機械外文文獻翻譯-運動學分析和優化設計3-PPR平面平行【中文3090字】【PDF+中文WORD】

機械外文文獻翻譯-運動學分析和優化設計3-PPR平面平行【中文3090字】【PDF+中文WORD】,中文3090字,PDF+中文WORD,機械,外文,文獻,翻譯,運動學,分析,優化,設計,PPR,平面,平行,中文,3090,PDF,WORD 【中文3090字】 運動學分析和優化設計3-PPR平面平行機械手 紀邦崔* 機器人與控制組，智能與精密機械部，韓國機械與材料研究所，Jang Dong，Yuseong Gu，Daejeon，305-343，韓國 摘要：本文提出了一種3-PPR平面并聯機械手，其中包括三個活性柱狀節理、三個被動的柱狀節理、三個被動轉動關節。分析了運動學和優化設計的機械手，對其進行了討論，提出機械手具有直接運動學的封閉式和無空隙隨著邊界的凸式空間。分析機械手的運動學和逆運動學，和逆雅可比矩陣推導機械手。改變旋轉限制和機械手的工作空間研究，對機械手的工作空間進行了仿真。此外，為優化設計的機械手，機械手的性能指標進行了研究，然后優化設計方法是使用最小最大理論。最后，用一個例子進行了優化設計。 關鍵詞：平面并聯機械手，運動學，雅可比矩陣，優化設計，最大最小 1 引言 并聯機器人組成的閉環有許多優勢，比串聯機器人有更高的精度和剛度。眾所周知，并行比串聯機器人有更高的有效載荷重量比、更高的精度和較高的結構剛度。最近一些機床已開發利用這些優點，機械手的精細運動的全鋼載重子午線也采用并行機制，從平價等位基因機制制造單片機。并聯機器人中，平面并聯機器人是平面機械手運動。平面并聯機器人有兩個自由度（DOF）的運動；這是兩個自由度的運動和一個旋轉的運動。這眾所周知，平面三自由度并聯機器人的存在，RRP，RPR，RPP，PRR，PRP，和PPR，取決于棱柱形接頭和旋轉的組合接頭，不包括PPP的組合，其中棱柱和旋轉接頭由P和R。解決方案的直接運動學系統架構的平面并聯機器人進行了已經提出了，但更多的控制—克里特島的解決方案和運動學分析架構要求。大多數的3-DOF平面并聯機器人有手有缺點復雜的直接運動學多項式類型和無用的空隙小工作區以及凹型邊界。直接運動學多項式的增加，求解方程以及選擇合適的溶液變成一個巨大的負擔。此外，凹型邊界誘導非直從鄰居的運動邊界其他人。因此，一個并行的是很重要的機械手具有封閉式直接運動學和一個凸型空隙泰伊工作區邊緣。在本文中，一種3-PPR平面平行機器人，其中P是一個活躍的棱柱關節，提出了克服上述的缺點，即該機器人有一個封閉式直接運動學和無空隙隨著邊界的凸式空間。該機械手的運動學分析，首先直接運動學，逆運動學，該機器人逆雅可比矩陣派生的。第二，旋轉限制和工作區進行調查。同時為優化設計該機械手的操作，性能指標機器人進行了研究并優化設計方法是使用最小最大值進行理論。最后，一個例子使用最佳的的設計方法。 2 對3-PPR平面描述并聯機器人 圖1顯示了由三個有源棱鏡接頭，三個被動棱柱接頭，三個被動旋轉接頭，移動板和連桿構成的3-I> _PR平面平行機械手的示意圖。主動接頭可通過電動旋轉電機和滾珠絲杠進行運動轉換。活動接頭運動的三個連桿固定在每個連桿兩端的基架上。平面制動器m的自由度（DOF）由（Merlet，2000）表示， 其中l是剛體的數量，n是關節的數量，d1是關節i的DOF。由于這個操縱器有八個剛體 圖。 1 3 _I> _PR平面噴嘴的示意圖 包括基地，九個關節共九個自由度，自由度三個;即平面上的兩個平移和一個旋轉。此外，當操縱器的主動關節被鎖定時，操縱器的DOF變為零，因為九個關節只有六個DOF。在這個人之前，當活動接頭被激活時，我的碎漿機有三個自由度，而當活動接頭被鎖定時，它是一個靜態結構。 3 直接運動學 該手動分配器的坐標和幾何參數如圖1所示。 移動板是一個圓，它包含一個半徑為r的等邊三角形。 旋轉關節的中心位于三角形的頂點。 活動的棱柱形關節可以在包含一個半徑為R的圓的外等邊三角形的邊上行進 主動關節A1為（爪，y; ），其中i = 1,2和3， 移動板具有平移的姿態，y）和從參考點0的旋轉rp。然后，無源鏈路的每個長度變為L，。 