基于幾何方法精確分析一類三自由度的平面并聯(lián)機(jī)器人外文文獻(xiàn)翻譯、中英文翻譯
基于幾何方法精確分析一類三自由度的平面并聯(lián)機(jī)器人外文文獻(xiàn)翻譯、中英文翻譯,基于,幾何,方法,法子,精確,分析,一類,自由度,平面,并聯(lián),機(jī)器人,外文,文獻(xiàn),翻譯,中英文
基于幾何方法精確分析一類三自由度的平面并聯(lián)機(jī)器人
Alexander Yu(a),Ilian A. Bonev(b),Paul Zsombor-Murray(a)
(a)機(jī)械工程學(xué)院,McGill大學(xué),817 Sherbrooke St.W ,蒙特利爾,加拿大H3A 2K6
(b)自動(dòng)化制造工程部,E′ cole de technologie supe′ rieure (E′ TS)
(a)1100 Notre-Dame St.W.,蒙特利爾,魁北克,加拿大 H3C 1K3
2006年6月12收到; 2006年12月11收到訂正形式; 2007年3月16號(hào)被接受
2007年5月3日出現(xiàn)在網(wǎng)上
摘要
并聯(lián)機(jī)器人正越來越多地被用于工業(yè)的精確定位和調(diào)整。他們的優(yōu)點(diǎn)是剛性好,快速,準(zhǔn)確。隨著并聯(lián)機(jī)器人被廣泛使用,我們有必要制定一種方法來比較不同的并聯(lián)機(jī)器人設(shè)計(jì)方案。但是不存在一個(gè)簡單的方法用來充分比較并聯(lián)機(jī)器人的精確度。過去已有某些被采用的指標(biāo)如靈活度,可達(dá)工作空間和總體調(diào)節(jié)指數(shù),但當(dāng)它們用于既有平移又有轉(zhuǎn)動(dòng)自由度的機(jī)器人時(shí)表現(xiàn)都不完美。針對(duì)這些問題,本文提出了一種簡單的幾何方法來精確計(jì)算由于驅(qū)動(dòng)元件的準(zhǔn)確性不夠而引起的局部最大位置度誤差和局部最大方向度誤差。這個(gè)方法是針對(duì)一類三自由度平面并聯(lián)機(jī)器人的,該類機(jī)器人的最大的工作空間是由圓弧和直線段構(gòu)成的,并且工作空間內(nèi)沒有奇異位置。三個(gè)特別設(shè)計(jì)的方案用于說明這種方法。
2007 Elsevier Ltd. 版權(quán)所有。
關(guān)鍵詞:并聯(lián)機(jī)器人; 靈活度;工作空間; 位置; 精確性;誤差分析
1. 引言
并聯(lián)機(jī)器人一度是實(shí)驗(yàn)室里純學(xué)術(shù)的東西,但是在最近幾年它已經(jīng)被越來越多地用于工業(yè)上的定位和調(diào)整。由于當(dāng)今市場有這些快速和敏捷的機(jī)器人的需求,因此新的并聯(lián)機(jī)器人正在被設(shè)計(jì)和制造。但是,仍然有許多設(shè)計(jì)新的并聯(lián)機(jī)器人的關(guān)鍵問題沒有解決,如優(yōu)化設(shè)計(jì)和性能標(biāo)指。怎么能證明一個(gè)新的并聯(lián)機(jī)器人設(shè)計(jì)是現(xiàn)有機(jī)器人的優(yōu)化方案呢?根據(jù)機(jī)器人的工作空間來評(píng)價(jià)該機(jī)器人的性能是否就足夠了呢?顯然,在當(dāng)前工業(yè)環(huán)境下,答案是否定的,因?yàn)樵谠S多應(yīng)用中定位精度已成為一個(gè)關(guān)鍵的問題。
有幾個(gè)很好的界定性能指標(biāo)已被廣泛的用于串聯(lián)和并聯(lián)的機(jī)器人。然而,最近的一項(xiàng)研究[ 1 ]再次審查這些指標(biāo),并討論其用于有平移和旋轉(zhuǎn)自由度的并聯(lián)機(jī)器人的嚴(yán)重不一致性。該研究審查了最常見的指標(biāo),優(yōu)化并聯(lián)機(jī)器人:靈活度[ 2 ] 和整體協(xié)調(diào)指數(shù)。用各種條件指數(shù)應(yīng)用于機(jī)器人以找出其靈活度,以提高其準(zhǔn)確性,如兩種標(biāo)準(zhǔn)指數(shù)或弗洛比尼斯指數(shù),并通過計(jì)算機(jī)器人的全部工作空間來了解它的整體協(xié)調(diào)指數(shù)。[ 3 ]論文的結(jié)論是,這些指標(biāo)不應(yīng)當(dāng)用于混合自由度類型的并聯(lián)機(jī)器人(平移和旋轉(zhuǎn)) 。
當(dāng)作者設(shè)計(jì)了一個(gè)新的三自由度( 3自由度)平面并聯(lián)機(jī)器人,他們用靈活度來比較類似的設(shè)計(jì) [ 4 ] 。這種是比較公平的,因?yàn)樵O(shè)計(jì)允許使用相同的層面。但是如果比較的是一個(gè)變化的單位(例如,從厘米級(jí)到毫米級(jí)) ,這些數(shù)字的指數(shù)將發(fā)生重大變化。更高價(jià)值的長度單位實(shí)質(zhì)上將使機(jī)器人的靈活度接近0。如圖1所示。
靈活度指標(biāo)對(duì)使用部件的類型和部件的重要性非常敏感,因?yàn)殪`活度依賴于雅可比矩陣。這個(gè)矩陣混合了可變的功能,如平移和旋轉(zhuǎn)功能。一種可能的解決這一問題的辦法是增加了條件數(shù),但是每個(gè)條件都是由特定的問題來描述的[ 1 ]。
整體協(xié)調(diào)指數(shù)( GCI )可以用來評(píng)估機(jī)器人在其工作空間里指標(biāo),它可用于機(jī)器人的優(yōu)化設(shè)計(jì)。但是,仍然存在著兩個(gè)問題,這指數(shù)。首先,它仍是依賴于一些問題的條件,這些問題必須已被概括描述出來[ 1 ] 。其次,它是計(jì)算密集型的。
顯然,工業(yè)并聯(lián)機(jī)器人最好的測量精度應(yīng)與局部最大位置度誤差和局部最大方向度誤差在一個(gè)數(shù)量級(jí)上,因?yàn)轵?qū)動(dòng)器誤差(輸入誤差) ,或一些廣義的誤差(例如,在特定工作空間里的均值和方差的誤差) 。一般方法可以在區(qū)間分析的基礎(chǔ)上近似計(jì)算這些誤差[ 5 ] 。但是,此方法計(jì)算密集型的,并沒有深入到此問題優(yōu)化設(shè)計(jì)的運(yùn)動(dòng)學(xué)分析。
圖 1 靈活度例子(a) PreXYT , (b) HephAist并聯(lián)機(jī)器人( c )星三角并聯(lián)機(jī)器人。
圖2 (a) PreXYT (正在申請(qǐng)專利)( b ) HephAist的NAF3型校準(zhǔn)臺(tái)(courtesy of HephAist Seiko Co., Ltd) 。
與此相反,本文提出了一種簡單的幾何方法用于準(zhǔn)確的計(jì)算一類三自由度平面完全并聯(lián)機(jī)器人局部最大位置度誤差和局部最大方向度誤差。該機(jī)器人的可達(dá)工作空間是由圓弧和直線組成的,并且工作空間內(nèi)沒有奇異位置。該方法不僅速度快于其他任何方法(針對(duì)特定類別的并聯(lián)機(jī)器人) ,而且也帶來了有價(jià)值的運(yùn)動(dòng)學(xué)分析。
這種做法體現(xiàn)在三個(gè)特別設(shè)計(jì),這些設(shè)計(jì)中最佳微定位的位置在相對(duì)較大的工作空間里的位置頗有不同:
1. 一種新的并聯(lián)機(jī)器人,名為PreXYT ,由E′ cole de technologie supe′rieure (E′ TS)設(shè)計(jì)和建造,它具有獨(dú)特2-PRP/1-PPR配置( P和R用于被動(dòng)棱柱和旋轉(zhuǎn)接頭,而P用于驅(qū)動(dòng)滑移鉸)。圖2(a)顯示的是PreXYT的CAD模型。
2. HephAist的3-PRP并聯(lián)機(jī)器人,由日本公司HephAist Seiko設(shè)計(jì) ,目前已投入商業(yè)使用。圖2顯示的是其工業(yè)模型的照片。
3. 星三角并聯(lián)機(jī)器人是另外一種在法國LIRMM設(shè)計(jì)的3-PRP并聯(lián)機(jī)器人 [ 6 ] 。這種機(jī)器人是雙三角并聯(lián)機(jī)器人的優(yōu)化設(shè)計(jì)方案 [ 7 ] 。
其余內(nèi)容安排如下。下一節(jié)介紹了擬議的幾何方法。然后,第3節(jié)介紹了所有三個(gè)并聯(lián)機(jī)器人逆運(yùn)動(dòng)學(xué)方程和直接運(yùn)動(dòng)學(xué)方程,其精確度將是本文研究重點(diǎn)。第4節(jié)簡要敘述了這三個(gè)機(jī)器人幾何形狀不變的工作空間。第5節(jié)適用于擬議的幾何方法計(jì)算局部最大位置度誤差和局部最大方向度誤差。最后一節(jié)給出了結(jié)論。
2. 幾何方法計(jì)算輸出誤差
考慮三自由度完全平行平面機(jī)器人在一個(gè)理想的(名義上)狀態(tài)下。設(shè)x, y和Φ表示移動(dòng)平臺(tái)的假設(shè)位置和假設(shè)方向,ρ1、ρ2和ρ3表示活動(dòng)關(guān)節(jié)的變量。由于驅(qū)動(dòng)器的誤差是±ε ,實(shí)際活動(dòng)關(guān)節(jié)變量的范圍大概在[ρ1-ε,ρ2+ε]。因此,移動(dòng)平臺(tái)的實(shí)際位置和方向分別是x+Δx,y+Δy,和Φ+ΔΦ?,F(xiàn)在的問題是給動(dòng)機(jī)器人的公稱尺寸(x,y,Φ,ρ1,ρ2,ρ3)和公差范圍 ±ε,怎樣確定最大位置誤差和最大方向誤差。例如,最大位置誤差 δmax=max?x2+?y2 ,最大方向誤差σmax=max?Φ。
