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黄河科技学院毕业设计(文献翻译) 第 10 页 三自由度模块化微并联机床的开发摘要本文介绍了三自由度微型并联机械臂在微细加工和微装配中的准确定位问题，并介绍了其研究和发展。同时对微并联机床的结构特点和平移、旋转运动特性进行了评估。基于这些鉴定，设计制造了三自由度微型模块化并联机床。这种微并联机床的运行和模拟原理与一个正常大小的Stewart平台大致相同。该平台整体的体积大约为300mm300mm300mm。这种微并联机床的建造采用了模块化的设计方法。本文阐述了这种三自由度微并联机床的校验结果。关键词：微并联机床，工作区模拟，模块化设计1引言人们对于高品质产品，降低产品成本，缩短产品开发周期的需求日益增加，导致人们急切要求机械加工，特别是高速加工的多功能性和准确性方面能得到更快速地提升。因此，高速加工最的新研究趋向于研究和发展全新类型的并联运动机床。并联机床是一种闭环机制，其中一个移动平台通过至少两个串行运动学链连接至基座。传统的Stewart平台（SP）的机械臂有六条可扩展的链，拥有非常严格的运动学结构。链的数目通常与自由度的数目相同。每条腿与平台上的一个三自由度球状接头和基座上的一个两自由度通用接头相连。与串行运动学机械手相比，SP有高有效载荷和高刚性的特点，而其缺点是工作范围有限，运动学控制算法更为复杂。此前，新加坡国立大学开发了一个传统的SP。这个sp由六个线性驱动器组成，每个线性驱动器分别被六台步进电机独立驱动。它可以进行平移运动和精密工程应用，如精细且要求多自由度控制的运动。该平台已经过校准，具有相当好的准确性和可重复性。不过，这个平台也有一些局限性，即复杂的直接运动学解决方案，运动定位和定向方面的一些问题，以及昂贵的高精度球形接头。该校正在着手进一步的研究，准备设计和开发一种改进的SP。并联机床发展的新趋势是将自由度从六个减少至三个。减少自由度的并联机床具有减小工作空间和降低成本的优势。然而，三自由度并联机床的刚性降低，自由度也有所减少。为了克服这些缺点，人们一直在积极研究少于六个自由度的并联机床。例如，克拉维尔和斯特恩汉姆报告的四自由度高速机床，机床。李和沙阿分析研究的三自由度并联机床。这两台机床具有相似的直接运动学解决方案，然而李和沙阿研究的机械臂包含三个球形接头，机床包含十二个球形接头。此外，李和沙阿研究的机床的定位和定向是复合的。一些三自由度并联机床架构，提供绕一个固定点做定转角转动的平台，该平台可作为导向设备，机器手手腕定向设备使用。最近，蔡介绍了一种仅有转动副组成的平移平台。它执行纯粹的平移运动，并且有相近的直接运动学和反向运动学解决方案。此外，多轴机床的使用是非常不合理的，它冗余的自由度导致一些轴没有得到充分的利用，同时也增加了它的复杂性和成本。纯粹的三自由度平移或旋转运动需要激活所有六条腿，这意味着能耗的增加。因此，在成本方面，三腿三自由度微并联机床是符合成本效益的，而且其简单的运动学机制使其更加易于控制。本研究开发的微并联机床采用模块化的设计理念，增加了其灵活性和功能。近年来，模块化设计理念已经越来越多地被提出作为开发可重构和自我修复的机床系统的手段。为了优化三足三自由度微并联机床的性能和自我修复能力，要求操作系统采用模块化设计且拥有自我重构能力。由许多自主单位或模块组成的模块化机床可以被重新配置成许多新的设计。理想状态下，各模块应该是统一又相互独立的。该机床可以手动或自动的从一种配置更改成另一种配置。因此，模块化的机床可以被重新配置或修改以适应新的环境。模块必须彼此互动和合作以实现自我配置。此外，模块化微并联机床可以通过拆除和更换故障模块实现自我修复。由于自重构模块化机床可以提供许多传统机械的功能，它们特别适合于各种任务，如精密工程业的高速加工。2 背景在这项研究中，微并联机床，即微并联机床的发展目标，是最大限度地减少系统的尺寸和增加系统的便携性，如便携式数控机床。考虑到要使微并联机床的尺寸最小化，平台的链接数从六个减少到三个。闭环并联机床的自由度数目检验使用Grber公式：其中Fe是集合或机械装置的有效自由度。机械运功空间自由度，i是链接的数量，j是关节数，fi是第i个关节的自由度，Id是闲置或被动的自由度。接头的数目是9（六个万向接头和三个柱状接头）。链接的数量是8个（每一个执行机构，末端效应器和基座各两个链接）。接头自由度的总和为15.因此，使用Grber公式，微并联机床的自由度可以做如下计算：F6（891）153.我们将使用蔡开发的系统枚举法，进行对各种系统配置的比较研究，以选择出一种符合要求的并联系统在本研究中使用。本研究的目的是建立一个具有微米精度的微型并联机床，该机床可承受高达三公斤的有效载荷，并能用于微加工和微装配。因此必须选用高品质的元件已达到所需的精度。即使一台微并联机床以高精度，高刚度，高速度，高有效载荷著称，在选用其相关部件时仍应该十分小心，因为该平台是用来进行微加工，这需要比以前的开发平台具有更高的刚性。因此，以先前开发的SP为基础，我们将大量的微并联机床设计以同比例的微观尺寸在Matlab上进行仿真模拟，例如六足SP，三足微并联机床和六足PUS SP,其中PUS表示一个具有链接的平台，其链接由棱柱接头，万向接头和球形接头组成。这些平台的工作空间被仿真模拟以选择出最适合的设计，已达到研究目标。此外，我们还对工作空间和移动平台，移动平台半径之间的关系进行了研究，以使微并联机床的工作空间得到优化。我们以执行器的行程为基础计算了该平台的的最大转角，并以此作为执行器选择的参考。3各种微并联机床的仿真3.1仿真模型我们通过Matlab对目前已报道的各种微并联数控机床数学模型进行了研究。通过比较这些数学模型的工作空间以确定一种适合本研究的一种。我们通过Matlab对六足的SP，三足的微并联机床和PUS微并联机床进行了模拟测试。选择这些机床的运动学模型进行测试的原因是因为它们有更高的有效载荷和较为简单的控制机构。所有三种模型如图一所示：图1 a 六足的Stewart平台，b 三足的Stewart平台,c PSUStewart平台我们进行了一系列的模拟测试，以确定基座和平台、被动式球形接头的合适半径，达到使微并联机床的工作空间最优化的目的。在这些测试中某些参数保持不变，如链接的长度被设为0.22米，行程为0.05米。万向节的倾角被设为45。利用Matlab进行仿真模拟可以记录并比较每种机床所能到达的三维空间位置（如图2所示）。从图2可以看出，六足微并联机床的工作空间高度较高，而三足微并联机床的工作空间范围更广。图2 三足（红色圆圈）和六足（蓝色十字）微并联机床工作空间比较3.2 仿真结果在这些模拟中，并联机床基座的半径被设置为0.075米。在仿真模拟测试中通过改变微并联机床移动平台的半径，得到了各种不同的工作空间，结果显示在表1和图3中。从结果中得知，微并联机床的工作空间大小取决于它的移动平台的半径。图3 工作区体积与工作平台半径间的关系因此，移动平台的半径越大，工作空间的范围就越大。然而，可以预见的是当移动平台的半径和基座的半径相等时会出现潜在的问题。当两者相等时可能产生突变。微并联机床的刚度可能会减少，因为所有的关节都垂直向上运动，而且，移动平台和基座之间的张力也会随之降低。此外，微并联机床的高度也会受到移动平台的影响。由于链接的长度是固定的，链接只能倾斜到一定的角度，而这样会减少平台的高度。表 1不同工作平台半径间下的工作空间体积微型工作平台的半径,r工作空间体积103m30.0353.00320.0403.28100.0505.73130.0606.71630.0709.99880.07511.0000与平台半径和工作区大小之间的关系相反，当基座的半径增加时，工作区范围减小，这两者之间的关系如表2和图4的仿真结果所示。在这个模拟中，平台的半径被设为0.04米。通过对两个模拟结果的分析发现，当基座和平台的尺寸差别越大，工作空间的范围就越小。因此，建议在设计中将平台和基座的半径之比设为1:2以得到合适的工作空间。图4 工作区体积与基座半径间的关系3.3 球形接头倾斜角间关系的仿真结果在这次研究中，我们设定平台半径为0.03米，基座半径为0.075米，通过在20到45范围内改变球形接头的活动角度，我们得到了工作空间的仿真测试结果。从图5的仿真结果中我们可以看出，在球形关节从20变化至45时，平台的工作空间大幅增加。通过安装一个45的球形关节，工作空间的体积可以从4.1263104m3增加至0.0014m3，即当安装一个45的球形关节时工作空间的体积增加了3.4倍。我们还进行了测试以得到当执行器达到最大行程时，所能得到的最大取向角。通过对不同执行器的不同行程进行模拟测试，我们分别得到了它们的旋转角度。从结果中得知，为了使移动平台得到45的旋转角，我们必须保证执行器的行程不少于50mm.表 2工作空间大小与基座半径之间的关系基座半径,r工作空间体积103m30.0750.41260.0850.27600.0950.16650.1050.0826通过研究三足MSP对于本项目的可行性，我们发现MSP的自由度的数量受制于活动关节的数目。因此，为了得到三个棱柱执行器，三足MSP必须只能有三个自由度。根据式1中的Grber公式三个棱柱执行器可以执行三自由度的运动。然而，我们需要采用以下措施提升腿的稳定性，使用万向接头代替球形接头，或者在球形接头的中间部位安装一定的制约部件，如刚性链接。从对并联机床的文献研究得知，三自由度微并联机床可分为三大类，即平面并联，球面并联和空间并联机床。在本文中，空间机床是研究重点，它们既可以执行平移运动，又能执行旋转运动。然而，虽然这两项运动的复合是可行的，但将会增加机构的复杂性。三自由度微并联机床具有吸引力的特征之一就是它们具有灵活的适应所需功能的能力。例如，通过耦合一个三足的导向装置和一个三足并联平移机械手，可以得到一个六自由度的微并联机床。图5被动关节角度为20（圆形点）和45（交叉点）时的工作区比较因此，在本文中我们探讨了纯粹的并联机床和平移机床的发展。然而，我们在设计和制造方面侧重于3UPU并联机床。一个混合的3UPU平台安装有棱柱链接的并联机器手，且该机器手通过万向接头连接在平台和基座上。此外，还有一个被动链接通过球形接头被安装在平台的中心。这一装置的一个有趣的特点是，它复合了平移并联机床和旋转并联机床的运动。它在Z轴上更大的工作空间使它可以作为一种旋转机械装置使用。这台并联机床的特征是，当提供一些额外方向的运动时，它能扮演很有趣的角色，而且，如果将它安装在串行运动学数控机床上时，它具有很好的灵活性。因此根据模拟的结果，我们构建了一个三自由度的MSP，该装置拥有一个行程为50mm的执行器，直径为125mm的平台和一个直径为250mm的基座，以此来实现平台的工作空间和功能的要求。在下一节中我们为MSP开发了数学模型，进行了工艺上的优化以最终确定平台的工作空间。参考文献1. Joshi SA, Tsai LW (2003) The kinematics of a class of 3 DOF,4legged parallel manipulators.Trans ASME 125:52 602. Yang G, Chen IM, Chen WH, Yeo SH (2003) Design and analysis of a 3RPRS modular parallel manipulator for rapid deployment.In Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Kobe, Japan,pp 125012553. Tsai LW, Walsh GC, Stamper RE (1996) Kinematics of a novel three DOF translational platform. In: Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, MN, 2228 April 1996, pp 3446 34514. Clavel R (1988) DELTA, a fast robot with parallel geometry. In:Proceedings of the 18th International Symposium on Industrial Robots, Lausanne, Switzerland, 2628 April 1988, pp 91 1005. Lee K, Shah DK (1987) Kinematic analysis of a three degrees of freedom in-parallel actuated manipulator. 