机械类外文翻译【FY209】具有2或3个自由度的对应机械手【中英文WORD】【中文4700字】
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具有 2 或 3 个自由度的对应机械手 Christiaan J。 J。 Paredis, H。 Benjamin Brown, Pradeep K。 Khosla 摘要 : 为了 更好的应用 ,机床工业的对应机械手获得了广泛的研究 ,然而 ,有限的工作空间 ,较差的灵活性 ,复杂对应机械手的难于设计 ,导致人们把目光投向于少于 6个自由度的对应机械手 ,本篇论文描述了几个在自由度的数量和类型上都不相同的对应机械手 ,这些对应机械手可被用语对应运动机器 ,运动模拟器和工业机器人 。 关键词 : 对应机械手 ,对应运动机械 ,机器人 。 引言 允许一个刚体可相对于固定底座运动的机构 ,在许多领域扮演着许多非常重要的角色 。 一个刚体可以有几个平移和旋转与动 ,该平移和旋转与动称为刚体的自由度 。 刚体的自由度最多 6 个 。 也就是说 ,对于 3 个互相垂直的坐标轴的 3 个平移 ,和 3 个旋转 。 一个机器人包含一个系统 ,用以控制末端受动器的数个自由度 。 最近几年 ,有证据显示 ,工业机器人 ,因其灵活性 ,在应用上得到了推广 。 然而 ,普通 的机器人却不能用来完成某些任务 ,因此 ,最近为了工业上的使用 ,其他类似的机构 ,包括对应机械手得到了发展 。 一种 对应机械手 ,即一个由几个连接杆构成的封 闭环机构 ,通常包含一个运动平台 ,若干下肢或腿 ,后者把运动平台连接到固定基础上 。 通常 ,下肢的数量等于自由度的数量 。 一个下肢受一个 驱 动装置控制 ,所有的驱动装置 安装在固定基础上或其附近 ,对应机械手有时也称作平台机构 ,因为外部载荷能被各驱动装置分担 ,对应机械手具有较大的承载能力 。 它常用于要求精度高 ,刚性好 ,和承载能力大的场合 ,他们在航天和航行模拟器上都得到了广泛的应用 ,在机床工业的知名度也越来越大 。 nts 图 1 对应机械手概念的提出可追溯到 1947 年 ,那时 ,高 夫创立了有一个封闭环的运动机构的基础原理 ,在那时 ,是为了在测试轮胎磨损时 ,控制运动平台的位置和方向 。 在 1955 年 ,他 创立 了一个模型 （ 图 1） ,在该模型里 ,受动器是一个六角平台 。他的 6 个顶点 ,分别以球型铰链形式 ,与连接杆相连 。 连接杆的另一端 ,以通用的铰链连接 ,连接到基础上 。 6 个线性驱动装置可调整连接杆的长度 。 斯图尔特 ,在 1965年 ,设计了一个平台机构 ,作为一个飞行起模拟器 （ 图 1b） ,在其中 ,受动器是一个三角平台 ,它的顶点同样 ,用球型铰链连一连接杆 ,连接到支持机构上 ,每个支持机构 喊又 2 个支撑杆 ,支撑机构是三角形布 置 。 在 1978 年 ,亨特 ,以空间 3-RPS 对应机械手为例 ,对对应机械手的运动机制 做了体统的研究 。 从那时起 ,又有很多人多次对对应机械手记性了广泛的研究 。 直到如今 ,所研究的大多数是 6-自由度对应机械手 ,他们都有 6 个可伸缩的下肢 。 这些对应机械手具有高的刚 度 ,低惯性 ,大的承载能力 。 然而 ,他们受到有限使用空间的限制 ,和难于设计的困扰 。 而且 ,对他们的运动也很难进行分析 。 因此 ,在工业领域 ,人们越来越关注少余个自由度的对应机械手 。 这篇论文引入了对应机械手的概念 ,机器分类 。 几扫了 3 种新对应机械手 ,新空间 3-自由度对应机械手 ,新 2-自由度平移平台 ,新平面 3-自由度系列对应机构 。 nts1 对应机械手的定义 对应机械手由一个具有 n 个自由度的受动器 ,基础组成 ,基础和受动器之间 用 2 或 3 个独立运动连接杆相连 ,驱动器担负驱动任务 。 这些机构有下列特性 : 至少有 2 个连接杆支持受动器 ,每个连接杆至少陪有一个简单的受动器 。 驱动器 的数量与受动器的自由度的数量相等 。 当受动器自锁时 ,机械手的可动度为零 。 对应机械手因下列原因引起人们的兴趣 : 载荷可被分配到多哥连接杆上 。 需要数个驱动器 。 驱动器自锁时 ,机械手停止运动 ,停在 原来的位置不动 ,从使 用的安全角度讲 ,这是很重要的 。 对应机械手 ,因其连接杆的数量完全等于受动器的自由度的数量 ,即一一对应而得名 。 2 机构的自由度 机构的自由度是所需的独立参数 ,或输入的参数 ,目的是为了定义机构的结构 。 然而 ,如亨特和 Lerbet 所说 :可动度的准则是很难定义的 。 典型的可动度公式忽略了某些自由度 。 现在 ,人们仍在使用 Grubler 的公式 : M=d（ n-g-1） +gnfi1（ 1） 式中 : M 可动度 ; D 螺钉系统顺序数 （ 对于平面和球型的运动 ,d=3;对于空间运动 ,d=6） ; n 连接杆的数量 ,包括框架 ; j 铰链的数量 ; fi 第 i 个连接杆的自由度 。 3。 对应机械手的分类 一个刚体有 6 个自由度 ,对应机械手的自由度在 2 到 6 之间 。 自从第一个机nts械手设计开始 ,又推出了许多个有 2 到 6 个自由度的机械手 。 对文献所记载的 87个驱动装置的调查显示 ,6 个自由度的对应机械手占 40%,5 个自由度的占 3。 5%,4个自由度的占 6%,3 个自由度的占 40%,余下的为 2 个自由度的 。 3.1 2-自由度对 应机械手 大多数现存的 2 个自由度的机械手是平面的 ,有 2 个平移自由度 。 这样的设计采用棱柱状的能移动的铰链 。 Mccloy 提出了 20 个不同的组合 。 驱动器系统在地上 的有 6 个 ,如图 2 所示 。 没有被动的棱柱铰链 ,任何一个驱动器也不承受另一个驱动器的重量 。 图 2 3.2 3-自由度对应接携手 有许多 3-自由度的机械手 。 这里只举几个典型的例子 。 图 3a,平面 3-移动铰链对应机械手 。 运动平台有 3 个平面自由度 ,沿 X、 Y 轴的平移和 Z 轴的旋转 。 图 3b,球面的 3-铰链连接杆对应机械手 。 三个铰链连接杆汇交于一点 。 机构上任何一点的运动是绕共 同焦点的旋转 。 图 3c,3-RPS 机械手 ,由亨特设计 ,有综合的自由度 ,很难定义 。 图 3d,DELTA 机械人 ,由 Clave 设计 ,由 Demaurex 公司和 ABB,以 IRB 340 Fpexpicker 的名字推向市场 ,DELTA 在工业上得到了广泛应用 。 图 3e,有被动腿的机械手的例子 ,运动平台有 4 条腿 ,第 4 条腿是被动的 ,也是主要的 ,决定运动平台的运动 ,例如 ,本图显示的球坐标对应机械手 。 该对应机械手被 Hannover 大学的 IFW 用于机床设计 。 nts 图 3 3.3 4-自由度对应机械手 对于一个 4-自由度全对应机械手 ,d=6,n=10,g=12,M=4。 