圓心就是G點。U軸與直線P2P3平行,v軸穿過點P1。W軸與移動平臺平行。 為了確定機(jī)制的工作空間，我們需要正運動學(xué)的一些知識。同樣，當(dāng)涉及到并聯(lián)機(jī)制時，就會用到逆運動學(xué)，但逆運動學(xué)需要使用數(shù)值解。如果已知機(jī)械手的位置和方向，那么上平臺的參考點就能確定工作區(qū)域內(nèi)的某點，前提是它必須在逆運動學(xué)所限定的條件下。當(dāng)給了一系列點后，就相應(yīng)的在上平臺上獲得了一系列點，這樣，工作空間就得到了所有的點組成的一個集合。 3.1 逆運動學(xué) 當(dāng)上平臺的位置和方向都知道時，通過坐標(biāo)傳遞，就能得到移動平臺上點的坐標(biāo)。通過歐拉的三個方向就確定了移動平臺的方向。而G點的坐標(biāo)則與靜參照系中的Xg，Yg，Zg位置對應(yīng)。坐標(biāo)傳遞矩陣是： 而P1，P2,P3點的坐標(biāo)和參照系Guvw中的點對應(yīng)： 其中，m =12cos 30°.同樣，對應(yīng)于靜參照系的Pi點是： 從圖1中機(jī)械手的幾何關(guān)系中不難得出，能同時得到兩個等式，第一個是： 其中，K就是Ri點的Z坐標(biāo)，而p,i,r中，i分別表示Pi和Ri。這是這個大圓的一個基本公式，第二個公式是這樣得到的：由于Pi點是旋轉(zhuǎn)關(guān)節(jié)的，而Ri點則是圓Eq的交匯點，這樣就得到了相互垂直的兩個矢量RiPi和Pi+1Pi+2。表示如下： 方程式是 聯(lián)立方程（6）和（8）解得Xr,i和Yr,i 因為(6)是二階多項式因式，所以x,y可以有兩組解，只要它們滿足聯(lián)動限制和干涉的條件，并在足跡范圍內(nèi)，那么這兩組解就是有效的。 3.2 運動學(xué)中的約束 為了確定3PPSR機(jī)械手的工作空間，我們需要考慮三種不同類型的運動學(xué)約束。它們是足跡圓的直徑，關(guān)節(jié)角度的限制和連接干涉。 足跡圓：圖一所示三個下肢的末端都必須在足跡圓之內(nèi)，即 其中，d是足跡圓的直徑，ORi則為矢量半徑 關(guān)節(jié)角度的限制：通過有物理極限的運動學(xué)配對，將上，下板連接。例如，一個球狀關(guān)節(jié)，理論上在正交坐標(biāo)軸中可以旋轉(zhuǎn)360度，但是實際上由于物理結(jié)構(gòu)上的限制，它的運動范圍要相對減小一點。因此有必要求出每個關(guān)節(jié)的最大旋轉(zhuǎn)角度。旋轉(zhuǎn)角度和它的極限能用如下公式表示： 其中Vi就是矢量直線Li,而用以平分對應(yīng)于靜參照系中運動學(xué)配對的旋轉(zhuǎn)范圍的直線就是Ui 連接干涉：由于物理尺寸的存在，因此可能會發(fā)生干涉。假設(shè)每條直線是圓柱的直徑d而D是兩條相鄰直線的最短距離，那么干涉極限可以這樣來表示： 則兩條中心線間的最短距離就是Ni的位移Dn,即 其中，單位矢量Ni位移方向在相鄰直線Li和Li+1之間，即 注意，兩條直線之間的最短距離不是總等于Dn的位移長度。有可能大于Dn。如果Li上的交匯點Ci與兩直線的位移超出了直線Li，或者Li上的交匯點Mi和垂線Pi+1Li超出了直線Li本身，那么最短距離就是兩個端點Pi和Pi+1之間的長度，如果Mi和Mi+1兩個交匯點都不在Li和Li+1上的話 三條直線，包括兩條相鄰直線之間的位移，定義了兩個平面，兩個平面向量是： 兩個平面方程是 一條直線在三維空間可以表示成： 等式中的兩條中心線可以用下面的方程組來求解： 其中，方程(19) (20)代表中心線Li,方程（21）（22）代表中心線Li+1。 同時聯(lián)立方程（17）（19）（20），就可以求得直線Li和位移之間的交匯點Ci。 同理，聯(lián)立（16）（21）（22）就能得到直線Li+1上的交匯點Ci+1. 正如上文所分析的那樣，通過兩個平面確定了6條直線，那么如果滿足和，干涉就不會發(fā)生 4. 數(shù)例和討論 關(guān)于工作空間的限制條件已經(jīng)在前邊探討過，假設(shè)每個鏈接都是圓柱形的，幾何參數(shù)是°r = 2.5 units, l = 1.0 units, d = 6 units, = 0.15 units, = 75°, = 60°, 和 k = 0, ,k其中 r代表PiRi的長度，l是移動三角形每條邊的長度，d是基本平臺上足跡圓的直徑， 和 是旋轉(zhuǎn)關(guān)節(jié)和球狀關(guān)節(jié)的最大角度，而k代表Ri點的z坐標(biāo)。 三維空間可以用兩個圖像來表達(dá)，比如一個二維的上視圖好一個三維的等距視圖，為了視圖的方便，不用表達(dá)上下的邊界部分，由于工作空間涉及到位置和方向，所以三個不變的歐拉角度必須指定為每一個如下情況，為了證明不同條件下對工作空間的形狀和大小限制的影響，我們列舉了下面5種情況： 注意，由于這個平臺關(guān)于移動平臺的v軸對稱，所以{ } = {0°, —20°, 0°} 就等于{ } = {0°, 20°, 0°}，另外，由于工作空間并沒有改變，所以5種情況下仍然是零。只要形狀不變，工作空間將會隨著不為零的改變而改變。 圖4表示了工作空間上視圖的第一種情況，{} = {0°, 0°, 0°}.在不考慮干涉和關(guān)節(jié)角度限制的條件下，圖5表達(dá)了理論工作空間的等距視圖，在不考慮運動學(xué)限制的情況下，圖6表達(dá)了上述相同的5種條件下的實際工作空間?？梢钥闯?，這種3-PPSR 并聯(lián)機(jī)械手的工作空間的形狀和結(jié)構(gòu)不同于傳統(tǒng)的并聯(lián)機(jī)械手，他允許在z方向上有較大的動作范圍。 應(yīng)當(dāng)指出，圖5和6所得到的工作空間是在為零的情況下，由于 能夠旋轉(zhuǎn)360度所以實際的工作空間是將如圖5和6繞z軸旋轉(zhuǎn)同樣的角度而得。對比同等的蘑菇帽形狀的 平臺，它的工作空間是圓柱形的，所以有較大的z軸范圍 工作空間通過詳細(xì)的檢查來得到它的構(gòu)成。我們根據(jù)不同類型的機(jī)械手來劃分工作空間的制憲地區(qū)，從而完成檢查。以圖4所示的工作空間作為例子。四個不同類型的制憲地區(qū)都能被確定 Z=1.0. 第一個區(qū)域：如圖8a所示，它的形狀對應(yīng)著一條腿朝著平臺，另外兩條則向外，同理也可以是3條腿之間相互輪換，很類似，只是在原來的 基礎(chǔ)上旋轉(zhuǎn)120度 第二個區(qū)域：如圖8c所示，它的形狀對應(yīng)著兩條腿向里，一條向外，同理也可以是三條腿之間的輪換，仍是在原來基礎(chǔ)上轉(zhuǎn)120度 第三個區(qū)域：如圖8e所示，它的形狀對應(yīng)著三條腿向里 圖5 圖6 第四個區(qū)域：它的形狀和圖e類似，只是三條腿都向外了這種區(qū)域?qū)τ诠ぷ骺臻g沒有實際的影響，因為它是類型3的子類型。 所有的這些區(qū)域和直徑為6的足跡圓都繪制在圖9中。很明顯，這些區(qū)域的交叉形成了一塊面積，就像圖4所確定的一樣，所以，他們都應(yīng)該是工作空間的制憲地區(qū)。 圖8 圖7 5.結(jié)論 本文主要分析了3-PPSR機(jī)械手的工作空間。工作空間包括三種區(qū)域，每個都對應(yīng)著不同類別的機(jī)械臂。運動學(xué)限制的影響則包括：旋轉(zhuǎn)關(guān)節(jié)和球狀關(guān)節(jié)，工作空間結(jié)構(gòu)的肢的干涉。