关 键 词：

火星 环境 探测 结构设计 SW 三维 CAD 开题

资源描述：

火星环境探测车结构设计含SW三维及7张CAD图带开题,火星,环境,探测,结构设计,SW,三维,CAD,开题

内容简介：

一、毕业设计（论文）的内容内容：火星是太阳系中除地球外唯一有可能进化出生命的行星,是人类在太阳系中有希望在不久的将来实现登陆、行走并以传统方式进行探索的唯一颗行星。人类近期还不会将人送上火星，但我们仍可用机器人来代替作业, 火星探测车机器人就是这种思路的体现。本次火星车的结构设计内容，为设计满足火星车环境探测包括行走、避障、钻探等功能需求，进行受力计算，设计出合理的部件结构：1. 分析火星环境探测车的功能需求及应用特点，完成探测车的整体结构规划和设计。2. 探测车的行走功能，能在凸凹不平/类似戈壁的地域行走，能翻越高度、宽度小于0.3m的沟坎。3. 钻探功能：能够钻探火星表面以下1m深处的状况。4. 进行必要的受力计算，详细设计出主要部件的结构。5. 三维数字模拟设计样机，并通过运动仿真关键装置来验证该装置设计方案的合理性。二、毕业设计（论文）的要求与数据要求：在火星车设计过程中具体要求及主要技术指标如下：1. 整个装置结构合理、易于制造、应用三维软件模拟出产品样机。2. 火星车的外形尺寸不大于1000mm*800mm*600mm。重量不大于200kg。3. 装置设计说明书（毕业设计说明书）应包含中英文摘要、探测车功能分析、设计方案比较、结构设计、理论计算、行走机构设计、运动模拟过程分析及结果说明等内容。4. 提交装置总装图、装置的主要零部件图及模拟三维样机文件。三、毕业设计（论文）应完成的工作指定整个毕业设计学生应该完成的所有工作，包括：1、完成二万字左右的毕业设计说明书（论文）；在毕业设计说明书（论文）中必须包括详细的300-500个单词的英文摘要；2、独立完成与课题相关，不少于四万字符的指定英文资料翻译（附英文原文）；3、完成绘图工作量折合A0图纸3张以上，其中必须包含两张A3以上的计算机绘图图纸；4、导师所指定的其它工作，所有毕业设计的工作量要满足16周的工作量要求。四、应收集的资料及主要参考文献（与上述文字空1行）（首行与标题空0.5行，小4号宋体，行距20磅）列出至少5篇以上的参考文献，提供1篇以上的外文参考文献（不包括学生用的教材）。序号放在方括号中。1. 杨叔子，机械加工工艺师手册M，北京：机械工业出版社，200235-172.2. Yeong-Maw Hwang,Taylan Altan.Finite element analysis of tube hydroforming process in a rectangular dieJ.Finite element in Analysis and Design, 2003, 39 (11): 1071-1082.3. (美) Neil Sclate, Nicholas P.Chironis 编 邹平 译. 机械设计实用机构与装置图册M. 北京：机械工业出版社，2002105-214.4. 机械设计手册编委会.机械设计手册 机架、箱体及导轨M北京：机械工业出版社，200711-222.5. 詹迪维SolidWorks产品设计实例精解M北京：机械工业出版社，200875-268.6. 袁剑雄 李晨霞. 机械结构设计禁忌M. 机械工业出版社，200883-192.7. 廖念钊. SolidWorks基础教程:工程图M. 北京：机械工业出版社，200827-241.8. 吴宗泽.机械设计师手册M. 北京：机械工业出版社，200853-185.9. 机械设计手册编委会. 机械设计手册单行本造型设计和人机工程M. 北京：机械工业出版社，200723-102.10. Y. Zhang, W. Hu and Y. Rong et al. Graph-based set-up planning and tolerance decomposition for computer-aided fixture design. International Journal of Production Research J, 2001, 39(14): 3109-3126.五、试验、测试、试制加工所需主要仪器设备及条件（与上述文字空1行）计算机一台CAD设计软件 1毕业设计的主要内容、重点和难点等火星是太阳系中除地球外唯一有可能进化出生命的行星,是人类在太阳系中有希望在不久的将来实现登陆、行走并以传统方式进行探索的唯一颗行星。火星也是唯一可能被地球化而成为类似于地球的行星。然而我们尚不具备将人送上火星的能力。首先也是最重要的原因在于我们探索火星的历史记录不容乐观；其次是成本，目前将一公斤重的机器人送上火星需要花费约五十万美元的设计和发射费用，而机器人不需要考虑复杂的生命保障系统，也不用担心返回的问题这将为飞行任务节省很大的重量，另外，机器人不需要在火星表面软着陆；第三个原因来自工程上的挑战。综合以上原因，人类近期还不会将人送上火星。但是我们仍然可以用机器人来代替作业，让我们进一步了解火星。火星探测车（MER）机器人就是这种思路的体现。本次火星车的结构设计内容，为设计满足火星车环境探测包括行走、避障、钻探等功能需求，进行受力计算，设计出合理的部件结构。内容：1. 分析火星环境探测车的功能需求及应用特点，完成探测车的整体结构规划和设计。2. 探测车的行走功能，能在凸凹不平/类似戈壁的地域行走，能翻越高度、宽度小于0.3m的沟坎3. 钻探功能：能够钻探火星表面以下1m深处的状况。4. 进行必要的受力计算，详细设计出主要部件的结构。5. 三维数字模拟设计样机，并通过运动仿真关键装置来验证该装置设计方案的合理性。重点：1. 整个装置结构合理、易于制造、应用三维软件模拟出产品样机；2. 月球车的外形尺寸不大于1000mm*800mm*600mm。重量不大于200kg；3. 行走、避障、钻探等功能的实现；4. 三维数字模拟设计样机，并通过运动仿真关键装置来验证该装置设计方案的合理性。难点：1、 火星车结构要小体积低质量、低功耗结构可靠、有高度自适应性和容错性；2、 结构的合理性以及受力分析；3、 在三维数字模拟设计平台上装置模型的建立及仿真判断合理性。2准备情况（查阅过的文献资料及调研情况、现有设备、实验条件等）准备情况:通过查阅相关的文献，了解火星探测车的背景和现状，尤其是分析现有已成功研制的火星车的结构特点。学会基本的三维软件设计平台的使用，及相应的仿真、分析方式与方法。虽然毕设题目有一定的难度，通过近段时间的了解文献，还是攻克了不少难点，随着了解的深入，相信可以达到毕设的要求。文献资料：列出至少5篇以上的参考文献，提供1篇以上的外文参考文献（不包括学生用的教材）。序号放在方括号中。1 杨叔子，机械加工工艺师手册M，北京：机械工业出版社，200235-172.2 2Yeong-Maw Hwang,Taylan Altan.Finite element analysis of tube hydroforming process in a rectangular dieJ.Finite element in Analysis and Design, 2003, 39 (11): 1071-1082.3 (美) Neil Sclate, Nicholas P.Chironis 编 邹平 译. 机械设计实用机构与装置图册M. 北京：机械工业出版社，2002105-214.4 机械设计手册编委会.机械设计手册 机架、箱体及导轨M北京：机械工业出版社，200711-222.5 詹迪维SolidWorks产品设计实例精解M北京：机械工业出版社，200875-268.6 袁剑雄 李晨霞. 机械结构设计禁忌M. 机械工业出版社，200883-192.7 廖念钊. SolidWorks基础教程:工程图M. 北京：机械工业出版社，200827-241.8 吴宗泽.