為了計(jì)算這些誤差,我們基本上需要找到活動(dòng)關(guān)節(jié)的變量中存在誤差的值。最大的錯(cuò)誤是假設(shè)無論什么類型的機(jī)器人,無論機(jī)器人的尺寸如何,當(dāng)機(jī)器人的活動(dòng)關(guān)節(jié)變量受最大輸入誤差影響時(shí),產(chǎn)生最大位置誤差,如誤差是+ε或-ε。事實(shí)上,在[ 8 ] ,這是代數(shù)證明的最大方向角的三自由度平面并聯(lián)機(jī)器人,在兩條支鏈平行時(shí),在兩條支鏈平行時(shí)可能會(huì)出現(xiàn)第1類(串聯(lián))或第2類(并聯(lián))奇異,這就是間隔輸入誤差。同樣,有人證明,并非所有活動(dòng)關(guān)節(jié)變量必須在間隔輸入誤差里才會(huì)產(chǎn)生最大位置誤差。
當(dāng)然,雖然也有例外,三自由度平面并聯(lián)機(jī)器人用于精確定位操作遠(yuǎn)離第1類奇異點(diǎn),當(dāng)然也遠(yuǎn)離第2類奇異(如果存在) 。另外,無論機(jī)器人得兩條支鏈?zhǔn)欠衿叫卸己苋菀淄茢喑鼍植孔畲蠓较蛘`差,并且很容易設(shè)計(jì)出這種沒有奇異點(diǎn)的機(jī)器人。因此,實(shí)際上當(dāng)八個(gè)活動(dòng)關(guān)節(jié)變量中有一個(gè)的輸入誤差是+ε或-ε時(shí),這種三自由度平面并聯(lián)機(jī)器人會(huì)產(chǎn)生局部最大方向誤差。
現(xiàn)在,找到局部最大位置誤差相當(dāng)于找到距離移動(dòng)平臺(tái)通常位置最遠(yuǎn)的不確定區(qū)域的中心點(diǎn)。這種不確定區(qū)域基本上是機(jī)器人通過改變活動(dòng)鉸鏈所能到達(dá)的最大工作空間,距離是[ρi-ε,ρi+ε] 。顯然正因?yàn)檫@一點(diǎn),我們正在尋求在邊界的最大工作空間。
在[ 9 ]中有一個(gè)計(jì)算邊界的幾何算法 ,但我們不會(huì)在這里討論細(xì)節(jié)。我們只需要提及的是組成這一邊界部份的曲線的形態(tài),至少有一條支鏈會(huì)出現(xiàn)第1類奇異(這不在我們的研究范圍之內(nèi)) ,或者解耦度較低。當(dāng)這些曲線是由直線段或圓弧組成時(shí),這將是非常容易距離移動(dòng)平臺(tái)通常位置最遠(yuǎn)的不確定區(qū)域的中心點(diǎn)。這一點(diǎn)將會(huì)是最大工作空間的交匯點(diǎn)。
下面,將三個(gè)例子加以研究,以說明擬議的幾何方法。
3 。運(yùn)動(dòng)學(xué)分析和逆運(yùn)動(dòng)學(xué)分析
從圖3a-c可以看出,基礎(chǔ)參考系Oxy固定在地面上,定義一個(gè)每個(gè)平面都是水平運(yùn)動(dòng)的并聯(lián)機(jī)器人。同樣,移動(dòng)參照系Cx’ y’是固定在移動(dòng)平臺(tái)上的,并且與Oxy在同一平面內(nèi)。定義Ai是轉(zhuǎn)動(dòng)支鏈i的轉(zhuǎn)軸點(diǎn)(本文中,i=1,2,3),并且在Oxy平面內(nèi)。
從圖3a和b中可以看出,基礎(chǔ)y軸沿著A2點(diǎn)的運(yùn)動(dòng)方向 ,而運(yùn)動(dòng)的x’ 軸沿著線A2A3的方向。從圖3a中可以看出,原點(diǎn)C與A1點(diǎn)重合。而在圖3b中,原點(diǎn)C 的擺放可以使A1沿著y’ 軸移動(dòng)。S是A2和A3兩點(diǎn)的平行移動(dòng)路徑之間的距離。而在HephAist的調(diào)整平臺(tái),H是基礎(chǔ)坐標(biāo)系x軸與A1移動(dòng)路徑之間的距離。
從圖3中可以看出,設(shè)Oi是等邊三角形的三個(gè)頂點(diǎn)。令O點(diǎn)與O1點(diǎn)重合,令基礎(chǔ)坐標(biāo)系x軸與線O1O2重合。令移動(dòng)平臺(tái)上三條交叉線的交點(diǎn)是C點(diǎn),讓C點(diǎn)隨著Ai點(diǎn)的移動(dòng)而移動(dòng)。這三條線之間的夾角相同。最后,移動(dòng)坐標(biāo)系y’ 軸與線A1C重合。
圖。 3 。示意圖(a) PreXYT (正在申請(qǐng)專利) , (b) HephAist的并聯(lián)機(jī)器人和( c )星三角并聯(lián)機(jī)器人。
令ρi是活動(dòng)鉸鏈的距離,定義如下。以PreXYT和HephAist的調(diào)整平臺(tái)(圖3a和b )為例,ρ1是基礎(chǔ)坐標(biāo)系y軸到A1點(diǎn)的距離 ,而ρ2和ρ3是基礎(chǔ)坐標(biāo)系x軸分別到A2和A3的距離。最后,對(duì)于星三角機(jī)器人(圖3C) ,ρi是Oi到Ai的距離。事實(shí)上,對(duì)于星三角機(jī)器人,從機(jī)械設(shè)計(jì)上保證了Ai不能與Oi重合。
3.1. PreXYT
給定活動(dòng)鉸鏈變量,可以非常好的直截了當(dāng)?shù)拇_定移動(dòng)平臺(tái)的位置和方向。方向角也是很容易得到的,如:
Φ=tan-1 ρ3-ρ2s (1)
而移動(dòng)平臺(tái)的位置如下確定
x=ρ1 (2)
y=ρ2+ρ1(ρ_3-ρ_2s) (3)
從這個(gè)例子可以看到,直接運(yùn)動(dòng)學(xué)PreXYT非常簡單,部分解耦。
逆運(yùn)動(dòng)學(xué)分析也是很簡單的。根據(jù)移動(dòng)平臺(tái)位置和方向可以如下獲得活動(dòng)鉸鏈變量:
ρ1=x (4)
ρ2=y-x tanΦ (5)
ρ3=y+s-xtanΦ (6)
Obviously, PreXYT has no singularities (provided that s is non-zero).
顯然, PreXYT沒有奇異點(diǎn)(前提是S是非零) 。
3.2 。 HephAist的并聯(lián)機(jī)器人
給定活動(dòng)鉸鏈變量,可以非常好的直截了當(dāng)?shù)拇_定移動(dòng)平臺(tái)的位置和方向。方向角度的方程和方程(1)非常相似。移動(dòng)平臺(tái)的位置是A_2A_3與過A_1且與A_2A_3正交的線的交點(diǎn)。因此可以由方程計(jì)算出x和y:
x=dρ1s+h-ρ2ρ3-ρ2s2+ρ3-ρ22 (7)
y=s2ρ2+hρ3-ρ22+sρ1(ρ3-ρ2)s2+ρ3-ρ22 (8)
我們可以看到,直接運(yùn)動(dòng)學(xué)HephAist的并聯(lián)機(jī)器人更為復(fù)雜而且耦合度高。
逆運(yùn)動(dòng)學(xué)是比較容易解決的。根據(jù)移動(dòng)平臺(tái)的位置和方向,可以獲得活動(dòng)鉸鏈變量:
ρ1=x-h-ytanΦ (9)
ρ2=y-x tanΦ (10)
ρ3=y+s-xtanΦ (11)
由方程((7)-(11))定義(假設(shè)s是50),很顯然,這并聯(lián)機(jī)器人也沒有奇異點(diǎn)。注意 ,這是很大的優(yōu)勢,通常大多數(shù)平面并聯(lián)機(jī)器人都有奇異點(diǎn)。
3.3 。星三角并聯(lián)機(jī)器人
給定活動(dòng)鉸鏈變量,通過這種直接的運(yùn)動(dòng)中使用的方法,如[ 10 ]中,可以非常好的直截了當(dāng)?shù)拇_定移動(dòng)平臺(tái)的位置和方向。在圖4中,通過下列幾何的基礎(chǔ)上定義的第一次費(fèi)爾馬點(diǎn)可以很容易的獲得C點(diǎn)的位置。
圖。 4 。星三角并聯(lián)機(jī)器人的運(yùn)動(dòng)學(xué)求解
由于三角形A1A2A3中,沒有任何一個(gè)角大于120度 (因?yàn)辄c(diǎn)Ai無法移動(dòng)到三角形O1O2O3外面),等邊三角形在它外面。用Qi表示三角形最外層的角(見圖4 )。然后使QiAi成120度并且相交于一點(diǎn),即所謂的第一費(fèi)爾馬點(diǎn)。這一點(diǎn)是移動(dòng)支架的原點(diǎn)C。
雖然目前移動(dòng)平臺(tái)的位置是唯一的,但有兩種可能的方向角(Φ和Φ+180°) 。但是,很明顯只兩個(gè)方案中只有一個(gè)是是可行的(那個(gè)-90°<Φ<90°是可行的) 。
要找到C點(diǎn)的坐標(biāo)和方向角度θ ,必須進(jìn)行以下簡單的計(jì)算。讓Qi表示Ai到Qi的矢量,ai=[xAiyAi]T表示O到Ai點(diǎn)的矢量。因此,Qi可以被很簡單的寫成:
qi≡xQiyQi=12aj+ak+32E(aj-ak) (12)
其中
E=-1001 (13)
而且(i,j,k)=(1,2,3),(2,3,1)或(3,1,2)?,F(xiàn)在拿A1Q1和A2Q2來舉例說明,它們的交點(diǎn)是C,坐標(biāo)如下:
x=xA1yQ1-xQ1yA1xA2-xQ2-xA2yQ2-xQ2yA2xA1-xQ1xA1-xQ1yA2-yQ2-xA2-xQ2yA1-yQ1 (14)
y=xA1yQ1-xQ1yA1yA2-yQ2-xA2yQ2-xQ2yA2yA1-yQ1xA1-xQ1yA2-yQ2-xA2-xQ2yA1-yQ1 (15)
移動(dòng)平臺(tái)的方向可通過測量線A_1C和基礎(chǔ)坐標(biāo)系y軸的夾角來獲得:
Φ=a tan2y-yA1,x-xA1 (16)
這個(gè)裝置的逆運(yùn)動(dòng)學(xué)也很容易解決的。讓矢量c=[x,y]T表示基礎(chǔ)坐標(biāo)系原點(diǎn)O到移動(dòng)平臺(tái)原點(diǎn)C的距離。bi和pi分別表示沿著線OiAi和CAi的單位長度。然后,它可以被很容易地表示:
