【机械类毕业论文中英文对照文献翻译】机械手的给定工作区内的一种 6 自由度并联关键点三维设计方法
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【机械类毕业论文中英文对照文献翻译】机械手的给定工作区内的一种 6 自由度并联关键点三维设计方法,机械类毕业论文中英文对照文献翻译,【机械类毕业论文中英文对照文献翻译】机械手的给定工作区内的一种,自由度并联关键点三维设计方法,机械类,毕业论文,中英文,对照,文献,翻译,机械手,给定,工作,区内,一种
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机械手的给定工作区内的一种 6 自由度并联关键点三维设计方法摘要：本文提出了在给定工作区内一种6自由度的新三维设计新方法 。许多关键特性已经进行运动学分析和拉格朗日乘数法。此外，在整个机械手的直接几何关系中导出了参数。提出了设计方法，关于这些关键点特性具有很高的效率和准确性。此外，避免了复杂机械手的工作空间和无量纲化推导分析从而可能让这种方法的广泛应用。 2014年爱思唯尔有限公司。版权所有1 .导言对并联机器人的关注主要是发现他们有更好的承载能力，更好的刚度，和比串联机器人更好的精度1-4。因此并联机器人的研究已成为一个热门的国际机器人研究领域5-9。并联机器人的设计过程是机械产品中最具有挑战性的问题。设计机器人10-12的配置，机械臂的几何参数应由三维设计决定。引用13，14中提出的参数设计方法分别用于6 自由度歌赋型机器人和3自由度并联机器人。 一般来说，最重要的设计目标之一是让机器人在给定工作区工作。到目前为止，有主要有两种方法，根据给定的工作区的并联机器人的几何参数优化设计。第一次使用多点来描述给定工作区，然后检查机械手的每个点的设计要求是否符合参数15-17，与另一个边界的机械手之间建立参数和工作区中的函数，然后确保给定工作区是机械手的工作空间边界内18-22。基于我们在这项研究发现的几个关键问题，本文试图探索给定的工作区6自由度并联机器人新的三维设计方法。这种设计方法是快速的，它的结果是准确的。 在我们以前的工作中，这种新型的6自由度并联机构中用到了3-3-PSS配置。与传统6-SPS并联机器人相比这3-3-PSS并联的机械臂性能允许更高的各向同性的、更大的旋转范围移动平台，减少了身体惯性。 若要开始设计，应清楚的描述所需的工作区。因为不能以图形方式表示6 维工作区，以人类可读的方式，没有一般的方法来分析确定的6-D工作区的边界6 自由度并联机器人，大多数文献23-27将6-D区划分为工作区的位置和方位工作空间。工作区的位置是指机械手的移动平台可以达到一定的取向的空间。它可以容易地描述。方位工作空间是移动平台可以实现在某一时刻的所有方向的集合。然而，由于旋转角度的复杂性，方位工作空间很难确定和代表。考虑到我们并联机械手的对称性，简明描述6-D区找到了种的三维设计。 本文的结构如下。第二节介绍了建模的设计问题及运动学分析。第3节介绍如何找到关键点特征。第4节中，讨论了设计方法及应用。最后，第5节中总结发言。2模型的设计问题和力学分析新的PSS 3-3并联机器人的结构如图1所示，它是由一个移动的平台，一个固定基座，和六个具有相同的几何结构支撑臂组成。四肢编号从1到6的每个肢体由一个棱柱形接头，一个球形接头和联合空间综合信息网络球系列连接到固定基地到所述移动平台。一个线性执行机构驱动的棱柱沿着固定轨道各肢的关节。关节Bi和关节Ai之间是长为Li的刚性连杆（I =1，.，6） 1，2，和3被设置成位于一水平面的PB它们的轴线四肢的三个线性致动器，且当这些轴不交于一点时它们的轴之间的夹角为120。这些轴与操纵器的对称轴之间的距离是相同的，在这里我们使用一个参数来表示该距离。其他三个线性执行器四肢4,5，和6被设置成垂直的轴线。关节的移动平台A1A6分布在中心对称的半径为a的一个圆上。这种操纵器的中心在平面PB的交叉点和操纵器的对称轴上，在其上连接有固定笛卡尔参考帧-O X，Y，Z。固定框架y轴和z轴都在平面PB上，并且与操纵器的对称轴的X轴重合。移动框架OX，Y，Z连接移动平台O点“，这是指向位于圆心上的A1A6。关于机械手是轴对称的事实，移动台处于初始位置时让点O与点O重合，从而操纵器的工作空间相对于固定框也是轴对称。设计的操纵器的几何参数前，所需的工作空间应明确说明。从前面的讨论中可以看出，简明地描述所需6-D的工作区是一个具有挑战性的问题。在这个研究中，对移动台的方向的说明，仅指示向量（显示在图2中），而不是绕其对称轴旋转而言。事实上，这是许多机床有着的同样的情况。在此基础上，我们使用一组特殊的欧拉角来表示的移动平台的方向。移动平台的首先由一个角度固定x轴，然后由角度固定z轴，最后由角固定x轴（图2）。我们可以把旋转矩阵简单的写成这种情况：3.在给定的工作空间机器人的关键特征 在这项研究中，通过大量的计算，我们发现在qi最大范围内，尽管给定的工作区和操纵器的尺寸在改变，Bi和Ai总是发生在一定位置。这一特点对尺寸设计非常有帮助，所以我们称这些位置为关键点。本节将证明理论上使用拉格朗日乘子的方法，建立关键点。 为了推广，我们做了三维设计的相关参数量通过让他们每个人用钢筋混凝土进行划分。因此，工作空间汽缸的无量纲半径为1，并且其无量纲高度为2H。其中，H= HC / Rc。因此，基于该无量纲工作空间的尺寸设计的结果不能被直接当作操纵器的几何参数，除非由RC乘以它们所有（应当注意的是，在此过程中角度不影响）。由于机械手的配置两肢体的人群有不同的关键特征。因此，两肢组的特性，应分别研究。4.基础的三维设计方法的关键点及其应用 在上一节找到对应的工作空间内操纵的一些重要关键点的特征。其要点是极端位置，这将导致在给定的工作空间中操纵器的最坏运动学条件。如果操纵器可在关键点达到所需的运动学性能，那么这个运动性能将在给定的工作空间中保证每个点。这些特性可以被用于确定所述操纵器的几何参数，从而在三维设计将具有非常高的效率和准确性。对于这个关键点的设计方法的主要步骤如下： 1.描述所需的工作空间。研究了操纵器的工作任务，并计算出所需要的空间和方向。然后选择与可以只达到要求的给定的工作空间有一定指向灵巧指数缸。如果所需的工作空间是复杂的，它可以被描述为多个同轴圆柱体具有不同指向灵巧指数与图4所示。在这种状态下，下面的设计步骤2-5，对于每个气缸都应进行，其结果应结合作为最终的解决方案。 2.给定的工作空间量纲。对于每个气缸，让其半径和高度由它自己的半径进行划分。 3.明确额外的设计要求和使用表1中找到所有需要的关键点。如果关节角的范围没有限制，可以与工作区保证的关键点或相应的直接关系建立所述几何参数的约束关系。（参考表1）。如果接头角度是有要求限制的，应与最大Bi和最大Ai的关键点或相应的直接关系建立所述几何参数的约束关系。（参考表 1） 4.确定的几何参数。找到能满足前面建立的步骤中的约束关系的适当的参数。这些约束关系，a和Li有许多可能的解决方案可以找到。一般最小的a和Li将导致操作者的最小量应被选择。应当注意的是，只有一个肢需要被确定给每个组，因为操作者是对称。在一些情况下，a和Li可能有具有因工作任务的额外的限制，并且步骤可用于进一步优化设计的约束关系。 5.获得的a和Li应应乘以圆柱的半径得到维数。然后他们可以作为机器人的几何参数。 6.确定其余的几何参数。 如果有多于一缸用于工作区说明，在第5步中得到的结果应该作为一个相结合解决方案。那就是，选择的最大值和李之间所有气缸的结果作为最后的解决办法。因此，联合解决方案： 能满足各种约束关系的每个气缸。在那之后，Bi和Ai的范围应当重新计算与最终解决方案的关键点船帆齐和最低气或（请参阅表1）的直接对应关系，可以确定李和练习场。应当指出：所有气瓶必须检查在此过程中，其结果应作为最后的结果相结合。在这里，我们的项目用来证明该设计方法的应用。我们所需的工作区可以用描述筒（缸1）与半径为600毫米，高度为800毫米和0 时，指向灵巧和气缸（缸2）与半径200毫米、高度为400毫米和30 的指点灵巧。各关节角度被限制为小于45 。此外，参数需求大于350毫米将在移动平台放置对象的尺寸和接头的尺寸。为缸1，与最大值Bi和最大值Ai的关键点，可以获得参数的最小的解作为Li=1050毫米（i=1 2、 3）和Li=850毫米（i=4,5,6）而不是参与。油缸2，最小的解的参数可以作为发现a=350毫米，Li=1050毫米（i=1 2、 3）和Li=1000毫米（i=4,5,6）与要点船帆Bi和最大值Ai。结合这两项结果，可以得到该机械手的最终解，作为a=350毫米，Li=1050毫米（i=1 2、 3）和Li=1000毫米（ia=4,5,6）。最后，为每个气缸带有计算的Bi、Ai和驾驶中风最后的范围相应的关键点，然后结合。设计结果如表2所示。和与该机械手的原型这些设计的几何参数如图5所示。为了验证这些设计结果的正确性，设计的机械手性能在给定工作区中有已检查。我们采取了一系列圆筒截面和离散他们成均匀离散点。每个这些离散点的取向也进行离散化处理。然后联合角度的值记录在移动平台达到每个位置和方向。为清楚起见，都会选择一些典型的数据并绘制在这部分中。当设计的机械手工作缸2顶块、分布的Bi和Ai组1所示图6和7分别。图8和图9显示了同样的情况，Bi和Ai2组。可以观察到所有关节角度都小于45 ，并只是接近45 腿各关节角度的最大值出现在的关键点。所有这些结果都是一致的。本文分析研究并满足要求5.结论本文对此提出了新的三维设计方法，为我们的新 3-3-PSS并联机构根据给定提出了工作区。这种方法基于几个关键点，避免了机械手的复杂分析自己6-D区实际上并没有一个统一的描述人类可读的方式。关键点建立简单的关系机械臂的几何参数与工作区的要求。在此基础，提出的设计方法已非常高的效率和准确性。很多关键点特征已发现并在表1中列出。 要点是极端的立场,将导致最严重的机械手的运动学条件给定的工作区。运动学性能可以保证在整个工作区,让机械手实现性能的关键点。此外,一些直接运动学和几何参数之间的关系已经建立的空间设计。简明地描述6 d工作区,使设计要求很明显,已经发现了对称描述给定的工作区。这个描述很容易理解和接近机械手的操作条件。因此,这种方法可以很容易地用在许多不同的情况。关键点是会导致极端的立场。 机械手在给定工作区中的最差运动学条件。运动学性能可以保证内给定工作区，让整个机械手实现性能的关键点。此外，一些直接的关系之间的运动学和几何参数已经被为三维设计建造。 简要描述6-D区和清楚的设计要求，对称的描述找到了给定工作区。此描述是机械手的非常容易理解和接近工况。其结果是，这种方法可轻松用于许多不同的情况。这种方法推导了特定类型的并行机制，但找到关键点的想法可能会用于其它并联机构的类型。核心问题是找到其职位订明的工作区中是独立的关键点。 随着规模的订明的工作区和机制。这通常需要订明的工作区的形状和机制的工作区有一些相似的特征如本例中的轴向对称。在此研究中，任何其他轴对称的形状可以用于描述形状的除了气缸的给定工作区。重写的约束方程拉格朗日方法，以及这些形状的关键点，可以发现与本文类似的程序。可能很难找到关键点，但三维设计的并行机制会变得非常方便一旦它做了。如果机制是不对称的,那么它应当指出的关键点应分别为每个肢体找到。 提出的设计方法基于运动学。其实，关节角Bi和Ai，本文主要研究有直接雅可比矩阵，然后动态的关系。基于这项工作，在不久的将来，将研究基于动力学的设计方法。确认这项工作部分支持主要国家基本研究中国的发展计划（973计划）（第2013CB035501号），和国家自然科学基金（批准号：51335007）。文献资料1 B. Dasgupta, T. Mruthyunjaya, The Stewart platform manipulator: a review, Mech. Mach. Theory 35 (2000) 1540.2 J. Gallardo-Alvarado, M. Garca-Murillo, L. Prez-Gonzlez, Kinematics of the 3RRRS+ S parallel wrist: a parallel manipulator free of intersecting revolute axes,Mech. Based Des. Struct. Mach. 41 (4) (2013) 452467.3 S. Zarkandi, Kinematics and singularity analysis of a parallel manipulator with three rotational and one translational DOFs, Mech. Based Des. Struct. Mach. 39(2011) 392407.4 M. Valls,M. Daz-Rodrguez, . Valera, V.Mata, . Page, Mechatronic development and dynamic control of a 3-DOF parallelmanipulator,Mech. Based Des. Struct.Mach. 40 (2012) 434452.5 D. Gan, J.S. Dai, J. Dias, L. Seneviratne, Constraint-plane-based synthesis and topology variation of a class of metamorphic parallel mechanisms, J. Mech. Sci.Technol. 