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清华科技ISSN 1007 - 0214 06/18 pp269 - 275第12卷,第3号,2007年6月最优运动设计的一个二自由度平面并联机械手* 吴军，李铁民，刘辛军，王立平制造工程研究所，精密仪器和机械学系，清华大学，北京100084,中国摘要：封闭了优化设计的运动学2自由度平面并联机械手的解决方案。基于工作空间提出了优化设计。与此同时，一个全球性的，全面介绍了空调指数评估运动设计。最好的并联机械手是纳入一个五自由度混合机工具包括一个二自由度旋转机头和一个长移动工作台。结果表明，平面并联机械手的基础试机工具可以成功地用于机叶片和导叶的水力涡轮机。关键词:平面并联机械手;全球调节指数;混合机工具介绍并行机制能够非常快速和准确的运动，具有较高的平均刚度，且特征是在他们的工作空间，具有较低的惯性,可以操纵重载荷比串行、同行。因此,并行机制在为平台的计算机数控加工的同时也被众多研究者全面研究和制造。高夫斯图尔特的该是最受欢迎的平行运动机配置，当并联运动机床首次开发。并应用在机器人技术社区易感性的范围，从高速操纵【1】来力扭矩传感【2】。但是高夫斯图尔特平台对制造业的应用程序也有一些缺点，比如它的相对小的有用的工作区, 复杂的直接运动学和设计困难。并联机构少于6自由度是相对容易的设计及其运动学可以用封闭的形式描述。因此，并行机械手少于6自由度，特别是2或3自由度,已经越来越引起关注【3-5】。而在2个或3个平移自由度扮演重要角色的行业里，平行机械手能够适用于并联机器【6】挑选和地方应用【7】和其他领域。大多数现有的2自由度平面平行机械手是著名的五杆机构，用的是棱镜执行机构或外卷的致动器【8、9】。这机械手输出是平移运动的一种点和末端效应器取向不能保持不变。 运动学设计方法的一个关键部分是平行机制运动的设计理论。优化设计可以基于不同评估标准，括刚度【10】、敏捷、或一个全球调节指数【11、12】。因此,没有一个是被广泛接受的设计方法。黄等、艾尔【13】，提出了一种混合的方法基于调节指数。而Liu X J【14】给了一个全球刚度索引类似全球调节指数。本文描述了一个平面并联机械手的分析。机械手的两个平移自由度不同，从传统的五杆机构在一个平行四边形结构用在每个链的运动学进行了分析，得到一个最佳的运动设计。尽量减少全局,全面调节指数，结果给最优链路长。一个五自由度混合机工具的开发是由一个串并联结构和并联机械手结合1自由度平移工作台和一个二自由度旋转铣头。为制造企业混合机工具使用平面并联机械手一直用磨叶片和导叶的水力涡轮表明混合机工具是合适的。1、结构描述二自由度并联机械手的图1所示。这个机制是由一个龙门框架,一个移动的平台,两个主动滑块,两个运动链。每个链建设作为一个平行四边形。按照设计,机械手是在约束一个平行四边形链接和另一个单一的链接的足够移动平台,拥有2平移自由度。两个平行四边形链被用来增加刚度,使结构对称。收到:2006-06-02;修订:2006-08-21*国家自然科学基金资助的中国(第50305016号)和国家高科技研究和振(863)计划的中国(Nos。2004 aa424120和2005 aa424223)*电子邮件:;电话:86-10-62792792 图1运动学模型运用P1 and P2(见图1)被添加到机制来提高负载能力和加速度的致动器。独立驱动的滑块由两个伺服电机在列上沿着导轨滑动安装在列,因此移动平台由纯粹的平移运动在一个平面组成。2、运动学和奇异性2.1逆运动学 因为运动的两个链接的每个运动链是相同的平行四边形结构,链模型可以简化为一个链接 Ai Bi( i= 1, 2)见图1。基坐标系O-xy附加到基地以其y轴垂直通过的中点。一个移动的坐标系统O-xy 是固定在移动平台。r Ai和r Bi分别是位置向量Ai和Bi的接头位置的。2 r是移动平台宽度和两列。 位置矢量的起源O与关于这个坐标系o-xy被定义为 r O= x y T (1) 位置矢量Ai的位置在O - x y是 r A1= r 0 T (2) r A2= r 0 T (3)然后这个位置向量Ai在基坐标系O- xy可以表示为 270 清华大学科学与技术,2007年6月,12(3):269 - 275 rAi =rO+ r Ai (4)各关节的位置矢量Bi位置在O-xy是 RB1= r y1T (5) RB2= r y2T (6)因此,约束方程关联第i运动链可以写成 rAi-rBi=li ni, i=1, 2b (7) 在l,n表示长度和链接的单位向量,分别。 以2规范双方的Eq。(7)给了 y l =y (8) y 2 =y (9) 为配置图1所示,逆运动学正解 y l =y+ (10) y 2 =y+ (11) 从方程式。(10)和(11),该解决方案直接运动学的机械手可以表示为 (12) 在 a=, b=, c=, f =Rr+b+c, x= ay+b+c (13) 为配置见图1,“”Eq。(12)应该只有“-”。 方程(7)-(13)表明,直接和逆运动学的机械手可以描述在封闭形式。2.2奇点分析以衍生品的Eq。(10)和Eq。(11)关于时间给了胡军(吴军)et al:最优运动设计的一个二自由度平面 271 (14) (15)方程(14)和(15)可以重新排列在矩阵形式 (16)在J是雅可比行列式表示为 (17) 因为奇异性导致损失的可控制性和退化的自然刚度的操纵者,他们必须避免在任务的工作区。奇点可以分为直接运动学奇异点,逆运动学奇异性,结合奇异性15,可以区分机械手的雅可比行列式。当其中一个链接是水平的,机械手逆运动学的经验奇点。 直接运动学奇异点时发生的一个链接链和一个链接的其他链是共线。因为l1+l2 2R结合奇异性不能发生在该机械手。图2显示了一个例子,每个类型的奇点。在实际应用中,通过限制避免了奇异性任务的工作区。 图2 奇异的配置3、工作空间分析工作区为二自由度平面的并联机械手是一个区域的平面所取得的工作空间参考点O的移动平台。方程(10)和(11)可以重写为 (x-r+R)2+(y-y1)2=l12 (18) (x+r-R)2+(y-y2)2=l22 (19)因此,可达工作空间的参考点O是交叉点的子工作区关联两个运动链见图3。 图3机械手工作空间任务工作空间是一个部分的可达工作空间。在实际应用中,这个任务的工作区通常定义为一个矩形区域的可达工作空间。 让最大限度的角度和,链接AiBi之间的夹角(i = 1,2)和垂直轴,用max和max。让yi,max和yi,min代表最大和最小的位置i-th滑块。当O到达点Q1滑块B1达到下限和的值最大,即yi=yi,min和=max。同样, 当O到达点Q4，即y2 =y2,min和=max。 一个垂直的线通过Q1与上界的可达工作空间点Q2。Q3在Q4的正上方(见图3)。该地区Q1 Q2 Q3 Q4 然后构成任务的工作区,作为一个矩形的宽度b和高度h,注明wt。4、最优运动设计4.1优化设计基于工作区 这个小节的目标是确定机械手参数所需的工作区。优化设计的范围可以表达为:给定r、b,和霍夫重量,确定R,l1、l2, 总长度为的滑块。