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1933 教学 立体仓库 提升 部分 设计

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1933_教学型立体仓库提升与送物部分设计,1933,教学,立体仓库,提升,部分,设计

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黄河科技学院毕业设计开题报告表课题名称教学型立体仓库提升与送物部分设计课题来源教师拟订课题类型AX指导教师杨汉嵩学生姓名张晓倩专 业机械设计制造及其自动化学 号080105501一、调研资料的准备根据任务书的要求，在做本课题之前，查阅了与该课题相关的资料，有：机械制图、机械设计、机电一体化技术、机械控制工程基础与毕业设计指导手册等一系列与设计相关的材料。二、设计的目的与要求 毕业设计是大学教学中最后一个实践性教学环节，通过该设计过程，可以检验学生对所学知识融会贯通的程度，同时培养学生处理工程中实际问题的能力，因此意义特别重大。立体仓库提升与送物部分主要利用步进电机作为动力源输出转矩，电机输出轴通过联轴器与滚珠丝杠相连接，最后通过滚珠螺母带动载物台Y向或者Z向运动。载物台在滚动导轨上滚动。图1 教学型立体仓库参考模型立体仓库提升与松物部分的结构包括：(1) 电机底座；(2) Y向步进电机；(3) 联轴器；(4) 载物台；(5) 滚珠丝杠；(6) 行程开关；(7) 滚动导轨；(8) Y向支座；(9) Z向支座；(10) Z向步进电机。 综合利用机械设计制造及其自动化方面的知识，对教学型立体仓库的提升与送物部分进行分析、计算，完成其总体设计；再利用Auto CAD软件先对立体仓库提升与送料部分进行数据布局设计，并完成其装配图与部分零件图的造型图。三、设计的思路与预期成果 1、设计思路首先要深刻理解立体仓库的整体结构，弄清楚各部分工作的原理及相互之间的作用关系；其次根据所选参数并结合实际因素，综合运用各方面的知识对各部分进行设计；最后参照已有的资料和多方经验，创造性的完成立体仓库提升与送物部分的制图工作。2、预期的成果 （1）完成文献综述一篇，不少与3000字，与专业相关的英文翻译一篇，不少于3000字；（2）绘制装配图和部分零件图；（3）完成内容与字数都不少于规定量的毕业设计说明书一份； （4）刻录包含本次设计所有内容的光盘一张。四、任务完成的阶段内容及时间安排 2月19日之前完成开题报告；3月 5 日之前完成文献翻译；3月15日之前完成文献综述及总体方案设计；4月10日之前完成装配图和部分零件图的绘制；4月25日之前完成设计说明书；4月30日之前完成所有设计任务并修改； 5月19日毕业答辩。五、完成设计所具备的条件因素 我已修完机械制图、机械设计、机电一体化技术、机械控制工程基础、机器人技术及其应用、机械制造技术基础、电工学及毕业设计指导等课程，同时借助图书馆的相关文献资料并查询相关的网络等资源。 指导教师签名： 日期： 课题来源：（1）教师拟订；（2）学生建议；（3）企业和社会征集；（4）科研单位提供课题类型：（1）A工程设计（艺术设计）；B技术开发；C软件工程；D理论研究；E调研报告 （2）X真实课题；Y模拟课题；Z虚拟课题要求（1）、（2）均要填，如AY、BX等。2 黄河科技学院毕业设计（文献翻译） 第10页使用CAD功能对空间3-，4-，5-，6-足并联机器人进行线性驱动的运动学分析摘要提出了一种用于设计的计算机模拟空间并联3-，4-，5-，6-驱动四肢的机器人的新的计算机辅助几何方法的机制。几个与3-，4-，5-，6-驱动四肢相关的新的空间平行机器人的合成。设计的模拟机制中的一些常见的计算机辅助几何制约因素和长度的驱动技术以及定义得以提出。根据一些新的和原有的空间并联3-，4-，5-，6-驱动的四肢机器人，分别应用这些技术，创建12种仿真机制。当通过使用尺寸驱动技术来修改驱动四肢的驱动尺寸，模拟机制的相应配置是多种多样的，运动学参数的移动平台得以解决了。计算机模拟的结果证明，计算机辅助几何的方法，不但相当地快速和简单，而且也从有利的观点说明了准确性和可重复性。关键词：计算机辅助几何，空间并联机器人，仿真机制1简介一些与3-或6-驱动四肢的1,2空间并联机器人已被用于许多实际应用中，这些良好的运动学和动力学性能被应用于机械臂，并联机床，漫步机的腿部，飞行模拟器，汽车或坦克模拟器，地震模拟器等4-6的高负荷生产能力。Tsai3证明了在一般情况下，驱动空间并联机器人四肢的数量等于其自由度数。进行空间并联机器人的合成，运动分析和优化设计的一些分析方法（如影响系数矩阵的方法7，螺丝的方法8以及空间矢量分析方法9,10）已用于空间3-或6-足并联机器人驱动肢体7-17运动学和动力学性能的研究。到现在为止，空间4-或5-足驱动的并联机器人还没有被用于许多实际应用程序中，主要是因为他们的研究还不是十分完善和成熟。这种分析方法适合于计算机编程和拥有精度和重复性的优势。然而，对计算分析空间并行机理的过程是相当复杂的，如进行尺寸合成，分析运动学和动力学，并确定奇异配置。在针对于CAD机制分析的基础上，对一些合适的方案进行了研究和编译18,19。由于这些分析方法是复杂的，其编程过程也是复杂的，而不是简单的，尤其是在多个解决方案的情况下的空间并行机制。因此，这种分析方法的应用受到限制。目前，计算机辅助几何技术是一种在功能设计、概念设计、三维建模以及对平面机理20-25的合成和分析方面都是极为有效的工具。然而，如何使用这个工具来解决运动学和空间3-，4-，5-，6-足驱动的并联机器人的SC分析问题，仍是一个尚未解决的关键问题。为了解决上述问题，通过使用有几何约束，等式约束和尺寸驱动功能的CAD软件分别设计空间3-，4-，5-，6-足驱动并行机器人的一些仿真机制。一种新型无需建模和三维实体组装的计算机辅助几何方法用以对动态运动学参数进行探索。2创建模拟机制的常用技术创建空间并联机器人的模拟机制之前，一些常见的技术和定义如下所述。第1步. 模拟机制中的尺寸分为驱动尺寸，传动尺寸，固定尺寸。驱动尺寸赋予驱动四肢的驱动移动平台。传动尺寸被赋予相应移动平台的位置和方向，以便在平等的基础上，解决运动学机制的参数。固定尺寸给出了移动平台的边线，边线的基础，并连接任何两个关节，以修改仿真机制的大小和配置。第2步. 模拟机制中的一些基本环节组成如下：(a)构成一条直线l，并给它一个初始的固定尺寸。这样，这条固定的尺寸线就相当于二进制中的一个环节。(b)在先进的CAD软件中的二维草图环境下，分别使用多边形命令，构成了等边三角形、等边四边形和正六边形。使用平面面域的命令，把它们分别转换成等边三角形平面，等边四边形平面和正六边形平面。举一个使用尺寸命令来初始固定多边形平面尺寸的例子。因此，这些多边形的平面可以作为基准或移动平台，但它们不能被同时使用，因为基准和移动平台，不能在二维草图环境下同时产生。因此，如果在二维草图环境下构成该基准，移动平台就必须采取如下2c，2d和2e的步骤在三维草图环境下产生，反之亦然。(c)构成三条线li（i=1，2，3），将它们采用点对点的同步命令连接起来，形成一个封闭的的三角形a1a2a3。接下来，给a1a2a3的每条边li赋初值。通过点点重合命令和点线重合命令将它的两端点a1和边l3C1分别连接而构成c1。构成c2线并连接它的两端以指向点a2和c1线上的a0点。设置c1、c2分别垂直于l3和l1，在这种方式中，相当于是在三维草图环境中构成的；同时，其中心点a0被确定，如图2.1a所示。(d)构成四条线li（i=1，2，3，4），通过使用点对点重合命令把它们连接起来，形成一个封闭的四边形（a1a2a3a4）。设置l1垂直于l2和l4，并设置l1平行于l3。因此，四点（a1，a2，a3，a4）总是在一平面上。分别连接线c1的两端a1和a3；分别连接线c2的两端a2和a4；使c1垂直于c2。举一个使用尺寸命令来初始四边形边的例子。在这种方式中，相当于在三维草图环境中构成了平面四边形(a1a2a3a4)的环节，中心点a0可以从c1和c2的交叉点来确定，如图2.1b。(e)构成六线li（i=1，2，.，6），采用点对点的同步命令把它们连接起来，形成一个封闭的六边形(a1a2a3a4a5a6)。分别连接它的两端a3和a6以构成一条线c1。采用点对点和点线约束命令，分别连接a1和c1线上的a0点以构成线c2。 设置c1分别平行l2和l5以及c2分别平行于l3和l6。给予相同的尺寸c2和边li。因此，这六个点a1（i=1，2，.，6）始终保留在平面a1a6a0上。在这种方式中，一个正六边形平面（a1a2a3a4a5a6）在三维草图环境中得以产生，其中心点a0可根据图2.1c中c1和c2的交叉点予以确定。（a）正三角形三元链接;（b）等边四边形链接以及（c）正六边形链接图2.1 相当于平面正三角形的三元，四边形，六边形的链接第3步. 一些在模拟机制中等同于接头的构成如下：（a）构成一条线l，并给它一个在长度上的驱动尺寸。这样，l就相当于一个棱柱的交点p或与棱柱交点P等同的驱动尺寸。（b）按照上述的第2步，构成一个链接和一条直线l，使用点对点重合命令将l的一端连接到链接（如基础，移动平台，或另一条线）的任一点P。因此，连接点p就相当于一个球形接头S。（c）按照上述的第2步，构成一个链接和一条直线l，通过使用点对点重合的命令把l的任一端或另一条线的末端连接到环节（如基准，或移动平台）上的顶点P，并且在这个环节上设置l垂直于另一条线。因此，连接点p就是等同于旋转的共同R。（d）构成两条线l1和l2，使用点线重合命令将l1的一端连接到l2上的p点，并设置l1垂直于l2。这样，连接点p就等同于一个圆柱体的共同点C。（e）构成两条线l1和l2，使用点线重合命令将l1的一端连接到l2上的p点，如图2.2a所示。这样，连接点P就相当于一个复合的交点PS。(a)等价的复合交点PS和（b）等价的UPU-驱动四肢r1图2.2 等价的平面等边三边形，四边形，六边形的连结第4步. 按照上述的第2步，分别构建一个基准B和一个移动平台M。按照上述的步骤3a和3b，构成一条线r1，并赋给R1在长度上的驱动尺寸，再通过使用点对点重合命令，把它的两端分别连接到点A1的B和m。这样，分别构成了两个在A1的等效球形接点S和a1点，这就相当于连接m和B的SPS驱动四肢r1。第5步. 按照上述的第4步，设置驱动四肢r1垂直于基准B的一条边线。这样，连接r1和B的等效的回转交点R得以构成。这样，等效的SPS驱动四肢r1就被转化成等价的SPR驱动四肢r1。第6步. 按照上述的第4步，构成辅助线B1并使其一端连接到点A1的基准B。采用垂直约束命令，设置B1垂直于两个驱动四肢r1和B的一条边线（L3用于针对3-足驱动四肢以及4-足驱动四肢C1的情况）。这样，在B处的点A1构成一个相当于常见的接点U。同样，构成一条辅助线b1，将它的一端连接到点a1的m并使b1垂直于r1和m的一条边线（3-足驱动四肢的L3，4-足驱动四肢的C1）。因此，一个相当于通用的交点U在点a1处予以点形成。接下来，设置B1平行或垂直于b1。这样，一个等效的SPS驱动四肢r1就转换成一个等价的UPU驱动四肢r1，如图2.2b所示。第7步. 按照第4步和第6步，构成一个等效球在B上的点A1的共同点S和在m上的点a1的等价万向点A1。这样，一个等价的SPS驱动四肢r1就转化为等价的SPU驱动四肢r1。事实上，SPU的驱动四肢r1就相当于在模拟机制下的SPS驱动四肢r1，因为一个本地冗余的自由度，这是从本身旋转轴的驱动四肢产生的，不影响该移动平台的运动。因此，UPS在模拟机制中的驱动四肢r1可以被替换为相当于SPS的驱动四肢r1。由以下的Kutzbach Grubler方程2-4可以计算出空间并联机构的自由度， （2.1）其中k是链接的数量，j为环节数，k是空间动态=6的机制运作内的尺寸；fi是第i个共同的自由程度，F0是本地的冗余自由度，这并不影响机制的运动。 4空间4-足驱动的并联机器人空间4-足驱动的并联机构的新型设计，如图4.1a所示。它包括一个四足的移动平台m，一个四足的基准B，两个扩展UPU的四肢，两个扩展SPU的四肢。其中，m是一个以a0为中心的等边四边形（a1a2a3a4），B是一个以a0为中心的平面等边四边形（A1A2A3A4）。分别通过一个在点ai的球形交点S、与棱柱的交点P相关的驱动尺寸ri、点Ai的万向节U，把两个相同的SPU驱动尺寸从m连接到B，其中i=2,4。分别通过点ai的万向节U、与棱柱相关的驱动尺寸ri、点Ai的万向节U，把两个相同的UPU驱动尺寸从m连接到B，其中i=1,3。我们定义它为一个2-SPU-UPU的机制I。当2-UPU尺寸和2-SPU的尺寸被安排在不同的位置时，可以分别合成2-SPU-UPU的机制II和2-SPU-UPU的机制III其他两种形式，如图4.1b和c所示。 （a）2-SPU-UPU的机制I；（b）2-SPU-UPU的机制II；（c）2-SPU-UPU的机制III。图4.1 空间4-足驱动尺寸的并行机制通过查看图4.1的整个机理，我们知道，对于移动平台、四个气缸以及四个活塞棒和一个基准的=10；六万向节U，四棱柱交点P和两个球体交点S的j=12；万向节的f1=2，棱柱集合的f2=1，球体集合的f3=3；F0=0。因此，根据公式（2.1），整个机制的自由度是： (2.2）空间2-SPU-UPU并联机制的仿真机理是建立在三维草图环境中的，如图4.2a所示，其创建过程描述如下。(1)按照第2步中的常用技术，分别构成了与长度边线（60cm）相关的一个移动平台（a1a2a3a4），以及与长度边线（100cm）的基准（A1A2A3A4）。 (2)按照第6步和第7步中的常用技术，分别构成两条相当于UPU驱动尺寸（r1和r3）和两个等价的SPU驱动尺寸（r2和r4）。通过这种方式，可以创造一个空间2-SPU-UPU并联机制I下的模拟机理。 (3)通过类似的过程，2-SPU-UPU模拟机制I的移动平台的位置-方向就得以确定。同样的，2-SPU-UPU模拟机制II也得以创建，如图4.2b所示。 （a）模拟机制I和（b）模拟机制II。图4.2 在驱动尺寸2-SPU-UPU模拟机制下的并行机理 8结论通过在先进的CAD软件的三维草图环境中，应用计算机辅助几何约束和尺寸驱动技术，一些新的仿真机制和空间具有3-，4-，5-，6-足驱动四肢的并联机器人得以设计。当通过使用尺寸驱动技术来修改驱动四肢的驱动尺寸时，相应的模拟机制的配置是多种多样的。这样，每个模拟机制的移动平台的运动参数就可以得到解决。当模拟机制的所有驱动尺寸和固定尺寸被修改后，在每个模拟机制中的所有几何约束和尺寸约束却被始终保留。因此，一旦创建了一个模拟机制，它就可以反复合成具有不同尺寸和配置的同类型的空间并联机构。当修改模拟机制的驱动尺寸时，导向尺寸也会相应地随之更改。这样，在平行空间机构自由度的基础上，模拟机制的驱动尺寸的数量就可以确定了。通过应用驱动和导向的尺寸技术和记录功能，在同等基础上的移动平台的位置-方向就可以得到解决。在Microsoft Excel环境下通过拟合曲线和拟合方程技术，可以解决在同等基础上的移动平台的近似速度（转速）和近似运动（旋转加速加速）问题。