机械设计师手册M. 北京：机械工业出版社，200853-185.9 机械设计手册编委会. 机械设计手册单行本造型设计和人机工程M. 北京：机械工业出版社，200723-102.10 Y. Zhang, W. Hu and Y. Rong et al. Graph-based set-up planning and tolerance decomposition for computer-aided fixture design. International Journal of Production Research J, 2001, 39(14): 3109-3126. 现有设备：计算机一台和SolidWorks、UG、AutoCAD等相关设计软件。3、实施方案、进度实施计划及预期提交的毕业设计资料实施方案:对整个毕设的思路、流程有所了解，收集关于火星探测车的相关资料；确定火星车结构的设计方案。学会使用三维数字模拟设计平台软件，对火星车零部件结构进行设计并装配，进行仿真分析装置合理性，进而完善装置的设计。完成一份两万字左右的毕业论文，以及相关的图和程序及答辩前准备。实施计划：1、 3.01-3.10 完成4万字符的相关英文资料翻译及开题报告并提交；2、 3.11-3.15 了解毕设的内容和方向，查阅Solidworks教学资料进行学习；从网上及其它可利用的资源上收集有关的毕设资料，为后期冲刺做好准备；3、 3.16-4.10 收集尽可能多的相关资料，确定结构设计方案；4、 4.10-4.25 完成火星环境探测车参数化的设计；5、 4.26-5.15 对设计出的模型进行仿真判断设计合理性，并修改完善设计；6、 5.16-5.31 完成毕业论文，做好答辩前的准备。所提交的资料：1、 二万字左右的毕业设计说明书（论文）；在毕业设计说明书（论文）中必须包括详细的300-500个单词的英文摘要；2、与课题相关，不少于四万字符的指定英文资料翻译（附英文原文）；3、绘制装配图A0图纸一张，主要零部件图A3图纸；指导教师意见指导教师（签字）： 年月日开题小组意见开题小组组长（签字）：年月日 院（系、部）意见主管院长（系、部主任）签字： 年月日- 3 - /Robotics ResearchThe International Journal of /content/20/4/312The online version of this article can be found at: DOI: 10.1177/02783640122067426 2001 20: 312The International Journal of Robotics ResearchBruno Monsarrat and Clment M. GosselinGrassmann Line GeometrySingularity Analysis of a Three-Leg Six-Degree-of-Freedom Parallel Platform Mechanism Based on Published by: On behalf of: Multimedia Archives can be found at:The International Journal of Robotics ResearchAdditional services and information for /cgi/alertsEmail Alerts: /subscriptionsSubscriptions: /journalsReprints.navReprints: /journalsPermissions.navPermissions: /content/20/4/312.refs.htmlCitations: What is This? - Apr 1, 2001Version of Record at Tsinghua University on January 8, 2012Downloaded from Bruno MonsarratClment M. GosselinMechanical Engineering Department,Laval UniversityQubec, Qubec, Canada, G1K 7P4gosselingmc.ulaval.caSingularity Analysisof a Three-LegSix-Degree-of-FreedomParallel PlatformMechanism Based onGrassmann LineGeometryAbstractThis paper addresses the determination of the singularity loci ofa six-degree-of-freedom spatial parallel platform mechanism of anew type that can be statically balanced. The mechanism consistsof a base and a mobile platform that are connected by three legsusing five-bar linkages. A general formulation of the Jacobian ma-trix is first derived that allows one to determine the Plcker vectorsassociated with the six input angles of the architecture. The lin-ear dependencies between the corresponding lines are studied usingGrassmann line geometry, and the singular configurations are pre-sented using simple geometric rules. It is shown that most of thesingular configurations of the three-leg six-degree-of-freedom par-allel manipulator can be reduced to the generation of a generallinear complex. Expressions describing all the corresponding sin-gularitiesarethenobtainedinclosedform. Thus,itisshownthatfora given orientation of the mobile platform, the singularity locus cor-respondingtothegeneralcomplexisaquadraticsurface(i.e., eithera hyperbolic, a parabolic, or an elliptic cylinder) oriented along thez-axis. Finally, three-dimensional representations that show the in-tersection between the singularity loci and the constant-orientationworkspace of the mechanism are given.KEY WORDSstatic balancing, parallel manipulator, sin-gularities, Grassmann line geometry, singularity loci1. IntroductionIn the context of manipulators and motion simulation mech-anisms, parallel architectures have particularly aroused theThe International Journal of Robotics ResearchVol. 