3-PPSR機(jī)械手的工作空間是圓柱形的，而 平臺則通常是蘑菇帽形的工作空間，它在z軸方向的動作受到較大的限制。 參考文獻(xiàn) 1. D. Stewart, “A platform with six degrees of freedom”，Proceedings Institution of Mechanical Engineers, 180, pp. 25-28, 1965. 2.F. Tahmasebi and L.-W. Tsai, “On the stiffness of a novel six-DOF parallel manipulator”, Intelligent Automation and Soft Computing, Proceedings of the First World Automation Congress (WAC，94), vol. 2, pp. 189-194, 1994. 3.F. Tahmasebi and L.-W. Tsai, “Closed-form direct kinematics solution of a new parallel manipulator”, Journal of Mechanical Design, 116, pp. 1141-1147, 1994. 4.L.-W. Tsai and F. Tahmasebi, “Synthesis and analysis of a new class of six-DOF parallel manipulators”, Journal of Robotic Systems, 10, pp. 561-580, 1993. 5.F. Tahmasebi and L.-W. Tsai, “Jacobian and stiffness analysis of a novel class of six-DOF parallel manipulators”, Proceedings of the 22nd Biennial Mechanisms Conference, ASME, DE-vol. 47, pp. 95-102, 1992. 6.F. Tahmasebi and L.-W. Tsai, “Workspace and singularity analysis of a novel six-DOF parallel manipulator”, Journal of Applied Mechanisms and Robotics, 1(2), pp. 31-40, 1994. 7.H. Asada and I. H. Ro, “A linkage design for direct drive robot arms”, Journal of Mechanisms, Transmissions and Automation in Design, 107, pp. 536-540, 1985. 8.A. Bajpai and B. Roth, “Workspace and mobility of a closed- loop manipulator”, International Journal of Robotics Research, 5(2), pp. 131-142, 1986. 9.C. Gosselin and J. Angeles, “The optimum kinematic design of a planar three-DOF parallel manipulator”, Journal of Mechanisms, Transmissions and Automation in Design, 110, pp. 35-41, 1988. 10.C. Gosselin and J. Angeles, “The optimum kinematic design of a spherical three-DOF parallel manipulator”, Journal of Mechanisms, Transmissions and Automation in Design, 111, pp. 202-207, 1989. 11. K. M. Lee and D. K. Shah, “Kinematic analysis of a three-DOF in-parallel actuated manipulator,” IEEE Journal of Robotics and Automation 4(3), 354-360, 1988. 12.K. J. Waldron, M. Raghavan and B. Roth, “Kinematics of a hybrid series of parallel manipulation system”, ASME Journal of Dynamic System Measurement and Control, 111, pp. 211221, 1989. 13.D. C. H. Yang and T. W. Lee, “Feasibility study of a platform type of robotic manipulators from a kinematic viewpoint”, Journal of Mechanisms, Transmissions and Automation in Design, 106, pp. 191-198, 1984. 14.E. F. Fichter, “A Stewart platform-based manipulator: general theory and practical construction”, International Journal of Robotics Research, 5(2), pp. 157-182, 1986. 15.J. P. Merlet, “Force-feedback control of parallel manipulator”, IEEE International Conference on Robotics and Automation, pp. 1484-1489, 1988. 16.C. Gosselin, “Determination of the workspace of six-DOF parallel manipulator”, Journal of Mechanical Design, 112, pp. 331-336, 1990. 17.O. Masory and J. Wang, “Workspace evaluation of Stewart platform”, Proceedings ASME Winter Annual Meeting, DE-45, pp. 337-352, 1992. 18.R. S. Ball, Theory of Screw, Cambridge University Press, 1900. 19. K. H. Hunt, Kinematic Geometry of Mechanisms, Oxford, Clarendon Press, 1978. 附錄B：外文資料原文 Int J Adv Manuf Technol (2000) 16:441-449 ? 