機(jī)械外文文獻(xiàn)翻譯-運(yùn)動(dòng)學(xué)分析和優(yōu)化設(shè)計(jì)3-PPR平面平行【中文3090字】【PDF+中文WORD】,中文3090字,PDF+中文WORD,機(jī)械,外文,文獻(xiàn),翻譯,運(yùn)動(dòng)學(xué),分析,優(yōu)化,設(shè)計(jì),PPR,平面,平行,中文,3090,PDF,WORD 【中文3090字】 運(yùn)動(dòng)學(xué)分析和優(yōu)化設(shè)計(jì)3-PPR平面平行機(jī)械手 紀(jì)邦崔* 機(jī)器人與控制組，智能與精密機(jī)械部，韓國機(jī)械與材料研究所，Jang Dong，Yuseong Gu，Daejeon，305-343，韓國 摘要：本文提出了一種3-PPR平面并聯(lián)機(jī)械手，其中包括三個(gè)活性柱狀節(jié)理、三個(gè)被動(dòng)的柱狀節(jié)理、三個(gè)被動(dòng)轉(zhuǎn)動(dòng)關(guān)節(jié)。分析了運(yùn)動(dòng)學(xué)和優(yōu)化設(shè)計(jì)的機(jī)械手，對(duì)其進(jìn)行了討論，提出機(jī)械手具有直接運(yùn)動(dòng)學(xué)的封閉式和無空隙隨著邊界的凸式空間。分析機(jī)械手的運(yùn)動(dòng)學(xué)和逆運(yùn)動(dòng)學(xué)，和逆雅可比矩陣推導(dǎo)機(jī)械手。改變旋轉(zhuǎn)限制和機(jī)械手的工作空間研究，對(duì)機(jī)械手的工作空間進(jìn)行了仿真。此外，為優(yōu)化設(shè)計(jì)的機(jī)械手，機(jī)械手的性能指標(biāo)進(jìn)行了研究，然后優(yōu)化設(shè)計(jì)方法是使用最小最大理論。最后，用一個(gè)例子進(jìn)行了優(yōu)化設(shè)計(jì)。 關(guān)鍵詞：平面并聯(lián)機(jī)械手，運(yùn)動(dòng)學(xué)，雅可比矩陣，優(yōu)化設(shè)計(jì)，最大最小 1 引言 并聯(lián)機(jī)器人組成的閉環(huán)有許多優(yōu)勢，比串聯(lián)機(jī)器人有更高的精度和剛度。眾所周知，并行比串聯(lián)機(jī)器人有更高的有效載荷重量比、更高的精度和較高的結(jié)構(gòu)剛度。最近一些機(jī)床已開發(fā)利用這些優(yōu)點(diǎn)，機(jī)械手的精細(xì)運(yùn)動(dòng)的全鋼載重子午線也采用并行機(jī)制，從平價(jià)等位基因機(jī)制制造單片機(jī)。并聯(lián)機(jī)器人中，平面并聯(lián)機(jī)器人是平面機(jī)械手運(yùn)動(dòng)。平面并聯(lián)機(jī)器人有兩個(gè)自由度（DOF）的運(yùn)動(dòng)；這是兩個(gè)自由度的運(yùn)動(dòng)和一個(gè)旋轉(zhuǎn)的運(yùn)動(dòng)。這眾所周知，平面三自由度并聯(lián)機(jī)器人的存在，RRP，RPR，RPP，PRR，PRP，和PPR，取決于棱柱形接頭和旋轉(zhuǎn)的組合接頭，不包括PPP的組合，其中棱柱和旋轉(zhuǎn)接頭由P和R。解決方案的直接運(yùn)動(dòng)學(xué)系統(tǒng)架構(gòu)的平面并聯(lián)機(jī)器人進(jìn)行了已經(jīng)提出了，但更多的控制—克里特島的解決方案和運(yùn)動(dòng)學(xué)分析架構(gòu)要求。大多數(shù)的3-DOF平面并聯(lián)機(jī)器人有手有缺點(diǎn)復(fù)雜的直接運(yùn)動(dòng)學(xué)多項(xiàng)式類型和無用的空隙小工作區(qū)以及凹型邊界。直接運(yùn)動(dòng)學(xué)多項(xiàng)式的增加，求解方程以及選擇合適的溶液變成一個(gè)巨大的負(fù)擔(dān)。此外，凹型邊界誘導(dǎo)非直從鄰居的運(yùn)動(dòng)邊界其他人。因此，一個(gè)并行的是很重要的機(jī)械手具有封閉式直接運(yùn)動(dòng)學(xué)和一個(gè)凸型空隙泰伊工作區(qū)邊緣。在本文中，一種3-PPR平面平行機(jī)器人，其中P是一個(gè)活躍的棱柱關(guān)節(jié)，提出了克服上述的缺點(diǎn)，即該機(jī)器人有一個(gè)封閉式直接運(yùn)動(dòng)學(xué)和無空隙隨著邊界的凸式空間。該機(jī)械手的運(yùn)動(dòng)學(xué)分析，首先直接運(yùn)動(dòng)學(xué)，逆運(yùn)動(dòng)學(xué)，該機(jī)器人逆雅可比矩陣派生的。第二，旋轉(zhuǎn)限制和工作區(qū)進(jìn)行調(diào)查。同時(shí)為優(yōu)化設(shè)計(jì)該機(jī)械手的操作，性能指標(biāo)機(jī)器人進(jìn)行了研究并優(yōu)化設(shè)計(jì)方法是使用最小最大值進(jìn)行理論。最后，一個(gè)例子使用最佳的的設(shè)計(jì)方法。 2 對(duì)3-PPR平面描述并聯(lián)機(jī)器人 圖1顯示了由三個(gè)有源棱鏡接頭，三個(gè)被動(dòng)棱柱接頭，三個(gè)被動(dòng)旋轉(zhuǎn)接頭，移動(dòng)板和連桿構(gòu)成的3-I> _PR平面平行機(jī)械手的示意圖。主動(dòng)接頭可通過電動(dòng)旋轉(zhuǎn)電機(jī)和滾珠絲杠進(jìn)行運(yùn)動(dòng)轉(zhuǎn)換。活動(dòng)接頭運(yùn)動(dòng)的三個(gè)連桿固定在每個(gè)連桿兩端的基架上。平面制動(dòng)器m的自由度（DOF）由（Merlet，2000）表示， 其中l(wèi)是剛體的數(shù)量，n是關(guān)節(jié)的數(shù)量，d1是關(guān)節(jié)i的DOF。由于這個(gè)操縱器有八個(gè)剛體 圖。 1 3 _I> _PR平面噴嘴的示意圖 包括基地，九個(gè)關(guān)節(jié)共九個(gè)自由度，自由度三個(gè);即平面上的兩個(gè)平移和一個(gè)旋轉(zhuǎn)。此外，當(dāng)操縱器的主動(dòng)關(guān)節(jié)被鎖定時(shí)，操縱器的DOF變?yōu)榱?，因?yàn)榫艂€(gè)關(guān)節(jié)只有六個(gè)DOF。在這個(gè)人之前，當(dāng)活動(dòng)接頭被激活時(shí)，我的碎漿機(jī)有三個(gè)自由度，而當(dāng)活動(dòng)接頭被鎖定時(shí)，它是一個(gè)靜態(tài)結(jié)構(gòu)。 3 直接運(yùn)動(dòng)學(xué) 該手動(dòng)分配器的坐標(biāo)和幾何參數(shù)如圖1所示。 移動(dòng)板是一個(gè)圓，它包含一個(gè)半徑為r的等邊三角形。 旋轉(zhuǎn)關(guān)節(jié)的中心位于三角形的頂點(diǎn)。 活動(dòng)的棱柱形關(guān)節(jié)可以在包含一個(gè)半徑為R的圓的外等邊三角形的邊上行進(jìn) 主動(dòng)關(guān)節(jié)A1為（爪，y; ），其中i = 1,2和3， 移動(dòng)板具有平移的姿態(tài)，y）和從參考點(diǎn)0的旋轉(zhuǎn)rp。然后，無源鏈路的每個(gè)長度變?yōu)長，。 