因為內外三角是等邊的，所以三角形的角度是 圖 2直接運動學坐標系 內三角形的邊長為e 內三角的傾斜度為9,通過移動板的旋轉，￠， 活動接頭的相對位移為 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）引入參數T如下： 代入方程 （14） （12）中，得到T中的二階多項式 等式（15）提供了閉式形式解? 從等式 （4）和（14），運動的旋轉板是 因此，移動板的平移是 可以理解的是，根據方程式，該操縱器具有至多兩個用于直接運動學的解決方案。 （16），而且等式的閉合形式解 （17）和（18） 4 反向運動學和逆雅可比 圖3顯示了一種用于描述3-P_P PR平行手動分配器的反向ki nematics的坐標系。 當移動板的中心從原點O移動到平移（x，y）和旋轉￠的0'時，板B的頂點; 表示為 并且i = I，2，3。有源棱鏡關節的起點0; 由原點0離開。只要U; 和v 是軸線辛i和t i的單位矢量，它們是活動棱柱形接頭的軸線，頂點B的坐標，由占有項表示; t i是 因此，活動棱柱關節的位置是 圖3逆運動學坐標系 方程的直接分化 （22）相對于姿勢（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。 對應于平移的前兩列是無量綱的，而對應于旋轉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.旋轉極限和工作空間 移動板的移動受到連桿和旋轉接頭之間的干擾的限制。 圖。 圖4示出了具有順時針（a）和逆時針（b）的旋轉極限的該操縱器的構造。 Assumi噸鏈接的寬度可以忽略不計，旋轉￠>被限制 假如ef> o是定義的初始旋轉 旋轉范圍（25）被修改為 得出結論，3- P_PR平面平行機械手的移動板從初始旋轉的界限為±5Jr / 6。 圖4旋轉極限 假設每個活動關節可以沿著等邊三角形的側面移動，從三角形的中心到活動關節的垂直距離R。 如圖2所示，活動關節的移動范圍為2T R然后，移動板的中心位置可以在旋轉范圍內只有一些旋轉中達到位置空間（271或可達到 旋轉“ange”（27）中的完全可旋轉的位置空間，前者被稱為可到達的工作空間，后者是一個靈巧的工作空間，圖5顯示了當r / R為0.3'0時，可達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到的工作空間，隨著r / R的增加，可達到的空間稍微增加，但是靈巧的工作空間也減少了，同時也顯示了工作空間 不含任何空隙，而且它們具有凸型的邊界。 6.本地業績指標使用 逆雅可比 逆Jacobian提供了關于并聯機械手的運動學結構的質量的信息。 在這個pap町，操縱性，電阻率。 并且使用逆雅可比的各向同性被認為是該并行機械手的性能指標操作性。 這是與奇異性的距離（中村，1991）。 評估運動質量，如操縱器的關節速度（Yoshikawa'1990）。 也就是說，操縱器的配置越遠離奇點，操縱器移動越快操縱性越小，表示在機械手的配置附近有奇異點。 因此，它具有最大的可操縱性。 平行機械手的操縱性由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板的位置無關。 但僅依賴于φ和α。 此外。 決定因素在￠=α處變為零，即S s =±Jr / 2。 3 E_PR平面并聯機械手在薩=士π/ 2處具有單一配置。電阻率Wr評估關節力，并由可操縱性的倒數定義如下 電阻率對應于力傳遞比，其是單位操作負載的致動器容量（Lee at al“2001”）。 從一個機制的堅固性的角度來說，“大”的主觀性是首選。 然而，在單一配置附近，電阻率突然增加，并且該機制具有重要的意義，在這種配置下不會發生故障。 因此，機械的設計必須考慮到可操縱性和電阻率之間的折中。 對應于條件數的倒數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n是逆雅可比的維數時。 可操縱性與可操縱性橢圓體的大小有關，而條件數則涉及橢圓體的形狀（Nakamura，1991）。各向同性評估工作空間質量，并喜歡像形狀即團結的球體。 圖6顯示了在r / R = 0.5時相對于α和薩的局部性能指標（可操縱性，電阻率和各向同性）的模擬結果。 在這項研究中，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