20, No. 4, April 2001, pp. 312-328,2001 Sage Publications, Ierest of researchers over the past 30 years for their prop-erties of better structural rigidity, positioning accuracy, anddynamic performances. However, in industrial applicationsof such mechanisms where displacements of heavy loads areinvolved, theoperatingcostsareincreasedsubstantially. Thismotivated the development of parallel architectures that canbe statically balanced. A mechanism is statically balanced ifits potential energy is constant for all possible configurations(i.e., zero actuator torques are required whenever the manip-ulator is at rest in any configuration). To the knowledge ofthe authors, static balancing of spatial six-degree-of-freedomparallel manipulators was first introduced by Streit (1991),Leblond and Gosselin (1998), J. Wang (1998), Gosselin andWang (2000), and Herder and Tuijthof (2000). Two static-balancing methods, namely, using counterweights and usingsprings, are used. The first method, introducing additionalmasses into the system, tends to increase the inertia con-siderably. Because many commercial applications involvelarge accelerations of the mobile platform, the use of coun-terweights is undesirable. On the other hand, the use of analternative architecture for the legs, similar to what was usedby Streit and Gilmore (1989), allows one to obtain an effi-cient static balancing using only springs. Moreover, Ebert-Uphoff, Gosselin, and Lalibert (2000) presented a class ofspatial parallel platform mechanisms of novel architecture,with three legs or more using five-bar linkages, that is suit-able for static balancing. A prototype with three legs was de-signed by Gosselin et al. (1999) in accordance with the static-balancing-relatedconstraints(seeFig.1). Akinematicanaly-sisofthatclassofmechanismwaspresentedbyEbert-Uphoffand Gosselin (1998). In the latter reference, the inverse kine-maticproblemwasresolvedandanexpressionoftheJacobian312 at Tsinghua University on January 8, 2012Downloaded from Monsarrat and Gosselin / Parallel Platform Mechanism313Fig. 1.Prototype of the statically balanced three-leg six-degree-of-freedom parallel platform mechanism (as reportedin Gosselin et al. 1999).matrix was obtained in a general form. Closed-form expres-sions of the corresponding singularity loci were determinedonly for the special case in which (i) the fixed and mobileplatforms are parallel and (ii) for each leg i, the rotation an-gle around the z-axis and the angle between the two proximallinks of the parallelogram are actuated.Thispaperfollowsupwiththecompletegeometricandan-alytical characterization of the singularity loci of the above-mentioned three-leg parallel manipulator. The determinationof such configurations is a critical issue to be taken into ac-count early in the design process, thus maximizing the inher-ent performance of the system during trajectory tracking. Inpractice, these degeneracies of the instantaneous kinematicslead to a change in the number of degrees of freedom of themechanism and to a degradation of the stiffness propertiesthat may lead to very high joint torques or forces. In bothcases, the accuracy of the control