因?yàn)閮?nèi)外三角是等邊的，所以三角形的角度是 圖 2直接運(yùn)動(dòng)學(xué)坐標(biāo)系 內(nèi)三角形的邊長為e 內(nèi)三角的傾斜度為9,通過移動(dòng)板的旋轉(zhuǎn)，￠， 活動(dòng)接頭的相對(duì)位移為 從等式 （5）和（6）中，長度La和L2為 另外，從等式 （7），（9）和（10），長度L3是 取代方程式 （9） - （11） （8）派生以下等式： 可以求解方程（12）引入?yún)?shù)T如下： 代入方程 （14） （12）中，得到T中的二階多項(xiàng)式 等式（15）提供了閉式形式解? 從等式 （4）和（14），運(yùn)動(dòng)的旋轉(zhuǎn)板是 因此，移動(dòng)板的平移是 可以理解的是，根據(jù)方程式，該操縱器具有至多兩個(gè)用于直接運(yùn)動(dòng)學(xué)的解決方案。 （16），而且等式的閉合形式解 （17）和（18） 4 反向運(yùn)動(dòng)學(xué)和逆雅可比 圖3顯示了一種用于描述3-P_P PR平行手動(dòng)分配器的反向ki nematics的坐標(biāo)系。 當(dāng)移動(dòng)板的中心從原點(diǎn)O移動(dòng)到平移（x，y）和旋轉(zhuǎn)￠的0'時(shí)，板B的頂點(diǎn); 表示為 并且i = I，2，3。有源棱鏡關(guān)節(jié)的起點(diǎn)0; 由原點(diǎn)0離開。只要U; 和v 是軸線辛i和t i的單位矢量，它們是活動(dòng)棱柱形接頭的軸線，頂點(diǎn)B的坐標(biāo)，由占有項(xiàng)表示; t i是 因此，活動(dòng)棱柱關(guān)節(jié)的位置是 圖3逆運(yùn)動(dòng)學(xué)坐標(biāo)系 方程的直接分化 （22）相對(duì)于姿勢（x，y，￠>）得出如下的“Jacobian J-1”： 方程的逆雅可比元素 （23）沒有sam巳維度。 對(duì)應(yīng)于平移的前兩列是無量綱的，而對(duì)應(yīng)于旋轉(zhuǎn)的最后一列具有長度的尺寸。 通過使第三列無量綱，通過（Byun，1997）獲得具有無量綱元素的均勻反Jacobian. 5.旋轉(zhuǎn)極限和工作空間 移動(dòng)板的移動(dòng)受到連桿和旋轉(zhuǎn)接頭之間的干擾的限制。 圖。 圖4示出了具有順時(shí)針（a）和逆時(shí)針（b）的旋轉(zhuǎn)極限的該操縱器的構(gòu)造。 Assumi噸鏈接的寬度可以忽略不計(jì)，旋轉(zhuǎn)￠>被限制 假如ef> o是定義的初始旋轉(zhuǎn) 旋轉(zhuǎn)范圍（25）被修改為 得出結(jié)論，3- P_PR平面平行機(jī)械手的移動(dòng)板從初始旋轉(zhuǎn)的界限為±5Jr / 6。 圖4旋轉(zhuǎn)極限 假設(shè)每個(gè)活動(dòng)關(guān)節(jié)可以沿著等邊三角形的側(cè)面移動(dòng)，從三角形的中心到活動(dòng)關(guān)節(jié)的垂直距離R。 如圖2所示，活動(dòng)關(guān)節(jié)的移動(dòng)范圍為2T R然后，移動(dòng)板的中心位置可以在旋轉(zhuǎn)范圍內(nèi)只有一些旋轉(zhuǎn)中達(dá)到位置空間（271或可達(dá)到 旋轉(zhuǎn)“ange”（27）中的完全可旋轉(zhuǎn)的位置空間，前者被稱為可到達(dá)的工作空間，后者是一個(gè)靈巧的工作空間，圖5顯示了當(dāng)r / R為0.3'0時(shí)，可達(dá)到的工作空間和靈活的工作空間 ，5，0.7和αisJf / 2，靈巧的工作空間顯示了一組可達(dá)到的工作空間，隨著r / R的增加，可達(dá)到的空間稍微增加，但是靈巧的工作空間也減少了，同時(shí)也顯示了工作空間 不含任何空隙，而且它們具有凸型的邊界。 6.本地業(yè)績指標(biāo)使用 逆雅可比 逆Jacobian提供了關(guān)于并聯(lián)機(jī)械手的運(yùn)動(dòng)學(xué)結(jié)構(gòu)的質(zhì)量的信息。 在這個(gè)pap町，操縱性，電阻率。 并且使用逆雅可比的各向同性被認(rèn)為是該并行機(jī)械手的性能指標(biāo)操作性。 這是與奇異性的距離（中村，1991）。 評(píng)估運(yùn)動(dòng)質(zhì)量，如操縱器的關(guān)節(jié)速度（Yoshikawa'1990）。 也就是說，操縱器的配置越遠(yuǎn)離奇點(diǎn)，操縱器移動(dòng)越快操縱性越小，表示在機(jī)械手的配置附近有奇異點(diǎn)。 因此，它具有最大的可操縱性。 平行機(jī)械手的操縱性由Wm定義 在非冗余的情況下。 可操縱性減少到 均勻逆雅可比的行列式是 從等式 （31）中，均勻逆雅可比的行列式被顯示為與移動(dòng)板的位置無關(guān)。 但僅依賴于φ和α。 此外。 決定因素在￠=α處變?yōu)榱?，即S s =±Jr / 2。 3 E_PR平面并聯(lián)機(jī)械手在薩=士π/ 2處具有單一配置。電阻率Wr評(píng)估關(guān)節(jié)力，并由可操縱性的倒數(shù)定義如下 電阻率對(duì)應(yīng)于力傳遞比，其是單位操作負(fù)載的致動(dòng)器容量（Lee at al“2001”）。 從一個(gè)機(jī)制的堅(jiān)固性的角度來說，“大”的主觀性是首選。 然而，在單一配置附近，電阻率突然增加，并且該機(jī)制具有重要的意義，在這種配置下不會(huì)發(fā)生故障。 因此，機(jī)械的設(shè)計(jì)必須考慮到可操縱性和電阻率之間的折中。 對(duì)應(yīng)于條件數(shù)的倒數(shù)的各向同性由最小奇異值與逆雅可比（Nakamura'199川）的最大值之比定義，定義如下（Liu at al。，2000） 當(dāng)n是逆雅可比的維數(shù)時(shí)。 可操縱性與可操縱性橢圓體的大小有關(guān)，而條件數(shù)則涉及橢圓體的形狀（Nakamura，1991）。各向同性評(píng)估工作空間質(zhì)量，并喜歡像形狀即團(tuán)結(jié)的球體。 圖6顯示了在r / R = 0.5時(shí)相對(duì)于α和薩的局部性能指標(biāo)（可操縱性，電阻率和各向同性）的模擬結(jié)果。 