28 (2014) 41794191.6 A. Karimi, M.T. Masouleh, P. Cardou, The Dimensional Synthesis of 3-RPR Parallel Mechanisms for a Prescribed Singularity-free Constant-orientationWorkspace,Advances in Robot Kinematics, Springer, 2014. 365373.7 K. Zhang, J.S. Dai, Y. Fang, Geometric constraint and mobility variation of two 3SvPSv metamorphic parallel mechanisms, J. Mech. Des. 135 (2013) 011001.8 M.T. Masouleh, C. Gosselin, M. Husty, D.R. Walter, Forward kinematic problemof 5-RPUR parallel mechanisms (3T2R)with identical limb structures,Mech. Mach.Theory 46 (2011) 945959.9 A. Chaker, A. Mlika, M.A. Laribi, L. Romdhane, S. Zeghloul, Robust Design Synthesis of Spherical Parallel Manipulator for Dexterous Medical Task, ComputationalKinematics, Springer, 2014. 281289.10 F. Gao, J. Yang, Q.J. Ge, Type Synthesis of Parallel Mechanisms Having the Second Class G Sets and Two Dimensional Rotations, ASME, 2011.11 K. Korkmaz, Y. Akgn, F. Maden, Design of a 2-DOF 8R linkage for transformable hypar structure, Mech. Based Des. Struct. Mach. 40 (2012) 1932.12 X. Meng, F. Gao, S. Wu, Q.J. Ge, Type synthesis of parallel robotic mechanisms: framework and brief review, Mech. Mach. Theory 78 (2014) 177186.13 J.P. Merlet, Designing a parallel manipulator for a specific workspace, Int. J. Robot. Res. 16 (1997) 545.14 T. Sun, Y. Song, Y. Li, L. Liu, Dimensional synthesis of a 3-DOF parallel manipulator based on dimensionally homogeneous Jacobian matrix, Sci. China Ser. E:Technol. Sci. 53 (2010) 168174.15 A. Kosinska,M. Galicki, K. Kedzior, Determination of parameters of 3-dof spatial orientationmanipulators for a specifiedworkspace, Robotica 21 (2003) 179183.16 E. Ottaviano, M. Ceccarelli, Optimal design of CaPaMan (Cassino Parallel Manipulator) with a specified orientation workspace, Robotica 20 (2002) 159166.17 A. Kosinska, M. Galicki, K. Kedzior, Design of parameters of parallel manipulators for a specified workspace, Robotica 21 (2003) 575579.18 M. Laribi, L. Romdhane, S. Zeghloul, Analysis and dimensional synthesis of the DELTA robot for a prescribed workspace, Mech.Mach. Theory 42 (2007) 859870.19 R. Di Gregorio, R. Zanforlin, Workspace analytic determination of two similar translational parallel manipulators, Robotica 21 (2003) 555566.20 F. Gao, B. Peng, W. Li, H. Zhao, Design of a novel 5-DOF parallel kinematic machine tool based on workspace, Robotica 23 (2005) 3543.21 A. Hay, J. Snyman, Optimal synthesis for a continuous prescribed dexterity interval of a 3dof parallel planar manipulator for different prescribed outputworkspaces, Int. J. Numer. Methods Eng. 68 (2006) 112.22 F. Gao, X.J. Liu, X. Chen, The relationships between the shapes of the workspaces and the link lengths of 3-DOF symmetrical planar parallel manipulators, Mech.Mach. Theory 36 (2001) 205220.23 C. Gosselin, Determination of the workspace of 6-dof parallel manipulators, J. Mech. Des. 112 (1990) 331.24 V. Kumar, Characterization of workspaces of parallel manipulators, J. Mech. Des. 114 (1992) 368.25 O. Masory, J. Wang, Workspace evaluation of Stewart platforms, Adv. Robot. 9 (1994) 443461.26 L.C.T. Wang, J.H. Hsieh, Extreme reaches and reachable workspace analysis of general parallel robotic manipulators, J. Robot. Syst. 15 (1998) 145159.27 J.-P. Merlet, C.M. Gosselin, N. Mouly, Workspaces of planar parallel manipulators, Mech. Mach. Theory 33 (1998) 720.A key point dimensional design method of a 6-DOF parallelmanipulator for a given workspaceRui Cao1, Feng Gao,1, Yong Zhang1, Dalei Pan1State Key Lab of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, PR Chinaa r t i c l ei n f oa b s t r a c tArticle history:Received 3 April 2014Received in revised form 7 November 2014Accepted 8 November 2014Available online 25 November 2014This paper presents a new method of dimensional design for a 6-PSS parallel mechanismaccording to a given workspace. A symmetrical description has been found to describe the 6-Dworkspace concisely for the dimensional design. Many key point characteristics have beenfound and verified by the kinematic analysis and the method of Lagrange multipliers.Furthermore,thedirectrelationsbetweenthegivenworkspaceandthemanipulatorsgeometricalparameters have been derived. The proposed design method which is based on these key pointcharacteristics has very high efficiency and accuracy. Additionally, the avoiding of the complexanalysis of the manipulators workspace and the dimensionless derivation make the possibilityof wide use of this method. 