把这些值代入公式 （ 1） ,得出 F=22/4,即每个腿有 22/4 的自由度 。 因此 ,实际上 ,不可能有 4-自由度的全对应机械手 。 早期的 4-自由度机构并不是全对应机械手 ,也就是说 ,机械手的每个连接杆配有两个驱动器或者有被动约束 。 3.4 5-自由度对应机械手 5-自由度 全对应机械手的 F=29/5,所以 ,实际上 ,不可能有 5-自由度的全对应机械手 。 由 Austad 提出的 5-自由度对应机械手 ,实际上包含 2 个对应机械手 。 3。 5 6-自由度对应机械 手 6-自由度对应机械手是最流行的 ,并得到广泛的研究 。 图 4a 显示的是一个典型的6-自由度对应机械手 。 大多数 6-自由度对应机械手都有 6 个可伸缩的下肢 。 这些对应机械手具有高的刚 度 ,低惯性 ,大的承载能力 。 然而 ,他们受相对小的可用空间限制和难于设计的困扰 。 而且 ,对它们的运动很难进行分析 。 也有一些奇特的链机械手 ,这些机械手被一个平面机构驱动 ,例如 ,4 杆机构或 5 杆机构 ,或每条腿有两个驱动趋的 ,通常有 3 条腿 。 nts 图 4 4 对应机械手的评价 高夫在 1947 年建立了如图 1a 所示的 ,有封闭环的运动机构的原理之后 ,推出了许多自由度数量和类型不同的对应机械手 。 图 1b 的结构是 1965 年由 Steward设计的 。 图 4a 所示的通用机械手 ,从理论上讲 ,他的 6 个腿可以任意布置 ,设计成各种各样的 6-自由度对应机械手 ,例如 ,图 4b所示的 ,6条腿按 3-2-1配置 ,结构紧凑 ,可用在微观系统 。 图 4c 所示的 ,是机械手按规定的方向运动自如 ,在工业上应用广泛 。 图 4a 所示的通用机械手 ,类似于 Pierrot 提出的 ,每对的两条腿互相平行 。 即使每一对的两条腿的输入是相同的 ,机械手自由度的数量也是不同的 （ 机构不同所致 ） 。 与此相当的机械手显示在图 5。 该机械手输出的是 3 个平移 ,几乎是 DELTA机器人的初型 。 驱动的连接杆可以按众所周知的 ,如图 3d 所示的 ,快速机器人DELTA 的布置 。 DELTA 有几个变体 ,例如 ,Pollard 机构 ,Tsai 机器人 ,如图 3 f 所示的 ,也是 3-平移自由度对应机器人之一 。 虽然 Tsai 机器人与 DELTA 一样 ,也有 3个平移 ,准确的说 ,他不是 DELTA 的变体 。 它们的设计观念是不同的 ,Tsai 机器人是用来处理 UPU 链的问题的第一个设计 。 另一个 3-平移自由度对应机器人 Star是由 Herve 以群论为基础设计的 。 这些设计观念对设计新机械手提供了新的想法 ,后续的工作是 ,设计出把平移和旋转结合在一起的 ,自由度少于 6 个的机器人 。 例如 ,有不多的空间 3-自由度机器人 ,把两个空间平移和一个旋转结合在一起 ,如下一节提出的 。 nts 图 5 5 新空间 3-自由度对应机械手 5。 1 新空间 3-自由度 对应机械手的结构 新空间 3-自由度对应机械手 ,如图 6a 所示 ,包含 : 基础板 运动平台 （ 等腰三角形 ） 14 腿 1 12 8 铰链 15 和 3（ 被动 ） 13 和 11（ 被动 ） 16 和 5（ 被动 ） 主动滑套 4 10 6 导轨 （ 固定的 ） 2 9 7 运动平台的运动由 3 个滑套在导轨上滑动来实现 。 图 6 nts5。 2 新空间 3-自由度对应机械手的自由度 当约束起作用时 ,此机械手有 3 个自由度 。 腿 1 和 12 给出两个约束 。 即限制运动平套对 Z 轴的旋转和对 X 轴的平移 。 腿 8 的铰链 5 和 16 有 2 个平行轴 。 腿8 给出一个约束 ,即限制对 X 轴的旋转 。 因此 ,3 条腿结合在一起 ,限制了运动平台对 X、 Z 轴的旋转 ,限制了对 X 轴的平移 。 因此 ,运动平台仅剩下 3 个自由度 :即沿YZ 轴的平移 ,和对 Y 轴的旋转 。 5.3 新空间 3-自由度对应机械手的新颖之处和应用 驱动腿 8,如 Star Like 机器人 ,Tsai 机械手和 CaPaMan 一样 ,采用一个平面的4-杆平行四边形结构 ,机构设计显得很有趣 。 这个独特的空间 3-自由度对应机械手有 3 个特点 :（ a） 仅有转动铰链 ,（ b） 把空间平移和旋转结合在一个 对应机械手内 ,（ c） 绕 Y 轴转动的自由度具有灵活性 。 从 实用角度 ,此设计使用与对应机床 。 由于少于 6-自由度的对应机械手有低的可动度和好的适应性 ,越来越多的对应机床被作成混合结构 ,例如 ,Tricept 和 George 机床 ,他们都是基于 3-自由度对应机构 。 在将来 ,所提出的对应机械手将被应用与设计混合对应机床 。 该对应机构也能用于工业机器人 ,运动模拟器 ,或微机器人 。 图 6a 显示的设计 ,是约束过的 ,也就是说 ,加工的零件必须是高精度的 。 然而 ,球铰链可用容易加工的 ,精度较高上午转动铰链来代替 。 5.4 新空间 3-自由度对应机械手的逆 运算 逆运算即用输出平台的坐标计算输入变量的坐标 。 运动模型见图 6b。 输出平台的 3 个顶点用 P1,P2,P3 表示 。 基础平台的 3 个顶点用 b1,b2,b3 表示 。 固定球型参考系 R:O-xyz,原点 O 在边 b1b2 中心 ,z 轴垂直于基础平面 ,y 轴沿b1,b2。 另一个参考系 ,称为顶架 ,R:O-xyz,原点 O在边 P1P2 的中心 ,z轴垂直于输出平台 ,y轴沿 P1,P2。 3 个连接杆的长度 L1,L2,L3,相等 ,为 L,有时 L3 不等于 L1,L2。 假定 ,运动平台的笛卡儿参考系原点坐标在 R:O-xyz 是已知的 ,即 OR =（ x y z） T （ 2） 式中 ,x=0,方向由矩阵 Q 给出 , ntsQ= c o s0s in010s in0c o s （ 3） 式中 ,角度 是输出平台对 y 轴的转角 。 P1,P2,P3 在参考系 R的坐标用向量 P1R,P2R,P3R表示 , p1R=（ 0 r 0） T p2R=（ 0 r 0） T p3R=（ r 0 0） T （ 4） 向量 P1R,P2R,P3R 用基础铰链在参考系 R 的位置向量定义 , b1R=（ 0 R z1） T b2R=（ 0 R z2） T b3R=（ R 0 z3） T （ 5） 向量 P1R,P2R,P3R 在参考系 O-xyz 可写为 , pim=QpiR + OR （ 6） 那么 ,对应机械手的逆运动能用下列限定公式求解 , iRiR bp =L ,i=1,2,3 （ 7） 因 此 ,对于给顶的机械手 ,给顶的运动平台的方向值 ,需要的驱动器输入可从公式 （ 7） 直接计算出 , z1= zRyrl 22 )( （ 8） z2= zRyrl 22 )( （ 9） 3z= zryRrl s in)c o s( 222 （ 10） 从公式 （ 8） -（ 10） ,可以看出对于一个对应机械手的给顶方位 。 