在這項(xiàng)研究中，i / 1由方程 （27）和α受限制 528 KSME International Journal,Vol.17 No.4,pp.528537,2003 Kinematic Analysis and Optimal Design of 3-PPR Planar Parallel Manipulator Kee-Bong Choi*Robot&Control Group,Intelligence&Precision Machine Dept.,Korea Institute of Machinery and Materials 171,Jang-Dong,Yuseong-Gu,Daejeon,305-343,Korea This paper proposes a 3-PPR planar parallel manipulator,which consists of three active prismatic joints,three passive prismatic joints,and three passive rotational joints.The analysis of the kinematics and the optimal design of the manipulator are also discussed.The proposed manipulator has the advantages of the closed type of direct kinematics and a void-free workspace with a convex type of borderline.For the kinematic analysis of the proposed manipulator,the direct kinematics,the inverse kinematics,and the inverse Jacobian of the manipulator are derived.Alter the rotational limits and the workspaces of the manipulator are investigated,the workspace of the manipulator is simulated.In addition,for the optimal design of the manipulator,the performance indices of the manipulator are investigated,and then an optimal design procedure is carried out using Min-Max theory.Finally,one example using the optimal design is presented.Key Words:Planar Parallel Manipulator,Kinematics,Jacobian,Workspace,Optimal Design,Min Max I.Introduction Parallel manipulators consisting of closed-loop mechanisms have many advantages compared to serial manipulators in terms of payload,accuracy,and stiffness.It is well known that parallel mani-pulators have a higher payload to-weight ratio,higher accuracy,and higher structural rigidity than serial manipulators(Ben-Horin et al.,1998).Recently some machine-tools(Kim el al.,2001:Wang et al.,2001)have been developed utilizing these advantages.A manipulator tbr fine motion(Ryu et al.,1997)also adopted the parallel mec-hanism rather than the serial one,since the par-allel mechanism can be manufactured monolithic-ally.*E-mail:kbchoi kimmre.kr TEL:+82 42 868 7132:FAX:+82-42 868-7135 Robot&Control Group,Intelligence&Precision Ma-chine Dept.,Korea Institute of Machinery and Materials 171.Jang Dong,Yuseong-Gu,Daejeon.305 343.Ko-rea.(Manuscript Received May 22,2002;Revised De-cember 13,2002)Copyright(C)2003 NuriMedia Co.,Ltd.Among the parallel manipulators,the planar parallel manipulator is a manipulator for plane motion.Planar parallel manipulators have two degree-of-freedom(DOF)motion;that is two translations,or 3-DOF motion,consisting of two translations and one rotation.It is well known that(23-1)variations of 3-DOF planar parallel manipulators exist,which are RRR,RRP,RPR,RPP,PRR,PRP,and PPR,depending on the combinations of prismatic joints and rotational joints,excluding a PPP combination,where the prismatic and rotational joints are represented by P and R(Merlet,1996 and 2000).The solutions of the direct kinematics lbr possible architectures of the planar parallel manipulators were also already proposed(Merlet,1996),but more con-crete solutions and kinematic analyses of the architectures are stil!required.Most 3-DOF planar parallel manipulators have disadvantages that the manipulators have polynomial types of complex direct kinematics and small workspaces with useless voids as well as concave types of borderlines.As the order of Kinematic Analysis and Optimal Design of 3-PPR Planar Parallel Manipulator 529 the polynomials of the direct kinematics increase,solving equations as well as choosing a proper solution becomes a great burden.Moreover the concave types of borderlines induce non-straight motions from a neighbor of the borderline to the others.Therefore it is important that a parallel manipulator