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In: Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, MN, 2228 April 1996, pp 3446 345110. Badescu M, Mavroidis C, Morman J (2002) Workspace optimization of orientational 3legged UPS parallel platforms.In: Proceedings of the 2002 ASME International Design Engineering Technical Conferences and the Computers and Information in Engineering Conference (DETC/CIE), Mon-treal,Canada,29September2October2002,DETC2002/MECH34366, pp 1711. Mckee G, chenker P (1999) Sensor fusion and decentralized control in robotic systems II. SPIE Proc Ser 3839:270 27812. Tsai LW (1999) Robot analysis: the mechanics of serial and parallel manipulators. Wiley, New York, pp 11616413. Tsai LW (2000) Mechanism design: enumeration of kinematic structures according to function, CRC Press, Boca Raton, FL,ISBN 084930901814. Tsai LW, Joshi S (2002) Kinematic analysis of 3 DOF position mechanisms for use in hybrid kinematic machines. J Mech Des 124:24525315. Dash AK, Chen IM, Yeo SH, Yang GL (2003) Task-based configuration design for 3legged modular parallel robots using simplex methods. In: Proceedings of the 2003 IEEE Interna-tional Symposium on Computational Intelligence in Robotics and Automation, Kobe, Japan, 16 20 July 2003, pp 998 100316. MaO,Angeles J(1991) Architecture singularities of platform manipulators. In: Proceedings of the 1991 IEEE International Conference on Robotics and Automation, Sacramento, CA, 9 11 April 1991, pp 1542154717. Gregorio RD (2001) Statics and singularity loci of the 3UPU wrist. In: Proceedings of the 2001 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Como,Italy, 8 12 July 2001, pp 470 475Int J Adv Manuf Technol (2006) 31: 188200DOI 10.1007/s00170-005-0166-yORIGINAL ARTICLEC. C. Ng.S. K. Ong.A. Y. C. NeeDesign and development of 3DOF modular micro parallelkinematic manipulatorReceived: 4 November 2004 / Accepted: 13 April 2005 / Published online: 16 March 2006# Springer-Verlag London Limited 2006Abstract Thispaperpresentstheresearchanddevelopmentof a 3legged micro parallel kinematic manipulator (PKM)for positioning in micromachining and assembly operations.The structural characteristics associated with parallel ma-nipulators are evaluated and the PKMs with translationaland rotational movements are identified. Based on theseidentifications, a hybrid 3UPU (universal jointprismaticjointuniversal joint) parallel manipulator is designed andfabricated. The principles of the operation and modelling ofthismicroPKMarelargelysimilartoa normal-sizedStewartplatform (SP). The overall size of the platform is containedwithin a space of 300 mm300 mm300 mm. A modulardesign methodology is introduced for the construction ofthis micro PKM. Calibration results of this hybrid 3UPUPKM are discussed in this paper.Keywords Stewart platform.Micro parallel kinematicmanipulator.Workspace simulation.Modular design1 IntroductionThe increasing demand for high product quality, reducedproduct cost and shorter product development cycle resultsin a continuing need to achieve improvements in speed,versatility and accuracy in machining operations, especiallyhighspeedmachining.Hence,therecenttrendstowardshighspeed machining have motivated research and developmentof new novel types of parallel kinematics machines 1.A parallel manipulator is a closed-loop mechanismwhere a moving platform is connected to the base by atleast two serial kinematics chains (legs). The conventionalStewart platform (SP) manipulator has six extensible legsand hence a very rigid kinematics structure 2. Thenumber of legs is usually equal to the number of degrees offreedom (DOF). Each leg is connected to a 3DOF spher-ical joint at the platform and a 2DOF universal joint at thebase. Compared to serial kinematics manipulators, a SP hasthe desirable characteristics of high payload and rigidity,while the drawback is a much limited working envelope aswell as more complex direct kinematics and controlalgorithms.A conventional SP had earlier been developed at theNational University of Singapore. This platform consists ofsix linear actuators independently driven by six steppermotors. It can perform translational movements and beimplemented in precision engineering applications, such asfine motion control requiring multi-degrees freedom. Theplatform has been calibrated and found to be fairly accurateand repeatable. However, there are some limitations of thisplatform, namely, the complex direct kinematics solution,coupled problems of the position and orientation move-ments, as well as the expensive high precision sphericaljoints 3. Further research is being performed to designand develop an improved SP.A new trend in the development of parallel kinematicsmanipulators (PKM) is the reduction from six DOFs tothree. The decrease of DOFs of the PKM has advantages inworkspace and cost reduction. However, the 3DOF PKMprovides less rigidity and DOFs. To overcome these short-comings, PKMs with fewer than six DOFs have been ac-tively investigated. For example, Clavel 4 and Sternheim5 reported a 4DOF high speed