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XX设计(XX)外文资料翻译院 系专业学生姓名班级学号外文出处The International Journal of Advanced manufacturing Technology附件：1外文资料翻译译文（约3000汉字）； 2外文资料原文（与课题相关的1万印刷符号左右）。指导教师评语：指导教师签名： 年 月 日附录A：外文资料翻译译文对由棱柱体-棱柱-球形组合旋转并联而成的机械手的六个自由度的工作空间的分析M. Z. A. Majid, Z. Huang and Y. L. YaoDepartment of Mechanical Engineering, Columbia University, New York, USA摘要：本文主要讨论了一般棱柱体-棱柱-球形类型的并联机械手的6个自由度的工作空间。我们知道，并联机械手一个缺陷就是它有限的工作空间。所以，一个潜在的应用就是扩大它的有限工作区。在这里，首先对此类型的机制与机制的变化作了简单的分析，然后对工作空间作了研究，并且把工作空间的形状和大小对联合限制和肢体干扰限制作了大量的研究。结果表明，此种并联机械手的工作空间后远远大于同等的Stewart平台上的工作空间，特别是在竖直方向上。关键字：并联机械手；Stewart平台；工作空间1介绍：在最近几十年，许多研究者显示了对并联机械手的兴趣。相比于更加普遍和常用的串联机械手，并联机械手在精度，硬度、容量和装载对重量比率上有更大的优势。并联机械手由一个移动的平台，一个基本平台和几个通过恰当的运动关节和控制器来连接两个平台的支架构成。最著名的并联机械手是被广泛研究的Stewart平台。在此平台中，6个杠连接，并能移动，基本平台能够控制移动平台的位置与方向。许多不同的6自由度并联机械手也被提及。最近Tahmasebi and Tsai发明了一款新颖的并联机械手（图1）。它包括上下2个平台和3个可扩展的肢。每个肢的末端通过舵机的球状关节来连接。舵机是直线步进类型，但能在基本平台上实现x 和y方向的同时移动。上肢的末端则通过旋转关节与移动平台相连。因此这个机械手是一个3PPSR机制，其中，P代表柱体，S代表球体，R代表旋转体。2维步进电动机的输出动作在基本平台上和交叉的柱体相似，需要在上平台上获得的动作就通过移动基本平台上的舵机来实现。基本平台还连接着3个下肢的末端。除了前面提到的一般并联机械手相对于串联机械手的特性以外，此种3PPSR机制还有 些其他的优点，比如更简单的结构和更高的刚度。由于在Stewart 平台上用3个可扩展的肢代替了6个可扩展的肢，因此，肢之间发生干涉的可能性很小。Tahmasebi和Tsai察觉到，此种机制能够安装在串联机械手的手腕与结束效应之间，用于误差补偿和微小的位置变化以及力的控制。所以当工作空间被考虑的时候，由于只需要很小的工作空间，每台舵机的动作也只能限定在一块很小的圆面积之内。尽管如此，载着肢的这些舵机不用如此的严格受限，它们能在整个基本平台上运动从而获得较大的工作区域。所以，这种3PPSR机制能够作为独立的机械手来用。另外，它特殊的装配运动学配对让它能在同等的Stewart平台上获得截然不同的和更大的工作空间。对机械手的工作空间的研究是设计机器人手臂的基本问题。很多学者指出，并联体制的主要缺陷就是它有限的工作空间。而3PPSR这种并联机制克服了传统机械手的这个缺陷，并拓展了并联机制的应用。本文分析了这种3PPSR并联机械手的工作空间的大小，形状，构成和限制因素。 并联机械手的工作区域在过去十年中吸引了很大学者的注意。 有不少报道说2个或3个自由度的平面或球面机器人用到了并联机制工作空间。Asada and Ro 7 and Bajpai and Roth 8分析了闭合回路平面2个自由度5杆的并联机制。Gosselin and Angeles 9,10分析了平面和球面3自由度机制的工作空间。Lee and Shah 11 and Waldron et al.12则阐述了空间3自由度并联机制的工作空间。 关于6自由度并联机制的工作空间的研究较少，Yang and Lee 13, Fichter 14,and Merlet 15用一种基于笛卡尔空间的方法描述了6自由度并联机制。Gosselin 16从几何角度介绍了一种决定6自由度 平台工作区域的算法。他的结果表明，工作空间就是6个环形区域的交汇点。Masory and Wang17则更系统的研究了6自由度的Stewart平台。他们讨论了几个用于计算Stewart平台工作区域的限定条件，包括运动学配对的转角度的区域和机制里任意两个肢之间的干涉。另外，他们还分析了工作空间的形状和工作空间与机制几何参数之间的关系。Tahmasebi and Tsai 6还研究了一种新的3PPSR并联机械手，其中，由舵机连接的每个下肢的末端的动作都被限制在很小的圆面积范围内。在本文对工作空间的分析中，肢之间的干涉和联动限制都是考虑过的，而且舵机能在一个较大的直径范围内移动。通过构成机械手的不同类型来确定制宪地区，从而确定了工作空间的组成。2 机制分析 如上所述，3PPSR机制的上下平台通过三个确定的肢用如下的运动学配对来连接：两个棱柱，球形一双和一个 旋转体（图2a）。一个球形关节等同于三个旋转关节和一个共通点，用RRR来表达。因此，如图2b所示的这个PPRRRR系统同PPSR系统在运动学上是等价的。PPSR系统同图2c所示的肢也等价。这种肢在每个上肢和下肢的末端都有两个万向节，其中，上万向节的一根轴线与肢共线，而另一根轴线和下万向节的一根轴线则和肢相垂直。总之，图2a,b,c所示的PPSR, PPRRRR, 和 PPUU这3种结构在运动学上是等价的，其中，u代表万向节。如图2d所示的PPRS结构和先前的类似，通过改变PPSR的球形和旋转体而获得。尽管如此，它同PPSR系统在运动学上不能等同，与图2中的e,f在运动学上是等价的。因此a,b,c这三张图同d,e,f这三张图看起来像，但其实不一样。 螺旋理论18,19用于辨别这两个系列的结构。旋转关节的每根轴线都可以用零间距的螺丝来表达。我们用分别属于第一和第二系列的PPRRRR系统和PPRRRR作为例子。两个系统都定义了oxyz参照系，原点是由同2维舵机对应的两个方向的两条直线相交所得。x,y轴与两个棱柱的移动方向共线。Z轴则由右手法则确定，并与基本平台相垂直。