从方程式(10),(11)和(17),机械手的性能与Rr但不要R或r独立。实际上,r应该尽可能的小,因为较小的值r导致较小的机械手。通常,r取决于轴,轴承,和工具维度在他移动平台。因此,r应该是设计师给出的。 当移动平台到达下限,见图4,以下参数关系可以被获得 （20） （21）其中d的距离左列到左极限任务的工作区。在实际应用中,l1应该等于l2改善系统性能和刚度。因此 （22）272 清华大学科学与技术,2007年6月,12(3):269 - 275 图4机械手的优化设计因此, 对于l1=l2当移动平台将从Q1到Q4点沿x轴、滑动距离的滑块在导轨应 (24)当移动平台从点Q4到Q3传播沿y方向,滑动距离的滑块 (25) 因此,总旅程的滑块 (26)因为优化设计基于任务的工作区不考虑灵巧和刚度的机械手,链接长度不是最优。最优长度的链4.2 全球,综合指数条件数的雅可比行列式作为当地的性能指标评估速度、精度和刚度映射特征变量之间的联合和移动平台。这个条件数被定义为 (27)在 和 是最大和最小奇异值的导数与给定的姿势有关。和 可以由求解特征方程WU Jun (WU army) et al: Optimal Kinematic Design of a 2 - DOF Planar. 273 (28)I代表一个单位矩阵的订单2。这里研究了并联机械手, (29)在和因为K随机制配置,一个总体的能指标作为绩效测量的最优运动设计, （30）注意,本身不能充分描述总体的整体运动学性能由于其无法描述之间的差异的 最大值与最小值。黄et al7。提出了一种全局,综合条件指数为目标函数在运动设计。目标函数可以表示为 （31）在和max()和min()代表最大值与最小值的K在任务工作空间和w代表重量放在的比率到。 和可以计算的数值 （32）在 的价值评估在节点(m,n)(m1)(n1)同样网状的Wt。5、 应用程序5.1 XNZD2415混合机工具 一个二自由度平面并联机械手结合工作台有一个平移自由度在z轴和一个铣头和两个旋转自由度约x轴和z轴的创建一个五自由度混合串并联机床(XNZD2415)。与双轴铣头,该工具可以获得任何想要的取向在任务工作空间,它的高灵活性可以自由移动的工作台沿z轴,使机床制造长的工件。 任务的工作区Wt的并联机械手设计为一个长方形b = 1600毫米宽度和h = 1000毫米高。机械手也有max = 80,min = 10,r = 75毫米。奇点理论分析表明,并联机械手的奇异性不能发生任务工作空间。 最优设计基于任务的工作区结果在R = 1217.4毫米。然后通过计算最小化目标函数其他设计参数。图5表明,有最小值1.916当L1 = 2060毫米为w= 0.1。基于Eq。(26),它可以被获得为2345.2毫米。 图5为不同链接最小化长度 条件数的雅可比行列式的并联机械手在机床是图6所示。条件数的分布是对称的x轴。良好的运动性能是通过平衡和之间的矛盾。机床是由清华大学在中国轧机一系列叶片和导叶的水力透平发电机组并应用在哈尔滨电机有限公司。原型图7所示。274 清华大学科学与技术,2007年6月,12(3):269 - 275 图6条件数的雅可比行列式的XNZD2415机床 图7 XNZD2415混合机工具5.2 XNZD2430重型机床 XNZD2415之后成功的应用在对各种加工任务,哈尔滨电机有限公司需要一个混合机工具机非常大的叶片和导叶的水力涡轮机发电机组。 当时另一个重型机床(XNZD2430)建基于二自由度并联机械手。新的类似于第一个机在结构与一个更大的XNZD2430新工作区和一些其他的包括修改两个额外的盖表固定的列和一个辅助盘固定在顶部的铣头。双方的盖表联系辅助托盘覆盖着薄铜板。辅助盘接触两个盖板和幻灯片之间的铣头动作。因此,覆盖表限制变形的铣头和提高机床刚度正常的平面机械手运动。四个括号也添加到新机床提高刚度。 两个额外的非旋转铣头被设计来取代二自由度铣头,安装在侧面或底部的移动平台。这种重型机床是那么一个三自由度的机器可以机纵向或横向。 Wt任务的工作区在新的并联机械手别被设计为一个长方形宽度b = 3000毫米和高度h = 1800毫米与max =,min =。这个机械手不能达到奇异con形状在任务的工作区。 最优设计基于任务的工作区结果在R = 2342.5毫米有最小值l1= 3550毫米,当 w= 0.1。可以从Eq决定。(26)。主要设计参数表1中列出。 表1的结果XNZD2430试机优化工具 参数 数值 l1 3550 mm l2 3550 mm r 550 mm 4010.7 mm 1.99 的分布为优化设计显示在图8。分布表明,条件数也是对称所以x轴和运动性能是最优的。 图8条件数的雅可比行列式的XNZD2430机床机床的XNZD2430显示在图9是由清华大学正在很快的建造与机床的动力学和控制 胡军(吴军)et al:最优运动设计的一个二自由度平面 275的测试。 图9 XNZD2430机床的原型 6 、结论运动学设计的一个二自由度平面并联机械手已被调查。调查显示如下。(1) 两列之间的距离通常是通过优化设计基于工作区。(2) 一个全球性的,综合性能指标,包括均值和条件数的范围的雅可比行列式在任务工作空间,可以用来有效地优化运动设计。(3) 平面并联机械手建立机床被成功用于机叶片和导叶的水力涡轮机。另一个重型机床基于并联机械手被建立更大的加工任务。参考文献1 Beqon P, Pierrot F, Dauchez P. 模糊滑模控制的快速并行机器人。在:IEEE机器人与自动化国际会议上,1995:1178 - 1183。2 Ferraresi C, Pastorelli S, Sorli M, Zhmud N. 静态和动态行为的高刚度Stewart平台力/力矩传感器。机器人系统杂志,1995,12(12):883 - 893。3 吴J,李T M,唐XQ:鲁棒轨迹跟踪控制的一个平面并联机构。j .清华大学(Sci。&工艺。),2005年,45(5):642 - 646。(在中国)4 Kock S, Schumacher W. 平行x - y机制与驱动冗余应用为高速和活跃的刚度。在:IEEE机器人与自动化国际会议上。鲁汶。比利时,1998:2295 - 2300。5 刘X J,王QM,小王J s. 运动学、动力学和空间合成二自由度平动机械手的一本小说。智能和机器人系统杂志,2004年,41(4):205 - 2246 小王J S,唐XQ:分析和空间设计的一种新型的混合机工具。国际期刊的机床和制造,2003年,43(7):647 - 6557 黄T,李Z X,李M,Chetwynd D G, Gosselin C M.概念设计和空间合成的二自由度平动并联机器人小说为选择和地点操作。ASME杂志机械设计,2004,126(3):449 - 455。8 高F,刘X J,Gruver WA .绩效评估两个程度的自由平面并联机器人。机制和机理论,1998,33(6):661 - 668。9 McCloy D.一些比较串行驱动和并行驱动的机械手。Robotica,1990,8(4):355 - 362。