机械外文文献翻译-运动学分析和优化设计3-PPR平面平行【中文3090字】【PDF+中文WORD】

机械外文文献翻译-运动学分析和优化设计3-PPR平面平行【中文3090字】【PDF+中文WORD】,中文3090字,PDF+中文WORD,机械,外文,文献,翻译,运动学,分析,优化,设计,PPR,平面,平行,中文,3090,PDF,WORD 【中文3090字】 运动学分析和优化设计3-PPR平面平行机械手 纪邦崔* 机器人与控制组，智能与精密机械部，韩国机械与材料研究所，Jang Dong，Yuseong Gu，Daejeon，305-343，韩国 摘要：本文提出了一种3-PPR平面并联机械手，其中包括三个活性柱状节理、三个被动的柱状节理、三个被动转动关节。分析了运动学和优化设计的机械手，对其进行了讨论，提出机械手具有直接运动学的封闭式和无空隙随着边界的凸式空间。分析机械手的运动学和逆运动学，和逆雅可比矩阵推导机械手。改变旋转限制和机械手的工作空间研究，对机械手的工作空间进行了仿真。此外，为优化设计的机械手，机械手的性能指标进行了研究，然后优化设计方法是使用最小最大理论。最后，用一个例子进行了优化设计。 关键词：平面并联机械手，运动学，雅可比矩阵，优化设计，最大最小 1 引言 并联机器人组成的闭环有许多优势，比串联机器人有更高的精度和刚度。众所周知，并行比串联机器人有更高的有效载荷重量比、更高的精度和较高的结构刚度。最近一些机床已开发利用这些优点，机械手的精细运动的全钢载重子午线也采用并行机制，从平价等位基因机制制造单片机。并联机器人中，平面并联机器人是平面机械手运动。平面并联机器人有两个自由度（DOF）的运动；这是两个自由度的运动和一个旋转的运动。这众所周知，平面三自由度并联机器人的存在，RRP，RPR，RPP，PRR，PRP，和PPR，取决于棱柱形接头和旋转的组合接头，不包括PPP的组合，其中棱柱和旋转接头由P和R。解决方案的直接运动学系统架构的平面并联机器人进行了已经提出了，但更多的控制—克里特岛的解决方案和运动学分析架构要求。大多数的3-DOF平面并联机器人有手有缺点复杂的直接运动学多项式类型和无用的空隙小工作区以及凹型边界。直接运动学多项式的增加，求解方程以及选择合适的溶液变成一个巨大的负担。此外，凹型边界诱导非直从邻居的运动边界其他人。因此，一个并行的是很重要的机械手具有封闭式直接运动学和一个凸型空隙泰伊工作区边缘。在本文中，一种3-PPR平面平行机器人，其中P是一个活跃的棱柱关节，提出了克服上述的缺点，即该机器人有一个封闭式直接运动学和无空隙随着边界的凸式空间。该机械手的运动学分析，首先直接运动学，逆运动学，该机器人逆雅可比矩阵派生的。第二，旋转限制和工作区进行调查。同时为优化设计该机械手的操作，性能指标机器人进行了研究并优化设计方法是使用最小最大值进行理论。最后，一个例子使用最佳的的设计方法。 2 对3-PPR平面描述并联机器人 图1显示了由三个有源棱镜接头，三个被动棱柱接头，三个被动旋转接头，移动板和连杆构成的3-I> _PR平面平行机械手的示意图。主动接头可通过电动旋转电机和滚珠丝杠进行运动转换。活动接头运动的三个连杆固定在每个连杆两端的基架上。平面制动器m的自由度（DOF）由（Merlet，2000）表示， 其中l是刚体的数量，n是关节的数量，d1是关节i的DOF。由于这个操纵器有八个刚体 图。 1 3 _I> _PR平面喷嘴的示意图 包括基地，九个关节共九个自由度，自由度三个;即平面上的两个平移和一个旋转。此外，当操纵器的主动关节被锁定时，操纵器的DOF变为零，因为九个关节只有六个DOF。在这个人之前，当活动接头被激活时，我的碎浆机有三个自由度，而当活动接头被锁定时，它是一个静态结构。 3 直接运动学 该手动分配器的坐标和几何参数如图1所示。 移动板是一个圆，它包含一个半径为r的等边三角形。 旋转关节的中心位于三角形的顶点。 活动的棱柱形关节可以在包含一个半径为R的圆的外等边三角形的边上行进 主动关节A1为（爪，y; ），其中i = 1,2和3， 移动板具有平移的姿态，y）和从参考点0的旋转rp。然后，无源链路的每个长度变为L，。 