假设肢同z轴不共线，两个系列的螺丝系统可能已如下形式出现：拧紧系统1 (为PPRRRR)螺丝系统2 (为PPRRRR) 从这两个螺丝系统我们能看出：每个螺丝系统的六个螺丝是线性独立的，因为雅克比矩阵并没有去除这里，我们就能明白为什么上或下万向节的一根轴线必须与肢共线，如图2c,f所示。如果条件不满足，雅克比矩阵就变成奇异矩阵。例如，如果$R15与肢不共线（图2c），而与y轴平行，它的螺丝变成$R15 = (0 1 0; 21 0 0)。因此，由于线性独立，包含$9的新雅克比矩阵将会成为奇异矩阵。很容易看出，$9与$P11 和$R14.线性相关。如果所有并联机械手的三只腿都不满足这个条件，那么它至少会失去三个自由度。注意，如果最后一个旋转体轴线，在PPRRRR (或PPSR)系统中的$R16与z轴相交，则$R16的最后一部分就为零。也就是说，a,b,c,d为任何实数，而且a,b,c和d,e不能同时为零。这里有一组数据：(0 0 1; 0 0 0)同螺丝系统1的6个螺丝是对等的，也就是说螺丝系统1是线性独立的，等价于5个螺丝的系统。这个系统失去了一个自由度。整个3PPSR系统都将是奇异矩阵，即使3个肢中只有一个满足此条件。另一方面，由于最后3个不共面的旋转体与球形等价，因此，这种类型的奇异矩阵不会在螺丝系统2中出现。只可能3根轴线中的2根与z轴同时相交，但球形的第三根轴线就绝不可能与z轴相交。 虽然他们线性独立，并且都有六个自由度，这两个系统是不一样的。结果，包含这两个螺丝系统的并联机制，如图3所证。这两个机制有相同的几何属性，包括腿长和移动三角形，唯一的区别是一个用PPSR，另一个用PPRS。假上下两个平台最初是平行的，让两个上平台从最初位置旋转到同样的量，分别用B2,B4和b2,b4表示。容易看出，由这两个机制所产生的位置是不同的3 工作空间分析如图1所示，固定的参考系OXYZ处于基本平台上，原点就是以直径d为大圆的圆心。X,Y轴都在基本平台上，Z轴则与基本平台垂直。动参照系处于移动平台上。等边三角形的圆心就是G点。U轴与直线P2P3平行,v轴穿过点P1。W轴与移动平台平行。为了确定机制的工作空间，我们需要正运动学的一些知识。同样，当涉及到并联机制时，就会用到逆运动学，但逆运动学需要使用数值解。如果已知机械手的位置和方向，那么上平台的参考点就能确定工作区域内的某点，前提是它必须在逆运动学所限定的条件下。当给了一系列点后，就相应的在上平台上获得了一系列点，这样，工作空间就得到了所有的点组成的一个集合。3.1 逆运动学当上平台的位置和方向都知道时，通过坐标传递，就能得到移动平台上点的坐标。通过欧拉的三个方向就确定了移动平台的方向。而G点的坐标则与静参照系中的Xg，Yg，Zg位置对应。坐标传递矩阵是：而P1，P2,P3点的坐标和参照系Guvw中的点对应： 其中，m =12cos 30.同样，对应于静参照系的Pi点是： 从图1中机械手的几何关系中不难得出，能同时得到两个等式，第一个是： 其中，K就是Ri点的Z坐标，而p,i,r中，i分别表示Pi和Ri。这是这个大圆的一个基本公式，第二个公式是这样得到的：由于Pi点是旋转关节的，而Ri点则是圆Eq的交汇点，这样就得到了相互垂直的两个矢量RiPi和Pi+1Pi+2。表示如下： 方程式是 联立方程（6）和（8）解得Xr,i和Yr,i因为(6)是二阶多项式因式，所以x,y可以有两组解，只要它们满足联动限制和干涉的条件，并在足迹范围内，那么这两组解就是有效的。3.2 运动学中的约束为了确定3PPSR机械手的工作空间，我们需要考虑三种不同类型的运动学约束。它们是足迹圆的直径，关节角度的限制和连接干涉。足迹圆：图一所示三个下肢的末端都必须在足迹圆之内，即 其中，d是足迹圆的直径，ORi则为矢量半径关节角度的限制：通过有物理极限的运动学配对，将上，下板连接。例如，一个球状关节，理论上在正交坐标轴中可以旋转360度，但是实际上由于物理结构上的限制，它的运动范围要相对减小一点。因此有必要求出每个关节的最大旋转角度。旋转角度和它的极限能用如下公式表示： 其中Vi就是矢量直线Li,而用以平分对应于静参照系中运动学配对的旋转范围的直线就是Ui连接干涉：由于物理尺寸的存在，因此可能会发生干涉。假设每条直线是圆柱的直径d而D是两条相邻直线的最短距离，那么干涉极限可以这样来表示：则两条中心线间的最短距离就是Ni的位移Dn,即其中，单位矢量Ni位移方向在相邻直线Li和Li+1之间，即注意，两条直线之间的最短距离不是总等于Dn的位移长度。有可能大于Dn。如果Li上的交汇点Ci与两直线的位移超出了直线Li，或者Li上的交汇点Mi和垂线Pi+1Li超出了直线Li本身，那么最短距离就是两个端点Pi和Pi+1之间的长度，如果Mi和Mi+1两个交汇点都不在Li和Li+1上的话三条直线，包括两条相邻直线之间的位移，定义了两个平面，两个平面向量是：两个平面方程是一条直线在三维空间可以表示成：等式中的两条中心线可以用下面的方程组来求解：其中，方程(19) (20)代表中心线Li,方程（21）（22）代表中心线Li+1。同时联立方程（17）（19）（20），就可以求得直线Li和位移之间的交汇点Ci。同理，联立（16）（21）（22）就能得到直线Li+1上的交汇点Ci+1.正如上文所分析的那样，通过两个平面确定了6条直线，那么如果满足和，干涉就不会发生4. 数例和讨论关于工作空间的限制条件已经在前边探讨过，假设每个链接都是圆柱形的，几何参数是r = 2.5 units, l = 1.0 units, d = 6 units, = 0.15 units, = 75, = 60, 和 k = 0, ,k其中 r代表PiRi的长度，l是移动三角形每条边的长度，d是基本平台上足迹圆的直径， 和 是旋转关节和球状关节的最大角度，而k代表Ri点的z坐标。三维空间可以用两个图像来表达，比如一个二维的上视图好一个三维的等距视图，为了视图的方便，不用表达上下的边界部分，由于工作空间涉及到位置和方向，所以三个不变的欧拉角度必须指定为每一个如下情况，为了证明不同条件下对工作空间的形状和大小限制的影响，我们列举了下面5种情况：注意，由于这个平台关于移动平台的v轴对称，所以 = 0, 20, 0 就等于 = 0, 20, 0，另外，由于工作空间并没有改变，所以5种情况下仍然是零。只要形状不变，工作空间将会随着不为零的改变而改变。 图4表示了工作空间上视图的第一种情况， = 0, 0, 0.在不考虑干涉和关节角度限制的条件下，图5表达了理论工作空间的等距视图，在不考虑运动学限制的情况下，图6表达了上述相同的5种条件下的实际工作空间。