ρi=biTE(oi-c)biTEpi-d (17)
其中d是的三角形O1O2O3各頂點(diǎn)和相應(yīng)初始位置之間的線性調(diào)節(jié)器(見圖3c) 。
4 。恒方向工作空間分析
不變方向(位置)工作的平面并聯(lián)機(jī)器人[ 9 ]存在一個(gè)簡單的幾何計(jì)算方法。雖然,計(jì)算恒定方向工作的三個(gè)機(jī)器人時(shí)沒有必要精確計(jì)算。這個(gè)工作空間的分析表明,這些機(jī)器人沒有簡單的關(guān)系,精確的工作空間形狀和尺寸。
為了能夠公平比較,這將假定只有限個(gè)制約工作機(jī)器人驅(qū)動(dòng)器的限制。在這種情況下,恒定方向工作的所有三個(gè)平面并聯(lián)機(jī)器人可以很容易達(dá)到如圖5所示的幾何形狀,那里的恒定方向工作空間顯示為一塊區(qū)域。
很顯然,導(dǎo)向工作的PreXYT的工作空間為任何非零方向大于恒定方向工作的HephAist機(jī)器人的工作空間。此外,導(dǎo)向工作的PreXYT機(jī)器人始終有兩到三個(gè)運(yùn)動(dòng)方向 ,而位置不變的恒定方向工作的HephAist的機(jī)器人是非常不同的。這意味著,局部地區(qū)每個(gè)點(diǎn)在某一范圍內(nèi)至少有一個(gè)方向,Φ∈-Φmax,Φmax,并且比PreXYT更大 。然而,所謂的最大的工作空間(所有能到達(dá)的位置)HephAist的并聯(lián)機(jī)器人要大得多的。
不幸的是,圖5c不能明確表明恒定方向工作的平面星三角并聯(lián)機(jī)器人的工作空間是否大于其他兩個(gè)機(jī)器人,因?yàn)榉较虿灰粯?,也沒有限制的驅(qū)動(dòng)器。事實(shí)上,在零方向,導(dǎo)向工作的PreXYT工作空間大于星三角并聯(lián)機(jī)器人,但是除了某些特定的方向角,情況就會(huì)反過來。然而,很明顯,星三角并聯(lián)機(jī)器人的最大工作空間大于PreXYT (這是一個(gè)正方形,其副作用是致i動(dòng)器的運(yùn)動(dòng))。
直觀地看,一個(gè)機(jī)器人的工作空間越大,它的定位精度就越低。但是,如果只考慮最大位置度誤差(如文中所做) ,而沒有考慮它是否在準(zhǔn)確定義的最大工作空間內(nèi),就會(huì)出現(xiàn)極端的現(xiàn)象,它將會(huì)在下一節(jié)中出現(xiàn)。
5 。誤差分析
因?yàn)橛糜诒容^的原因,將被認(rèn)為所有這三個(gè)并聯(lián)機(jī)器人使用相同的驅(qū)動(dòng)器,其行程為500毫米。設(shè)這些驅(qū)動(dòng)器的精確度是ε=0.05毫米,這意味著如果活動(dòng)鉸鏈i的形式參數(shù)是Qi,那 么它的實(shí)際參數(shù)再[ρi-ε,ρi+ε]范圍內(nèi)。設(shè)星三角并聯(lián)機(jī)器人的偏距是100毫米?;谶@些假設(shè),它現(xiàn)在將顯示,可以使用簡單幾何方法來分析最大的方向和位置誤差。
圖5 恒方向工作空間(a) PreXYT , (b) HephAist的并聯(lián)機(jī)器人和( c )星三角并聯(lián)機(jī)器人。
5.1. PreXYT
5.1.1 。最大方向度誤差
圖6a顯示,很明顯,最大方向度誤差并不取決于名義位置的移動(dòng)平臺(tái)并且最大方向度誤差發(fā)生在2號(hào)和3號(hào)驅(qū)動(dòng)器各自在ρ_2和ρ_3的位置(取決于名義方向角的符號(hào),這將是兩個(gè)組合) 。為了分析方向度誤差,應(yīng)分析一個(gè)簡單的系統(tǒng)的類似三角形。跳過詳細(xì)分析,可以表示最大方向角:
σmax=sin-12εcosΦs2+s tanΦ-2ε2 (18)
Φ是名義上的方向角。圖7顯示了PreXYT用作表示的名義方向角時(shí)的最大的方向誤差。顯然,這清楚地表明,當(dāng)Φ增加, σmax下降。
5.1.2 。最大位置度誤差
再次談到圖6a,陰影區(qū)域是活動(dòng)鉸鏈的角度在[ρi-ε,ρi+ε]范圍內(nèi)的最大工作空間。事實(shí)上,這是該地區(qū)在C點(diǎn)。其幾何計(jì)算[ 9 ]是相當(dāng)簡單的,三自由度并聯(lián)機(jī)器人的腿關(guān)節(jié)有兩個(gè)平動(dòng)關(guān)節(jié)和一個(gè)轉(zhuǎn)動(dòng)關(guān)節(jié)。最大工作空間的邊界是,當(dāng)兩個(gè)驅(qū)動(dòng)器正處于一個(gè)極限點(diǎn)時(shí),C點(diǎn)描述的曲線。請(qǐng)注意,對(duì)于一般3自由度平面并聯(lián)機(jī)器人的界限,最大的工作空間的邊界可能還包括其中一條支鏈?zhǔn)窃谄娈愇恢脮r(shí)的曲線,在奇異位置時(shí)計(jì)算很復(fù)雜。
圖。 6 。獲得的最大位置度誤差 (a) PreXYT , (b) HephAist的并聯(lián)機(jī)器人和( c )星三角并聯(lián)機(jī)器人。
圖。 7 。最大方向度誤差函數(shù)的名義方向角PreXYT 。
圖。 8 。最大位置度誤差函數(shù)的名義方向角PreXYT 。
但幸運(yùn)的是,所有三個(gè)并聯(lián)機(jī)器人的所有支鏈都沒有奇異位置(在不拆卸機(jī)器人的情況下)。
要找的最大位置度誤差,就應(yīng)該找到名義位置與最大工作空間邊界的最大距離。分析一個(gè)通用3自由度平面并聯(lián)機(jī)器人很困難,最大工作空間的邊界可能是橢圓的一個(gè)片斷或者是連續(xù)狀態(tài)。然而,三自由度并聯(lián)機(jī)器人的腿關(guān)節(jié)有兩個(gè)平動(dòng)關(guān)節(jié)和一個(gè)轉(zhuǎn)動(dòng)關(guān)節(jié)(或者是交叉形的)。而且由于這些環(huán)形交叉口的中心明顯在最大工作空間外(與驅(qū)動(dòng)器的行程相比ε非常?。畲笪恢枚日`差發(fā)生在最大工作空間的一個(gè)角落。即,當(dāng)三個(gè)驅(qū)動(dòng)器正處于一個(gè)極限位置時(shí)。因此,只有8個(gè)可能性(8個(gè)角)應(yīng)檢查移動(dòng)平臺(tái)的每個(gè)名義位置和方向。
然而對(duì)PreXYT來說,最大工作空間是一個(gè)平行四邊形,最大位置度誤差δmax較長對(duì)角線的一半。因此,下面的簡單方程表示PreXYT的最大位置度誤差:
δmax=ε1+1+tanΦ2 (19)
請(qǐng)注意, PreXYT的最大位置度誤差并不取決于機(jī)器人的尺寸(驅(qū)動(dòng)器的距離和行程s) 。這一誤差在圖8中顯示為關(guān)于Φ的函數(shù)。
5.2 。 HephAist的并聯(lián)機(jī)器人
5.2.1 。最大方向度誤差
HephAist并聯(lián)機(jī)器人的最大方向角的解析表達(dá)式顯然和PreXYT的一樣 ,即方程(18)。
5.2.2 。最大位置度誤差
HephAist并聯(lián)機(jī)器人最大工作空間的邊界是由四個(gè)圓弧和兩個(gè)線段的一部分組成的(圖6b)。可以看出,最大位置度誤差出現(xiàn)時(shí)最大方向度誤差已經(jīng)出現(xiàn)了。因此,不難看出,要找出最大位置度誤差只有兩種配置需要測試:ρ1-ε,ρ2+ε,ρ3-ε和ρ1+ε,ρ2-ε,ρ3+ε。經(jīng)過進(jìn)一步分析后確定,如果Φ是確定的,那么最大位置度誤差發(fā)生在δmax和ρ1-ε處,否則最大位置度誤差發(fā)生在δmax和ρ1+ε處。從本質(zhì)上講,對(duì)于給定移動(dòng)平臺(tái)名義位置和方向的HephAist并聯(lián)機(jī)器人只要一次計(jì)算就能找到的最大位置度誤差。
通過一個(gè)解析表達(dá)式可以很容易的找到最大位置度誤差,就不在這里推導(dǎo)了。對(duì)于一個(gè)給定的名義位置和方向,我們只需要解決活動(dòng)鉸鏈的正確組合的正運(yùn)動(dòng)學(xué)問題,并且計(jì)算出名義位置和新位置的距離。利用這一過程,就能找到最大位置度誤差與名義位置和兩個(gè)給定方向之間的關(guān)系曲線(圖9)。
圖。 9清楚地表明, HephAist的并聯(lián)機(jī)器人的移動(dòng)平臺(tái)遠(yuǎn)離它的最大高度時(shí),最大位置度誤差顯著上升。同樣明顯的是,從總體上來說, HephAist的并聯(lián)機(jī)器人比PreXYT的最大位置度誤差要大。
5.3 。星三角并聯(lián)機(jī)器人
5.3.1 。最大方向度誤差
直觀來看,最大方向度誤差發(fā)生在ρ1-ε,ρ2-ε,ρ3-ε和ρ1+ε,ρ2+ε,ρ3+ε處。進(jìn)一步檢查發(fā)現(xiàn),最大的方向度誤差發(fā)生在,Φ≥0時(shí)ρ1-ε,ρ2-ε,ρ3-ε,Φ≤0時(shí)ρ1+ε,ρ2+ε,ρ3+ε處。
數(shù)值分析表明,在星三角并聯(lián)機(jī)器人的工作空間中對(duì)任何名義的位置和方向來說,最大方向度誤差幾乎是相同的。因此,只有均值和方差是這里的最大方向度誤差,而不是表現(xiàn)出輪廓線,或是曲線:
Φ=0°時(shí),
σmaxmean=14.117°×10-3 σmaxvar=9.71°×10-27
Φ=15°時(shí)
σmaxmean=13.228°×10-3 σmaxvar=6.86°×10-27