10 Arsenault M, Boudreau R.合成一个通用平面并联机械手与棱镜关节最佳刚度。在:第11届世界大会在机制和机科学。北京:中国机械出版社,2004:1633 - 1637。11 Gosselin C M. 优化设计的机器人机械手使用灵巧指数。机器人和自治系统,1992,9(4):213 - 226。12 Gosselin C M, Angeles J. 一个全球性能指数的运动优化机器人机械手。ASME杂志机械设计,1991,113(3):220 - 226。13黄T, 李M, 李Z X, Chetwynd D G, Whitehouse D J. 最优运动设计的二自由度并联机构工作空间边界的形状规整与指定条件指数。IEEE机器人和自动化,2004,20(3):538 - 543。14 刘X J、金Z L,高f .优化设计的三自由度球面并联机制对于调节和刚度指标。机制和机理论,2000,35(9):1257 - 1267。15 Gosselin C M, Angeles J. 奇异性分析的闭环运动链。IEEE机器人和自动化,1990,6(3):281 - 290。 。 TSINGHUA SCIENCE AND TECHNOLOGY ISSN 1007-0214 06/18 pp269-275 Volume 12, Number 3, June 2007 Optimal Kinematic Design of a 2-DOF Planar Parallel Manipulator* WU Jun (吴 军), LI Tiemin (李铁民), LIU Xinjun (刘辛军), WANG Liping (王立平) Institute of Manufacturing Engineering, Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, China Abstract: Closed-form solutions were developed to optimize kinematics design of a 2-degree-of-freedom (2-DOF) planar parallel manipulator. The optimum design based on the workspace was presented. Meanwhile, a global, comprehensive conditioning index was introduced to evaluate the kinematic designs. The optimal parallel manipulator is incorporated into a 5-DOF hybrid machine tool which includes a 2-DOF rotational mill-ing head and a long movement worktable. The results show that the planar parallel manipulator-based ma-chine tool can be successfully used to machine blades and guide vanes for a hydraulic turbine. Key words: planar parallel manipulator; global conditioning index; hybrid machine tool Introduction Parallel mechanisms are capable of very fast and accu-rate motion, possess higher average stiffness character-istics throughout their workspace, have lower inertia, and can manipulate heavier payloads than their serial counterparts. Therefore, parallel mechanisms have been studied extensively by numerous researchers in manufacturing, primarily as platforms for computer numerical control machining. The Gough-Stewart plat-form was the most popular parallel kinematic machine configuration when parallel kinematic machine were first developed. Applications within the robotics com-munity range from high-speed manipulation1 to force-torque sensing2. But the Gough-Stewart platform has some disadvantages for manufacturing applications such as its relatively small useful workspace, com-plex direct kinematics, and design difficulties. Parallel manipulators with less than 6 DOFs are relatively easy to design and their kinematics can be described in closed form. Therefore, parallel manipula-tors with less than 6 DOFs, especially 2 or 3 DOFs, have increasingly attracted attention3-5. Parallel ma-nipulators with 2 or 3 translational DOFs play impor-tant roles in industry and can be applied in parallel kinematics machines6, pick and place applications7, and other fields. Most existing 2-DOF planar parallel manipulators are the well-known five-bar mechanism with prismatic actuators or revolute actuators8,9. The manipulator output is the translational motion of a point on the