可以看出，这种3-PPSR并联机械手的工作空间的形状和结构不同于传统的并联机械手，他允许在z方向上有较大的动作范围。 应当指出，图5和6所得到的工作空间是在为零的情况下，由于 能够旋转360度所以实际的工作空间是将如图5和6绕z轴旋转同样的角度而得。对比同等的蘑菇帽形状的 平台，它的工作空间是圆柱形的，所以有较大的z轴范围 工作空间通过详细的检查来得到它的构成。我们根据不同类型的机械手来划分工作空间的制宪地区，从而完成检查。以图4所示的工作空间作为例子。四个不同类型的制宪地区都能被确定Z=1.0. 第一个区域：如图8a所示，它的形状对应着一条腿朝着平台，另外两条则向外，同理也可以是3条腿之间相互轮换，很类似，只是在原来的 基础上旋转120度第二个区域：如图8c所示，它的形状对应着两条腿向里，一条向外，同理也可以是三条腿之间的轮换，仍是在原来基础上转120度第三个区域：如图8e所示，它的形状对应着三条腿向里 图5 图6 第四个区域：它的形状和图e类似，只是三条腿都向外了这种区域对于工作空间没有实际的影响，因为它是类型3的子类型。所有的这些区域和直径为6的足迹圆都绘制在图9中。很明显，这些区域的交叉形成了一块面积，就像图4所确定的一样，所以，他们都应该是工作空间的制宪地区。图8图75.结论本文主要分析了3-PPSR机械手的工作空间。工作空间包括三种区域，每个都对应着不同类别的机械臂。运动学限制的影响则包括：旋转关节和球状关节，工作空间结构的肢的干涉。3-PPSR机械手的工作空间是圆柱形的，而 平台则通常是蘑菇帽形的工作空间，它在z轴方向的动作受到较大的限制。参考文献1. D. Stewart, “A platform with six degrees of freedom”，Proceedings Institution of Mechanical Engineers, 180, pp. 25-28, 1965.2.F. Tahmasebi and L.-W. Tsai, “On the stiffness of a novel six-DOF parallel manipulator”, Intelligent Automation and Soft Computing, Proceedings of the First World Automation Congress (WAC，94), vol. 2, pp. 189-194, 1994.3.F. Tahmasebi and L.-W. Tsai, “Closed-form direct kinematics solution of a new parallel manipulator”, Journal of Mechanical Design, 116, pp. 1141-1147, 1994.4.L.-W. Tsai and F. Tahmasebi, “Synthesis and analysis of a new class of six-DOF parallel manipulators”, Journal of Robotic Systems, 10, pp. 561-580, 1993.5.F. Tahmasebi and L.-W. Tsai, “Jacobian and stiffness analysis of a novel class of six-DOF parallel manipulators”, Proceedings of the 22nd Biennial Mechanisms Conference, ASME, DE-vol. 47, pp. 95-102, 1992.6.F. Tahmasebi and L.-W. Tsai, “Workspace and singularity analysis of a novel six-DOF parallel manipulator”, Journal of Applied Mechanisms and Robotics, 1(2), pp. 31-40, 1994.7.H. Asada and I. H. Ro, “A linkage design for direct drive robot arms”, Journal of Mechanisms, Transmissions and Automation in Design, 107, pp. 536-540, 1985.8.A. Bajpai and B. Roth, “Workspace and mobility of a closed- loop manipulator”, International Journal of Robotics Research, 5(2), pp. 131-142, 1986.9.C. Gosselin and J. Angeles, “The optimum kinematic design of a planar three-DOF parallel manipulator”, Journal of Mechanisms, Transmissions and Automation in Design, 110, pp. 35-41, 1988.10.C.Gosselin and J. Angeles, “The optimum kinematic design of a spherical three-DOF parallel manipulator”, Journal of Mechanisms, Transmissions and Automation in Design, 111, pp. 202-207, 1989.11. K. M. Lee and D. K. Shah, “Kinematic analysis of a three-DOF in-parallel actuated manipulator,” IEEE Journal of Robotics and Automation 4(3), 354-360, 1988.12.K. J. Waldron, M. Raghavan and B. Roth, “Kinematics of a hybrid series of parallel manipulation system”, ASME Journal of Dynamic System Measurement and Control, 111, pp. 211221, 1989.13.D.C. H. Yang