will be critically affected.As shown by Gosselin and Angeles (1990), Gosselin andSefrioui (1992), Gosselin and Wang (1997), and Mayer St-Onge and Gosselin (2000), an efficient approach consists inobtaining the equations describing the singularity loci fromthe closed-form expressions of the determinant of the Jaco-bian matrix. The resulting singular configurations have beenclassified in three main groups: type I, where the end effec-tor lies at the boundary of the Cartesian workspace; type II,where the output link gains one or more degrees of freedom(i.e., the end effector is movable when all the input joints arelocked); and type III, where the chain can undergo finite mo-tions when the actuators are locked or a finite motion of theactuators produces no motion at the end effector. This clas-sification was recently generalized by Zlatanov, Fenton, andBenhabib (1995, 1998), who developed a unified frameworkforthesingularityanalysisofnonredundantmechanisms. Sixtypes of singularities reflecting different possibilities for theoccurrence of degeneracies of the instantaneous forward andinverse kinematics are defined. The determinant-based ap-proach was illustrated by Gosselin and Sefrioui (1992) andCollins and McCarthy (1998) with the analysis of planar3-RPR parallel manipulators. The approach was also usedby Gosselin and Wang (1997) and more recently by Bonevand Gosselin (2001) to determine the singularity loci of 3-RRR parallel manipulators. Moreover, the same procedurewas implemented by G. Wang (1998) and St-Onge and Gos-selin (2000) to obtain the equations of the singularity loci ofthe well-known Gough-Stewart platform.However, the structure of parallel architectures requires insome cases that the closed-form expression of the determi-nant of the Jacobian matrix depend not only on the Cartesianbut also on the joint coordinates. After substituting the ac-tive and passive joint coordinates with the Cartesian coordi-nates using the equations of the inverse kinematic problem,the closed-form expression of the corresponding determinantis of a very complex form. Therefore, the determinant-basedmethod is not adapted to the singularity analysis of the three-leg six-degree-of-freedom parallel mechanism that uses five-bar-linkagesforacaseofactuationidenticaltotheoneshownin Figure 1.In this context, the approach used here is the one intro-duced by Merlet (1988, 1989) and Mouly and Merlet (1992)in which the singularities of spatial 6-(RR)PS and 6-P(RR)Sparallel manipulators were studied using Grassmann line ge-ometry. Theprocedureleadstoanexhaustivelistofgeometricconditions that correspond to singularities. It is known thatsuch an analysis allows the characterization of the degenera-cies of the Jacobian matrix that correspond to singularitiesof type II, following the classification given by Gosselin andAngeles (1990). A recent work by Hao and McCarthy (1998)allowedonetospecifythedesignfeaturesofparallelplatformmechanisms that guarantee that only line-based singularitiesexist. A complete classification of linearly dependant sets oflines is also provided using Merlets notation.However, it was shown by Zlatanov, Fenton, and Ben-habib (1995) that the existence of an invertible 66 Jacobianmatrix is not a sufficient condition for nonsingularity unlessthevelocityequationsbetweentheactiveandpassivejointve-locities are