2014 Elsevier Ltd. All rights reserved.Keywords:Parallel manipulatorDimensional designWorkspace6-PSSKey point1. IntroductionThe interest for parallel manipulators arises from the fact that they have better load-carrying capacity, better stiffness, and betterprecision than serial manipulators 14. Thus the research on designing parallel manipulators has become a hot topic in theinternational robotic research area 59. The design of parallel manipulators is a challenging problem in the machinery productdesign process. The type synthesis is for designing the configuration for manipulators 1012. And then the geometrical parametersofmanipulatorsshouldbedetermined bythedimensionaldesign.Becausethetypesof parallelmechanisms arealmostunlimited,thedimensionaldesignmustbebasedonacertaintypeofmechanisms.Theparameterdesignmethodspresentedinreference13,14arebased on 6-DOF Gough-type manipulators and 3-DOF parallel manipulators, respectively.Generally, one of the most important design objectives is to let the manipulator work in a given workspace. Therefore, thedimensional design of parallel manipulators for a given workspace is an important problem, which has not gained too much interest.So far, there are mainly two ways to design the geometrical parameters of parallel manipulators according to a given workspace. Thefirst one uses many points to describe the given workspaceand then check whether the manipulator with certain parameters fits thedesign requirements at each point 1517. The other one establishes a function between the parameters and the workspaceboundaries of the manipulator, then make sure that the given workspace is within the manipulators workspace boundaries1822.Basedonseveralkeypointsthatwehavefoundinthisstudy,thispaperattemptstoexploreanewwayofdimensionaldesignfor a new 6-DOF parallel manipulator according to a given workspace. This design method is fast and its result is accurate.In our previous work, a new type of 6-DOF parallel mechanism with an orthogonal 3-3-PSS configuration has been proposed.Compared with the traditional 6-SPS parallel manipulators, this 3-3-PSS parallel manipulator allows higher isotropy of themanipulators performance, larger rotation range of the moving platform and less body inertia.Mechanism and Machine Theory 85 (2015) 113 Corresponding author.E-mail addresses: azuresilent (R. Cao), (F. Gao), (Y. Zhang), (D. Pan).1P.O. Box ME290, Mechanical Building, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Shanghai 200240, PR China./10.1016/j.mechmachtheory.2014.11.0040094-114X/ 2014 Elsevier Ltd. All rights reserved.Contents lists available at ScienceDirectMechanism and Machine Theoryjournal homepage: /locate/mechmtTobeginthedesign,therequiredworkspaceshouldbeclearlydescribed.Becausethe6-dimensionalworkspacecannotberepresent-edgraphicallyinahuman-readablewayandtherearenogeneralwaytoanalyticallydeterminetheboundariesofthe6-Dworkspacefor6-DOFparallelmanipulators,mostliteratures2327dividethe6-Dworkspaceintopositionworkspaceandorientationworkspace.Theposition workspace refers to a space that the manipulators moving platform can reach with a certain orientation. And it can be easilydepicted.Theorientationworkspaceisthecollectionofalltheorientationsthatthemovingplatformcanachieveatacertainpoint.How-ever, due to the complexity of the rotating angles, the orientation workspace is difficult to be determined and represented. Consideringthe symmetry of our parallel manipulator, a concise way of describing the 6-D workspace is found for the dimensional design.The paper is organized as follows. Section 2 introduces the modeling of the design problem and the kinematics analysis. Section 3shows how the key point characteristics are found. The design method and its application are discussed in Section 4. Finally,concluding remarks are presented in Section 5.2. Modeling of the design problem and kinematic analysisThearchitectureofthenew3-3-PSSparallelmanipulatorisshowninFig.1,whichiscomposedofamovingplatform,afixedbase,and six supportinglimbswith identical geometrical structure. The limbs are numbered from 1 to 6. Each limbconnects the fixed basetothemovingplatformbyaprismaticjoint,asphericaljointBiandasphericaljointAiinseries.Alinearactuatoractuatestheprismaticjoint of each limb along a fixed rail. Between the joint Biand joint Aiis a rigid link of length Li(i=1,6).The three linear actuators of the limbs 1, 2, and 3 are arranged with their axes located in a horizontal plane PB, and the angles be-tween each of their axes are 120 while these axes do not intersect at one point. The distances between these axes and the symmetryaxis of the manipulator are the same, and here we use the parameter a to represent this distance. The other three linear actuators ofthe limbs 4, 5, and 6 are arranged with their axes vertically. The centers of the joints A1 A6of the moving platform are distributedsymmetrically on a circle of radius a. The center of this manipulator is at the intersection of the plane PBand the symmetry axis ofthe manipulator, on which attached a fixed Cartesian reference coordinate frame Ox, y, z. The fixed frames y-axis and z-axis are inthe plane PB, and its x-axis coincides with the symmetry axis of the manipulator. A moving frame O x , y , z is attached on themoving platform at point O which is the center of the circle that points A1 A6located on. Considering the fact that the manipulatoris axisymmetric, let point O coincides with point O when the moving platform is at the initial position. Thus the workspace of themanipulator is also axisymmetric with respect to the fixed frame O.Before designing the geometrical parameters of the manipulator, the required workspace should be clearly described. As can beseen from the previous discussion, concisely describing the required 6-D workspace is a challenging problem. In this research, forthe orientation description of the moving platform, only the pointing vector (showed in Fig. 2) rather than the rotation about itssymmetry axis is concerned. In fact this has the same situation for many machine tools. Based on this, we use a special set of Eulerangles to represent the orientation of the moving platform. The moving platform first rotates about the fixed x-axis by an angle-,thenaboutthefixedz-axisbyanangle,andfinallyaboutthefixedx-axisbytheangle(Fig.2).Andwecansimplywritetherotationmatrix