有 8 个逆运算解 。 为了得到如图 6 的逆结构 ,公式中的三个 “ ” 符号都应该取 “ +” 6 其他对应机构 6.1 新 2-自由度平移平台 图 7a 显示一个新 2-自由度平移平台 ,图 7b 是他的运动简图 。 基础 1,运动平台 2,运动平台有两条一样的退 。 每个腿包含一个平面 4-杆平行四边形 。 第一个腿有四个杆 :2,3,4,5;第二个腿有四个杆 :2,6,7,8。 每个平面 4-杆平行 四nts边形结构 ,运动平台对于基础有两个平移自由度 。 系统是过约束的 ,因有一个平面4-杆平行四边形杆 ,就能实现一个刚体仅有 2-自由度 ,用两个平面 4-杆平行四边形杆 ,是为了增加系统的硬挺行和保持对称性 。 该机构正被用于 与齐齐哈尔第二机床厂的合作 开发 新型 6-轴机床 。 图 7 图 8 6.2 平面 3-自由度系列对应机构 图 8 显示该机构运动平台有两条腿 。 右边的腿 ,下部与转动铰链连接 ,上部与被动转动铰链连接 。 被动铰链通过棱柱铰链与基 础连接 。 左边的腿 ,完全不同于右边的 ,是一个可变的四边形 ,其一边是可伸缩的 ,目的是变更运动平台的方位 。 四条边 ,通过转动铰链互相连接 。 四边形与基础的连接是靠棱柱铰链 。 该机构正被用于与江东机床厂合作开发新型 6-轴机床 。 7 结论 这篇论文给出了对应机械手和其自由度的定义 。 讨论了三种类型的新对应机械手 ,新空间 3-自由度对应机械手 ,新兴 2-自由度平移平台 ,3-自由度平面系列 对应机构 ,这些设计的后两个正被机床工业的开发设计所采用 。 ntsParallel Mechanisms with Two or Three Degrees of Freedom Christiaan J.J. Paredis, H. Benjamin Brown, Pradeep K. Khosla Abstract: Parallel manipulators for the machine tool Industry have been studied extensively for various industrial applications. However, limited useful workspace areas, the poor mobility, and design difficulties of more complex parallel manipulators have led to mare interest in parallel manipulators with less than six degrees of freedom (DoFs). Several parallel mechanisms with various numbers and types of degrees of freedom are described in this paper, which can be used in parallel kinematics machines, motion simulators, and industrial robots. Key words: parallel manipulator; parallel kinematic machine; degree of freedom; robot Introduction Mechanical systems that allow a rigid body to move with respect to a fixed base play a very important role in numerous applications. A rigid body can move in various translational or rotational directions which are called degrees of freedom (DoFs). The total number of degrees of freedom for a rigid body cannot exceed six, for example, three axes, A robot includes a system to control several degrees of freedom of an end effector. The last few years have witnessed important developments in the use of industrial robots, mainly due to their flexibility. However, the mechanical architecture of the most common robots is not well adapted to certain tasks. Other types of architectures have, therefore, recently been developed for industrial use, including parallel manipulators. A parallel manipulator, which is a closed-loop mechanism, typically consists of a moving platform that is connected to a fixed base by several limbs or legs. Typically, the number of limbs is equal to the number of degrees of freedom such that every limb is controlled by one actuator and all the actuators can be mounted at or near the ntsfixed base. For this reason, parallel manipulators. Because the external load can be shared by the actuators, parallel manipulators tend to have a large load-carrying capacity. Parallel manipulators are always presented as having very good performance in terms of accuracy, rigidity and the ability to manipulate large loads. They