defined. Therefore, the occurrence of additionalsingular configurations is studied here through an analysis ofthe cases in which the 36 matrix relating the actuated jointvelocities and the passive joint velocities is not defined.The organization of this paper is as follows. In Section 2,we will briefly review the design of the prototype and in-troduce the corresponding notation. In Section 3, a generalexpression of the Jacobian matrix will be obtained using theprinciple of virtual work, and the corresponding Plcker vec-tors associated with the six input angles will be derived. InSection4,thelineardependenciesbetweenthecorresponding at Tsinghua University on January 8, 2012Downloaded from 314THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2001Fig. 2. Parallel platform mechanism with three legs usingfive-bar linkages (CAD model by Jiegao Wang).Grassmann lines are studied and the singular configurationsare described using simple geometric rules. The closed-formequations of the singularity loci are then obtained in Sec-tion 5. It is shown that for a given orientation of the mobileplatform, the singularity locus corresponding to the generallinear complex is a quadratic surface (i.e., either a hyper-bolic, a parabolic, or an elliptic cylinder) oriented along thez-axis. In Section 5, additional singularities corresponding tothenonexistenceofthe36matrixdefinedabovearestudied.To clearly illustrate the results, three-dimensional represen-tations that show the intersection between the singularity lociand the constant-orientation workspace of the mechanism aregiven.2. Description of the MechanismThe type of mechanism considered in this paper is shownin Figure 2. The architecture consists of a fixed base anda mobile platform connected by three legs. The ith leg isattached to the base at point Pi0and to the mobile platformat point Pi5(see Fig. 3). The attachment points on the baseand mobile platforms form equilateral triangles. We define ageometric parameter r that is used to describe the position ofpoints Pi0, making the assumption that the attachment pointsofthethreelegsareequallyspacedonacircleofradiusr withcenter at point O; that is, r =?pi0?, i = 1,2,3.At point Pi0, the two links of the parallelogram aremounted using revolute joints. These provide two degreesof freedom, associated with angles i2and i3. In addition,the whole leg can rotate about the vertical axis at Pi0, whichprovides a third degree of freedom for each leg, associatedwith angle i1. This rotation includes the mounting points ofFig. 3. Kinematic parameters of one leg of the mechanism.the springs at the base, so that the springs always remain inthe plane of the parallelogram. The upper end of the leg, Pi5,is attached to the mobile platform using a spherical joint. Inpractice, a universal joint combined with an additional rev-olute joint has been used in the design of the prototype (seeFig. 