2000 Springer-Verlag London Limited The International Journal of Advanced manufacturing Technology Workspace Analysis of a Six-Degrees of Freedom, Three- Prismatic-Prismatic-Spheric-Revolute Parallel Manipulator M. Z. A. Majid, Z. Huang and Y. L. Yao Department of Mechanical Engineering, Columbia University, New York, USA Abstract：This paper studies the workspace of a six-degrees-of-freedom parallel manipulator of the general three-PPSR (prismatic- prismatic-spheric-revolute) type. It is known that a drawback of parallel manipulators is their limited workspace. To develop parallel mechanisms with a larger workspace is of use to potential applications. The mechanism of a three-PPSR manipulator and its variations are briefly analysed. The workspace is then investigated and the effects of joint limit and limb interference on the workspace shape and size are numerically studied. The constituent regions of the workspace corresponding to different classes of manipulator poses are discussed. It is shown that the workspace of this parallel manipulator is larger than that of a comparable Stewart platform, especially in the vertical direction. Keywords: Parallel manipulator; Stewart platform; Workspace 1.Introduction In the past decade, many researchers have shown an interest in parallel manipulators. Compared with the more commonly used serial manipulators, the parallel ones have advantages in accuracy, rigidity, capacity, and load-to-weight ratio. A parallel manipulator consists of a moving platform, a base platform and several branches connecting both platforms through appropriate kinematic joints with appropriate actuators. The best known parallel manipulator is the Stewart platform [1], which has been widely studied. In a Stewart platform, six bars connecting moving and base platforms are extensible to control the position and orientation of the moving platform. Many different 6-degrees-of-freedom (DOF) parallel manipulators have been proposed. Recently, Tahmasebi and Tsai [25] introduced and studied a novel parallel manipulator (Fig. 1). Correspondence and offprint requests to: Dr Y. Lawrence Yao, Department of Mechanical Engineering, Columbia University, 220 Mudd, MC 4703, New York, NY 10027, USA. E-mail: ylyl @columbia.edu This mechanism consists of an upper and a lower platform and three inextensible limbs. The lower end of each limb connects through a ball-and-socket joint to an actuator. The actuator is of a linear stepper type but is capable of moving in both x- and y-directions simultaneously on the base platform. The upper end of each limb is connected to the moving platform by a revolute joint. The manipulator is therefore a 3PPSR mechanism, where P denotes the prismatic pair, S the spherical pair, and R the revolute pair. The output motion of the 2D linear stepper motors is similar to that of two cross- prismatic pairs on the base platform. The desired motion of the upper platform is obtained by moving the actuators on the base platform, to which the lower ends of the three limbs are attached. Besides the merits of general parallel mechanisms over their serial counterparts mentioned before, this 3PPSR mechanism has added advantages, including simpler structure and higher stiffness. It is also less likely that its limbs will interfere with each other, since it has only three inextensible limbs instead of six extensible limbs as in a Stewart platform. Fig. 1. A 3-PPSR parallel manipulator. 