仿真结果证明，计算机辅助几何方法相当于结构的合成和运动学分析的分析任务，不仅相当的便捷，还有也来自准确性和可重复性的优势观点。参考文献1D.Stewart，与六自由度相关的Proc平台.英国：机械工程研究所第一部分180期，1965,（15）：371-386.2K.H.Hunt.并行驱动机器人手臂的跨结构运动学，德国：ASME研究反式自动售货机的力学105期，1983，（4）：705-712.3Z.Huang，Q.Li.轻型移动并联机器人的类型合成原则，中国：北京科学45期，2002，（3），241-248.4L.W.Tsai，F.Tahmasebi.六自由度并联微型机器人的一类新的合成与分析.研究机器人10期，1993，（5），561-580.5F. Gao, W. Li, X. Zhao.2-,3-,4-,5-足自由度并联机器人的新运动结构设计.马赫机械理论37期，2002,：1395-1411.6L.W. Tsai, S. Joshi.空间3-UPU并联机械手的运动学和优化.德国：ASME反式力学研究122期，2000，（4）：439-446.7Z. Huang, Y.F. Fang.并联驱动棱锥机制的3自由度运动学特征分析.马赫机械理论31期，1996，（8）：1009-1018.8Z. Huang, J. Wang.通过二次变性来鉴定3自由度并联机器人的主要螺丝.马赫机械研究理论36期，2001，（8）：893-911.9C.M. Gosselin.在球面三自由度并联机器人上的直接运动学.国际机器人研究12期，1993，（4）：394-402.10C.M. Gosselin, J. Angeles.球面三自由度并联机器人的最佳运动学设计.ASME研究反式自动发射讯号116期，（4），1141-1147.11H.Y. Lee, B. Roth.一类平行机制正向位移分析的封闭形式的解决方案.IEEE国际机器人自动化会议上的议程，1993，720-724.12S.V. Sreentvasan, K.J. Waldron.6-6型Stewart平台的封闭形式的直接位移分析.马赫机械研究理论29期，1994，（2）：855-864.13B. Dasgupta, T.S. Mruthyunjaya.常用6-6型Stewart平台的一个建设性的预估校正算法的直接位置运动问题.马赫机械研究理论31期，1996，（6）：799-811.14R.I. Alizade, N.R. Tagiyev, J. Duffy.6自由度并联机器人的正向和反向位移分析.马赫机械研究理论29期，1994，（1）：115-124.15K. Luck.以Burmester理论为基础的计算机辅助机制的合成.马赫机械研究理论29期，1994，（6）：877-886.16R. Matone, B. Roth.并联机器人：如何模拟驱动计划和它们在奇异姿势上的影响的研究框架.德国：ASME反式力学研究121期，1999，（1）：2-8.17P.N. Sheth.基于数字电脑的自由度几何约束的机械系统的多维模拟程序.美国:美国威斯康星大学，大学的博士论文，1972，大学缩微胶片：第73-2565号.18C.M. Gosselin, L. Pereault, Ch. Vaillancourt.3自由度球面并联机器人的模拟和计算机辅助运动学设计.SYST机器人研究12期，1995,（12）：857-869.19T.J. Furlong, J.M. Vance, P.M. Larochelle.球形机制在虚拟现实中的合成.德国：ASME反式机械研究121期，1999，（4），515-520.20Y.M. Deng, G.A. Britton, S.B. Tor.基于约束功能设计验证的概念设计.计算机辅助设计32期，2000，（14）：889-899.21Q.J. Ge, M. Sirchia.两个参数自由度形式运动的计算机辅助几何设计.ASME反式机械研究121期，1999，（4）：9-14.22Y. Lu, T. Leinonen.近似平面四杆连锁的三维合成的计算机辅助几何方法.德国：J.计算机辅助设计14图，2002，（6）：547-552.23Y. Lu, T. Leinonen.为了更高的秩序和结合点合成的计算机仿真机制的方法.德国：J.计算机辅助设计14图,2002,(8):845-851.24Y. Lu, T. Leinonen.制造形状复杂的正交6杆虚拟机床的计算机模拟方法.国际马赫机械工具制造42期，2002，（3）：441-447.25Y. Lu.3维自由表面上正交的3杆的虚拟机工具的计算机模拟制造.国际马赫机械工具制造43期，2002，（11）：734-742.Using CAD functionalities for the kinematics analysisof spatial parallel manipulators with 3-, 4-, 5-, 6-linearlydriven limbsYi Lu*Robotics Research Center, Department of Mechanical Engineering, Yanshan University, Qinhuangdao,Hebei 066004, PR ChinaAbstractA novel computer aided geometric approach is put forward for designing the computer simulationmechanisms of spatial parallel manipulators with 3-, 4-, 5-, 6-driving limbs. Several new spatial parallelmanipulators with 3-, 4-, 5-, 6-driving limbs are synthesized. Some common computer aided geometryconstraints and dimension driving techniques and definitions for designing the simulation mechanisms arepresented. Based on some new and original spatial parallel manipulators with 3-, 4-, 5-, 6-driving limbs, the12 types of simulation mechanisms are created, respectively, by applying these techniques. When the drivingdimensions of driving limbs are modified by using the dimension driving technique, the configurations ofthe simulation mechanisms are varied correspondingly, and the kinematic parameters of the movingplatform are solved. The results of computer simulation prove that the computer aided geometric approachis not only fairly quick and straightforward, but is also advantageous from viewpoint of accuracy andrepeatability.? 2003 Elsevier Ltd. All rights reserved.Keywords: Computer aided geometry; Spatial parallel manipulator; Simulation mechanism1. IntroductionSome spatial parallel manipulators with 3- or 6-driving limbs 1,2 have been utilized for manypractical applications, in which the good kinematic and dynamic performance are adopted for therobot manipulator, the parallel machine tool, and the legs of walking machine, high load car-rying capacity is used for the flight simulator, the automobile or tank simulator, the earthquake*Tel.: +86-35-8057070; fax: +86-335-8061449.E-mail address: luyip0736 (Y. Lu).0094-114X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0094-114X(03)00103-4Mechanism and Machine Theory 39 (2004) 4160/locate/mechmtsimulator, and so on 46. Tsai 3 proved that in general, the number of driving limbs of thespatial parallel manipulator equals to the number of its DOF. In conducting the synthesis, ki-nematic analysis, and optimum design of the spatial parallel manipulators, some analytic ap-proaches (such as the influence coefficient matrix approach 7, the screw approach 8, and thespatial vector analytic approach 9,10) have been applied for studying the kinematic and dynamicperformances of the spatial parallel manipulators with 3- or 6-driving limbs 717. Up to now, thespatial parallel manipulators with 4- or 5-driving limbs have not been used for many practicalapplications, since the studies on them have not been perfect and mature.The analytic approaches are suitable for computer programming and have the advantages ofaccuracy and repeatability. However, the calculation in the analytic processes of the spatialparallel mechanism is quite complicated, such as conducting the dimension synthesis, analyzingthe kinematic and dynamics, and determining the singularity configuration. In terms of CADmechanism analyses, based on the analytic approaches, some suitable programs are studied andcompiled 18,19. Since these analytic approaches are complex, the programming processes arecomplicated and not straightforward, especially in the case of multiple-solutions and the spatialparallel mechanisms. Therefore, the application of analytic approaches is limited. Currently, thecomputer aided geometric technology is an effective tool for feature designing, concept designing,3D modeling, and synthesis and analysis of planar mechanism 2025. However, how to use thistool to solve kinematic and the SC analyses problems for spatial parallel manipulators with 3-, 4-,5-, 6-driving limbs is still a key question, which has not been solved.In order to solve the above problems, some simulation mechanisms of the spatial parallelmanipulators with 3-, 4-, 5-, 6-driving limbs are designed, respectively, by using a CAD softwarewith the functions of geometric constraint, equation constraint, and dimension driving functions.A novel computer aided geometric approach without modeling and assembling of 3D solid isexplored for dynamically solving kinematic parameters.2. The common technique for creating the simulation mechanismsBefore creating the simulation mechanisms of spatial parallel manipulators, some commontechniques and definitions are described below.Step 1. The dimensions in the simulation mechanisms are classified into the driving dimension,the driven dimension, and the fixed dimension. The driving dimensions are given to the drivinglimbs for driving the moving platform. The driven dimensions are given to the position andorientation of the moving platform in the respect to the base in order to solve kinematic pa-rameters of mechanism. The fixed dimensions are given to the sideline of the moving platform, thesideline of the base, and the line for connecting any two joints in order to modify the size andconfiguration of the simulation mechanism.Step 2. Some basic equivalent links in simulation mechanism are constituted as follows:(a) Constitute a line l, and give it an initial fixed dimension. Thus, a line with fixed dimensionsequivalent to a binary link.