have been used in a large number of applications ranging from astronomy to flight simulators, and are becoming increasingly popular in the machine-tool industry, The conceptual design of parallel manipulators can be dated back to 1947, when Gough established the basic principles of a mechanism with a closed-loop kinematic structure to control the position and orientation of a moving platform to test tire wear and damage. He built a prototype in 1955 (Fig, 1a) where the moving element was a hexagonal platform whose vertices were all connected to links by ball-and-socket joints. The other end of the link was attached to the base by a universal joint. Six linear actuators modified the total link length. Stewart designed a platform manipulator as an aircraft simulator in 1965 (Fig, 1b), in which the moving element was a triangular platform whose vertices were all connected by ball-and-socket joints to support mechanisms each constituting of two jacks, also placed in a triangle. In 1978, Hunt made a systematic study of kinematic structure of parallel manipulators, with the planar three-RPS parallel manipulator as typical example. Since then, parallel manipulators have been studied extensively by numerous researchers. nts Most of the six-DoFs parallel manipulators studied to date have included six extendible limbs. These parallel manipulators possess the advantages of high stiffness, low inertia, and large payload capacity. However, they suffer the problems of relatively small useful workspace and design difficulties. Furthermore, their direct kinematics is very difficult to analyze. Therefore, parallel manipulators with less than six-DoFs have increasingly attracted attention for industry applications. This paper introduces parallel manipulators and the classification of parallel manipulators. Three types of new parallel manipulators are introduced: a spatial three-DoFs parallel manipulator, a two- DoFs parallel manipulator, and a planar three- DoFs serial-parallel manipulator. 1 Definition of parallel Manipulator A parallel manipulator is made of an end-effector with n degrees of freedom with a fixed base linked together by at least two independent kinematic linkages. Actuation takes place through n-simple actuators. These mechanisms have the following characteristics. At least two linkages support the end-,effector. Each of those linkages contains at least one simple actuator. The number of actuators is the same as the number of degrees of freedom of the end-effecto. nts The mobility of the manipulator is zero when the actuators are locked. Parallel mechanisms are of interest for the following reasons: The load can be distributed on the multiple linkages. Few actuators are needed. When the actuators are locked, the manipulator remains in position, which is an important safety concern for certain applications. Parallel manipulators for which the number of linkages is strictly equal to the number of degrees of freedom of the end-effector are called fully parallel manipulators. 