因为内外三角是等边的，所以三角形的角度是 图 2直接运动学坐标系 内三角形的边长为e 内三角的倾斜度为9,通过移动板的旋转，¢， 活动接头的相对位移为 从等式 （5）和（6）中，长度La和L2为 另外，从等式 （7），（9）和（10），长度L3是 取代方程式 （9） - （11） （8）派生以下等式： 可以求解方程（12）引入参数T如下： 代入方程 （14） （12）中，得到T中的二阶多项式 等式（15）提供了闭式形式解Ť 从等式 （4）和（14），运动的旋转板是 因此，移动板的平移是 可以理解的是，根据方程式，该操纵器具有至多两个用于直接运动学的解决方案。 （16），而且等式的闭合形式解 （17）和（18） 4 反向运动学和逆雅可比 图3显示了一种用于描述3-P_P PR平行手动分配器的反向ki nematics的坐标系。 当移动板的中心从原点O移动到平移（x，y）和旋转¢的0'时，板B的顶点; 表示为 并且i = I，2，3。有源棱镜关节的起点0; 由原点0离开。只要U; 和v 是轴线辛i和t i的单位矢量，它们是活动棱柱形接头的轴线，顶点B的坐标，由占有项表示; t i是 因此，活动棱柱关节的位置是 图3逆运动学坐标系 方程的直接分化 （22）相对于姿势（x，y，¢>）得出如下的“Jacobian J-1”： 方程的逆雅可比元素 （23）没有sam巳维度。 对应于平移的前两列是无量纲的，而对应于旋转的最后一列具有长度的尺寸。 通过使第三列无量纲，通过（Byun，1997）获得具有无量纲元素的均匀反Jacobian. 5.旋转极限和工作空间 移动板的移动受到连杆和旋转接头之间的干扰的限制。 图。 图4示出了具有顺时针（a）和逆时针（b）的旋转极限的该操纵器的构造。 Assumi吨链接的宽度可以忽略不计，旋转¢>被限制 假如ef> o是定义的初始旋转 旋转范围（25）被修改为 得出结论，3- P_PR平面平行机械手的移动板从初始旋转的界限为±5Jr / 6。 图4旋转极限 假设每个活动关节可以沿着等边三角形的侧面移动，从三角形的中心到活动关节的垂直距离R。 如图2所示，活动关节的移动范围为2T R然后，移动板的中心位置可以在旋转范围内只有一些旋转中达到位置空间（271或可达到 旋转“ange”（27）中的完全可旋转的位置空间，前者被称为可到达的工作空间，后者是一个灵巧的工作空间，图5显示了当r / R为0.3'0时，可达到的工作空间和灵活的工作空间 ，5，0.7和αisJf / 2，灵巧的工作空间显示了一组可达到的工作空间，随着r / R的增加，可达到的空间稍微增加，但是灵巧的工作空间也减少了，同时也显示了工作空间 不含任何空隙，而且它们具有凸型的边界。 6.本地业绩指标使用 逆雅可比 逆Jacobian提供了关于并联机械手的运动学结构的质量的信息。 在这个pap町，操纵性，电阻率。 并且使用逆雅可比的各向同性被认为是该并行机械手的性能指标操作性。 这是与奇异性的距离（中村，1991）。 评估运动质量，如操纵器的关节速度（Yoshikawa'1990）。 也就是说，操纵器的配置越远离奇点，操纵器移动越快操纵性越小，表示在机械手的配置附近有奇异点。 因此，它具有最大的可操纵性。 平行机械手的操纵性由Wm定义 在非冗余的情况下。 可操纵性减少到 均匀逆雅可比的行列式是 从等式 （31）中，均匀逆雅可比的行列式被显示为与移动板的位置无关。 但仅依赖于φ和α。 此外。 决定因素在¢=α处变为零，即S s =±Jr / 2。 3 E_PR平面并联机械手在萨=士π/ 2处具有单一配置。电阻率Wr评估关节力，并由可操纵性的倒数定义如下 电阻率对应于力传递比，其是单位操作负载的致动器容量（Lee at al“2001”）。 从一个机制的坚固性的角度来说，“大”的主观性是首选。 然而，在单一配置附近，电阻率突然增加，并且该机制具有重要的意义，在这种配置下不会发生故障。 因此，机械的设计必须考虑到可操纵性和电阻率之间的折中。 对应于条件数的倒数的各向同性由最小奇异值与逆雅可比（Nakamura'199川）的最大值之比定义，定义如下（Liu at al。，2000） 当n是逆雅可比的维数时。 可操纵性与可操纵性椭圆体的大小有关，而条件数则涉及椭圆体的形状（Nakamura，1991）。各向同性评估工作空间质量，并喜欢像形状即团结的球体。 图6显示了在r / R = 0.5时相对于α和萨的局部性能指标（可操纵性，电阻率和各向同性）的模拟结果。 在这项研究中，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