(b) In 2D sketch environment of advanced CAD software, constitute an equilateral triangle, anequilateral quadrangle, and an equilateral hexagon, respectively, by using the polygon command.Transform them into an equilateral triangle plane, an equilateral quadrangle plane, and an equi-42Y. Lu / Mechanism and Machine Theory 39 (2004) 4160lateral hexagon plane, respectively, by using the planar area command. Give one sideline ofpolygon plane an initial fixed dimension by using dimension command. Thus, these polygon planescan be used as either the base or the moving platform, but they cannot be used as both, because thebase and the moving platform cannot be constituted in 2D sketch environment at the same time.Therefore, if the base is constituted in 2D sketch environment, the moving platform must beconstituted in 3D sketch environment by adopting steps 2c, 2d and 2e below, and vice versa.(c) Constitute three lines lii 1;2;3, and connect them to form a closed triangle Da1a2a3byadopting the pointpoint coincident command. Next, give each sideline liof Da1a2a3an initialdriving dimension. Constitute a line c1connect its two ends to point a1and sideline l3by adoptingthe pointpoint coincident command and the pointline coincident command, respectively.Constitute a line c2and connect its two ends to point a2and line c1at point a0. Set c1perpen-dicular to l3and set c2perpendicular to l1In this way, an equivalent planar ternary link in 3Dsketch environment is constituted, and its center point a0is determined, as shown in Fig. 1a.(d) Constitute four lines lii 1;2;3;4, and connect them to form a closed quadrangle(a1a2a3a4) by using pointpoint coincident command. Set l1perpendicular to both l2and l4andset l1parallel to l3. Thus, the four points (a1;a2;a3;a4) are always retained onto the same plane.Constitute a line c1and connect its two ends to a1and a3, respectively. Constitute a line c2andconnect its two ends to a2and a4, respectively. Set c1perpendicular to c2. Give one sideline of thequadrangle an initial dimension by using dimension command. In this way, an equivalent planarquadrangle link (a1a2a3a4) in 3D sketch environment is constituted, and the central point a0of thelink can be determined from the crossover point of c1and c2as shown in Fig. 1b.(e) Constitute six lines lii 1;2;.;6, and connect them to form a closed hexagon(a1a2a3a4a5a6) by adopting the pointpoint coincident command. Constitute a line c1and connectits two ends to a3and a6, respectively. Constitute a line c2connect its two ends to a1and line c1ata0, respectively, by using the pointpoint and pointline constraint command. Set c1parallel toboth l2and l5and set c2parallel to both l3and l6. Give same dimension to line c2and each of thesideline li. Thus, the six points aii 1;2;.;6 are always retained on the plane Da1a6a0. In thisway, an equilateral hexagon plane (a1a2a3a4a5a6) in 3D sketch environment is constituted, and itscentral point a0can be determined from the crossover point of c1and c2as shown in Fig. 1c.Step 3. Some basic equivalent joints in simulation mechanism are constituted as follows:(a) Constitute a line l, and give it a driving dimension in length. Thus, l is equivalent to aprismatic joint p or a driving limb with a prismatic joint P.a1 a0 a2 a3 l1 l3 l2 c1 c2 a2 a1 a6 a3 a4 a5 c2 a0 c1 l1 l6 l3 l5 l4 l2 a4 a3 a2 a1 a0 l3 l2 l1 c2 c1 l4 (a) (b) (c)Fig. 1. (a) Equilateral ternary link; (b) equilateral quadrangle link and (c) equilateral hexagon link. The equivalentplanar equilateral ternary, quadrangle, and hexagon links.Y. Lu / Mechanism and Machine Theory 39 (2004) 416043(b) Follow the step 2 above, constitute a link and a line l, connect one end of l to any point p onthe link (such as the base, the moving platform, or another line) by using the pointpoint coin-cident command. Thus, the connecting point p is equivalent to a spherical joint S.(c) Follow the step 2 above, constitute a link and a line l, connect one end of l to any vertex pon the link (such as the base, or the moving platform) or the end of another line by using thepointpoint coincident command, and set l perpendicular to another line on this link. Thus, theconnecting point p is equivalent to a revolving joint R.(d) Constitute two lines l1and l2connect one end of l1onto l2at point p by using the pointlinecoincident command, and set l1perpendicular to l2. Thus, the connecting point p is equivalent toa cylinder joint C.(e) Constitute two lines l1and l2connect one end of l1onto l2at point p by using the pointlinecoincident command, as shown in Fig. 2a. Thus, the connecting point p is equivalent to a com-posite joint PS.Step 4. Follow the step 2 above, constitute a base B and a moving platform m, respectively.Follow the steps 3a and 3b above, constitute a line r1give r1a driving dimension in length, andconnect its two ends to B at point A1and m at point a1respectively, by using the pointpointcoincident command. Thus, the two equivalent spherical joints S at point A1and point a1areconstituted, respectively, and one equivalent SPS driving limb r1for connecting m and B is alsoconstituted.Step 5. Follow the step 4 above, set the driving limb r1perpendicular to one sideline of the baseB. Thus, an equivalent revolving joint R for connecting r1and B is constituted. In this way, oneequivalent SPS driving limb r1is transformed into an equivalent SPR driving limb r1.Step 6. Follow the step 4 above, constitute an auxiliary line B1and connect its one end to thebase B at point A1. Set B1perpendicular to both driving limb r1and a sideline of B (L3for thesituation of the 3-driving limbs, and C1for the 4 driving limbs) by using perpendicular