end-effector and the end-effector orienta-tion cannot remain constant. The kinematic design methodology is one of the key parts of kinematic design theory for parallel mecha-nisms. The optimum design can be based on various evaluation criteria involving stiffness10, dexterity, or a global conditioning index11,12. Hence, there has not been one widely accepted design method. Huang et al.13 presented a hybrid method based on a condition-ing index, while Liu et al.14 gave a global stiffness Received: 2006-06-02; revised: 2006-08-21 Supported by the National Natural Science Foundation of China (No. 50305016) and the National High-Tech Research and Devel-opment (863) Program of China (Nos. 2004AA424120 and 2005AA424223) To whom correspondence should be addressed. E-mail: ; Tel: 86-10-62792792 Tsinghua Science and Technology, June 2007, 12(3): 269-275 270 index similar to the global conditioning index. This paper describes the analysis of a planar parallel manipulator with two translational DOFs which differs from the conventional five-bar mechanism in that a par-allelogram structure is used in each chain. The kinemat-ics are analyzed to get an optimum kinematic design by minimizing a global, comprehensive conditioning index. The results give the optimal link lengths. A 5-DOF hybrid machine tool was developed with a serial-parallel architecture with a parallel manipulator combined with a 1-DOF translational worktable and a 2-DOF rotational milling head. The hybrid machine tool using the planar parallel manipulator has been used to mill the blades and guide vanes of a hydraulic turbine to show that the hybrid machine tool is suitable for the manufacturing industry 1 Structure Description The 2-DOF parallel manipulator is shown in Fig. 1. The mechanism is composed of a gantry frame, a mov-ing platform, two active sliders, and two kinematic chains. Each chain is built as a parallelogram. As de-signed, the manipulator is over-constrained since one parallelogram link and another single link are enough for the moving platform to posses 2 translational DOFs. Two parallelogram chains were used to increase the stiffness and make the structure symmetric. Fig. 1 Kinematic model Counterweights 1P and 2P (see Fig. 1) were added to the mechanism to improve the load capacity and ac-celeration of the actuator. The sliders are driven inde-pendently by two servo motors on the columns to slide along the guide ways mounted on the columns, thus moving platform with a 2-DOF purely translational motion in a plane. 2 Kinematics and Singularities 2.1 Inverse kinematics Since the motions of two links of each kinematic chain are identical due to the parallelogram structure, the chain model can be simplified as a link iiAB(1,2i =) as illustrated in Fig. 1. The base coordinate system -O xy is attached to the base with its y axis vertical through the midpoint of 12B B. A moving coordinate system -O x y is fixed on the moving platform. iAr and iBr are the position vectors of the joint positions iA and iB, respectively. 