has a closed type direct kinematics and a void-tYee workspace with a convex type borderline.In this paper,a 3-PPR planar parallel mani-pulator,in which P is an active prismatic joint,is proposed to overcome the aforementioned dis-advantages,i.e.,the proposed manipulator has a closed type direct kinematics and a void free workspace with a convex type of borderline.For the kinematic analysis of this manipulator,first the direct kinematics,inverse kinematics,and inverse Jacobian of the proposed manipulator are derived.Second,rotational limits and workspaces are investigated.Also,for the optimal design of this manipulator,performance indices of the mani-pulator are investigated and then an optimal design procedure is carried out using Min-Max theory.Finally,one example using the optimal design is presented.2.Description of 3-PPR Planar Parallel Manipulator Figure I shows the schematic configuration of a 3-PPR planar parallel manipulator that consists of three active prismatic joints,three passive prismatic joints,three passive rotational joints,a moving plate,and links.The active joints can be actuated by electric rotational motors and ball screws for motion transformation.The three links for motion of the active joints are fixed to a base frame with two ends of each link.The degree of freedom(DOF)of the planar manipulator,m,is represented by(Merlet,2000)?z m=3(l-n-1)+di(1)i=1 where l is the number of rigid bodies,n is the number of joints,and dl is DOF of joint i.Since this manipulator has eight rigid bodies Copyright(C)2003 NuriMedia Co.,Ltd.Fig.1 A:tlve prisrtic joirt Schematic configuration of 3 PPR planar manipulator including the base,and nine joints with a total of nine DOF,its DOF is three;i.e.,two translations and one rotation on the plane.In addition,when the active joints of the manipulator are locked,the DOF of the manipulator becomes zero be-cause the nine joints have only six DOF.There-fore this manipulator has three DOF when the active joints are activated,whereas it becomes a static structure when the active joints are locked.3.Direct Kinematics The coordinates and the geometric parameters of this manipulator are shown in Fig.2.The moving plate is a circle,which contains an equi-lateral triangle,with a radius r.The centers of the rotational joints are on the vertices of the triangle.The active prismatic joints can travel on the sides of the outer equilateral triangle which contains a circle with radius R.When the coordinate of each active joint Ai is(xi,yi),where i=1,2,and 3,the moving plate has the pose of translation(x,y)and rotation from the reference point O.Then each length of the passive link becomes Li.Because the inner and outer triangles are equi-lateral,the angle of the triangles,00,is 00=zc/3(2)530 Kee Bong Choi A:,(x/777 1 ol Aj(r I,)Fig.2 Coordinate system for direct kinematics The side length of the inner triangle,e,is e=f 3 r(3)The incline of the inner triangle,9,is expressed by the term of rotation of the moving plate,9=+3(4)The relative displacements of the active joints are Lx cos a+e cos 9+L2 cos(a-0o)=x2-x(5)Lsina+esin p+L2sin(a-Oo)=yz-y,(6)L1 cos a+e cos(00+9)+L3cos(a+Oo)=xa-xl(7)Ltsin a+esin(00+9)+L3sin(a+00)=y3-y(8)From Eqs.