robot called the Deltarobot. Lee and Shah 6 analyzed a 3DOF parallel ma-nipulator. Although these two robots possess closed-formdirect kinematics solutions, the Delta robot contains 12spherical joints while Lee and Shahs manipulator containsthree spherical joints. In addition, the position and ori-entation of Lee and Shahs manipulator are coupled. Some3DOF parallel manipulator architectures provide pureC. C. Ng.A. Y. C. Nee (*)Singapore-MIT Alliance,National University of Singapore,9 Engineering Drive 1,117576 Singapore, Singaporee-mail: mpeneeyc.sgS. K. Ong.A. Y. C. NeeMechanical Engineering Department,Faculty of Engineering,National University of Singapore,9 Engineering Drive 1,117576 Singapore, Singaporerelative rotation of the mobile platform about a fixed pointand are used as pointing devices, wrists of manipulatorsand orienting devices 7, 8. Recently, Tsai 9 introduced anovel 3DOF translational platform that is made up of onlyrevolute joints. It performs pure translational motion andhas a closed-form solution for the direct and inversekinematics.Furthermore, the use of multi-axis robots is highlyunjustified as several of the axes remain underutilizedbecause of the redundancy in DOFs, which increases thecomplexity and cost. In addition, a purely 3DOF trans-lational or rotational motion would require the activation ofall six legs, which means an increase in energy consump-tion 10. Hence, in terms of cost and complexity, a 3DOF3legged micro PKM is cost effective and the kinematicsof the mechanism is further simplified for the purpose ofcontrol.To increase the flexibility and functionality of the microPKM developed in this research, the concept of modulardesign has been introduced. In recent years, modular robotshave increasingly been proposed as a means to developreconfigurable and self-repairable robotic systems 11. Tooptimize the performance and self-repairability of a 3legged 3DOF PKM, the manipulation system needs toincorporate modularity and self-reconfiguration capabil-ities. Modular robots consisting of many autonomous unitsor modules can be reconfigured into a large number ofdesigns. Ideally, the modules should be uniform and self-contained. The robot can change from one configuration toanother manually or automatically. Hence, a modular ma-nipulator can be reconfigured or modified to adapt to a newenvironment. The modules must interact with one anotherand cooperate to realize self-configuration. In addition, amodular micro PKM can have self-repairing capability byremoving and replacing failed modules. Since self-reconfigurable modular robots can provide the function-ality of many traditional mechanisms, they are especiallysuited for a variety of tasks, such as high speed machining1 in the precision engineering industry.2 BackgroundIn this research, the objectives of the development of themicro PKM, i.e. micro parallel kinematic manipulator(MSP), are the minimization of the dimensions of thesystem and the portability of the system, e.g. portable on aCNC machine. With regards to the minimization of thedimensions of the MSP, the number of links of the platformis reduced from six to three. The DOF for a closed-loopPKM is examined using Grblers formula:Fe l ? j ? 1 Xji1fi? Id(1)where Feis the effective DOF of the assembly or mech-anism, is the DOF of the space in which the mechanismoperates, l is the number of links, j is the number of joints, fiis the DOF of the ith joint and Idis the idle or passiveDOFs.The number of joints is nine (six universal joints andthree prismatic joints). The number of links is eight (twolinks for each actuator, the end effectors and the base). Thesum of all the DOFs of the joints is 15. Hence, usingGrblers formula, the DOF of the micro PKM is computedas F 68 ? 9 ? 1 15 3: Using a systematic enu-meration methodology developed by Tsai 12, a compar-ison study of the configurations is performed to select aconfiguration that meets the requirement of the parallelkinematics system to be constructed in this research.The research objective is to build a micro PKM withmicron precision and that can withstand a load of up to 3 kgfor the purpose of micromachining and assembly. Thus,high quality components need to be searched for to achievethe required accuracy. Even though a PKM is well knownfor its high accuracy, rigidity, speed as well as payload, theselection of the relevant parts needs to be carefully per-formed since the platform to be developed is used formicromachining, which requires a higher rigidity as com-pared to the previously developed platform. Therefore,based on the previously developed SP, various designs ofPKMs are simulated on a micro scale using Matlab, such asthe 6legged SP, 3legged PKM and 6legged PUS SP,where PUS denotes a platform with links of prismatic joint,universal joint and spherical joint. The workspaces of theseplatforms are simulated and compared to select the mostsuitable design to achieve the research objectives.In addition, the relationships between the workspace andthe radii of the mobile platform and the fixed base arestudied so that the workspace of the micro PKM is op-timized. The maximum angle that the platform can beoriented is also deliberated based on the stroke of theactuators