2 Degrees of Freedom of a Mechanism The degrees of freedom of a mechanism are the number of independent parameters or inputs needed to completely specify the configuration of the mechanism. However, a general mobility criterion cannot be easily defined for closed-loop kinematic linkages， as Hunt and Lerbet already noted. Classical mobility formulae can indeed neglect some degrees of freedom. Grublers formulae is nevertheless generally used, which may be written as M=d(n-g-1)+gnfi1(1) where M is system mobility (degrees of freedom); d is screw system order (d=3 for planar and spherical motion, d=6 for spatial motion); n is number of links including the frame; g is number of joints; and 关 are degrees of freedom associated with the i-th joint. nts3 Classification of Parallel Manipulators The total number of degrees of freedom of a rigid body cannot exceed 6; therefore， the number of DoFs of a parallel manipulator will be between 2 and 6. Since the first parallel mechanism design， many mechanical designs have been proposed for parallel manipulators with 2 to 6 DoFs. A survey of 87 actuators proposed in the literature showed that 40% has six DoFs, 3.5% five DoFs, 6% four DoFs， 40%three DoFs， and the remaining two DoFs. 3. 1 Two-DoFs parallel manipulators Most existing two-DoFs parallel manipulators are planar manipulators with two-translational DoFs. Such designs use only prismatic and revolute joints. McCloy showed that there are 20 different combinations. This number is reduced to 6 as shown in Fig. 2 if the actuators are assumed to be attached to the ground. There is no passive prismatic joint and no actuator is supporting the weight of another actuator. 3. 2 Three-DoFs parallel manipulators There are many three-I?oFs parallel manipulators, so only the classical designs will be presented here. One example is the planar three-RRR (R stands for revolving joint) parallel manipulator as shown in Fig. 3a. The moving platform has three planar DoFs， which are two translations along the x and y axes and one rotation around the axis perpendicular to the O-xy plane. Another example is the spherical three-RRR parallel manipulator as shown in Fig. 3b， in which all the joint axes intersect at a common vertex. The motion of any point in the mechanism is rotation about the vertex. The moving platform has only rotational DoFs with respect to the base. Hunt presented the three-RPS parallel manipulator shown in Fig. 3c, which has complex DoFs， which cannot be strictly defined. The most famous robot with three translations is the DELTA (Fig.3d)， proposed by Clavel and marketed by the Demaurex Company and ABB under the name IRB 340 FpexPicker. DELTA has been widely used in industry. Another type of three-DoFs parallel manipulator has the moving platform connected to the base through four legs，where the fourth leg is passive and is also the leading leg， which means that the leg determines the motion of the moving platform, for example， in the spherical coordinate parallel manipulator shown in Fig. 3e.This parallel manipulator is used for the machine tool design by IFW of the ntsUniversity of Hannover. 