2r is the moving platform width and 2R is the width between the two columns. The position vector of the origin O with respect to the coordinate system -O xy is defined as TOxy=r (1) The position vector of joint position iA in -O x y is 1T0Ar = r (2) 2T0Ar =r (3) Then the position vector of iA in the base coordi-nate system -O xy can be expressed as iiAOA=+rrr (4) The position vector of each joint position iB in -O xy is 1T1BRy= r (5) 2T2BRy=r (6) Thus, the constraint equation associated with the i-th kinematic chain can be written as iiABiil=rrn, 1,2i = (7) where il and in denote the length and the unit vector of the i-th link, respectively. Taking the 2-norm of both sides of Eq. (7) gives ()2211yylxrR=+ (8) ()2222yylxrR=+ (9) For the configuration shown in Fig. 1, the inverse solutions of the kinematics are ()2211yylxrR=+ (10) WU Jun (吴 军) et al：Optimal Kinematic Design of a 2-DOF Planar 271()2222yylxrR=+ (11) From Eqs. (10) and (11), the solutions for the direct kinematics of the manipulator can be expressed as 222221112(22)4(1)()2(1)afyfalyfya+=+ (12) where 12,2()yyaRr= 2221,4()yybRr= 2212,4()llcRr= ,fRrbc=+ xaybc=+ (13) For the configuration as shown in Fig. 1, the “” of Eq. (12) should be only “”. Equations (7)-(13) show that the direct and inverse kinematics of the manipulator can be described in closed form. 2.2 Singularity analysis Taking the derivatives of Eq. (10) and Eq. (11) with re-spect to time gives ()1221xrRyyxlxrR+=+? (14) ()2222xrRyyxlxrR+=+? (15) Equations (14) and (15) can be rearranged in matrix form as 12yxyy = J? (16) where J is the Jacobian expressed as ()()22122211xrRlxrRxrRlxrR+=+J (17) Because singularities lead to a loss of controllability and degradation of the natural stiffness of the manipu-lators, they must be avoided in the task workspace. Singularities can be classified as direct kinematic sin-gularities, inverse kinematic singularities, and com-bined singularities15, and can be distinguished by the manipulator Jacobian. When one of the links is hori-zontal, the manipulator experiences an inverse kine-matic singularity. Direct kinematic singularities occur when one link of a chain and a link of the other chain are collinear. Since 122llR+, combined singularities cannot oc-cur in this manipulator. Figure 2 shows one example of each kind of singularity. In practical applications, sin-gularities are avoided by limiting the task workspace. Fig. 2 Singular configurations 3 Workspace Analysis The workspace for the 2-DOF planar parallel manipu-lator is a region of the plane derived by the workspace of the reference point O of the moving platform. Equations (10) and (11) can be rewritten as ()22211()xrRyyl+= (18) ()22222()xrRyyl+= (19) Therefore, the reachable workspace of the reference point O is the intersection of the sub-workspaces associated with the two kinematic chains as shown in Fig. 3. Fig. 3 Manipulator workspace The task workspace is a part of the reachable work-space. In practical applications, the task workspace is usually