in order that it can be used as a reference for theselection of the actuators.3 Simulation of various PKMs3.1 Simulation modelsMathematical models of various kinds of PKMs that havecurrently been reported are investigated through perform-ing simulations using Matlab. The workspaces of thesePKMs are compared to identify a suitable PKM kinematicsmodel for the present research. A 6legged SP, 3leggedPKM and a PUS PKM are simulated using Matlab. Thereasons for choosing the kinematics models of these ma-nipulators are because they have a higher payload and aresimple to control. All three models are shown in Fig. 1.A series of simulations has been performed to determinethe optimized workspace of the PKMs with suitable radii ofthe base and the platform, as well as passive sphericaljoints. In these simulations, certain parameters are kept asconstant, such as the length of the link is set as 0.22 m witha stroke of 0.05 m. Besides, the tilting angle of theuniversal joint is set as 45.189Using Matlab to simulate the motion of the each of themanipulators, the 3D position that the manipulator canreach is recorded (as shown in Fig. 2) and compared for allthree PKMs. From Fig. 2, it can be observed that theworkspace of the 6legged parallel manipulator can reach ahigher position. However, the volume of the workspace ofthe 3legged parallel manipulator is larger.3.2 Simulation resultsIn these simulations, the radius of the base of the PKM isset as 0.075 m. By varying the radius of the mobileplatform of the PKM throughout these simulations, variousworkspaces are found, and the results are shown in Table 1and Fig. 3. From these results, the workspace of the PKM isdependent on the radius of the mobile platform. Thus, thebigger the radius of the mobile platform, the larger is theworkspace that can be achieved. However, a potentialproblem can be foreseen if the radius of the mobile plat-form is equal to the radius of the base. Singularity mighthappen at the point when the radii of mobile platform andthe base are equal. The stiffness of the PKM might bereduced because all the joints are vertically upwards andthe tension of the links between the mobile platform andthe base may be reduced. In addition, the height of thePKM is also affected by the radius of the mobile platform.Since the length of the links is fixed, with a smaller mobileplatform, the links can only be tilted to a certain angle toreduce the height of the platform.In contrast to the relationships between the radius of theplatform and the workspace, the workspace of the platformdecreases while the radius of the base is increased, asshown in Table 2 and Fig. 4. In this simulation, the radiusof the platform is set as 0.04 m.Fig. 1 a 6legged micro Stewart platform, b 3legged micro Stewart platform and c PSU micro Stewart platformFig. 2 Comparison ofthe workspace of 3legged(red circle) and 6legged(blue cross) parallelmanipulators190Based on the analysis of the results of both simulations,when the difference between the size of the base and theplatform becomes bigger, the workspace of the platformwill become smaller. Thus, a ratio of 1:2 is suggested forthe design of the radius of the mobile platform and the base.3.3 Simulation result of relationshipsbetween the tilting angle of the spherical jointsInthisinvestigation,theresultsoftheworkspacesimulationare obtained by setting the base radius as 0.075 m and theplatform radius as 0.03 m with varying passive sphericaljoint angles of 20 and 45. From the results in Fig. 5, theworkspace of the platform increased drastically when thetilting angle of the spherical joint varied from 20 to 45.By installing a 45 spherical joint, the volume of the work-spacecanreachupto0.0014m3insteadof4.1263104m3,i.e.a 3.4-fold increase, when a spherical joint of 20 is installed.Simulation was also carried out to determine the maxi-mum stroke that an actuator has made to achieve the largestorientation angle. Different actuators with various strokesare simulated to obtain the angles they could rotate. Fromthe results, to achieve a 45 rotation for the mobile plat-form, the stroke of the actuators must be at least 50 mm.From the study on the feasibility of the 3legged MSP toachieve the objectives of this research, it is found that theDOF of the MSP is limited by the number of active joints.Thus, by having three prismatic actuators, there can only bethree DOFs for the 3legged MSP. According to Grblersformula in Eq. 1, three prismatic actuators can performthree DOF movements. The stability of the legs, however,needs to be improved, such as by replacing the sphericaljoints with universal joints, or by installing certain con-straining components, such as a rigid link connected to aspherical joint in the middle.From literature studies of the parallel manipulators 10,13, 3DOF PKMs can be divided into three major types,namely, the planar parallel, spherical parallel and spatialparallel manipulators. In this paper, the study of spatialmanipulators is focused, and they can perform either purelytranslational movements or rotational movements. How-ever, the hybrid of both motions is feasible but thecomplexity will be increased. One of the attractive char-acteristics of 3DOF PKMs is their flexibility and canadapt to the required function that they need to perform14. As an example, by coupling a 3legged orientationmechanism PKM and a 3-legged translational mechanismparallel manipulator, a 6DOF PKM can be achieved.Hence, in this paper, the development of the purelytranslational PKM and rotational PKM is explored. How-ever, the design and fabrication of a hybrid 3UPU PKM isfocused. A hybrid 3UPU platform is a parallel manip-ulator that is installed with prismatic links, which are con-nected to the base and the platform with universal joints.Furthermore, a passive link is installed at the center of theplatform using a spherical joint. One of the interestingFig. 