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运动学分析和优化设计3-PPR平面平行机械手摘要：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 一种3-PPR平面并联机械手的组成三个活动的柱状节理，三个被动柱状节理，三个被动转动关节，一个移动板和链接。主动关节可以电动旋转电机和滚珠式运动变换螺钉。三个环节为主动关节运动是固定的基地与每个链路两端架。这个程度自由（自由度）的平面机械手m，是由 其中L为刚性体的数目，n是接头的数目，和DL是自由度关节I。3 直接运动学 坐标和几何参数本机械手。移动板是一个圆圈，它包含一个设备外侧三角，一个半径为R的中心旋转接头在三角形的顶点。有源棱柱形接头两侧的旅行在外的等边三角形，其中包含一个当坐标与半径为R的每圈主动关节的AI（Xi，Yi），其中i1，2和3，移动板（X，Y）和旋转从参考点。然后每个长度的被动链接，由于内部和外部的三角形等侧的三角形，角：因为这个机器人有八个刚体内部的三角形，边长E： 内部三角形： 源关节的相对位移： 4 逆运动学和逆雅可比矩阵 图3显示了DES的坐标系统主要研究一种3-PPR平面运动学逆解并联机器人。当中心的移动板从原点O的动作翻译（x，y）和旋转，顶点B表示为： i= 1，2，3。有源棱柱的起源是由联合r从成因分提供的用户界面，VI是单位向量轴，E是主动轴移动关节，B是顶点的坐标： 对方程的逆雅可比矩阵的元素不具有相同的尺寸，相应的最后一列旋转具有长度。由第三列的无量纲，一个均质水逆雅可比矩阵与无量纲元素主要是通过 5 旋转的限制和工作区 动片的转动受到限制通过链接之间的干扰旋转关节。图4显示了配置这种机械手的旋转限制顺时针和逆时针的案例（a）和案例（b）。假设该链接的宽度是可以忽略的： 它的结论是移动板3PPR平面并联机器人是有界的通过从最初的旋转。6 局部性能指标的使用逆雅可比矩阵 逆雅可比矩阵提供了对PAR的运动学结构瓦的质量等位基因操纵。在本文中可操作性电阻率，并通过逆雅科各向同性边被认为是该性能指标并联机器人。 可操作性，这是从一个距离奇异，评估的母牛产品质量如关节速度的机械手。那是，更远的机械手的结构是从奇异性更快的机械手移动。小的可操作性，表明有一个奇点附近的摩尼的配置机器人。因此，最好是有最大质量，并联机器人的可操作度，是指由 在非冗余的情况下，可操作性的Wm降低到 相逆的行列式雅可比矩阵 7 结论 一种3-PPR平面并联机器人已提出的，和运动学分析和机械手的优化设计进行了探讨。直接运动学，逆像数学和玛尼逆雅可比矩阵推导了机器人。模拟同样的工作区机械手，为优化设计机械手的迹象，性能指标作为灵巧的工作尺寸空间，在可达大小的差异工作空间和灵巧的工作区，可操作性，电阻率和各向同性。优化设计参数进行了实况转播得到的极大极小原理，和一个例子说明了优化的有效性设计。从机器人的运动学分析，我们可以得出以下结论：（1）该机械手的自由度为三；即两个平移和一个旋转的飞机上。（2）这个机器人有一个简单的封闭式直接运动学，逆运动学和雅可比。特别是直接的机械手，最多有两个根。（3）由于各环节之间的干扰和旋转接头，该移动板机械手控制程序是被旋转的。它也表明，工作空间该机械手不包含任何空隙，而且他们有凸型边界。（4）该机械手的奇异配置理性。全球性能是a ZC / 2对称。可操作性随远离N / 2和R / R增加。可操作度急剧增加，作为一个“远从，7 / 2和R / R增加。相反的可操作性，电阻率降低。各向同性的，这取决于R / R比，R / R增加增加。（5）该机械手可以优化设计通过选择适当的分配权重，每性能指标的最小最大理论。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 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- 3+c0tal r(c0s-c0tasin) (23) I - - (l+,c0ta) g:-3 +c0ta; r(c0s 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-PPR planar parallel manipulator has been proposed, and the analysis of the kinematics and the optimal design of the manipulator were dis- cussed. The direct kinematics, the inverse kine- matics, and the inverse Jacobian of the mani- pulator were derived. Also the workspace of the manipulator was simulated. For the optimal de- sign of the manipulator, the performance indices were chosen as the size of the dexterous work- space, the difference in the sizes of the reachable workspace and the dexterous workspace, the manipulability, the resistivity, and the isotropy. The optimal design parameters were then ob- tained by the Min-Max theory, and one example was presented to show the validity of the optimal design. From the kinematic analysis of the manipula- tor, we can draw the following conclusions : (1) The DOF of this manipulator is three ; i.e., two translations and one rotation on a plane. (2) This manipulator has a simple closed type of direct kinematics, inverse kinematics, and in- verse Jacobian. In particular, the direct kine- matics of this manipulator has at most two roots. (31) Due to the interference between the links and the rotational joints, the moving plate of this rnanipulator is hounded by +57c/6 from the ini- tial rotation. Also it is shown that the workspaces of this manipulator do not contain any voids, and Kinematic Analysis and Optimal Design of 3 PPR Planar Parallel Manipulator 537 moreover they have convex types of borderlines. (4) This manipulator has a singular configu- ration at ,=7r,/2. The global performance in- dices are symmetrical to a=zc/2. Manipulability increases as a is far from n/2 and r/R increases. The manipulability increases steeply as a is far from ,7/2 and r/R increases. In contrast to the manipulability, resistivity decreases as a is far from /2 and r/R increases. Isotropy, which depends on r/R rather than a, increases as r/R increases. (5) This manipulator can be optimally desi- gned by choosing proper weights for the per- formance indices by Min-Max theory. References Ben-Horin. R., Shoham, M. and Djerassi, S., 1998, “Kinematics, Dynamics and Construction of a Planarly Actuated Parallel Robot,“ Robotics and Computer-Integrated Manufacturing, Vol. 14, No. 2, pp. 163-172. Byun, Y.-K., 1997, Analysis and Design of a 6 Degree of Freedom 3-PPS(RRR) P Parallel Manipulator, Ph. D. Dissertation in Korea Ad- vanced Institute of Science and Technology. Kim, J., Park, F. C., Ryu, S. J., Kim, J., Hwang, J. C., Park, C. and lurascu, C. C., 2001, “Design and Analysis of a Redundantly Actuated Parallel Mechanism for Rapi