請(qǐng)注意,最大位方向度誤差很可能是關(guān)于名義方向角Φ的函數(shù),這將解釋幾乎為零的方差。需要作進(jìn)一步分析,以驗(yàn)證這一有趣的假設(shè),這將極大地簡化了最大方向角的計(jì)算。
應(yīng)當(dāng)指出的是,星三角并聯(lián)機(jī)器人的最大方向度誤差是其他兩個(gè)平行的機(jī)器人的近兩倍。
5.3.2 。最大位置度誤差
如前所述,最大位置度誤差將出現(xiàn)在所有驅(qū)動(dòng)器正處于極端的位置(最大工作空間的轉(zhuǎn)彎處,如圖6c所顯示的)。只要有一種舍棄8種配置中幾種的方法,就可以簡單地測試所有這些配置,并且不需要擔(dān)心計(jì)算時(shí)間。的確,對(duì)于每一個(gè)配置來說,通過正運(yùn)動(dòng)學(xué)方程(14)和(15)可以達(dá)到相同的位置。然后這一位置和名義位置的距離可以用于計(jì)算,最大位置度誤差是所有8個(gè)距離中最大的距離。
星三角機(jī)器人的最大位置度誤差的等高線圖(圖10)清楚地表明,其平均最大位置度誤差與PreXYT 是類似的。然而,這不是什么在圖1中建議的巧妙方法,這說明靈巧度的不足保證了其精確度。
圖。 9 。HephAist的并聯(lián)機(jī)器人的最大位置度誤差的等高線圖(在μm級(jí)別):(a)Φ=0°和(b)Φ=15°。
圖。 10 。星三角并聯(lián)機(jī)器人最大位置度誤差(在μm級(jí)別)的等高線圖:(a)Φ=0°和(b)Φ=15°。
顯然,對(duì)一個(gè)特定的工作空間來說,一個(gè)更詳細(xì)的比較應(yīng)基于均值和方差的最大位置和最大方向角。由于這些設(shè)備是最有可能像圓片一樣對(duì)齊,這樣的工作空間可能是一個(gè)直徑300毫米的圓形區(qū)域,在這圓形區(qū)域中±5度角內(nèi)的任何方向角都是可能的。但是,這并不是主題,本文并不會(huì)詳細(xì)敘述。
6 。結(jié)論
本文是失敗古典精度指標(biāo)的直接響應(yīng),如局部靈巧指標(biāo),在處理并聯(lián)機(jī)器人的平移和取向的自由度,在優(yōu)化設(shè)計(jì)和比較。而不是使用非物理概念,如靈活性,作者提出了簡單的幾何辦法來精確的確定在某一特定的名義位置和方向時(shí)的由于驅(qū)動(dòng)器誤差而引起的最大位置度誤差和最大的方向度誤差。顯然,這種方法還可以提供最大線速度和最大角速度。
這種方法只適用于簡單的三自由度平面完全并聯(lián)機(jī)器人,并且沒有奇異位置。不過,這種機(jī)器人肯定是最佳的微定位和調(diào)整儀器。作者說明擬議幾何方法中引用了三個(gè)特殊的例子,其中有一個(gè)是商用的。簡單的比較揭示了大多數(shù)商用設(shè)計(jì)上的缺點(diǎn)。
鳴謝
作者要感謝Natural Sciences and Engineering Research Council of Canada (NSERC)和Fonds Que′becO_is de la recherche sur la nature et les technologies (FQRNT)的經(jīng)濟(jì)支持。
翻譯人:蔣林
翻譯文獻(xiàn): Geometric approach to the accuracy analysis of a class of 3-DOF planar parallel robots。Available online at www.sciencedirect.com。Mechanism and Machine Theory 43 (2008) 364–375。
Received 12 June 2006; received in revised form 11 December 2006; accepted 16 March 2007 Keywords: Parallel robots; Dexterity map; Workspace; Positioning; Accuracy; Error analysis has become a key issue in many applications. * Corresponding author. E-mail address: ilian.bonev@etsmtl.ca (I.A. Bonev). Available online at Mechanism and Machine Theory 43 (2008) 364–375 Mechanism and Machine Theory 0094-114X/$ - see front matter C211 2007 Elsevier Ltd. All rights reserved. 1. Introduction Parallel robots which were once constructed solely in academic laboratories have increasingly been used in industry for positioning and alignment in recent years. With such demand in the market today for these fast and agile machines, new parallel robots are being designed and manufactured. However, there are still many key issues regarding the design of new parallel robots, such as optimal design and performance indices. How could one prove that a new parallel robot design is an improvement over existing designs? Is it enough to eval- uate a robot based on its workspace? Clearly in the current industrial climate it is not, as positioning accuracy Available online 3 May 2007 Abstract Parallel robots are increasingly being used in industry for precise positioning and alignment. They have the advantage of being rigid, quick, and accurate. With their increasing use comes a need to develop a methodology to compare di?erent parallel robot designs. However no simple method exists to adequately compare the accuracy of parallel robots. Certain indices have been used in the past such as dexterity, manipulability and global conditioning index, but none of them works perfectly when a robot has translational and rotational degrees of freedom. In a direct response to these problems, this paper presents a simple geometric approach to computing the exact local maximum position error and local maximum orientation error, given actuator inaccuracies. This approach works for a class of three-degree-of-freedom planar fully-par- allel robots whose maximal workspace is bounded by circular arcs and line segments and is free of singularities. The approach is illustrated on three particular designs. C211 2007 Elsevier Ltd. All rights reserved. Geometric approach to the accuracy analysis of a class of 3-DOF planar parallel robots Alexander Yu a , Ilian A. Bonev b, * , Paul Zsombor-Murray a a Department of Mechanical Engineering, McGill University, 817 Sherbrooke St.W., Montreal, Canada H3A 2K6 b Department of Automated Manufacturing Engineering, E ′ cole de technologie supe′rieure (E ′ TS), 1100 Notre-Dame St.W., Montreal, Quebec, Canada H3C 1K3 doi:10.1016/j.mechmachtheory.2007.03.002 Several well defined performance indices