defined as a rectangular area in the reachable workspace. Tsinghua Science and Technology, June 2007, 12(3): 269-275 272 Let the maximum limit of the angles and , which are the angles between link iiAB (1,2i =) and the vertical axis, be denoted by max and max. Let ,maxiy and ,miniy represent the maximum and mini-mum positions of the i-th slider. O reaches point 1Q when slider 1B reaches its lower limit and the value of is the maximum, namely 11,minyy= and max=. Similarly, O reaches point 4Q when 22,minyy= and max=. A vertical line through 1Q intersects with the upper bound of the reachable workspace at point 2Q. 3Q is directly above 4Q (see Fig. 3). The region 1234QQ Q Q then makes up the task workspace, as a rectangle of width b and height h, denoted by tW. 4 Optimal Kinematic Design 4.1 Optimal design based on the workspace The objective of this section is to determine the ma-nipulator parameters for a desired workspace. The scope of optimal design can be stated as: given r, b, and h of tW, determine ,R 1,l 2l, and the total jour-ney ,max,miniiyy of the slider. From Eqs. (10), (11), and (17), the manipulator per-formance is related to Rr but not to r or R alone. Practically, r should be as small as possible since smaller values of r lead to smaller manipulator volumes. Usually, r depends on the shaft, bearing, and tool dimensions on the moving platform. Therefore, r should be given by the designer. When the moving platform reaches the lower limit, as shown in Fig. 4, the following parametric relation-ships can be obtained max1sindbrl+= (20) min2sindrl= (21) where d is the distance from the left column to the left limit of the task workspace. In practical applications, 1l should equal 2l to improve the system performance and stiffness. There- fore maxminmaxminsinsin()sinsinrbrd+= (22) Fig. 4 Optimal design of the manipulator Thus, 2bRd=+ (23) For 12ll=, when the moving platform moves from point 1Q to 4Q along the x axis, the sliding dis-tance of the slider in the guide way should be 122221122BddylRrlRr=+ (24) When the moving platform travels from point 4Q to 3Q along the y direction, the sliding distance of the slider is 2Byh= (25) Hence, the total journey ,max,miniiyy of the slider is 12,max,miniiBByyyy=+ (26) Because the optimum design based on the task workspace does not consider the dexterity and stiffness of the manipulator, the link lengths are not optimal. The optimal lengths of the links and ,max,miniiyy for the slider are determined in the next subsection. 4.2 Global, comprehensive index The condition number of the Jacobian is used as the local performance index for evaluating the velocity, accuracy, and rigidity mapping characteristics between the joint variables and the moving platform. The condi-tion number is defined as 211= (27) where 1 and 2 are the minimum and maximum sin-gular values of the Jacobian associated with a given WU Jun (吴 军) et al：Optimal Kinematic Design of a 2-DOF Planar 273posture. 