3 Workspace versus radiusof mobile platformTable 1 Workspaces of platform with varying platform radiiRadius of the micro SP, rWorkspace104m30.0353.00320.0403.28100.0505.73130.0606.71630.0709.99880.07511.0000191features of this mechanism is that it has a hybrid movementof both the pure translational PKM and rotational PKM. Ithas a bigger workspace along the Zaxis while it canperform as a rotational mechanism. The specialty of thisPKM is that it is able to play an interesting key role formanipulating some extra directions of movement andproviding more flexibility if it is added to serial kinematicsNC machines.Hence, based on the results of the simulations, a 3DOFMSP has been constructed with a 50 mm stroke actuator, a125 mm diameter platform and a 250 mm diameter base, toachieve the requirements of the workspace and functions ofthe platform. A mathematical model of the MSP is devel-oped in the next section. An optimization process has alsobeen performed to finalize the workspace of the platform.4 Mathematical model of hybrid 3UPU PKMTo analyze the kinematics model of the parallel mecha-nism, two relative coordinate frames are assigned, asshown in Fig. 6. A static Cartesian coordinate frame XBYBZBis fixed at the center of the base while a mobileCartesian coordinate frame XPYPZPis assigned to thecenter of the mobile platform. Pi, i=1,2,3 and Bi,i=1,2,3 arethe joints that are located in the center of the base and theplatform passive joints, respectively. A passive middle linkLmis installed at the middle of the platform. The purpose ofthe link is to constrain the kinematics of the platform toperform translation along the zaxis and rotational aroundthe x and yaxes.Let rBand rPbe the radii of the base and the platformpassing though joints Piand Bi(i=1,2,3), respectively. Theposition of Biwith reference to the fixed coordinate frameXBYBZBcan be expressed as in Eq. 2.B1 rB0 0?TB2 ?12rBffiffiffi3p2rB0?TB3?12rB?ffiffiffi3p2rB0?T(2)The positions of joints Pi, (i=1,2,3) are expressed withrespect to the mobile frame XPYPZPas in Eq. 3.P1 rP0 0?TP2 ?12rPffiffiffi3p2rP0?TP3?12rP?ffiffiffi3p2rP0?T(3)Since the MSP is proposed to be built within a space of300 mm300 mm300 mm, the actuators with length of216 mm are too long to be assembled directly to the basejoints. Instead of assembling the actuators on top of thejoints, the actuators are aligned parallel to the joints asshown in Fig. 11. Therefore, the calculation of the length ofTable 2 Workspace of platform with varying base radiiThe radius of the base, rWorkspace103m30.0750.41260.0850.27600.0950.16650.1050.0826Fig. 4 Workspace versusradius of base192the link is not as direct as the inverse kinematics of anormal SP. As shown in Fig. 6, the length of the link can becalculated using Eq. 4.l!i ?B!i t! R ? P!i; i 1.3(4)where liis the dotted link length, and t!and R are thetranslation and orientation of Piwith respect to XPYPZPT.However, for this present hybrid PKM, extra calculationsteps in Eqs. 5, 6, 7 and 8 are needed as shown in Fig. 7.x2 a2 y2(5)k ? xxzaL ? yy) y xLk(6)5 ) 6 ) x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2L2k2 ? 1s(7)7 ) 6 ) z a ?k ? xxk ?ffiffiffiffiffiffiffiffia2L2k2?1rffiffiffiffiffiffiffiffia2L2k2?1r0BBB1CCCA? a(8)By determining the length of the dotted link, Liusinginverse kinematics, the length of Z can be determined usingthe similarity triangular theory as shown in Eq. 8. Hencethe strokes of the links are found indirectly based on themanipulation of the platform. After analysing the strokethrough inverse kinematics methods, the required rota-tional angle of the base joints and platform joints can befurther affirmed through determining the angles of rotationas shown in Fig. 8.The geometry matrix method of analysis was used sincefor parallel manipulators, it is often more convenient toemploy the geometric method 13. Generally, a vectorloop equation is written for each limb, and the passive jointvariables are eliminated among these equations as shown inFig. 8.Let X=, Y=, Z= and Li+H=Mi,since Liis knownfrom the similarity triangle equation. Furthermore, Zi(i=1,2,3) is known based on the coordination of each ofthe base joints, which are 0, 120 and 240. Besides,Birxiryi0?T, i=1,2,3 is also known from Eq. 3. Basedon inverse kinematics of the platform, one can determineFig. 6 Schematic diagram of the PKMFig. 5 Workspace comparisonfor passive joint angles of 20(circular point) and 45 (crosspoint)322193the pointPxiPyiPzi2435; i=1,2,3 from Eq. 4. Thus, for a knownplatform position, one can calculate the rotation angles, Xiand Yibased on the known actuator stroke length usingEq. 