constraintcommand. Thus, one equivalent universal joint U at point A1on B is constituted. Similarly,constitute an auxiliary line b1connect its one end to m at point a1and set b1perpendicular to bothr1and a sideline of m (l3for the 3-driving limbs, and c1for the 4-driving limbs). Thus, oneequivalent universal joint U at point a1is constituted. Next, set B1parallel or perpendicular to b1.In this way, one equivalent SPS driving limb r1is transformed into one equivalent UPU drivinglimb r1as shown in Fig. 2b.b1 a1 r1 A1 B1 l3 L3 B m p l1 l2 P S l1 l2 P U U m B c1C1 a1 C1 r1 b1 A1 c1 B1 B m A1 a1 r1 (a)(b) Fig. 2. (a) The equivalent a composite joint PS and (b) the equivalent UPU-driving limb r1. The equivalent planarequilateral ternary, quadrangle, and hexagon links.44Y. Lu / Mechanism and Machine Theory 39 (2004) 4160Step 7. Follow steps 4 and 6, constitute an equivalent spherical joint S at point A1on B and anequivalent universal joint U at point a1on m. In this way, one equivalent SPS driving limb r1istransformed into an equivalent SPU driving limb r1. In fact, the SPU driving limb r1is equivalentto the SPS driving limb r1in simulation mechanism, because one local redundant DOF, which isproduced from the driving limb rotating about itself axis, does not influence the motion of themoving platform. Therefore, the UPS driving limb r1in simulation mechanism can be replaced bythe equivalent SPS driving limb r1.The DOF of the spatial parallel mechanism can be calculated by Kutzbach Grubler equation24 belowF kk ? j ? 1 Xni1fi? F01where k is the number of links, j is number of joints, k is the degrees of the space within which themechanism operates for spatial motions k 6; fiis the degree of freedom of the ith joint, F0is thelocal redundant DOF, which do not influence the motion of mechanism.3. The spatial parallel manipulator with 3-driving limbs3.1. The 3-RPS parallel manipulator and its simulation mechanismAn existing spatial 3-RPS parallel manipulator has three DOF 2,3. It includes a movingplatform m, a base B, and three extendable driving limbs riwith their hydraulic cylinders andpiston-rods, as shown in Fig. 3a. Where, m is a regular triangle Da1a2a3with a0as its center, and B(a) (b) R, A1S, a1P4BS, a2a3, SR, A2A3, Rm13625PP130(130)(130)85103103(95.28)(37.87)(64.41)(79.18)(96.22)(192.52)A2a3dya1a2A1A3mr3r2r1dzdxa0d0l3L3l2l3L2L1nn1xyNn2BzFig. 3. (a) The spatial 3-SPR parallel manipulator and (b) its simulation mechanism. The spatial 3-SPR parallelmanipulator and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 416045is a regular triangle DA1A2A3with A0as its center. Three identical limbs connect m to B by aspherical joint S at point ai, a driving limb with a prismatic joint P, and a revolute joint R at pointAi(i 1;2;3), respectively.In the 3D sketch environment, a simulation mechanism of the 3-SPR spatial manipulatoris created, as shown in Fig. 3b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute the moving platform Da1a2a3witha sideline (80 cm) in length, and the base DA1A2A3with a sideline (130 cm) in length.2. Follow step 5 in common technique, constitute an equivalent SPR driving limb r1.3. Repeat step 2 above, but (r1and L3) are replaced by (r2and L1) and (r3and L2), respectively.Thus, the other two equivalent SPR driving limbs (r2r3) can be constituted. In this way,a 3-SPR parallel simulation mechanism is created.3.2. The position-orientation of the moving platformThe position-orientation of the moving platform in the respect to the base of the 3-SPR sim-ulation mechanism can be determined by following processes.1. Constitute a line n, connect its one end to the center point a0on the moving platform m, and setline n perpendicular to m by adopting the perpendicular constraint command. In this way, anormal line n of m is constituted.2. Constitute a line N, set it perpendicular to both sideline L1and sideline L2of the base B, andconnect its one end to point a0on m by adopting the pointpoint coincident command.3. Constitute a line dy, connect its two ends to line N at point d0and sideline L1, respectively, byadopting the pointline coincident command. Next, set dyperpendicular to both N and L1.4. Give the distance from point a0to point d0an initial driven dimension dz(95.28 cm), give thedistance from point A1to line dyan initial driven dimension dx(64.41 cm), and give line dyaninitial driven dimension (37.87 cm) in length, respectively, by using dimension command. Thus,the three translation components (dxdy;dz) of m are constituted, as shown in Fig. 3b.5. Constitute line n1and line n2and connect their one ends to point A1on B, set n1parallel to n,and set n2perpendicular to both L1and N. Give the angle between n1and L1an initial drivendimension bx(79.18?), give the angle between n1and n2an initial driven dimension by(96.22?),give the angle between n and N an initial driven dimension bz(192.52?), respectively. Thus, thethree rotation components (bx;by;bz) of m are constituted, as shown in Fig. 3b.6. When varying or modifying each driving dimension of the driving limb ri, the driven dimen-sions of the three translation components (dxdy;dz) and the three rotation components(bx;by;bz) of m can be varied correspondingly. In this way, the position-orientation of the mov-ing platform in respect to the base of the 3-SPR simulation mechanism can be solved dynam-ically.3.3. The spatial 3-UPU parallel manipulator with 3-driving limbsAn existing spatial 3-UPU parallel manipulator has three DOF 3,4. It includes a movingplatform m, a base B, and three extendable driving limbs riwith their hydraulic cylinders and46Y. Lu / Mechanism and Machine Theory 39 (2004) 4160piston-rods as shown in Fig. 4a. Where, m is a regular triangle Da1a2a3with a0as its center, and Bis a regular triangle A1A2A3with A0as its center. Three identical UPU limbs connect m to B by auniversal joint U at point ai, a driving limb riwith a prismatic joint P, and a universal joint U atpoint Aifor i 1, 2, and 3, respectively.In the 3D sketch environment, a simulation mechanism of the spatial 3-UPU manipulator iscreated, as shown in Fig. 4b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute a moving platform Da1a2a3with sideline(80 cm) in length, and a base DA1A2A3with sideline (130 cm) in length, respectively.2. Follow the step 6 in the common technique, constitute an equivalent UPU driving limb r1. Sim-ilarly, replace (r1and L3) by (r2and L1) and (r3and L2), respectively. Thus, the other two equiv-alent UPU driving limbs (r2and r3) are constituted. In this way, a 3-UPU parallel simulationmechanism is created.3. The position-orientation of the moving platform of the 3-UPU simulation mechanism can besolved by adopting similar processes in the Section 3.2.3.4. The spatial 3-URPR parallel manipulator with 3-driving limbsA new type of the spatial 3-URPR parallel manipulator is designed, as shown in Fig. 5a. Itincludes a moving platform m, a base B, three binary links, and three extendable driving limbswith their hydraulic cylinders and piston-rods. Where, m is a regular triangle Da1a2a3with a0asits center, and B is a regular triangle A1A2A3with A0as its center. Three identical RPRU drivinglimbs connect m to B by a universal joint U at point ai, a binary link gi, a revolute joint R atpoint ei, a driving rod riwith a prismatic joint P, and a revolute joint R at point Aii 1;2;3,respectively.