have been developed extensively and applied to serial and parallel robots. However, a recent study [1] reviewed these indices and discussed their severe inconsistencies when applied to parallel robots with translational and rotational degrees of freedom. The study reviewed the most common indices to optimize parallel robots: the dexterity index [2], the various condition numbers applied to it to increase its accuracy such as the two-norm or the Frobenius number, and the global conditioning index that is computed over the complete workspace of the robot [3]. The conclusion of the paper is that these indices should not be used for parallel robots with mixed types of degrees of freedom (translations and rotations). When the authors designed a new three-degree-of-freedom (3-DOF) planar parallel robot, they compared it to a similar design using the dexterity index [4]. This comparison was somewhat fair, because the designs allow the use of identical dimensions. However should one change the magnitude of the units (e.g., from cm to mm), the numbers within the index would change dramatically. The higher value for the length units would essen- tially make the dexterity of a robot closer to 0 as shown in Fig. 1. The dexterity indices are very sensitive to both the type of units used and their magnitudes because of the dependence on the Jacobian matrix. This matrix also mixes non-invariant functions such as translational and rotational capabilities. A possible solution to this problem is the addition of condition numbers, however with each condition number there are particular problems as described in [1]. The global conditioning index (GCI) can be used to evaluate a robot over its workspace, which can be used for the optimal design of a robot. However, there remain two problems with this index. Firstly, it is still depen- dant on a condition number whose problems were outlined in [1]. Secondly, it is computationally-intensive. Obviously, the best accuracy measure for industrial parallel robots would be the local maximum position error and local maximum orientation error, given actuator inaccuracies (input errors), or some generalization of this (e.g., mean value and variance of the errors over a specific workspace). A general method that can be used for calculating these errors based on interval analysis was proposed recently in [5]. However, this method is computationally-intensive and gives no kinematic insight into the problem of optimal design. 0. 0038 0. 00380. 0. 004 0. 0042 0. 0042 0. 0044 0 4 4 0. 0046 4 6 0. 0048 0. 0048 0. 0049 0. 0049 1 500 0 . 0 4 1 0 . 0. 00 49 3 500 ab A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 365 0. 0038 0. 0038 0. 0038 0. 0038 004 0. 004 0. 004 0. 004 0 . 0 0 4 0. 0042 0 . 0 0 4 2 0. 0042 0. 0042 0. 0044 0. 0044 0 . 0 0. 0044 0. 0044 0. 0046 0. 0046 0 .0 0 0. 0046 0. 0046 0. 0048 0. 0048 0. 0048 0. 0049 0. 0049 0 . 0 0 4 9 0 . 0 0 4 9 0. 004941 0. 004941 0 . 0 0 4 9 4 0. 004941 x (mm) y( m m ) 0 100 200 300 400 500 600 700 0 100 200 300 400 0 .00 2 5 0 .0 0 2 50 .0 0 27 0 0 027 0 .0 0 2 7 0 . 0 0 2 9 0. 00 29 0. 00 29 0 . 0 0 2 9 0 . 0 0 3 1 0. 00 31 0. 00 31 0 . 0 0 3 1 0 . 0 0 3 3 0. 00 33 0. 00 33 0 . 0 0 3 3 0 . 0 0 3 5 0 .0 0 3 5 0.0 03 5 0 . 0 0 3 5 0. 003 7 0 . 0 0 3 7 0.0037 0 . 0 0 3 7 0 . 00 37 0 3 9 0 . 0 0 3 9 0.0039 0 . 0 0 3 9 0 . 0 0 4 1 0. 00 41 0 .0 0 4 1 0 . 0 0 0 .0 0 4 3 0. 00 43 0 . 0 0 4 3 0 0 4 5 0.0045 0 . 0 0 4 5 0 .0 0 4 7 0 .0 0 4 7 0. 00 49 x (mm) y( m m ) 0 100 200 300 400 500 600 700 0 100 200 300 400 0 . 0 0 5 0.0 05 0 . 0 0 5 0. 0 05 5 0. 00 55 0 . 0 0 5 5 0.0 05 5 0 . 0 0 5 5 0 . 0 0 6 0 .0 0 6 0 .0 0 6 0 .00 6 0 .0 0 6 5 0 . 0 0 6 5 0. 0065 0 .0 0 7 0 . 0 0 7 0 .0 0 7 2 50 . 0 0 7 4 x (mm) y( m m ) 0 100 200 300 400 500 600 700 0 100 200 300 400 500 c Fig. 1. Example of dexterity maps for (a) PreXYT, (b) Hephaist’s parallel robot and (c) the Star-Triangle parallel robot. 