3. 3 Four-DoFs parallel manipulators. A four-DoFs fully parallel manipulator has d-=6, n=10, g=I2, and M=4.Substituting these coefficients into Eq. (1) gets F=22/4 which is the degrees of freedom for each leg. Therefore, there axe， actually, no four-DoFs fully parallel manipulators. The early mechanisms with four-DoFs were not fully parallel manipulators, i, e.， manipulators with two actuators per linkage or with passive constraints. 3. 4 Five-DoFs parallel manipulators Five-DoFs fully parallel manipulators must have F=29/5， so there are no five-DoFs fully parallel manipulators. A five-DoFs parallel manipulator proposed by Austad consists of two parallel manipulators. 3. 5 Six-DoFs parallel manipulators Six-DoFs parallel manipulators are the most popular manipulators so they have been studied extensively. The architecture shown in Fig. 4a is a classical six-DoFs parallel manipulator. Most six- DoFs parallel manipulators have six extendible limbs. These parallel manipulators possess the advantages of high stiffness, low inertia, and large payload capacity. However, they suffer the problems of relatively small useful workspace and design difficulties. Furthermore， analysis of ntstheir direct kinematics is very difficult. There are also Some exotic chain manipulators in which the manipulator is actuated by a planar mechanism, such as a four-bar mechanism, or a five-bar mechanism, or which have two actuators per leg and which usually have three legs 4 Evolution of Parallel Manipulators After Gough established the basic principles of mechanisms with closed-loop kinematic structures in 1947， as shown in Fig. 1a, many other parallel manipulators with a specified number and type of degrees of freedom have also been proposed. The architecture designed by Stewart in 1965 is shown in Fig, 1b. As shown in Fig. 4a, theoretically speaking, the six legs can be arranged at will to design various six-DoFs parallel manipulators, such as the manipulator shown in Fig. 46, where the legs are arranged in a 3-2-1 style which is a very compact structure that can be used in microsystem. The arrangement of the six legs shown in Fig. 4c makes the manipulator move freely along a specified direction， which is very useful for the industrial applications. A six-DoFs parallel manipulator similar to that proposed by Pierrot has each pair of legs in the manipulator shown in Fig. 4a parallel to each other. The number of DoFs of the manipulator will be different if the inputs to the two legs in each pair are the same. The equivalent manipulator architecture is shown in Fig.5. The manipulator output will be three translations， which is probably the origin of the DELTA