(a) (b) 1010101010(130)130(130)100103103(99.84)(31.67)(75.15)10A2a3dya1a2A1A3mr3r2r1dzdxBb2b1b3B1B3B2a0A0l3L3l2l3L2L1mBa3, UU, a2U,a1A3, UU, A1U, A2PPPFig. 4. (a) The spatial 3-UPU parallel manipulator and (b) its simulation mechanism. The spatial 3-UPU parallelmanipulator and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 416047By inspecting the whole 3-URPR parallel mechanism, we know that k 11 for one movingplatform, three binary links, three cylinders, and three piston-rods, and one base; j 12 for threeuniversal joints U, six revolute joints R, three prismatic joints P; f1 2 for the universal joint,f2 1 for the prismatic joint, f3 1 for the revolute joint; F0 0. Therefore, the DOF of thewhole mechanism isF kk ? j ? 1 Xni1fi? F0 6 ? 11 ? 12 ? 1 3 ? 2 6 ? 1 3 ? 1 ? 0 32In the 3D sketch environment, a simulation mechanism of the spatial 3-URPR parallel manip-ulator is created, as shown in Fig. 5b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute a moving platform Da1a2a3withsideline (80 cm) in length, and a base DA1A2A3with sideline (120 cm) in length, respec-tively.2. Follow the step 3c in the common technique, set line r1perpendicular to line g1and set line r1perpendicular to a sideline L3of B. Thus, the two revolute joints R at point e1and point A1areconstituted, respectively.3. Follow the step 6 in the common technique, set an auxiliary line b1perpendicular to both l1andg1and set b1parallel to r1. Thus, one equivalent universal joint U at point a1on m is consti-tuted. In this way, an equivalent URPR driving limb is constituted.4. Similarly, follow the steps 2 and 3 above, constitute the other two equivalent URPR drivinglimbs. In this way, a spatial 3-URPR simulation mechanism is created.5. The position-orientation of the moving platform of the 3-URPR simulation mechanism can bedetermined by adopting the similar processes in Section 3.2.(a) (b) A3R, A1PR, e1U, a1A2e2Bma2a3e3PPg3g1g2706540404080658080(8.44)(4.87)(73.63)101010120A1A3a2a3A2b2b1b3c1c2a0A0e1e2e3dxdydznr1r2r3a1d0g3g2g1l3l1l2L3L1L2Fig. 5. (a) The spatial 3-URPR parallel manipulator and (b) its simulation mechanism. The spatial 3-URPR parallelmanipulator and its simulation mechanism.48Y. Lu / Mechanism and Machine Theory 39 (2004) 4160From the 3-URPR simulation mechanism, we obtain two interesting conclusions:(1) If the three lines gii 1;2;3 are simultaneously reduced to 0 in length, then the spatial 3-URPR simulation mechanism is transformed into the spatial 3-SPR simulation mechanism.(2) If the three lines giare exchanged with the 3-driving rods rii 1;2;3 with a prismatic P,respectively, then the spatial 3-URPR simulation mechanism is transformed into a spatial3-UPRR parallel manipulator and its DOF is not change. In this case, if the three linesgii 1;2;3 are simultaneously reduced to 0 in length, then the spatial 3-UPRR simulationmechanism is transformed into a spatial 3-UPU simulation mechanism.3.5. The spatial 3-SRP parallel manipulator with 3-driving limbsA new type of spatial 3-SRP parallel manipulator is designed, as shown in Fig. 6a. It includes amoving platform m, a base B, three binary links, and three extendable driving limbs with theirhydraulic cylinders and piston-rods, as shown in Fig. 5a. Where, m is a regular triangle Da1a2a3with a0as its center, and B is a regular triangle DB1B2B3with B0as its center. Three identical SRPdriving limbs connect m to B by a spherical joint S at point ai, a binary link gi, a revolute joint R atpoint Aiand a driving limb riwith a prismatic joint P, for i 1;2, and 3, respectively. The drivinglimb riwith the prismatic joint P are retained coincident with the axis of revolute joint Riand thesideline Liof B i 1;2;3, respectively.By inspecting the whole mechanism of Fig. 6a, we know that k 8 for one moving platform,three cylinders, and three piston-rods, and one base; j 9 for three sphere joints S, three revolutejoints R, and three prismatic joints P; f1 3 for the sphere joint, f2 1 for the prismatic joint,(a) (b) S2, a1S2, a2S3, a3R2, A2R1, A1R3, A3g1P1, r1P3, r3P2, r2B1B3B2mB808080403050707070(69.17)(10)(5.78)(89.94)(90.85)(179.15+)S, a1a2a3g1g2g3B1A1B2B3AA3d0mBa0dzdydxnn1xyzr2r3r1L1L2L3Fig. 6. (a) The spatial 3-SPR parallel manipulator and (b) its simulation mechanism. The spatial 3-SPR parallelmanipulator and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 416049f3 1 for the revolute joint; F0 0. Therefore, based on Eq. (1), the DOF of the whole mech-anism isF kk ? j ? 1 Xni1fi? F0 6 ? 8 ? 9 ? 1 3 ? 3 3 ? 1 3 ? 1 ? 0 33In the 3D sketch environment, a simulation mechanism of the spatial 3-SPR manipulator iscreated, as shown in Fig. 6b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute the moving platform Da1a2a3with side-line (80 cm) in length, and the base DB1B2B3with sideline (130 cm) in length, respectively.2. Follow the step 4 in common technique, constitute a line g1connect its two ends to point a1onm and sideline L1at A1on B, and set g1perpendicular to L1. Give the distance from point B1topoint A1an initial driving dimension (40 cm). Thus, an equivalent SRP driving limb g1r1is con-stituted.3. Repeat the step 2 above, but (r1, g1and L1) are replaced by (r2, g2and L2) and (r3, g3and L3),respectively. Thus, other two equivalent SPR driving limbs (g2r2and g3r3) are constituted. Inthis way, a 3-SRP parallel simulation mechanism is created.4. The position-orientation of the moving platform of the 3-SRP simulation mechanism can bedetermined by adopting the similar processes in Section 3.2.4. Spatial parallel manipulator with 4-driving limbsA new type of spatial parallel mechanism with 4-driving limbs is designed, as shown in Fig. 7a.It includes a quaternary moving platform m, a quaternary base B, two extendable UPU limbs, andtwo extendable SPU limbs. Where, m is an equilateral quadrangle (a1a2a3a4) with a0as its center,and B is an equilateral quadrangle plane (A1A2A3A4) with A0as its center. Two identical SPUdriving limbs connect m to B by a spherical joint S at point ai, a driving limb riwith a prismaticjoint P, and a universal joint U at point Aifor i 2;4, respectively. Two identical UPU driving(a) (b) (c) PU, A1U, a1mBc2C2C1S, a2PPa3,US, a4A3,UPA4,UU, A2SPUmBUUUPPSUPUL1L3UUPUUSPPBmUUPSFig. 