for this case as:R Rot x;Rot z;Rot x;ccssscss2 c2cs sccsscs ccsc2 s2c2435;1Fig. 1. The configuration of the proposed 3-3-PSS parallel manipulator.2R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113where c stands for cosine, s stands for sine, , and 0, , respectively. For the convenience of description, Eq. (1) can beabbreviated asR r11r12r13r21r22r23r31r32r332435:2And it can be observed thatr32 r23sin 2 1cos 2:3This special set of Euler angles gives anintuitive representation of themovingplatforms orientation.The pointingvector is decid-ed byand . Duetothesymmetryof themanipulator, itis easytofindoutthat therange ofis unlimitedwhile isnot.Thus,all thepossiblepointingvectorsthatthemovingplatformcanachieveatacertainpointconstituteacone.Andtheapertureoftheconeisonlyrelated to the maximum range of which is represented by m. We call mas the pointing dexterity index of the moving platform.To take advantage of the symmetry of the manipulator, we restrict the required workspace as a symmetric space. Hence, wedescribethegivenworkspaceasacylinderwithradiusofRc,andheightof2Hc.Additionally,themanipulatorshouldhavethepointingdexterityof mat any pointwithin this cylinder. This human-readable workspacedescription fitsforthe manipulators symmetry andmakes the design objective clearly. Knowing that this workspace description is actually 5-DOF, to represent a 6-DOF workspace, anadditionaldexterityindexoftherotationaboutthemovingplatformssymmetryaxis isneeded.In thissituation,themovingplatformshould first perform an additional rotation about the fixed x-axis by an angle , and the rotation matrix can be written as Rot(x,)Rot(z,)Rot(x, -)Rot(x,). However, 5-DOF is enough for our current study and most multi-DOF machine tools.After the analysis of the required workspace, what parameters of the manipulator need to be determined should be clarified. Thefollowing part will find this out by analyzing the kinematics of the manipulator. As the six limbs of the manipulator have identicalgeometrical structure, we can choose one typical limb for the analysis and its vectors are described in Fig. 3. The linear actuatorsaxis is represented by eiwhich is a unit vector. The direction of the rigid link is represented by liwhose magnitude is Li. The vectorbetween O and the center of the joint Aiis represented by ai with respect to the moving frame O , and aiwith respect to thefixed frame O. It can be found from the previous part that the magnitude of ai/aiis a. When the manipulator at the initial positionthatmentionedabove,ei(i= 1,2,3)isperpendiculartoai,itshouldbenoted.AndtheinitialpositionofBiinthissituationisrepresent-ed by point Ciwhose position vector is ci. With the special set of Euler angles, the transformation from the moving frame to the fixedframe can be described by the position vector of the moving platform p = PxPyPzT, and the rotation matrix R. Thus the generalizedcoordinates of the moving platform can be described as (Px, Py, Pz, , , 0).Let qirepresent the stroke of the linear actuator. Then we can simply get the following relation from Fig. 3:li p Ra0iqieici:4In some cases, the joints Biand Aiwhose stiffness are the lowest of the manipulator need a strong structure to increase theirstiffness. However, the strong structure always limits the rotation ranges of these joints. Therefore, the swing amplitude of theFig. 2. The pointing dexterity and the special set of Euler angles.3R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113rigid link should be studied. We define the angle between liand eias the joint angle Biof joint Bi. As joint Aiis fixed on the movingplatform,thedefinitionshouldwithrespecttothemovingframeOx,y,z.ThusthejointangleAiofjointAiisdefinedastheanglebetween liand Rei. Biand Aiare depicted in Fig. 3. The following equations about Biand Aican be achieved by their defini-tion:li? ei LicosBi5li? Rei LicosAi:6According to these definitions, the rotation of the rigid link about its own axis liis not involved. So Biand Airepresent the swingamplitudeoftherigidlinkwithrespecttotheconnectingjoint.ThemaximumvaluesofBiandAiareveryimportantforthedesignofthe spherical joints and meaningful for avoiding the interference between the rigid links.The six limbs canbedividedintotwogroupsaccordingtotheconfiguration ofthemanipulator.Thelimbs1,2,and3are containedin group 1, and the limbs 4, 5, and 6 in group 2. These two groups have different kinematic characteristics, thus need to be studiedseparately. For the sake of symmetry, the rigid links in one group should have the same length. In group 1 for i = 1, 2 and 3, afixed Cartesian reference coordinate frame Oaix, y, z is attached at the point O. For simplicity and without losing the generality,we let its y-axis point in the negative direction of the vector eiand let its x-axis coincide with the x-axis of the frame Ox, y, z.With respect to the frame Oai, it can be known from the architecture of the manipulator that ei 010?T, a0i 00a?Tand ci 0Lia?T. Assume that lilxlylz?T. Substituting all the known variables into Eq. (4) yields the followingequations:lx ar13 px7ly ar23 pyLi qi8lz ar33 a pz:9Furthermore, the following relation can be achieved with the fact that Liis the magnitude of vector li:ly ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2xl2zq:10According to Eq. (10), lyhas two possible solutions. When a coordinate of the moving platform makes Li2 lx2 lz2b 0, lyhas nosolution, which means that this coordinate is out of the manipulators reachable workspace. The situation Li2 lx2 lz2= 0 meansthat the moving platform reaches the boundary of the reachable workspace. This situation is singular and should be avoided inFig. 