442 M. Z. A. Majid etal. Fig. 2. Variations in pair sequence and type. (a), (b), and (c) form a kinematically equivalent set, whereas (d), (e), and (f) form another. Tahmasebi and Tsai perceived this mechanism as being used as a minimanipulator, which can be mounted between the wrist and the end-effector of a serial manipulator for error compensation as well as for delicate position and force control. Therefore, the required workspace is rather small so that the motion of each actuator is limited to within a small circular area on the base platform when its workspace is considered [6]. The actuators carrying the limbs, however, do not have to be so restricted, they can move over the entire base platform, resulting in a much larger workspace. As a result, this 3PPSR mechanism can be used as a stand-alone manipulator. In addition, its special assembly of kinematic pairs makes it possible to have a workspace that is very different from and larger than that of a comparable Stewart platform. The study of workspace of a manipulator is one of the fundamental problems in the design of robot arms. As many researchers have pointed out, the major drawback of parallel mechanisms is their limited workspace. This 3PPSR parallel mechanism can help overcome the limitations of traditional parallel manipulators and extend the applications of parallel mechanisms. This paper analyses the size, shape, composition, and constraints of the workspace of the 3PPSR parallel manipulator. workspace of parallel manipulators has attracted the attention of many researchers over the past decade. Much reported work on parallel mechanism workspace dealt with 2DOF or 3DOF planar and spherical manipulators. Asada and Ro [7] and Bajpai and Roth [8] analysed the workspace of a closed-loop planar 2DOF 5-bar parallel mechanism. Gosselin and Angeles [9,10] studied the workspace of planar and spherical 3DOF mechanisms. Lee and Shah [11] and Waldron etal. [12] demonstrated the workspace of a spatial 3DOF in-parallel manipulator. Much less work has been reported for the workspace of 6DOF parallel manipulators. Yang and Lee [13], Fichter [14], and Merlet [15] described the workspace of 6DOF parallel manipulators, using a method based on discretisation of the Cartesian space. Gosselin [16] used geometric properties to introduce an algorithm for determining the workspace of a 6DOF Stewart platform. His results showed that the workspace was the intersection of six annular regions. Masory and Wang [17] more systematically studied the workspace of a 6DOF Stewart platform. Their report discussed several constraint conditions for calculating its workspace, including the region of the angle of rotation of kinematic pairs and the interference between any two limbs of the mechanism. In addition, they analysed the shape of the workspace and the relationship between the workspace and the geometric parameters of the mechanism. Tahmasebi and Tsai [6] studied the workspace of this new 3PPSR parallel manipulator, where the motion of each of the three actuators attached to the lower end of each limb is limited to a small circular area. In the workspace analysis presented in this paper, limb interference and joint limitations are considered, and the actuators are allowed to move within a larger circle of diameter d (Fig. 1). The composition of the workspace is also studied by identifying the constituent regions according to different classes of manipulator poses. 