1 and 2 can be determined by solving the characteristic equation ()2Tdet0=IJ J (28) where I denotes a unit matrix of order 2. For the parallel manipulator studied here, 22222222222()842()84stststststst+=+ (29) where 1xrRsyy+= and 2xrRtyy+=. Since varies with the mechanism configuration, a global performance index is used as the performance measure in the optimal kinematic design, ttttddWWWW= (30) Note that itself cannot give a full description of the overall global kinematic performance due to its in-ability to describe the difference between the maxi-mum and minimum values of . Huang et al.7 pro-posed a global, comprehensive condition index as the objective function in kinematic designs. The objective function can be expressed as 22()w=+? (31) where max( ) min( )=? with max( ) and min( ) representing the maximum and minimum values of in the task workspace and w representing the weight placed upon the ratio of to ?. and ? can be numerically calculated by 111MNmnmnMN=, max()min()mnmn=? (32) where mn is the value of evaluated at node ( , )m n of (1)M (1)N equally meshed tW . 5 Applications 5.1 XNZD2415 hybrid machine tool A 2-DOF planar parallel manipulator was combined with a worktable having a translational DOF in the z axis and a milling head with two rotational DOFs about the x and z axes to create a 5-DOF hybrid serial-parallel machine tool (XNZD2415). With the two-axis milling head, the tool can obtain any desired orientation in the task workspace, which provides high dexterity. The worktable can move freely along the z axis, which enables the machine tool to manufacture long workpieces. The task workspace tW of the parallel manipulator is designed as a rectangle b=1600 mm in width and h= 1000 mm in height. The manipulator also has max80=?, min10 ,=? and 75r= mm. The singular-ity analysis shows that singularities cannot occur in the task workspace of the parallel manipulator. The optimal design based on the task workspace re-sults in 1217.4R= mm. The other design parameters are then calculated by minimizing the objective func-tion . Figure 5 shows that has a minimum value of 1.916 when 1l=2060 mm for 0.1w=. Based on Eq. (26), it can be obtained that ,max,min|iiyy is 2345.2 mm. Fig. 5 Minimization of for various link lengths The condition number of the Jacobian of the parallel manipulator in the machine tool is shown in Fig. 6. The distribution of the condition number is symmetrical about the x axis. The good kinematic performance is achieved by balancing the conflict between and ?. Fig. 6 Condition number of the Jacobian of XNZD2415 machine tool The machine tool was built by Tsinghua University and was used in the Harbin Electric Machinery Co, Ltd in China to mill a series of blades and guide vanes for hydraulic turbine generator sets. The prototype is Tsinghua Science and Technology, June 2007, 12(3): 269-275 274 shown in Fig. 7. Fig. 7 XNZD2415 hybrid machine tool 5.2 XNZD2430 heavy duty machine tool After the successful application of XNZD2415 for various machining tasks, Harbin Electric Machinery Co needed another hybrid machine