9.:Rz? Rxy?0kiMi2435rxiryi02435PxiPyiPzi2435(9)cosZi?sinZi0sinZicosZi00012435?cosYisinYisinXisinYicosXi0cosXi?sinXi?sinYicosYisinXicosYicosXi2435?0kiMi2435rxiryi02435PxiPyiPzi2435)?cosYisinZi sinYisinXicosZiki sinYicosXi? Mi rxicosXicosZi? Ki? sinXi? Mi ryisinYisinZi cosYisinXicosZiKi cosYicosXi? Mi2435PxiPyiPzi2435(10)Fig. 8 Geometry method diagramFig. 9 Modified SP with a passive prismatic middle linkFig. 7 Calculation of the actual stroke of the linkFig. 10 Relationship between the point P and the spherical joint ScosZi?sinZi0sinZicosZi00012435?cosYisinYisinXisinYicosXi0cosXi?sinXi?sinYicosYisinXicosYicosXi2435?0kiMi2435rxiryi02435PxiPyiPzi2435)?cosYisinZi sinYisinXicosZiki sinYicosXi? Mi rxicosXicosZi? Ki? sinXi? Mi ryisinYisinZi cosYisinXicosZiKi cosYicosXi? Mi2435PxiPyiPzi2435194Hence,Xi 2 ? tan?12Mi?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Mi2? 4 ryi? Pyi? cosZi? Ki?cosZi? Ki ryi? Pyi?q2 ryi? Pyi? cosZi? Ki?2435(11)Yi 2 ? tan?12sinZi? Ki?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2sinZi? Ki2? 4 Pzi sinXicosZi? Ki cosXi? Mi Pzi? sinXicosZi? Ki? cosXi? Miq2 Pzi sinXicosZi? Ki cosXi? Mi2435(12)Fig. 12 The PI M235.5DGactuator (http:/www.pi.ws),Huco Miniature Acetal ForkType universal joint (http:/) and HephaistSeiko spherical joint (http:/www.hephaist.co.jp/e/pro/ball.html)Fig. 11 Workspace of thesurface point of the hybrid PKM195Through Eqs. 11 and 12, the solutions for the rotationangle Xi, Yiwhere i=1,2,3 can be found. Hence, throughdetermining the rotation angle of each universal joint of thelinks, the stroke of the actuators can be determined. Theknown rotation angle is also used as a geometry constraintto determine the workspace of the platform. After the de-termination of the stroke and angle of rotation of all theuniversal joints of the actuators, a simulated hybrid 3UPUPKM can be plotted using Matlab as shown in Fig. 9. Apassive middle link is installed and attached via a sphericaljoint to the platform as shown in Fig. 10. The passivemiddle link acts as a constraint for the extra DOF of theplatform. The inverse and forward kinematics algorithmsof this hybrid PKM are implemented using Matlab, and themovement of the platform is simulated to verify the mo-bility of the platform.5 Optimization method and analysis of workspaceBased on the inverse and forward kinematics algorithms ofthe platform, and due to the fact that all legs are equal andthe distance between the joints at the base and platform areequilateral, the workspace of the hybrid 3UPU platformdepends on three geometric parameters, namely, (1) thestroke of the prismatic link, Li, (2) the limitation of theuniversal joint angle of the base and platform, which is 45and (3) the distances K between the spherical joint of themiddle link to the middle point of the moving platform asshown in Fig. 10.The stroke Liof the actuator is the travel range of eachprismatic link, which is 50 mm. As shown in Fig. 11, theworkspace describes the manipulation of point P, which isat the middle of the mobile platform, with respect to themiddle point B of the base, as shown in Fig. 10. For eachorientation of the platform, the middle point P is rotatedwith respect to point S at the center of the spherical joint,which is located at the middle link of the platform as shownin Fig. 10.During the optimization process, each position of thepoint P at the mobile platform is verified with its geometryconstraints for every manipulation of the platform. Thelengths of the legs are determined using inverse kinemat-ics, i.e. Eq. 8. Next, the lengths of the legs are checked todetermine if they are within the allowable intervalLminLiLmaxand the interference of the legs is alsochecked. In addition, for rotation movements of theplatform, the joint angles are determined and verified.After all the geometry constraints have been verified, theverified Cartesian coordinates of point P are recorded inthe workspace database.The workspace optimization programme is implementedusing Matlab. The workspace data points collected usingthe optimization programme are plotted using Matlab asshown in Fig. 11. Figure 11 shows the workspace of theoptimal constrained configuration. The points represent theposition that can be achieved by point P of the mobileplatform. The workspace of point P describes a portion of asphere with its center at point S and a radius K. Theworkspace volume of point P increases gradually duringthe incremental motion of point P along the zaxis. Theworkspace volume increases to its maximum when theplatform is located at z=310 mm. Overall, the workspaceforms a diamond shape. The volume of the workspace isFig. 13 The connectors, linksand the mobile platformFig. 