366 A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 In contrast, this paper presents a simple geometric approach for computing the exact local maximum posi- tion error and local maximum orientation error for a class of 3-DOF planar fully-parallel robots, whose max- imal workspace is bounded by circular arcs and line segments and is free of singularities. The proposed approach is not only faster than any other method (for the particular class of parallel robots) but also brings valuable kinematic insight. The approach is illustrated on three particular designs that are arguable among the best candidates for micro-positioning over a relatively large workspace: 1. A new parallel robot, named PreXYT, designed and constructed at E ′ cole de technologie supe′rieure (E ′ TS) that has a unique 2-PRP/1-PPR configuration (P and R stand for passive prismatic and revolute joints, respectively, while P stands for an actuated prismatic joint). A CAD model of PreXYT is shown in Fig. 2a. 2. Hephaist’s 3-PRP parallel robot, designed by the Japanese company Hephaist Seiko and currently in com- mercial use. A photo of the industrial model is shown in Fig. 2b. 3. Star-Triangle parallel robot, another 3-PRP parallel robot designed at LIRMM in France [6]. This robot is a more optimal design of the double-triangular parallel manipulator [7]. The remainder of this paper is organized as follows. The next section presents the proposed geometric approach. Then, Section 3 presents the inverse and direct kinematic equations for all three parallel robots whose accuracy will be studied in this paper. Section 4 briefly describes the geometry of the constant-orienta- tion workspace for each of the three robots. Finally, Section 5 applies the proposed geometric method for computing the local maximum position and orientation errors. Conclusions are given in the last section. Fig. 2. (a) PreXYT (patent pending) and (b) Hephaist’s NAF3 alignment stage (courtesy of Hephaist Seiko Co., Ltd). 2. Geometric method for computing output errors Consider a 3-DOF fully-parallel planar robot at a desired (nominal) configuration. Let x, y, and / denote the nominal position and nominal orientation of the mobile platform and q 1 , q 2 , and q 3 denote the nominal active-joint variables. Due to actuator inaccuracies of up to ±e, the actual active-joint variables are somewhere in the ranges ?q i C0 e; q i t eC138 (i ? 1; 2;3). Therefore, the actual position and orientation of the mobile platform are x tDx, y tDy, and / tD/, respectively. The question is, given the nominal configuration of the robot ex; y; /; q 1 ; q 2 ; q 3 T and the actuator inaccuracy ±e, how much is the maximum position error, i.e., d max ? maxe ????????????????????? Dx 2 tDy 2 p T, and the maximum orientation error, i.e., r max ? maxejD/jT. In order to compute these errors, we basically need to find the values of the active-joint variables for which these errors occur. The greatest mistake would be to assume that whatever the robot and its nominal config- uration, the maximum position error occurs when each of the active-joint variables is subjected to a maximum input error, i.e., +e or C0e. Indeed, in [8], it was proven algebraically that the maximum orientation angle of a 3-DOF planar parallel robot may occur at a Type 1 (serial) or a Type 2 (parallel) singularity, or when two leg wrenches are parallel, for active-joint variables that are inside the input error intervals. Similarly, it was pro- ven that not all active-joint variables need to be at the limits of their input error intervals for a maximum posi- tion error. Naturally, though there are exceptions, 3-DOF planar parallel robots that are used for precision position- ing operate far from Type 1 singularities and certainly far from Type 2 singularities (if such even exist). Fur- thermore, it is simple to determine whether configurations for which two leg wrenches are parallel correspond to a local maximum of the orientation error and to design the robot in such a way that no such configurations exist. Therefore, for such practical 3-DOF planar parallel robot, the local maximum orientation error occurs at one of the eight combinations of active-joint variables with +e or C0e input errors. Now, finding this local maximum position error is equivalent to finding the point from the uncertainty zone of the platform center that is farthest from the nominal position of the mobile platform. This uncertainty zone is basically the maximal workspace of the robot obtained by sweeping the active-joint variables in the set of intervals ?q i C0 e; q i t eC138. Obviously, the point that we are looking for will be on the boundary of the maximal workspace. A geometric algorithm