3. One typical limb of the manipulator.4R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113practice. Because of these, the sign of lyshould be constant during the operation of the manipulator. Let p = 0 and R = I when themoving platform at the initial position. Substituting them into Eq. (8) yieldsly a ? 0 0Li0 Lib0:11Therefore, Eq. (10) should take a negative sign. Then substitute Eq. (10) into the left side of Eq. (8) and we can get the inverse so-lution of the actuating stroke qiof group 1qi ar23py LiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 ar13 px2 ar33 a pz2q12lican be written with Eq. (7), Eq. (9), and Eq. (10) asliar13 pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2xl2zqar33 a pz264375:13Then we can obtainli? eiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i ar13 px2 ar33 a pz2q;14li? Rei my mx mz:15Where, my= r22li ei, mx= ar12r13+ r12pxand mz= ar32r33+ r32a + r32pz.In each limb of group 2 (i = 4, 5, 6), for simplicity and without losing the generality, a fixed Cartesian reference coordinate frameOaix, y, z is also attached at the point O with its z-axis intersecting eiand its x-axis coinciding with the x-axis of the frame Ox, y, z.Hence, it can be observed from the architecture of the manipulator thatei 100?T,a0i 00a?Tandci Li0a?Twith respect to the frame Oai. Substituting all the known variables into Eq. (4) yields the following equations.lx ar13 pxLi qi16ly ar23 py17lz ar33 a pz18and the following relation can also be obtained with the fact that Liis the magnitude of vector lilx ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2yl2zq:19Similartotheanalysis of group 1, we can obtain that Eq. (19) should take a negative sign.Substitute it into the leftside of Eq. (16),and we can get the inverse solution of qifor group 2.qi ar13px LiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i ar23 py?2 ar33 a pz2r:20With Eq. (16), Eq. (17), and Eq. (18), lican be written asliffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2yl2zqar23 pyar33 a pz264375:21Then we can obtainli? eiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i ar23 py?2 ar33 a pz2r;22li? Rei m2x m2y m2z?:235R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113Here, m2x= r11li ei, m2y= ar21r23+ r21pyand m2z= ar31r33+ r31a + r31pz.Through the analysis above, it can be found that among the manipulators geometrical parameters, only a and Liare independentand need to be determined. And the maximum ranges of qi, Bi, and Aineed to be found out for the manufacture of the manipulator.Thus the design problem can be summarized as follows.Requirements:1. The manipulator should achieve the given workspace which is a cylinder with radius of Rc, and height of 2Hc. And the manip-ulator should have the pointing dexterity of mat any point of the given workspace. In some cases, when the requiredworkspace is complicated, the given workspace can be described as many coaxial cylinders with different pointing dexterityindices as Fig. 4 shows.2. Insomesituations,themaximumrangesofBiandAiarelimitedforthepurposeofincreasingthestiffnessofthejointsoravoidingthe interference.Design task:1. Find out appropriate geometrical parameters of a and Lithat can let the manipulator meet with all the requirements listed above.2. Afterthedeterminationofthegeometricalparametersabove,findoutthemaximumrangesofqi,Bi,andAiforthemanufactureofthe actuators and joints.3. The key point characteristics of the manipulator within the given workspaceIn this research, though a large amount of calculations, we have found that the maximum ranges of qi, Bi, and Aialwaysoccur at some certain locations in spite of the dimension changing of the given workspace nor the manipulator. In otherwords, there are some certain relations between these locations and the given workspace. This characteristic is very helpfulfor the dimensional design, thus we call these locations as key points. This section will prove the existence of these key pointstheoretically using the method of Lagrange multipliers and establish the relations between the key points and the givenworkspace.Forthesakeofgeneralization,wemaketherelatedparametersofthedimensionaldesigndimensionlessbylettingeachofthembedivided by Rc. Thus the workspace cylinders dimensionless radius is 1, and its dimensionless height is 2H. Where H = Hc/Rc. As a re-sult, the results of the dimensional design based on this dimensionless workspace cant be treated as the geometrical parameters ofthe manipulator directly, unless multiply each of them by Rc(it should be noted that the angles are not affected in this procedure).The two limb groups have different key point characteristics due to the configuration of the manipulator. So the characteristics ofthe two limb groups should be studied separately.Fig. 4. A description of the given workspace.6R. Cao et al. / Mechanism and Machine Theory 85 (2015) 1133.1. Group 1 (i = 1,2,3)As the given workspace is a cylinder, the points in the given workspace must meet the following equations:p2y p2z1;24HpxH:25From theanalysis in Section 2, we know that lyinEq. (10)should have a solution to let themanipulator reach the current positionand orientation. So it can be derived thatL2iN ar13 px2 ar33 a pz2:26The maximum value of therightpart of Eq. (26) represented by is studied. Andthe parameters involved are px, pz, and . Theirconstraint equations can be written asg1 p2y p2z1g2 pxHg3 pxHg4 mg5 8:27Thus the Lagrange function can be written as 1g1 21?2g2 22?3g3 23?4g4 24?5g5 25?:28The extreme values of occur where the gradient of K is zero. The partial derivatives arepx 0;py 0; 0; 0k 0;k 1;5k 2kk 0;k 1;58:29By solving the equation system (Eq. (29) and comparing the extreme values, we can obtain that reaches its maximum valuewhen px H;pz 1; 2; m, or px H;pz 1; 2; m. Substituting these two solutions into Eq. (26) yieldsL2iN asm H2 acm a 12:30Toensurethatthemanipulatorcanreacheverypointofthegivenworkspace,aandLishouldbechosentolettheEq.