and T. W. Lee, “Feasibility study of a platform type of robotic manipulators from a kinematic viewpoint”, Journal of Mechanisms, Transmissions and Automation in Design, 106, pp. 191-198, 1984.14.E.F. Fichter, “A Stewart platform-based manipulator: general theory and practical construction”, International Journal of Robotics Research, 5(2), pp. 157-182, 1986.15.J. P. Merlet, “Force-feedback control of parallel manipulator”, IEEE International Conference on Robotics and Automation, pp. 1484-1489, 1988.16.C. Gosselin, “Determination of the workspace of six-DOF parallel manipulator”, Journal of Mechanical Design, 112, pp. 331-336, 1990.17.O. Masory and J. Wang, “Workspace evaluation of Stewart platform”, Proceedings ASME Winter Annual Meeting, DE-45, pp. 337-352, 1992.18.R. S. Ball, Theory of Screw, Cambridge University Press, 1900.19. K. H. Hunt, Kinematic Geometry of Mechanisms, Oxford, Clarendon Press, 1978.附录B：外文资料原文Int J Adv Manuf Technol (2000) 16:441-449 2000 Springer-Verlag London LimitedThe International Journal of Advanced manufacturing TechnologyWorkspace Analysis of a Six-Degrees of Freedom, Three- Prismatic-Prismatic-Spheric-Revolute Parallel ManipulatorM. Z. A. Majid, Z. Huang and Y. L. YaoDepartment of Mechanical Engineering, Columbia University, New York, USAAbstract：This paper studies the workspace of a six-degrees-of-freedom parallel manipulator of the general three-PPSR (prismatic- prismatic-spheric-revolute) type. It is known that a drawback of parallel manipulators is their limited workspace. To develop parallel mechanisms with a larger workspace is of use to potential applications. The mechanism of a three-PPSR manipulator and its variations are briefly analysed. The workspace is then investigated and the effects of joint limit and limb interference on the workspace shape and size are numerically studied. The constituent regions of the workspace corresponding to different classes of manipulator poses are discussed. It is shown that the workspace of this parallel manipulator is larger than that of a comparable Stewart platform, especially in the vertical direction.Keywords: Parallel manipulator; Stewart platform; Workspace1.IntroductionIn the past decade, many researchers have shown an interest in parallel manipulators. Compared with the more commonly used serial manipulators, the parallel ones have advantages in accuracy, rigidity, capacity, and load-to-weight ratio. A parallel manipulator consists of a moving platform, a base platform and several branches connecting both platforms through appropriate kinematic joints with appropriate actuators. The best known parallel manipulator is the Stewart platform 1, which has been widely studied. In a Stewart platform, six bars connecting moving and base