for computing this boundary is presented in [9], but we will not discuss it here in detail. We need only mention that this boundary is composed of segments of curves that correspond to con- A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 367 figurations in which at least one leg is at a Type 1 singularity (which we exclude from our study) or at an active-joint limit. When these curves are line segments or circular arcs, it will be very simple to find the point that is farthest from the nominal position of the mobile platform. This point will be generally an intersection point on the boundary of the maximal workspace. In what follows, three examples will be studied in order to illustrate the proposed geometric approach. 3. Inverse and direct kinematic analysis Referring to Fig. 3a–c, a base reference frame Oxy is fixed to the ground and defines a plane of motion for each planar parallel robot. Similarly, a mobile reference frame Cx 0 y 0 is fixed to the mobile platform and in the same plane as Oxy. Let A i be a point on the axis of the revolute joint of leg i (in this paper, i ? 1; 2; 3) and in the plane of Oxy. Referring to Fig. 3a and b, the base y-axis is chosen along the path of motion of point A 2 , while the mobile x 0 -axis is chosen along the line A 2 A 3 .InFig. 3a, the origin C coincides with point A 1 , while in Fig. 3b, the origin C is placed so that point A 1 moves along the y 0 -axis. For both robots, s is the distance between the par- allel paths of points A 2 and A 3 , while in the Hephaist’s alignment stage, h is the distance between the base x- axis and the path of point A 1 . Referring to Fig. 3c, let points O i be located at the vertices of an equilateral triangle fixed in the base. Let the origin O of the base frame coincide with O 1 , and let the base x-axis be along the line O 1 O 2 . Let also the origin C be at the intersection of the three concurrent lines in the mobile platform, along which points A i move. These three lines make up equal angles. Finally, the mobile y 0 -axis is chosen along the line A 1 C. x y y ’ x ’ s O A 2 A 3 C A 1 C y ’ x ’ x s y A 2 A 1 A 3 O 1 ρ 2 ρ 3 ρ 2 ρ 3 ρ 1 ρ h C y’ x’ x O O 1 y O 2 O 3 A 1 A 2 A 3 2 ρ 3 ρ 1 ρ Fig. 3. Schematics of (a) PreXYT (patent pending), (b) Hephaist’s parallel robot and (c) the Star-Triangle parallel robot. Give mobile As one Give platform 3 2 The the active gularities. Note, that this is quite an advantage over most planar parallel robots, which have singularities. 368 A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 3.3. Star-Triangle parallel robot Given the active-joint variables, we are able to uniquely define the position and orientation of the mobile platform through this direct kinematic method used in [10]. Referring to Fig. 4, the position of C can be easily q 1 ? x C0eh C0 yTtan/; e9T q 2 ? y C0 xtan/; e10T q 3 ? y tes C0 xTtan/: e11T Since Eqs. ((7)–(11)) are always defined (assuming s50), it is evident that this parallel robot too has no sin- obtaine inverse kinematics are easier to solve for. Given the position and orientation of the mobile platform, -joint variables are obtained as As one can observe, the direct kinematics of Hephaist’s parallel robot are more complex and highly coupled. s 2 teq 3 C0 q 2 T 2 y ? s 2 q 2 t heq 3 C0 q 2 T 2 t sq 1 eq 3 C0 q 2 T s 2 teq C0 q T 2 : e8T is the intersection between line A 2 A 3 and the line passing through point A 1 and normal to A 2 A 3 . The resulting equations for x and y are therefore x ? seq 1 s teh C0 q 2 Teq 3 C0 q 2 TT ; e7T ’s parallel robot n the active-joint variables, it is simple to uniquely define the position and orientation of the mobile . The equation of the orientation angle is the same as Eq. (1). The position of the mobile platform The inverse kinematic analysis is also trivial. Given the position and orientation of the mobile platform, the active-joint variables are obtained as q 1 ? x; e4T q 2 ? y C0 xtan/; e5T q 3 ? y tes C0 xTtan/: e6T Obviously, PreXYT has no singularities (provided that s is non-zero). 