14 Parallel manipulatorsystems fabricated using thesame modular components19693,875 mm3, which is relatively small as compared withother modular PKM configurations. However, the objec-tive of this hybrid micro PKM is for fine machiningmovements along the zaxis as well as rotational move-ments around the x and yaxes. Hence, the simulatedworkspace shows that the platform is constrained fromover traveling. Although this system has a limitedCartesian workspace, in terms of angular workspace, itcan rotate up to 32 around the xaxis and 28.2 aroundthe yaxis when z=310 mm. Furthermore, the platform cantravel 450 mm along the zaxis. From the results of theworkspace analysis, the workspace is affected signifi-cantly by the limitation of the stroke of the actuators andthe imposed angle of the universal joints.6 Design of the micro parallel manipulatorA set of design criteria that has been formulated based onthe literatures review of PKMs and the results of theworkspace analysis is applied in the fabrication and designof the hybrid 3UPU PKM. To shorten the developmentcycle 2, the modular design methodology is employed inthe prototype development.Basically, a modular parallel robot consists of a set ofindependently designed modules, such as actuators, pas-sive joints, rigid links (connectors), mobile platforms andend-effectors that can be rapidly assembled into variousconfigurations with different kinematics characteristics anddynamic behaviors. There are many configurations of amodular platform. Hence, the development of a methodol-ogy for design and dimensional synthesis of a parallel robotsystem for a specific task is a basic and important require-ment 15. Based on the research reported by Dash et al.15, a modular parallel robot may have an unlimitednumber of configurations depending on the inventory ofthe modules. Principally, the modules for assembly into amodular micro PKM can be divided into two groups:1.Fixed dimension modulesFig. 15 a Pure translationalplatform, b pure rotationalplatformFig. 16 Hybrid 3UPU PKMFig. 17 Calibration of micro PKM using CMM197This group includes actuator modules, passive jointmodules (rotary, pivot and spherical joints) and end-effector joints, such as the PI M235.5DG actuators,Hephaist spherical joints and miniature acetal fork typeuniversal joints (Fig. 12).2.Variable-dimension modulesThis group provides the end users with the ability torapidly fine-tune the kinematic and dynamic perfor-mance of the manipulators, such as rigid links, jointconnectors and platform modules, which can berapidly assembled into various layouts with differentkinematic and dynamic characteristics.As shown in Fig. 13, a set of links and a mobile platformhave been designed and fabricated. Through a combinationof these modules, different configurations of modularPKMs can be assembled based on the functionality andpurpose of the required manipulator. Based on the modulardesign concept, three PKMs are configured virtually usingSolidworks and the same modular components as shown inFig. 14. In Fig. 14, platforms with only rotational move-ments, translational movements, as well as a hybrid UPUplatform can be assembled by interchanging the sphericaland the universal joints or adding extra rigid links.7 Prototype developmentAs stated earlier, as the actuators and joint modules areinterchangeable, and the connectors can be fabricated in avery short period due to the simple design as shown inFig. 13, this modular design approach has helped to reducethe complexity of design problems to a manageable level.Therefore, based on these concepts, three PKM prototypeshave been developed and assembled to compare theircharacteristics and functions in an actual working environ-ment. Two interchangeable prototypes of 3DOF PKMsare assembled as shown in Fig. 15, which shows a puretranslational PKM and a pure rotational PKM.To maintain a high rigidity of the platforms, the DOFs ofthese systems can be constrained through the installation ofextrajointsorthereductionoftheDOFofthejoints.Forthe4legged platform in Fig. 15a, a fixed rigid link with aspherical joint is installed in the middle of the platform sothat the motion of the platform is limited to angular rotationonly. Thus, it can perform pure rotational movements. Forthe system in Fig. 15b, the spherical joints of the platformhave been replaced with universal joints.Hence, this plat-form is limited to translational movements through the axesof the universal joints. Another 3DOF PKM is configuredTheoretical Position and Orientation Actual Position and Orientation Position and Orientation Error Rotation X Rotation Y Position Z (mm)Z-10.426 (mm) Actual X Actual Y Z error(mm)Angle X errorAngle Y error 0279 278.8502 -0.0452256 -0.365286 -0.1498 0.045225568 0.3652860304 304.157 -0.0397824 -0.372792 0.157 0.039782411 0.3727920279 278.887 -0.0545076 -0.367864 -0.113 0.054507585 0.3678640 304 304.1532 -0.0385792 -0.377662 0.1532 0.038579187 0.3776620279 278.8512 -0.0448245 -0.369812 -0.1488 0.044824494 0.3698120 328 328.4152 -0.0470018 -0.395424 0.4152 0.047001757 0.3954240 279 278.7997 -0.0487779 -0.372276 -0.2003 0.048777945 0.3722760 314 314.2583 -0.0551378 -0.382016 0.2583 0.055137845 0.3820165 314 314.4069 4.9566289 -0.478564 0.4069 0.043371086 0.47856410314 314.1093 10.005325 -0.708506 0.1093 -0.0053253 0.70850615314 313.1804 15.051127 -1.091752 -0.8196 -0.05112695 1.09175220310 309.0273 20.087883 -1.563315 -0.9727 -0.08788319 1.563315-5310 309.7469 -5.0196374 0.3754471 -0.2531 0.019637435 -0.37545-10310 309.1461 -10.025349 -0.128134 -0.8539 -0.02535 -0.12813-15310 307.8201 -15.026953 0.8498897 -2.1799 -0.02695 0.84989-20310 307.3897 -19.908848 2.1461184 -2.6103 0.091152 2.1461180 5 310 309.9521 0.3101281 4.5267511 -0.0479 0.310128 -0.473250 10 310 309.5268 0.5645316 9.5351477 -0.4732 0.564532 -0.464850 15 310 309.1953 0.8663929 14.535792 -0.8047 0.866393 -0.464210 20 310 306.751 1.1710863 19.495447 -3.249 1.171086 -0.504550 -5 310 309.8145 -0.0267188 -5.48063 -0.1855 -0.02672 -0.480630 -10 310 310.0537 -0.0847028 -10.51162 0.0537 -0.0847-0.511620 -15 310 310.9903 -0.25665 -15.54496 0.9903 -0.25665 -0.544965 5 310 309.9001 5.1334433 4.4381289 -0.0999 0.133443 -0.5618710 10 310 310.2003 9.8857264 9.6356161 0.2003 -0.11427 -0.36438-5 -5 310 310.3308 -5.0607633 -5.386758 0.3308 -0.06076 -0.38676-10 -10 310 310.0731 -10.365204 -10.03466 0.0731 -0.3652-0.03466-15 -15 310 308.0145 -15.678278 -14.33112 -1.9855 -0.67828 0.668877 = Maximum Error 0000000000000000Table 3 Calibration result of the micro SP198as shown in Fig. 16. It is a hybrid 3DOF platform thatconsists of translation movements along the zaxis androtation movements around the x and yaxes.Nevertheless, there are advantages of a 3DOF PKM.The control of a 3DOF PKM is less complex and therigidity can be increased. Furthermore, due to the intro-duction of the modular design concept through fabricatinginterchangeable parts, the 3DOF PKM can be easilymodified from a translational platform to a rotational plat-form. An extra link or active joints can be introduced infuture to increase the DOF of the micro PKM.Calibrations and various tests have been performed onthe modular PKMs. Architectural singularities 16 havebeen detected during the calibration of the pure translation-al PKM while it is in the static position. It is found that in astatic position, extra DOFs are introduced as some ge-ometry conditions are not met, such as all the revolute pairaxes at the ends of the links do not converge towards asingle point, and each link has two intermediate resolutepair axes that are not parallel to one another and per-pendicular to the straight line through the centre of theuniversal joint 17. Hence, a careful assembly of the mod-ular units of the platform satisfying certain geometryconditions is needed to attain a controllable pure transla-tional motion.For the rotational PKM, the platform is able to perform3DOF rotational movements in the roll, pitch and yawdirections. Since all the links are composed of 2DOFuniversal joints, prismatic links and 3DOF sphericaljoints, by installing a fixed link with a spherical joint at thecentre of the platform, the motion of the platform is limitedto purely rotational movements around a fixed point. Themajor disadvantage of this type of configuration is that theplatform cannot perform zaxis movements, which is acrucial requirement for micromachining. The over-con-strained design of this platform with a fixed middle linkcauses a high risk of damage to the platform when it ismanipulated outside the defined workspace.Hence, a hybrid PKM has been assembled by installing apassive prismatic link at the centre of the pure translationalplatform as shown in Fig. 16, and the singularity problem issolved. By installing a passive prismatic link in the middleof the platform and attached to the platform by a sphericaljoint, the extra DOF caused by the universal joints of thelinks is constrained. Thus, the hybrid PKM is able toperform movements along the zaxis and rotation about thex and yaxes. Among the three PKMs architectures thathave been configured, the hybrid platform is further elab-orated as it can be coupled with the present normal-sizedSP to perform as a micro manipulator for a tool holder toperform machining operations on a workpiece that isfixtured on the SP. Calibration of the hybrid PKM wasperformed and the results showed that the accuracy of themovement is 0.5 mm and 0.02.8 Error calibration with CMMAfter the development of the simulation and interfacingprogrammes, error calibration was performed using aCMM machine. The accuracy and repeatability of the plat-form was determined to be 100 micron. The set-up of themicro PKM on the CMM machine is shown in Fig. 17.The calibration was performed by manipulating theplatform to the theoretical positions and orientations usingthe visual C+ interface. Next, the data of the coordinatesfrom the surface of the platform were collected to de-termine the actual surface plane of the platform. Using theCMM machine, the measurement variations between theactual angle and position and the theoretical angle andposition were compared. The result of the calibration isshown in Table 3.From Fig. 18, the maximum error of the roll angle is1.171, the maximum error of the pitch angle is 2.1461and the maximum error of the zaxis displacement is3.249 mm. At least two errors occurred in the same dataset when the platform was translating and rotating simul-taneously within the allowable maximum rotation angles.However, when the platform was performing purelytranslation movements, the error of the displacement waswithin 0.42 mm.The variations of the angular rotation and the translationmovement of the zaxis are acceptable. The overall av-erage errors of the angular rotation and translation are0.063942 for the roll rotation, 0.186244 for the pitchrotation and 0.42854 mm for the zaxis movement. FromPosition and Rotation Error Analysis-4-3-2-10123051015202530Set Po