2. Mechanism Analysis As mentioned before, the upper and lower platforms of a 3PPSR mechanism are connected by three identical limbs each with the following kinematic pairs: double prismatic pairs, one spherical pair and one revolute pair (Fig. 2(a)). A spherical joint is kinematically equivalent to three non-coplanar revolute joints with a common point denoted as RRR. Thus the PPRRRR system shown in Fig. 2(b) is kinematically equivalent to the PPSR system. The PPSR arrangement is also kinematically equivalent to the limb shown in Fig. 2(c). This limb has a 2DOF universal joint at each of its lower and upper ends, where one of the axes of the upper universal joint is collinear with the limb, whereas another axis of the upper universal joint as well as one of the axes of the lower universal joint are always perpendicular to the limb [6]. In summary, the three structures: PPSR, PPRRRR, and PPUU, as shown in Figs 2(a), 2(b) and 2(c), are kinematically equivalent, where U denotes the universal joint. A similar structure, PPRS (Fig. 2(d)), can be obtained by exchanging the spherical pair and the revolute pair of the PPSR system. It is, however, kinematically different from the PPSR system, as shown below; but the PPRS system is kinematically equivalent to the two systems shown in Figs 2(e) and 2(f). Therefore, the set shown in Figs 2(a), 2(b), and 2(c) look similar but different from the set shown in Figs 2(d), 2(e) and 2(f). Screw theory [18,19] is used in determining the difference between the two sets of structures. Every axis of the revolute joints can be expressed as a screw with zero pitch. The PPRRRR system (Fig. 2(b)), which belongs to the first set, and PPRRRR (Fig. 2(e)), which belongs to the second set are taken as an example. A reference frame oxyz is defined for both PPRRRR and PPRRRR. The origin o is located at the intersecting point of two lines corresponding to the two directions of the 2D actuators. The x and y-axes lie collinear with the moving directions of the two prismatic pairs. The z-axis is defined by the right-hand-rule perpendicular to the base platform. Assuming that the limb is not collinear with the z-axis, two sets of screw systems may be given as follows: Screw system 1 (for PPRRRR) Screw system 2 (for PPRRRR) From these two screw systems one can see that the six screws of each screw system are linearly independent, since their Jacobian matrices do not vanish Here, one can see why one of the axes of the upper (lower) universal joint must be collinear with the limb, as shown in Figs 2(c) (2(f)). If this condition is not satisfied, the Jacobian matrix will become singular. For instance, if $R15 is not collinear with the limb (Fig. 2(c)) but is parallel with the y-axis, its screw becomes . The new Jacobian matrix involving instead of in the screw system 1 will be singular owing to linear dependence. It is easy to see that is a linear combination of and . If all three legs of the parallel mechanism do not satisfy this condition, it will lose at least two degrees of freedom. Note that, if the axis of the last revolute pair, ，of system PPRRRR (or PPSR) intersects the Z-axis, the last component of will be zero. That is,