7. (a) 2-SPUUPU mechanism I; (b) 2-SPUUPU mechanism II and (c) 2-SPUUPU mechanism III. The spatialparallel mechanism with 4-driving limbs.50Y. Lu / Mechanism and Machine Theory 39 (2004) 4160limbs connect m to B by a universal joint U at point ai, a driving limb riwith a prismatic joint P,and a universal joint U at point Aifor i 1;3, respectively. We define it as a 2-SPUUPUmechanism I.When 2-UPU limbs and 2-SPU limbs are arranged in different positions, the other two types of2-SPUUPU mechanism II and 2-SPUUPU mechanism III can be synthesized, respectively, asshown in Fig. 7b and c.By inspecting the whole mechanisms of Fig. 7, we know that k 10 for one moving platform,four cylinders, and four piston-rods, and one base; j 12 for six universal joints U, four prismaticjoints P, and two sphere joints S; f1 2 for the universal joint, f2 1 for the prismatic joint,f3 3 for the sphere joint; F0 0. Therefore, based on Eq. (1), the DOF of the whole mechanismisF kk ? j ? 1 Xni10fi? F0 6 ? 10 ? 12 ? 1 6 ? 2 4 ? 1 2 ? 3 ? 0 44A simulation mechanism I of the spatial 2-SPUUPU parallel mechanism is created in the 3Dsketch environment, as shown in Fig. 8a, and its creation processes are described below.1. Follow the step 2 in the common technique, constitute a moving platform (a1a2a3a4) with side-line (60 cm) in length, and a base (A1A2A3A4) with sideline (100 cm) in length, respectively.2. Follow the steps 6 and 7 in the common technique, constitute two equivalent UPU drivinglimbs (r1and r3) and two equivalent SPU driving limbs (r2and r4), respectively. In this way,a simulation mechanism of the spatial 2-SPUUPU parallel mechanism I can be created.3. The position-orientation of the moving platform of the 2-SPUUPU simulation mechanism Ican be determined by adopting similar processes in Section 3.2.Similarly, a 2-SPUUPU simulation mechanisms II is also created, as shown in Fig. 8b.(a) (b)60607270727010101020(85.03)(46.45)(50)10101060A1A3A4A2C1dxdza2a1b1B1B2B3B4b3anaA0ac2c1L4C2dyr1r2r4r3Bm60601010101020108083801083(76.20)(50)(56.11)mBdydxdzFig. 8. (a) The simulation mechanism I and (b) the simulation mechanism II. The simulation mechanisms of the2-SPUUPU parallel mechanism with 4-driving limbs.Y. Lu / Mechanism and Machine Theory 39 (2004) 4160515. Spatial parallel manipulator with 5-driving limbs5.1. The 4-SPU and 1-UPU parallel manipulator and its simulation mechanismA new type of the spatial 4-SPU and 1-UPU parallel manipulator with 5-driving limbs is de-signed, as shown in Fig. 9a. It includes a pentagonal moving platform m, a pentagonal base B, oneextendable UPU driving limb with a hydraulic cylinder and a piston-rod, and four extendableSPU driving limbs with their hydraulic cylinders and piston-rods. Where, m is an equilateralquadrangle (a1a2a3a4) with a0as its center, and B is an equilateral quadrangle plane (A1A2A3A4)with A0as its center. Four identical SPU driving limbs connect m to B by a spherical joint S atpoint ai, a driving limb riwith a prismatic joint P, and a universal joint U at point Ai(i 1;2;3;4), respectively. 1-UPU driving limb connects m to B by a universal joint U at point a0a driving limb r5with a prismatic joint P, and a universal joint U at point A0.By inspecting this whole mechanism, we know that k 12 for one moving platform, fivecylinders, and five piston-rods, and one base; j 15 for six universal joints U, five prismaticjoints P, and four sphere joints S; f1 2 for the universal joint, f2 1 for the prismatic joint,f3 3 for the sphere joint; F0 0. Therefore, based on Eq. (1), the DOF of the whole mech-anism isF kk ? j ? 1 Xni1fi? F0 6 ? 12 ? 15 ? 1 6 ? 2 5 ? 1 4 ? 3 ? 0 55A simulation mechanism of the spatial 4-SPU and 1-UPU parallel manipulator is created inthe 3D sketch environment, as shown in Fig. 9b. Its creation processes are described as fol-lows.1=i(a) (b)PU, A1S, a1mBc2C2C1S, a2PPa3, SS, a4A3,UPA4, UU, A2U, A0nPa06060106999890301313941013(91.26)(53.40)(72.29)131 3(92.15)(99.61)(170.15,)100A1A3A4A2a1a3a4a2B2B1B3B4BA5a0d0dydzdxc1c2C2C1xyznn1r1r2r3r4r5mFig. 9. (a)The spatial 4-SPU and 1-UPU parallel manipulator and (b) its simulation mechanism. The 4-SPU and1-UPU parallel manipulator with 5-driving limbs and its simulation mechanism.52Y. Lu / Mechanism and Machine Theory 39 (2004) 41601. Follow the step 2 in the common technique, constitute a moving platform (a1a2a3a4) with side-line (60 cm) in length, and a base (A1A2A3A4) with sideline (100 cm) in length, respectively.2. Follow the steps 6 and 7 in the common technique, constitute four equivalent SPU drivinglimbs (r1, r2, r3and r4) and one equivalent UPU driving limb r5, respectively. In this way, a sim-ulation mechanism of the spatial 4-SPU and 1-UPU parallel manipulator is created.3. The position-orientation of the moving platform of the spatial 4-SPU and 1-UPU parallelmanipulator can be determined by adopting similar processes in Section 3.2.5.2. The 4-PSU and 1-UPU parallel manipulator and its simulation mechanismA new type of spatial 4-PSU and 1-UPU parallel manipulator with 5-driving limbs is designed,as shown in Fig. 10a. It includes a moving platform m, a base B, 1-UPU driving limb, and 4-PSUdriving limbs. Where, m is an equilateral quadrangle (a1a2a2a4) with a0as its center. B is cubicframe, in which an equilateral quadrangle plane (A1A2A2A4) is connected onto another equilateralquadrangle plane (B1B2B2B4) by the four vertical columns AiBii 1;2;3;4, respectively. Thefour prismatic joints Piare retained coincident with the four vertical columns AiBii 1;2;3;4,respectively.By inspecting the whole mechanism, we know that k 12 for one moving platform, five cyl-inders, and five piston-rods, and one base; j 15 for six universal joints U, five prismatic joints P,and four sphere joints S; f1 2 for the universal joint, f2 1 for the prismatic joint, f3 3 for thesphere joint; F0 0. Therefore, based on Eq. (1), the DOF of the whole mechanism isF kk ? j ? 