(30)establish.Andthegeneralizedcoordinateofthekeypointsforthiscaseare H;1;2;m;0?and H;1;2;m;0?.Where meansthis valueis arbitrary.Inordertofindthemaximumrangeofqi,weneedtofindboththeminimumandmaximumvaluesofqi.AndtheLagrangefunctionfor finding its minimum value can be written as qi1g1 21?2g2 22?3g3 23?4g4 24?5g5 25?:31Notethatthiscaseisalittlebitcomplex,wefirstpresumepxandasknownvalues.Thentheminimumvalueofqiaboutpy,pzand can be achieved by the partial derivatives of Eq. (31) and the solution is py= 1, pz= 0, and = m. Substituting this solution intoEq. (12) yieldsqia 1cms 22 1 LiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i assm px2a21cm2s4 q:327R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113When|H|assm|,itiseasytofindthattheminimumvalueofEq.(32)occurswhenassequalsto px.Substitutethisrelationinto Eq. (32) and take the derivative with respect to .a 1cmc 2 2a21cm2cs3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i a21cm2s4 q 0:33The solution of Eq. (33) iss ?ffiffiffiffiLipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Li a 1cmp:34In this situation,the key points for theminimum qiare (asm, 1, 0, arcsin(), m, 0) and (asm, 1, 0, arcsin(), m, 0).Where ?ffiffiffiLipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Li a 1cmp.When|H|assm|=|asm|,itcanbefoundthattheminimumvalueofEq.(32)occurswhenpx=H,= arcsin(T1)bythemethod of Lagrange multipliers. Where T1is the root of the following equation.D5x5 D4x4 D3x3 D2x2 D1x D0 035whereD5 4Ha 1cm2sm;D4 4 1cm2H2L2ia2cm?;D3 4Ha 1cm2sm2Has3m;D24H2 4L2i a2s2m?1cm2H2s2m a2s4m;D1 4Hasm1 cm;D0 1cm 2H2L2i L2icm?:AsEq.(35)hasfivesolutions,T1shouldchoosetheonewhichleadstothesmallestqiamongthefivesolutions.Inthiscondition,thekey points are (H, 1, 0, arcsin(T1), m, 0) and (H, 1, 0, arcsin(T1), m, 0).For the purpose of finding the maximum value of qi, the related Lagrange function can be written as qi1g1 21?2g2 22?3g3 23?4g4 24?5g5 25?:36Similarlyduetothecomplex,we firstpresumeasa known value.Then itcanbesolvedthattheminimumvalueof qiaboutpx,py,pzand occurs when px= H, py= 1, pz= 0, = m. Substituting these results into Eq. (12) yieldsqia 1cms 22 1 LiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i assm? H2 a21cm2s4 q:37To find the maximum value of Eq. (37), we take the derivative with respect to . Then comes out the same result as Eq. (35).The one which leads to the largest qiamong the five solutions of Eq. (35) is chosen as T2. And it can be concluded that the key pointsare (H, 1, 0, arcsin(T2), m, 0) and (H, 1, 0, arcsin(T2), m, 0) for this condition.The condition when Bireaches its maximum value means that lieigets the minimum value. Then the Lagrange function for find-ing the minimum value of lieican be written as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2xl2zq1g1 21?3g3 23?4g4 24?5g5 25?:38WithEq.(38),thekeypointsofthemaximumBicanbesolvedas(H,01,/2,m,0)and(H,0,1,/2,m,0).Substitutingthesekey points into Eq. (14), the maximum Bican be obtained asBijmax arccosffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i asm H2 acm a 12q=Li?:39ThecaseforfindingthemaximumAiiscomplicated.First,ispresumedasaknownvalue,andthentheminimumvalueofli(Rei)about px, py, pzand can be found by the method of Lagrange multipliers. And the key points can be solved as (H, 0, 1, , m, 0) and(H, 0, 1, , m, 0), where means this value is undetermined. It needs to solve a very complicated equation to find the value of that causes the minimum value of li(Rei). In this condition, it is much more efficient to find the maximum Aiby numerical methodthan solving this complicated equation. This numerical method let change gradually from to , then substitute it together with8R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113thekeypointsintoEq.(15).Amongtheseresults,theminimumvalueofli(Rei)canbefounddirectly.Althoughthenumerical methodis needed in this case, the amount of variables is reduced by the key points from five to one, which has great significance for theefficiency.3.2. Group 2 (i = 4,5,6)The procedures for seeking the key point characteristics of group 2 are similar to group 1. To make sure that the manipulator canreach all the positions and orientations in the given workspace, we can obtain the following constraint equation from Eq. (19).L2iN ar23 py?2 ar33 a pz2:40By searching for the maximum value of the right part of Eq. (40), the key points of this case can be solved as ;0;1;2;m;0?and;0;1;2;m;0?. Where means the value is arbitrary. Substituting the key points into Eq. (40) yieldsL2iN acm a 12:41Tofindtheminimumvalueofqiwithinthegivenworkspace,theminimumvalueofEq. (20)iscalculatedwiththeLagrangemeth-od.Andthekeypointsare H;0;a cm1;2;m;0?whena(1cm)1,and H;0;1;2;m;0?whena(1cm)N1.Substitutingthis into Eq. (20) yieldsqijminasmHa 1cm1asmH LffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 acm a 12qa 1cmN1(:42Find the maximum value of Eq. (20) with the Lagrange method, and the key point for the maximum qican be solved asH;0;1;2;m;0?. And the maximum qican be written asqijmax asm H LffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 acm a 12q:43For the finding of the maximum Biwithin the given workspace, the minimum value of Eq. (22) is calculated by the Lagrangemethod.Andthekey point of thiscase is ;0;1;2;m;0?. Substitutingthis keypointinto Eq. (22), themaximum Bican be writtenasBijmax arccosffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i1 acm a 12q=Li?:44Find the minimum value of Eq. (23) with the Lagrange method, and the key point for the maximum Aican be solved as;0;1;2;m;0?. And the maximum Aican be obtained asAijmax acoscmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i1 acm a 12qsmacm a 1?