platforms are extensible to control the position and orientation of the moving platform.Many different 6-degrees-of-freedom (DOF) parallel manipulators have been proposed. Recently, Tahmasebi and Tsai 25 introduced and studied a novel parallel manipulator (Fig. 1).Correspondence and offprint requests to: Dr Y. Lawrence Yao, Department of Mechanical Engineering, Columbia University, 220 Mudd, MC 4703, New York, NY 10027, USA. E-mail: ylyl columbia.eduThis mechanism consists of an upper and a lower platform and three inextensible limbs. The lower end of each limb connects through a ball-and-socket joint to an actuator. The actuator is of a linear stepper type but is capable of moving in both x- and y-directions simultaneously on the base platform. The upper end of each limb is connected to the moving platform by a revolute joint. The manipulator is therefore a 3PPSR mechanism, where P denotes the prismatic pair, S the spherical pair, and R the revolute pair. The output motion of the 2D linear stepper motors is similar to that of two cross- prismatic pairs on the base platform. The desired motion of the upper platform is obtained by moving the actuators on the base platform, to which the lower ends of the three limbs are attached. Besides the merits of general parallel mechanisms over their serial counterparts mentioned before, this 3PPSR mechanism has added advantages, including simpler structure and higher stiffness. It is also less likely that its limbs will interfere with each other, since it has only three inextensible limbs instead of six extensible limbs as in a Stewart platform.Fig. 1. A 3-PPSR parallel manipulator.442 M. Z. A. Majid etal.Fig. 2. Variations in pair sequence and type. (a), (b), and (c) form a kinematically equivalent set, whereas (d), (e), and (f) form another.Tahmasebi and Tsai perceived this mechanism as being used as a minimanipulator, which can be mounted between the wrist and the end-effector of a serial manipulator for error compensation as well as for delicate position and force control. Therefore, the required workspace is rather small so that the motion of each actuator is limited to within a small circular area on the base platform when its workspace is considered 6. The actuators carrying the limbs, however, do not have to be so restricted, they can move over the entire base platform, resulting in a much larger workspace. As a result, this 3PPSR mechanism can be used as a stand-alone manipulator. In addition, its special assembly of kinematic pairs makes it possible to have a workspace that is very different from and larger than that of a comparable Stewart platform. The study of workspace of a manipulator is one of the fundamental problems in the design of robot arms. As many researchers