1 Xni1fi? F0 6 ? 12 ? 15 ? 1 6 ? 2 5 ? 1 4 ? 3 ? 0 56A simulation mechanism of the spatial 4-PSU and 1-UPU parallel manipulator is created in the3D sketch environment as shown in Fig. 10b, and its creation processes are described below.(a)(b) A3A2A1P3S, e3A4a4a3U, a1r2e1S, e1P2P1P4e4S, A0BC1a2r3r1r4mr5a0B4B3B1B2ggggP560606060404070707070120(57.61)(50.12)(119.76)10050e2e3e1A3A2A1a1a4e4a3a2A4A0d0C1a0mBdzdxdyr1r2gr3gr4ggr5Fig. 10. (a) The spatial 4-PSU and 1-UPU parallel manipulator and (b) its simulation mechanism. The 4-PSU and1-UPU parallel manipulator with five DOF and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 4160531. Follow the step 2 in the common technique, constitute a moving platform (a1a2a3a4) with side-line (50 cm) in length, and a base plane (A1A2A3A4) with sideline (100 cm) in length, respectively.2. Constitute four lines Aiei, connect their one ends to point Aii 1;2;3;4 of the base plane(A1A2A3A4), respectively, by using the pointpoint coincident command, and set Aieiperpendic-ular to the base plane.3. Follow the steps 3e, 6, and 7 in the common technique, constitute four equivalent PSU drivinglimbs (r1, r2, r3and r4) and an equivalent UPU driving limb r5, respectively. In this way, a sim-ulation mechanism of the 4-PSU and 1-UPU spatial parallel manipulator is created.4. The position-orientation of the moving platform of the 4-PSU and 1-UPU spatial parallelmanipulator can be determined by adopting similar processes in Section 3.2.6. Spatial parallel manipulator with 6-driving limbs6.1. The 3/6-SPS parallel manipulator and its simulation mechanismAn existing Stewart 3/6-SPS spatial parallel manipulator has six DOF 1,13,14. It includes amoving platform m, a base B, and six extendable driving limbs riwith their hydraulic cylinders andpiston-rods, as shown in Fig. 11a. The moving platform is a regular triangle Da1a2a3with a0as itscenter. The base is a regular hexagon (A1A2A3A4A5A6) with A0as its center. Six identical SPSdriving limbs riconnect m to B by a spherical joint S at point akk 1;2;3, a driving limb riwitha prismatic joint P, and a spherical joint S at point Aii 1;2;.;6, respectively.A simulation mechanism of the spatial 3/6-SPS parallel manipulator is created as shown inFig. 11b. The creation processes are explained as follows.(a) (b)S, a3S, a1S, a2A4A3A2S, A1A5A6Bmr1r2r3r4r5PPPPP5050808080909090(78.91)(38.64)(0.14)(98.08)(89.81)8.081a3a1a2A4A2A1A5A6a0r1r4r3r6r5r2c2c1n1ndzdxdyxzyxd0A0mBr6PA3Fig. 11. (a) The 3/6-SPS parallel manipulator and (b) its simulation mechanism. The 3/6-SPS parallel manipulator andits simulation mechanism.54Y. Lu / Mechanism and Machine Theory 39 (2004) 41601. Follow the step 2 in the common technique, constitute an equilateral hexagon plane(A1A2A3A4A5A6) with sideline (50 cm) in length, and take it as the base. Constitute an equilateraltriangle Da1a2a3with sideline (50 cm) in length, and take it as the moving platform.2. Follow the step 5 in the common technique, constitute 6-SPS driving limbs rit 1;2;.;6and give each driving limb an initial driving dimension (90 cm) in length. In this way, a simu-lation mechanism of the 3/6-SPS parallel manipulator is created.3. The position-orientation of the moving platform of the 3/6-SPS simulation mechanism can bedetermined by adopting similar processes in Section 3.2.By inspecting this whole mechanism, we know that k 14 for one moving platform, six cyl-inders, and six piston-rods, and one base; j 18 for six prismatic joints P, six sphere joints S on B,and six sphere joints S on M which are transformed into three composite sphere joints S; f1 1for the prismatic joint, f2 3 for the sphere joint; F0 6 for 6-driving limb rotating aboutthemselves axis. Therefore, based on Eq. (1), the DOF of the whole mechanism isF kk ? j ? 1 Xni1fi? F0 6 ? 14 ? 18 ? 1 6 ? 1 12 ? 3 ? 6 676.2. The 6-SPS spatial parallel manipulator and its simulation mechanismAn existing StewartGough 6-SPS spatial parallel manipulator has six DOF 1,13,14. It in-cludes a moving platform m, a base B, and six extendable driving limbs riwith the hydrauliccylinder and the piston-rod, as shown in Fig. 12a. Where, m is a regular hexagon (a1a2a3a4a5a6)with a0as its center, and B is the regular hexagon (A1A2A3A4A5A6) with A0as its center. Sixidentical SPS driving limbs riconnect m to B by a spherical joint S at point ai, a driving limb riwith a prismatic joint P, and a spherical joint S at point Aii 1;2;.;6, respectively.(a)(b)A2S, A1A5A6A3a2A4a4a5S, a1a0a6r6r1r2A0r3r4cbr6Pr5a3PPPPPmB82838250505050505082828310030(74.96)(57.86)(15.94)(93.28)(197.06#)106.72#25A2A1A5A6A3a2A4a4a3a5a1a0a6dznxzyn1ydydxxcd0A0mBFig. 12. (a) The 6-SPS parallel manipulator and (b) its simulation mechanism. The StewartGough 6-SPS parallelmanipulator and its simulation mechanism.Y. Lu / Mechanism and Machine Theory 39 (2004) 416055By inspecting this whole mechanism, we know that k 14 for one moving platform, six cyl-inders, and six piston-rods, and one base; j 18 for six prismatic joints P, six sphere joints S on B,and six sphere joints S on M; f1 1 for the prismatic joint, f2 3 for the sphere joint; F0 6.Therefore, the DOF of the whole mechanism is the same as that of Stewart 3/6-SPS spatial parallelmanipulator.A simulation mechanism of the 6-SPS spatial parallel manipulator with 6-driving limbs iscreated, as shown in Fig. 12b. The creation processes are explained as follows.1. Follow the step 2 in the common technique, constitute an equilateral hexagon (A1A2A3A4A5A6)with sideline (50 cm) in length, and take it as the base. Constitute an equilateral hexagon plane(a1a2a3a4a5a6) with sideline (25 cm) in length, and take it as the moving platform.2. Follow the step 5 in the common technique, constitute 6-SPS driving limbs rii 1;2;.;6and give each driving limb an initial driving dimension (82 cm) in length. In this way, a simu-lation mechanism of the 6-SPS parallel manipulator with 6-driving limbs is created.3. The position-orientation of the moving platform of the 6-SPS simulation mechanism can be de-termined by adopting similar processes in Section 3.2.6.3. The 6-SSP spatial parallel manipulator and its simulation mechanismA new type of the 6-SSP spatial parallel manipulator is designed, as shown in Fig. 13a. Itincludes a moving platform m, a base B, six binary links, and six extendable driving limbs riwiththe hydraulic cylinder