(5)and(6),the lengths La and L2 are LI=L2-cos(a-&)sin 0o(Y2-yl-e sin 9)(9)sin(a-&)(x2-Xl-e cos 9)sin00 1(xz-x-e cos 9)cos(a-00)COS a sin Oo(yz-yl-e sin 9)(10)COS a.,tama-0o)(x2-xl-e cos 9)Also,from Eqs.(7),(9),and(10),the length L3 is Copyright(C)2003 NuriMedia Co.,ltd.1 Ls=cos(a+0o)x3-Xl-e Cos(19o-9)cos(a-&)cos a,.,-y2-yl-e sin 9j(ll)sin(a-0o)cos a 4 cos(a+0o)sin 0o(X2-xl-e cos 9)Substitution of Eqs.(9)-(11)into Eq.(8)deri-ves the following equation:C+C2 cos 9+C3sin 9=0(12)where Cl=cos(a+00)(y3-y)-sin(a+00)(x3-x)+cos(a-00)(y2-yl)-sin(a-00)(x2-xx)C2=e sin a+sin(a_Oo)(13)C3=-e cos a+cos(a-00)Equation(12)can be solved introducing a para-meter T as follows:1-T z cos 9=I+T 2T sin 9=l+T 2 Substituting Eq.(14)into Eq.(14)order polynomial in T is obtained as(C1-C2)Tz+2C3T+(CI+C2)=0(15)Equation(15)offers the closed form solutions of T C3+C+2 2-C2-Cz(16)T-C1-C2 From Eqs.(4)and(14),the rotation of the mov-ing plate is 2T x(17)=tan-1(I-T z)3 Thus,the translation of the moving plate is x=xl+LI cos a-r sin (18)y=-R+Lxsin a+r cos It is remarkable that this manipulator has at most two solutions for direct kinematics according to Eq.(16),and moreover the closed form solutions of Eqs.(17)and(18).(12),a second Kinematic Analysis and Optimal Design of 3-P_PR Planar Parallel Manipulator 531 4.Inverse Kinematics and Inverse Jacobian Figure 3 shows a coordinate system for des-cribing the inverse kinematics of a 3-PPR planar parallel manipulator.When the center of the moving plate moves from origin O to O with translation(x,y)and rotation b,the vertex of the plate B is expressed as xBi=x+r sin(&+b)(19)ysi=y+r COS(0g+b)where 2(i-1)0,-r(20)3 and i=1,2,3.The origin of the active prismatic joint Oi is departed by R from the origin O.Provided that ui and vi are the unit vectors of the axes e,and,which are the axes of the active prismatic joint,the coordinate of the vertex B,expressed by the terms of,and i,is eBi=OBi ui(21)gi=R+OBivi Thus the position of the active prismatic joint,a,is Y B,(,L 0 R v o,Fig.3 Coordinate system for inverse kinematics=Bi-,cot a(22)Direct differentiation of Eq.(22)with respect to the pose(x,y,b)derives the inverse Jacobian j-i as follows:I 1-c0ta rt,cos4-cotasm4)j4=.5-(-1%3 c0ta)2(;3+c0tal r(c0s-c0tasin)(23)I-(l+,c0ta)g:-3+c0ta;r(c0s&c0tasin)The elements of the inverse Jacobian of Eq.(23)do not have the same dimension.First two co-lumns corresponding to translation are dimen-sionless,whereas the last column corresponding to rotation has the dimension of length.By mak-ing the third column dimensionless,a homogen-eous inverse Jacobian with non-dimensional ele-ments is obtained by(Byun,1997)la 1 J=J 0 1(24)o o I/R 5.Rotational Limit and Workspace The rotation of the moving plate is restricted by the interference between the links and the rotational joints.Fig.4 shows the configuration of this manipulator with rotational limits of the clockwise case(a)and counter clockwise case(b).Assuming the width of the links is negligible,the rotation b is bounded by _ 4 zc+cz3+a(25)3 Provided that b0 is the initial rotation defined by 0=te-(26)The rotation range(25)is modified as where 5 7c 56 zr 6(27)=qS-b0(28)It is concluded that the moving plate of the 3-PPR planar parallel manipulator is bounded by+-57c/6 from the initial rotation.Copyright(C)2003 NuriMedia Co.,Ltd.532 