=Li?:45Table 1The key points of the given workspace.CasesThe key pointsDirect relationsGroup 1Group 2Group 1Group 2WorkspaceguaranteeH;1;2;m;0?andH;1;2;m;0?;0;1;2;m;0?and;0;1;2;m;0?Eq. (30)Eq. (41)Minimum qiWhen |H| |asm|:(asm, 1, 0, arcsin(), m, 0) and(asm, 1, 0, arcsin(), m, 0)When |H| |asm|:(H, 1, 0, arcsin(T1), m, 0) and(H, 1, 0, arcsin(T1), m, 0)When a(cm 1) 1:H;0;a cm1;2;m;0?Whena(cm 1) N 1:H;0;1;2;m;0?Need to solveEq. (35)Eq. (42)Maximum qi(H, 1, 0, arcsin(T2), m, 0) and(H, 1, 0, arcsin(T2), m, 0)H;0;1;2;m;0?Need to solveEq. (35)Eq. (43)Maximum Bi(H, 01, /2, m, 0) and(H, 0, 1, /2, m, 0);0;1;2;m;0?Eq. (39)Eq. (44)Maximum Ai(H, 0, 1, , m, 0) and(H, 0, 1, , m, 0);0;1;2;m;0?Too complexfor usingEq. (45)9R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113Thepreviouspartisthesearchingandcertificationprocessofthekeypoints.Forthesakeofclarity,allthekeypointsaresortedoutin Table 1. Once the workspaceis given, people can findoutthekey points directly from this table. Furthermore, somedirectrelationsbetween the manipulators parameters and the kinematic restrictions are also listed in this table for the convenience.It can be observed that some cases of Table 1 have two key points. These two points are equivalent, so it only needs to choose anarbitraryoneforthedimensionaldesign.Fromthederivationinthissection,itisknownthatthekeypointsofonelimbaresolvedwithrespectto the fixed limb frame Oaix, y, z. Itis also known that thegiven workspace, thelimbs in one groupand thefixedlimbframesOaiare all axisymmetric. Therefore the limbs in the same group have the same characteristic. Moreover, the direct relations betweenthe kinematics and the geometrical parameters are irrelevant to the selection of the reference frame. In practice, we only need tochoose an arbitrary limb from each group to determine the geometrical parameters of the manipulator.4. The key point based dimensional design method and its applicationTheprevioussectionfindssomeimportantkeypointcharacteristicsofthemanipulatorwithinthegivenworkspace.Thekeypointsaretheextremepositionsthatwillcausetheworstkinematicconditionsofthemanipulatorinthegivenworkspace.Ifthemanipulatorcould achieve the required kinematic performance at the key points, then this kinematic performance will be guaranteed at everypoint within the given workspace. These characteristics can be used to determine the geometrical parameters of the manipulator,thus the dimensional design will have very high efficiency and accuracy. The main steps for this key point design method are asfollows:1. Describe the required workspace. Study the working task of the manipulator, and figure out the space and orientations thatrequired.Thenchooseacylinderwithacertainpointingdexterityindexthatcanjustmeettherequirementasthegivenworkspace.If the required workspace is complicated, it can be described as many coaxial cylinders with different pointing dexterity indices asFig. 4 shows. In this condition, the following design steps 25 should be carried out for each cylinder, and the results should becombined as the final solution.2. Make the given workspace dimensionless. For each cylinder, let its radius and height be divided by its own radius.3. Clarify the additional design requirements and use Table 1 to find all the key points that needed.If the ranges of the joint angles are not limited, the constraint relations of the geometrical parameters can be established with thekey points of Workspace guarantee or the corresponding direct relations (refer to Table 1).If the joint angles are limited by the requirements, the constraint relations of the geometrical parameters should be establishedwith the key points of Maximum Biand Maximum Aior the corresponding direct relations (refer to Table 1).Fig. 5. The prototype of the proposed manipulator.Table 2The design results.Limb number123456Link length (m)1.051.051.051.01.01.0Driving stroke (m)0.600.670.600.670.600.670.400.600.400.600.400.60Ai|max(deg)44.644.644.644.344.344.3Bi|max(deg)43.443.443.436.936.936.910R. Cao et al. / Mechanism and Machine Theory 85 (2015) 1134. Determine the geometrical parameters. Find the appropriate parameters that can meet the constraint relations that established inthe previous step. With these constraint relations, many possible solutions of a and Lican be found. Usually, the smallest a and Liwhich will lead to the smallest volume of the manipulator should be selected.It should be noted that only one limb needs to be determined for each group, because the manipulator is symmetrical. In somecases, a and Limay have additional constraints due to the working task, and the constraint relations of the previous step can beused for the further optimum design.5. The obtained a and Lishould all be multiplied by the cylinders radius to be dimensional. Then they can be treated as the geomet-rical parameters for the