robot. The actuated links can be arranged as the well-known, fast robot DELTA shown in Fig. 3d. DELTA has been made in several versions， such as the Pollard mechanism Tsars manipulator, Fig.3f, is also among three translational parallel manipulators. Although Tsars manipulator has translations identical with that of DELTA， it is not exactly a version of DELTA. Their design concepts are different and Tsars manipulator is the first design to ntsdeal with the problem of a UPU chain. Another three-translational DoFs parallel manipulator, Star, was design by Herve based on the group theory. Although these design concepts provide ideas to design a new manipulator, additional work is needed to design a robot combining translational and rotational DoFs with less than six DoFs. For example, there are few spatial three-DoFs parallel manipulators combining two spatial translations and one rotation, as will be presented in the following section. 5 New Spatial Three-DoFs 1 Manipulator 5. 1 Manipulator structure The spatial three-DoFs parallel manipulator shown in Fig.6a consists of a base plate, a movable platform， and three legs that connect the two plates. Each connecting leg has four degrees of freedom. Two of the three legs have identical chains with a two-DoFs joint (or two 1-DoF joints) and two 1-DoF joints.The third leg consists of a planar four-bar parallelogram and three 1-DoF joints. One 1-DoF joint in each leg is actuated. The moving platform is an isosceles triangle. The vertices of the platform are connected to a fixed-base plate through legs (1)， (8) and (12). Legs (1) and (12) have identical chains with a constant link connected to a universal joint (or two revolute joints) (15) or (13) at the bottom end and a passive revolute joint (3) or (11) at the other end. The revolute joint is then attached to an active slider (4) or (10)， which is mounted on the guide way (2) or (9) The third leg (8) has a constant link, a planar four-bar parallelogram, which is connected to a revolute joint (16) at the bottom end and a passive revolute joint (5) at the other end. The revolute joint is attached to an ntsactive slider (6) which is mounted on the guide way (7).The movement of the moving platform is accomplished by sliding the three sliders on the guide ways. 5. 2 Manipulator capability The proposed manipulator is a general manipulation device that must have three degrees of freedom when the input elements are active. In the arrangement of the links and joints shown in Fig. 6, legs (1) and (12) provide two constraints on the rotation of the moving platform about the z axis and the translation along the x axis. The revolute joints (5) and (16) for the third leg (8) have parallel axes as shown in Fig.6a.The third leg can provide one constraint on the rotation of the moving platform about the axis. Hence， the combination of the three legs constrains the rotation of the moving platform with respect to the