Kee-Bong Choi(a)Clockwise directional rotation limit(b)Counter clockwise directional rotation limit Fig.4 Rotational limit Provided that each active joint can move along the side of the equilateral triangle with a per-pendicular distance R fiom the center of the triangle to the active joint,as shown in Fig.2,and the moving range of the active joints is 24-R.Then,the center of the moving plate can reach a positional space within only some rotations in the rotational range(271,or can reach a fully rota-table positional space in the rotational range(27).The lbrmer is referred to as a reachable workspace and the latter is a dexterous work-space.Fig.5 shows the reachable workspaces and the dexterous workspaces when r/R is 0.3,0.5,0.7,and a is r/2.The dexterous workspaces Copyright(C)2003 NuriMedia Co.,Ltd.25.I 115 41-L 2 25-L5-2-I.5 41.5 tl 0 5 1.5 2 2 5 x:R Fig.5 Workspaces at ff=r/2 Reachable workspace I)exlca,us wr kspae-I,egend-.rR 0.3.r/R 0.5-rRl).7 show a set of the reachable workspaces.As r/R increases,the reachable workspace increases sli-ghtly but the dexterous workspace decreases.Also,it is shown that the workspaces do not contain any voids,and moreover they have con-vex types of borderlines.6.Local Performance Indices Using Inverse Jacobian The inverse Jacobian provides inlbrmation on tile quality of the kinematical structure of a par-allel manipulator.In this paper,manipulability,resistivity,and isotropy by using the inverse Jaco-bian are considered as performance indices of this parallel manipulator.Manipulability,which is the distance from a singularity(Nakamura,1991),evaluates the kine-matic quality like the articular speed(Yoshi-kawa,1990)of a manipulator.That is,the farther the configuration of the manipulator is from the singularity,the faster the manipulator moves.Smaller manipulability indicates that there is a singularity near the configuration of the mani-pulator.Thus,it is better to have maximal mani-pulability.The manipulability of a parallel manipulator,win,is defined by Wm=/det(JiJh)(29)In the non-redundant case,the manipulability Wm reduces to w,=det(J1)I(30)Kinematic Anal.vsis and Optimal Design of 3-PPR Planar Parallel Manipulator 533 The determinant of the homogeneous inverse Jacobian is det(jl)_33 r 2 R(cosqJ-cotasinCJ)(l+cotZa)(31)From Eq.(31),the determinant of the homo-geneous inverse Jacobian is shown to be inde-pendent of the position of the moving plate,but only dependant on b and a.In addition,the determinant becomes zero at qS=a,that is=-Fzr/2.The 3-PPR planar parallel manipulator 11 has a singular configt,ration at=-+-zr/2.Resistivity Wr evaluates the articular tbrce and is defined by the inverse of the manipulability as-lbllows(Byun,1997):1(32)Wr=I det(J)The resistivity is correspondent to the force tran-smission ratio,which is the actuator capacity lbr a unit operational load(Lee at al.,2001).In the view of robustness of a mechanism,large resi-stivity is preferred.However,the resistivity in-creases abruptly at the neighborhood of a singular configuration,and the mechanism has a signifi-cant risk of breakdown at such a configuration.Therelbre the design of the mechanism must be considered in terms of a compromise between,v manipulability and resistivity.lsotropy