manipulator.6. Determine the rest geometrical parameters.If there is more than one cylinder used for the workspace description, the results obtained in step 5 should be combined as onesolution. That is, choose the biggest values of a and Liamong all the cylinders results as the final solution. So the combined solutioncan meet with all the constraint relations of each cylinder.After that, the ranges of Biand Aishould be recalculated with the final solution of a and Liand the driving ranges can be deter-minedbythekeypointsofMaximumqiandMinimumqi,orthecorrespondingdirectrelations(refertoTable1).Itshouldbenotedthatall the cylinders must be checked in this procedure and their results should be combined as the final result.Here, our project is used to demonstrate the application of this design method. Our required workspace can be described with acylinder (Cylinder 1) with radius of 600 mm, height of 800 mm and pointing dexterity of 0, and a cylinder (Cylinder 2) with radiusof 200 mm, heightof 400 mm and pointing dexterity of 30. The joint angles are limited to be less than 45. Additionally, parameter aneedsto bigger than350 mmdue tothesizeof theobjectthat will beplaced on themovingplatformandthesizeof thejoints. For theCylinder 1, with the key points of Maximum Biand Maximum Ai, the minimum solution of the parameters can be obtained as Li=1050mm(i= 1,2,3)andLi=850mm(i= 4,5,6)whileaisnotinvolved.FortheCylinder2,theminimumsolutionoftheparameterscan be found as a = 350 mm, Li= 1050 mm (i = 1,2,3) and Li= 1000 mm (i = 4,5,6) with the key points of Maximum Biand Max-imum Ai. Combining these two results, the final solution for the manipulator can be obtained as a = 350 mm, Li= 1050 mm (i =1,2,3) and Li= 1000 mm (i = 4,5,6). Finally, the final range of Bi, Aiand driving strokes are calculated for each cylinder with thecorresponding key points and then combined. The design results are shown in Table 2. And the prototype of this manipulator withthese designed geometrical parameters is shown in Fig. 5.In order to verify the correctness of these design results, the designed manipulators performances in the given workspace havebeen checked. We took a series of cross sections of the cylinders and discretized them into uniform discrete points. And the orienta-tions at each of these discrete points were also discretized. Then the values of the joint angles were recorded while the movingFig. 7. The distribution of Biin group 1 when the designed manipulator works in a cross section of Cylinder 2.Fig. 6. The distribution of Aiin group 1 when the designed manipulator works in a cross section of Cylinder 2.11R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113platformreaches every position and orientation. For clarity, some typical data was selected and plotted out in this part. When the de-signed manipulator works in the top cross sectionof Cylinder 2, thedistributionsof Biand Aiof group1 are depictedin Figs. 6 and 7respectively. Figs. 8 and 9 display the same situation for Biand Aiof group 2. It can be observed that all the joint angles are less than45,andthemaximumvaluesofthejointangleswhicharecloseto45justoccuratthekeypoints.Alltheseresultsareconsistentwiththe analysis in this study and meet the requirements.5. ConclusionsIn this paper, a new dimensional design method for our new proposed 3-3-PSS parallel mechanism according to a givenworkspace has been presented. This method is based on several key points and avoids the complex analysis of the manipulatorsown 6-D workspacewhichin fact does not have a unifieddescription in a human-readable way. The key points build simple relationsbetweenthemanipulators geometrical parameters andtheworkspacerequirements.Based on this,the proposed design method hasvery high efficiency and accuracy.Many key point characteristics have been found and listed in Table 1. The key points are the extreme positions that will cause theworst kinematic conditions of the manipulator in the given workspace. The kinematic performance can be guaranteed within theentire given workspace by letting the manipulator achieve the performance at the key points. Additionally, some direct relationsbetween the kinematics and the geometrical parameters have been built for the dimensional design.To concisely describe the 6-D workspace and make thedesign requirements clearly, a symmetrical description has been found forthe given workspace. This description is very easy to understand and close to the operating condition of the manipulator. As a result,this method can easily be used in many different situations.This method is derived for a particular type of parallel mechanisms, but the idea of finding the key points might be used for othertype of parallel mechanisms. The core issue is to find the key points whose positions in the prescribed workspace are independentwith the size of the prescribed workspace and the mechanism. And this usually needs the shapes of the prescribed workspace andthe mechanisms own workspace have some similar characteristics, such as axial symmetry in our case. In this study, any other axi-symmetric shapes can be used for describingthe shape of the given workspace besides thecylinder. Rewrite theconstraint equationsoftheLagrangemethod,andthekeypointsfortheseshapescanbefoundwiththesimilarproceduresofthispaper.Itmightbedifficultto find the key points, but the dimensional design of the parallel mechanism will become very convenient once it has been done. Itshould be noted that
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