have pointed out, the major drawback of parallel mechanisms is their limited workspace. This 3PPSR parallel mechanism can help overcome the limitations of traditional parallel manipulators and extend the applications of parallel mechanisms. This paper analyses the size, shape, composition, and constraints of the workspace of the 3PPSR parallel manipulator.workspace of parallel manipulators has attracted the attention of many researchers over the past decade. Much reported work on parallel mechanism workspace dealt with 2DOF or 3DOF planar and spherical manipulators. Asada and Ro 7 and Bajpai and Roth 8 analysed the workspace of a closed-loop planar 2DOF 5-bar parallel mechanism. Gosselin and Angeles 9,10 studied the workspace of planar and spherical 3DOF mechanisms. Lee and Shah 11 and Waldron etal. 12 demonstrated the workspace of a spatial 3DOF in-parallel manipulator.Much less work has been reported for the workspace of 6DOF parallel manipulators. Yang and Lee 13, Fichter 14, and Merlet 15 described the workspace of 6DOF parallel manipulators, using a method based on discretisation of the Cartesian space. Gosselin 16 used geometric properties to introduce an algorithm for determining the workspace of a 6DOF Stewart platform. His results showed that the workspace was the intersection of six annular regions. Masory and Wang 17 more systematically studied the workspace of a 6DOF Stewart platform. Their report discussed several constraint conditions for calculating its workspace, including the region of the angle of rotation of kinematic pairs and the interference between any two limbs of the mechanism. In addition, they analysed the shape of the workspace and the relationship between the workspace and the geometric parameters of the mechanism. Tahmasebi and Tsai 6 studied the workspace of this new 3PPSR parallel manipulator, where the motion of each of the three actuators attached to the lower end of each limb is limited to a small circular area. In the workspace analysis presented in this paper, limb interference and joint limitations are considered, and the actuators are allowed to move within a larger circle of diameter d (Fig. 1). The composition of the workspace is also studied by identifying the constituent regions according to different classes of manipulator poses.2.Mechanism AnalysisAs mentioned before, the upper and lower platforms of a 3PPSR mechanism are connected by three identical limbs each with the following kinematic pairs: double prismatic pairs, one spherical pair and one revolute pair (Fig. 2(a). A spherical joint is kinematically equivalent to three non-coplanar revolute joints with a common point denoted as RRR. Thus the PPRRRR system shown in Fig. 2(b) is kinematically equivalent to the PPSR system. The