corresponding to the inverse of the condition number is defined by the ratio of the minimum singular value to the maximum one of the inverse Jacobian(Nakamura,1991),and is defined as follows(Liu at al.,2000):1(33)II Jh II II J;II 0J I I when n:,u,where II Jh II=tr(JhNJ r)and N=n-is the dimension of the inverse Jacobian.Mani-0 pulability is related to the magnitude of the mani-0 pulability ellipsoid,whereas the condition hum-i ber concerns the shape of the ellipsoid(Naka-mura,1991).lsotropy evaluates workspace quali-ty and prefers a sphere-like shape,i.e.,unity.Fig.6 shows the simulation results of local performance indices(manipulability,resistivity,Fig.6 and isotropy)with respect to o,and at r/g=Copyright(C)2003 NuriMedia Co.,Ltd.0.5.In this study,is bounded by Eq.(27)and,is restricted by zc 5 _a.rc(34)Tile local performance indices are symmetrical to 0=0 and a=n/2.Manipulability is zero at!/r=-I-n/2,increases as r is far from+n/2,and.i 150(a)Manipulability Infinity IN3(b)Resistivity?.,.,u.Io-W uoj,(c)Isotropy Local perlbrmance indices using inverse Jacobian at r/R=0.5 534 Kee-Bong Choi is maximal at=0 in the range-;,r/2!/r n/2.This means that,in the view of manipulability,the manipulator prefers configurations far from=_zc/2.Conversely,resistivity is infinite at lk-_+,n/2,and abruptly decreases as it is far from,n/2.Also,it has maximal values at a=n/2 and decreases as a is far from n/2.lsotropy depending on only rather than a is zero at=_n/2,increases as Ik is far from+/2,and is maximal at=0 in the range-n/2 zr/2.7.Optimal Design From the analyses of the above indices,tile range of rotation k is agreeable to be-a/2,n/2,even though the rotational limit is ex-pressed by the range of Eq.(27),because the singularity is at _+n/2.In this study,the range of k is limited by 17 17-3 n 36,n(35)for realistic implementation.Therefore the rota-tion range lbr the dexterous workspace is also modified by the range of Eq.(35).The perlbrm-ance indices on the workspace are chosen as the dexterous workspace within the rotation range of Eq.(35)and the difference in the size of the reachable workspace and the dexterous work-space along a and r/t?.Figure 7 shows the size of the reachable work-space,the size of the dexterous workspace and the difference in the size of the two workspaces with respect to a and r/l?based on a constraint as follows:0.1-_1.0(361)The size of the reachable workspace decreases as a keeps away from n/2 and increases slowly as r/l?increases.The size of the dexterous work-space decreases,as a is far from n/2 and r/l?increases The difference in the size of the two workspaces decreases as t is far from 7r/2,and increases as r/l?increases.In this study,a large size of the dexterous workspace and a small difference in the size of two workspaces are preferred.Copyright(C)2003 NuriMedia Co.,Ltd.A global performance index using an inverse Jacobian is expressed by an average local per-tbrmance index on the total pose in the work-space.In this study,since the inverse Jacobian of this manipulator is not the function of translation but the function of rotation,the global perform-ance index can be expressed by the average value on the range of Eq.(35).Fig.8 shows the global!g 8(a)Reachable workspace.:!i i i(b)Dexterous workspace.12,10,o 8 o 6.