3.2. Hephaist x ? q 1 ; e2T y ? q 2 t q 1 q 3 C0 q 2 s C16C17 : e3T can observe, the direct kinematics of PreXYT are very simple and partially decoupled. / ? tan C01 q 3 C0 q 2 s ; e1T while the position of the mobile platform is given by n the active-joint variables, it is straightforward to uniquely define the position and orientation of the platform. The orientation angle is easily obtained as C16C17 Let q i be the active-joint variables representing directed distances, defined as follows. For PreXYT and Hephaist’s alignment stage (Fig. 3a and b), q 1 is the directed distance from the base y-axis to point A 1 , while q 2 and q 3 are the directed distances from the base x-axis to points A 2 and A 3 , respectively. Finally, for the Star- Triangle robot (Fig. 3c), q i is the directed distance from O i to A i minus a constant positive o?set d. Indeed, in the Start-Triangle robot, no mechanical design would allow point A i to reach point O i . 3.1. PreXYT d through the following geometric construction based on the notion of the first Fermat point. Since sides are denoted which To perfor "# ???p A. Yu et al. / Mechanism and Machine Theory 43 (2008) 364–375 369 where E ? C010 01 C20C21 ; e13T and ei; j; kT?e1; 2; 3T or (2, 3, 1) or (3, 1, 2). Now, taking lines A 1 Q 1 and A 2 Q 2 for example, their intersection point is C and has the following coordinates: x ? ex A 1 y Q 1 C0 x Q 1 y A 1 Tex A 2 C0 x Q 2 TC0ex A 2 y Q 2 C0 x Q 2 y A 2 Tex A 1 C0 x Q 1 T ex A 1 C0 x Q 1 Tey A 2 C0 y Q 2 TC0ex A 2 C0 x Q 2 Tey A 1 C0 y Q 1 T ; e14T ex A 1 y Q 1 C0 x Q 1 y A 1 Tey A 2 C0 y Q 2 TC0ex A 2 y Q 2 C0 x Q 2 y A 2 Tey A 1 C0 y Q 1 T The orient axis The inverse origin Then, where ear a q i C17 x Q i y Q i ? 1 2 ea j t a k Tt 3 2 Eea j C0 a k T; e12T point O to point A i . Therefore, it can be easily shown that vector q i can be written as C090C176 < / <90C176). find the coordinates of point C and the orientation angle h, the following simple calculations need to be med. Let q i denote the vector connecting point A i to point Q i , and a i ??x A i ; y A i C138 T is the vector connecting Fermat point. This point is the mobile frame’s origin C. While there is only solution for the position of the mobile platform, there are two possibilities for the ori- entation angle (/ and / + 180C176). Obviously, however, only one of these two solutions is feasible (the one for in triangle A 1 A 2 A 3 , none of the angles is greater that 120C176 (because points A i cannot move outside the of triangle O 1 O 2 O 3 ), equilateral triangles are drawn outside of it. The outmost vertices of these triangles as Q i (see Fig. 4). Then lines Q i A i make 120C176 angles and intersect at one point, the so-called first 1 O 2 A 1 Q 2 Fig. 4. Solving the direct kinematics of the Star-Triangle parallel robot. x O O 3 C x' y A 3 Q y' A 2 O 3 Q 1 y ? ex A 1 C0 x Q 1 Tey A 2 C0 y Q 2 TC0ex A 2 C0 x Q 2 Tey A 1 C0 y Q 1 T : e15T ation of the mobile platform can be found by measuring the angle between line A 1 C and the base y- / ? a tan 2ey C0 y A 1 ; x C0 x A 1 T: e16T kinematics of this device is also easily solved for. Let c ??x; yC138 T be the vector connecting the base O to the mobile frame origin C, b i be the unit length along O i A i and p i be the unit length along CA i . it can be easily shown that q i ? b T i Eeo i C0 cT b T i Ep i C0 d; e17T d is the o?set between the vertices of triangle O 1 O 2 O 3 and the initial positions of the corresponding lin- ctuators (see Fig. 3c). 4. Constant-orientation workspace analysis There exists a simple geometric method for computing the constant-orientation (position) workspace of planar parallel robots [9]. Although, calculating the constant-orientation workspace for the three robots is not necessary for computing the accuracy of these robots, this workspace analysis shows that there is no sim- ple relationship between accuracy and workspace shape and dimensions. To allow for fair comparison, it will be assumed that the only limits restraining the workspace of the robots are the actuator limits. Under these conditions, the constant-orientation workspaces for all three planar par- allel robots can be easily obtained geometrically, as shown in Fig. 5, where the constant-orientation work- spaces for a given orientation are shown as the hatched regions. From here, it is obvious that the constant-orientation workspace of PreXYT is greater than the constant- orientation workspace of Hephaist’s robot, for any non-zero orientation. Furthermore, the constant- orientatio
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