z and z axes and the translation along the axis. Therefore, the mechanism has two translational degrees in the O-yz plane and one rotational degree of freedom about the y axis. 5.3 Novelties and applications The mechanical design is interesting because of the third actuating leg mechanism which uses a planar four-bar parallelogram, as used in other parallel mechanisms, such as Star Like robot, the Tsai manipulator, CaPaMan. This unique spatial three -DoFs parallel manipulator (a) has only revolute joints， b) combines spatial translational and rotational degrees of freedom in a spatial three-DoFs parallel manipulator, and (c) has high mobility of the rotational DoF. The design can be practically applied to parallel machine tools Because of the low mobility and flexibility of six- DoFs parallel mechanisms， more and more parallel machine tools are built as hybrid structures， such as those of Tricept and George V.， which are usually based on three-DoFs parallel mechanisms The proposed parallel manipulator will be designed as a hybrid parallel machine tool in future work. The parallel mechanism can also be used as an industrial robot, a motion simulator, or a micromanipulator. The design shown in Fig. 6a is over-constrained, which means that the machined components must be accurate. However, the universal joints can be replaced by two revolute joints which are not difficult to fabricate accurately. nts 5. 4 Inverse kinematics problem The inverse kinematics problem involves mapping a known pose (position and orientation) of the output platform to a set of input joint variables that will achieve that pose. A kinematics model of the manipulator is shown in Fig.6b.The output platform vertices are denoted as the platform joints， pi( i=1,2,3),with the vertices of the base platform denoted as bi(i = 1， 2,3). A fixed global reference system X: O-xyz is located at the center of side b1b2 with the z axis normal to the base plate and the y axis directed along b1b2. Another reference frame， called top frame X:O-xyz, is located at the center of side P1P2.The z axis is perpendicular to the output platform and the y axis is directed along P1P2. The length of each leg link is denoted by L, where P;B;=L, i=1,2,3. In some cases， the length of Link P3B3 can differ from that of P1B1and P2B2. The objective of the inverse kinematics solution is to define a mapping from the pose of the output platform in Cartesian space to the set of actuated inputs that achieve that pose. The pose of the moving platform is considered known, with the position given by the position vector OR, OR =(x y z)T (2) where x=0. The orientation is given by a matrix Q, ntsQ= c o s0s in010s in0c o s (3) where the angle is the rotation of the output platform with respect to the y axis. The coordinates of point Pi in the frame R can be described by the vector PiR- (i =1,2,3): p1R=(0 r 0)T p2R=(0 r 0
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