PPSR arrangement is also kinematically equivalent to the limb shown in Fig. 2(c). This limb has a 2DOF universal joint at each of its lower and upper ends, where one of the axes of the upper universal joint is collinear with the limb, whereas another axis of the upper universal joint as well as one of the axes of the lower universal joint are always perpendicular to the limb 6. In summary, the three structures: PPSR, PPRRRR, and PPUU, as shown in Figs 2(a), 2(b) and 2(c), are kinematically equivalent, where U denotes the universal joint. A similar structure, PPRS (Fig. 2(d), can be obtained by exchanging the spherical pair and the revolute pair of the PPSR system. It is, however, kinematically different from the PPSR system, as shown below; but the PPRS system is kinematically equivalent to the two systems shown in Figs 2(e) and 2(f). Therefore, the set shown in Figs 2(a), 2(b), and 2(c) look similar but different from the set shown in Figs 2(d), 2(e) and 2(f).Screw theory 18,19 is used in determining the difference between the two sets of structures. Every axis of the revolute joints can be expressed as a screw with zero pitch. The PPRRRR system (Fig. 2(b), which belongs to the first set, and PPRRRR (Fig. 2(e), which belongs to the second set are taken as an example. A reference frame oxyz is defined for both PPRRRR and PPRRRR. The origin o is located at the intersecting point of two lines corresponding to the two directions of the 2D actuators. The x and y-axes lie collinear with the moving directions of the two prismatic pairs. The z-axis is defined by the right-hand-rule perpendicular to the base platform. Assuming that the limb is not collinear with the z-axis, two sets of screw systems may be given as follows:Screw system 1 (for PPRRRR) Screw system 2 (for PPRRRR)From these two screw systems one can see that the six screws of each screw system are linearly independent, since their Jacobian matrices do not vanish Here,one can see why one of the axes of the upper (lower) universal joint must be collinear with the limb, as shown in Figs 2(c) (2(f). If this condition is not satisfied, the Jacobian matrix will become singular. For instance, if $R15 is not collinear with the limb (Fig. 2(c) but is parallel with the y-axis, its screw becomes . The new Jacobian matrix involving instead of in the screw system 1 will be singular owing to linear dependence. It is easy to see that is a linear combination of and . If all three legs of the parallel mechanism do not satisfy this condition, it will lose at least two degrees of freedom.Note that, if the axis of the last revolute pair, ，of system PPRRRR (or PPSR) intersects the Z-axis, the last component of will be zero. That
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