tool to machine very large blades and guide vanes for hydraulic turbine generator sets. Another heavy duty machine tool (XNZD2430) was then built based on the 2-DOF parallel manipulator. The new is similar to the first machine in structure with a much larger new workspace of XNZD2430 and some other modifications including two additional cover sheets fixed on the columns and an auxiliary tray fixed on top of the milling head. The sides of the cover sheets contact the auxiliary tray which is covered with copper sheet. The auxiliary tray contacts the two cover sheet and slides between them as the milling head moves. Therefore, the cover sheets limit the deforma-tion of the milling head and improve the machine tool stiffness normal to the plane of the manipulator motion. Four brackets are also added to the new machine tool to improve the stiffness. Two additional non-rotating milling heads were de-signed to replace the 2-DOF milling head for mounting on the side or underside of the moving platform. This heavy duty machine tool is then a 3-DOF machine which can machine vertically or horizontally. The task workspace tW of the new parallel manipulator was designed as a rectangle b=3000 mm in width and h= 1800 mm in height with max79=?, min5=?. The manipulator can not reach singular con-figurations in the task workspace. The optimal design based on the task workspace re-sults in 2342.5R=mm. has a minimum value when 1l=3550 mm and 0.1w=. ,max,min|iiyy can then be determined from Eq. (26). The main design parameters are listed in Table 1. Table 1 Results of optimization for XNZD2430 ma-chine tool ParametersValuesl1 l2 r ,max,min|iiyy 3550 mm 3550 mm 550 mm 4010.7 mm 1.99 The distribution of for the optimal design is shown in Fig. 8. The distribution shows that the condi-tion number is also symmetrical with respect to the x axis and the kinematic performance is optimal. Fig. 8 Condition number of the Jacobian of XNZD2430 machine tool The XNZD2430 machine tool shown in Fig. 9 is currently being built by Tsinghua University with the machines dynamics and controls to be tested soon. Fig. 9 Prototype of XNZD2430 machine tool WU Jun (吴 军) et al：Optimal Kinematic Design of a 2-DOF Planar 2756 Conclusions The kinematic design of a 2-DOF planar parallel ma-nipulator has been investigated. The investigation is shown as follows. (1) The distance between the two columns is deter-mined by the optimum design based on the workspace. (2) A global, comprehensive performance index, which includes both the mean and the range of the condition number of the Jacobian in the task work-space, can be used to effectively optimize the kine-matic design. (3) The planar parallel manipulator-based machine tool was successfully used to machine blades and guide vanes for a hydraulic turbine. Another heavy duty machine tool based on the parallel manipulator is being built for larger machining tasks. References 1 Beqon P, Pierrot F, Dauchez P. Fuzzy sliding mode
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