and the piston-rod. Where, m is a regular hexagon (a1a2a3a4a5a6) with a0asits center, and B is an equilateral quadrangle link (B1B3B4B6) with three translation paths. Sixidentical SSP driving limbs connect m to B by a spherical joint S at point ai, binary link gi, a(a)(b)S, A1PPPS, A6S, A3A2A5S, A4PPPS, a3S, a2S, a1S, a6S, a5S, a4mBL1L2L3150302819202830100100100106100100110(22.63)(88.58)(96.44)50(86.26)(114.09k)(24.419)A6A2A5A1A3A1a5a6a4a3a2a1g4r2g1g5g6g3dydxdza0L1L2L3xyzmBB6B1B5B2B3B4nn1Fig. 13. (a) The 6-SSP parallel manipulator and (b) its simulation mechanism. The 6-SSP parallel manipulator and itssimulation mechanism.56Y. Lu / Mechanism and Machine Theory 39 (2004) 4160spherical joint S at point Ai, and a driving limb riwith a prismatic joint P at point Ai,i 1;2;.;6, respectively.By inspecting the whole mechanism of Fig. 13a, we know that k 14 for the six cylinders, sixpiston-rods, one base, and one moving platform; j 18 for the 12 spherical joints S, and sixprismatic joints P; f1 3 for spherical joint, f2 1 for prismatic joints; F0 6 for the six binarylinks rotating about their axis. Therefore, the DOF of the 6-SSP parallel manipulator is deter-mined as follows.F kk ? j ? 1 Xni1fi? F0 6 ? 14 ? 18 ? 1 12 ? 3 6 ? 1 ? 6 68A simulation mechanism of the spatial 6-SSP parallel manipulator with 6-driving limbs is createdas shown in Fig. 13b. The creation processes are explained as follows.1. Follow the step 2b in the common technique, constitute an equilateral hexagon plane(a1a2a3a4a5a6) with sideline (50 cm) in length, and take it as the moving platform.2. Follow the step 2d in the common technique, constitute an equilateral quadrangle link(B1B3B4B6) with sideline (100 cm) in length, and take it as the base. Constitute a line L2andconnect its two ends to the middle point B2of sideline B1B3and the middle point B5of sidelineB4B6, respectively, by using the pointline coincident command.3. Follow the steps 3e and 7 in the common technique, constitute six lines gi, (i 1;2.;6), giveeach of them an initial fixed dimension (100 cm) in length. Connect their two ends to the mov-ing platform at point aiand the base at point Ai, respectively. Thus, 6-SSP driving limbs areconstituted. In this way, a simulation mechanism of the spatial 6-SSP parallel manipulator with6-driving limbs is created.4. The position-orientation of the moving platform of the 6-SSP simulation mechanism can be de-termined by adopting similar processes in Section 3.2.6.4. The spatial 6-SPRP parallel manipulator and its simulation mechanismFrom the spatial 3-SRP parallel manipulator of Fig. 6a, replace the three binary links giby theextendable driving limbs riwith the hydraulic cylinder and the piston-rod (i 1;2;3), respectively.In this way, a new type of the spatial 6-SPRP parallel manipulator is designed. It includes amoving platform m, a base B, and six extendable driving limbs with their hydraulic cylinders andpiston-rods, as shown in Fig. 14a. Moreover, this mechanism is partially parallel and partiallyserial, which is often referred to as hybrid structure.By inspecting the whole mechanism of Fig. 14, we know that k 11 for the six cylinders,three piston-rods, one base, and one moving platform; j 12 for the three spherical joints S,and six prismatic joints P, three revolute joints R; f1 3 for spherical joint, f2 1 for pris-matic joints, f3 1 for revolute joints; F0 0. Therefore, the DOF of the 6-SSP parallelmanipulator isF kk ? j ? 1 Xni1fi? F0 6 ? 11 ? 12 ? 1 3 ? 3 6 ? 1 3 ? 0 69Y. Lu / Mechanism and Machine Theory 39 (2004) 416057A simulation mechanism of the spatial 6-SPRP parallel manipulator with 6-driving limbs, asshown in Fig. 14b is similar to that of spatial 3-SRP parallel manipulator, as shown in Fig. 6b,except that the fixed dimension of the three binary links gii 1;2;3 are transformed intodriving dimension.7. Solving displacement of the moving platformIn the light of the simulation mechanism of the spatial 3-UPU parallel manipulator as shown inFig. 4b, when set li 50 cm, Li 120 cm (i 1;2;3), its original configuration is changed into anew one as shown in Fig. 14a. Therefore, once a simulation mechanism is created, it can be usedrepeatedly for synthesizing same type of spatial parallel mechanism with different sizes andconfigurations. The three position components of the moving platform of the 3-UPU simulationmechanism can be solved, and the solving processes are explained below.In the environment of the 3-UPU simulation mechanism, a data sheet of Microsoft Excel isembedded into the sketching interface of the CAD software. Next, click each driving dimension ofdriving limbs, thus their dimension names (r1, r2and r3) are automatically listed in the first row ofthe data sheet. After that, fill the three driving dimensions (r1, r2and r3) from 80 to 120 cm withdifferent increments (1, 1.5 and 2) and the number of configurations into their correspondingcolumn, respectively, by using automatically filling function. Thus, the 20 different configurationsof the simulation mechanism are constituted. Each configuration corresponds to each row drivingdimensions of (r1, r2and r3). When any one of all configurations is active, the 3-UPU simulationmechanism will be varied and can be directly visualized on screen. Thus the position componentsof the moving platform of spatial 3-UPU parallel manipulator can be solved automatically. The(a) (b)808080403050707070(69.17)(10)(5.78)(89.94)(90.85)(179.15+)S, a1a2a3g1g2g3B1A1B2B3AA3d0mBa0dzdydxnn1xyzr2r3r1L1L2L3S2, a1S2, a2S3, a3R2, A2R1, A1R3, A3P3,g1P1, r1P3, r3P2, r2B1B3B2mBP6,g3P2, g2Fig. 14. (a) The 6-SPRP parallel manipulator and (b) its simulation mechanism. The 6-SPRP parallel manipulator andits simulation mechanism.58Y. Lu / Mechanism and Machine Theory 39 (2004) 4160simulation results of position components of the spatial 3-UPU parallel manipulator are retainedthe same as that of analytics? approach, as shown in Fig. 15.8. ConclusionsBy applying the computer aided geometry constraint and dimension driving techniques in 3Dsketch environment of the advanced CAD software, the simulation mechanisms of the some newand original spatial parallel manipulators with 3-, 4-, 5-, 6-driving limbs can be designed. Whenmodifying the driving dimensions of driving limbs by using the dimension driving technique, theconfigurations of the simulation mechanisms are varied correspondingly. In this way, the kine-matic parameters of the moving platform of each simulation mech