-4.c:0-T|T q,(c)Difference in the size of the reachable workspace and the dexterous workspace Fig.7 Workspaces Kinematic Analysis and Optimal Design of 3 PPR Planar Parallel Manipulator 535 performance indices using the inverse Jacobian with respect to a and r/R.The global perfor-mance indices are symmetrical to a=z/2.Mani-pulability increases as a is far from 7c/2 and r/R increases.The manipulability increases steeply as a is far from z/2 and r/R increases.In contrast(a)Manipulability.:.i.!-.8.-:.i.-(b)Resistivity.7.i O.0 Fig.8(c)lsotrophy Global performance indices using inverse Jacobian to the manipulability,resistivity decreases as a is far from re/2 and r/R increases.Isotropy,depen-ding on r/R rather than a,increases as r/R increases.A set of optimal design parameters is obtained using the Min-Max theory of fuzzy theory(Tera-no at al.,1992,Yi at al.,1994,and Lee at al.,1996)in a constraint space with respect to a and r/R.in addition,performance indices for the design parameters a and r/R are the dexterous workspace wo,the difference in the size of the reachable workspace and dexterous workspace Ws,the global manipulability WM,the global resistivity WR,and the global isotropy Wr Let z,max(w),and min(w3)be the normalized per-formance index,the maximum,and the minimum values of wj in the given constraint space.Then the aforementioned performance indices are nor-malized by w-mi n(wj)u.b-(37)max(w)-min(w/)where the subscript j is D,M,R,and I,and max(wj)-w(38)max(wj)-min(wj)where the subscript j is S.When the performance indices are weighted,the normalized performance indices are modified by zj-=1+g)(uj-1)(39)where j is the weighted performance index and g is weight of the index with O_g_ 1(40)In the given constraint space,a composite global performance index,We,can be found by the mini-mum set of the weighted performance indices as follows Wc=wDAwsA WMA WRA tVl(41)where A means the fuzzy intersection.Then,the optimal design parameters are correspondent to the maximal position of the composite global performance index.In summary,the described optimization prob-lem can be written by Copyright(C)2003 NuriMedia Co.,Ltd.536 Kee-Bong Choi Find(a,r/R),1 minimize(UD A tsA ti;M A A t21)subject to a-_a 5;r 6 g,R oe t,.)Lo o-fll.(a)Composite global performance index Reachable works)ace Dexterous workspace 25 I !I 1 2i.-.:.1 i i _-L.=I,I os.:_-.C.i_ tr o i-Ii.-a i i i i!i,i i-Z 5 -25-2-15-I 4)5 0 05 I 15 2 5(b)Workspace 81-T r-6-.;.,-:.-.-=i i i i i,i /i i Resistivity i i;3.-+.+i,!:i!i f.,;i a.:Manipulabdity Isotropy /i2-_-.:.-,-:7-;-.-gO 410-40-20 0 20 40 O0 80(deg)Ic)Performance index Fig.9 Simulalion results for optimal design Copyright(C)2003 NuriMedia Co.,Ltd.and O.I _-1.0.Figure 9 shows an optimal design example.Fig.9(a)is the composite global performance index when the weights of the indices gD,gs,gl,gM,and gR are 1.0,0.9,0.8,0.7,and 0.6,respectively.Two sets of maximums in the map of the composite global performance index exist at(a,r/R)=(7 a-/18,0.5)and(11,7/18,0.5),because the map is symmetric to a=rc/2.The manipulator with these optimal design parameters a,and r/R has the workspace shown in Fig.9(b),and the perform-ance index using the inverse Jacobian shown in Fig.9(c).8.Conclusions A 3