2). If six of the joints are actuated, it results in a mecha-nism with six degrees of freedom.We define a fixed-base reference frame with its origin atpoint O and with axes x, y, and z such that the base z-axiscoincides with the axis of symmetry of the mechanism. Amobile frame is chosen fixed to the mobile platform at pointC with axes xp, yp, and zpsuch that the mobile zp-axis coin-cides with the axis of symmetry. The position of the mobileplatform is described by the vector p = x,y,zT, whichdenotes the coordinates of the point C in the reference frame.Glossary of Termsij: angles describing the configuration of the ith leg (j =1,2,3)O: origin of the fixed frame located at equal distance fromthe three points Pi0, i = 1,2,3C: centroid of the mobile platformp: vector connecting the origin of the fixed frame O to thecentroid C (position vector of the mobile platform)Q: rotation matrix representing the orientation of theplatform and defined by Euler angles (, , )pi0: vector connecting the origin of the fixed frame to theattachment point Pi0of leg i at the basepi5: vector connecting the origin of the fixed frame to theattachment point Pi5of leg i at the mobile platformbi: vector from centroid C to the attachment point Pi5ofthe ith leg with respect to the mobile frame at Tsinghua University on January 8, 2012Downloaded from Monsarrat and Gosselin / Parallel Platform Mechanism315b: distance between point C and the attachment pointsPi5of the ith leg to the mobile platform; that is, b =?bi?, i = 1,2,3?ir: length of the rth link of leg i.In the reference orientation of the mobile platform, the ori-entation of the mobile frame coincides with that of the baseframe. This orientation will be represented by the standardEuleranglesthataredefinedbyfirstrotatingthemobileframeaboutthebasez-axisbyanangle, thenaboutthemobilexp-axis by an angle , and finally about the mobile yp-axis by anangle . For this choice of Euler angles, the rotation matrixis defined asQ = Qz()Qx()Qy()=(cc sss)sc(cs+ ssc)(sc+ css)cc(ss csc)csscc,(1)where c= cos, s= sin, c= cos, s= sin, c=cos, and s=sin.In this context, (x,y,z,) denote the generalizedcoordinates of the three-leg six-degree-of-freedom parallelmanipulator. The reader should refer to Figure 3 and to theassociated glossary of terms for a complete description of themechanism and its configuration.3. Jacobian Matrix and Resulting PlckerCoordinatesIn this section, a general expression of the Jacobian matrixJ for a manipulator with D legs, D = 3 to 6, is obtainedusing the principle of virtual work when the manipulator isin its equilibrium state. The following study will address thedetermination of the matrix J in the case where, for each leg,any arbitrary subset of the joints corresponding to i1, i2,i3, and ?i3is actuated. This particular formulation of theJacobian matrix will allow the determination of the Plckervectors associated with the input angles of the manipulator.3.1. General Expression of the Jacobian MatrixThe Jacobian matrix derived here relates the velocities of theactuated joints and the platform velocity in the form = J? p ?,(2)where contains the velocities of all actuated joints, p is thevelocityofthecentroidC ofthemobileplatform, and istheangular velocity vector corresponding to the skew-symmetricmatrixQQT; that is, = vect(QQT). Let Cijbe the jointtorque associated with the actuated angle ij, and let? ? ? be thecorresponding six-dimensional joint force vector. Let F bethe force vector applied on the mobile platform, and let Mbe the torque vector acting on the centroid C of the mobileplatform. If f denotes the generalized force vector (i.e., theexternal wrench acting on the mobile platform and defined atpoint C), we thus obtainf =?FM?.(3)Let us define the auxiliary vectors gir= 0?ir0T, i =1,2,3, and the auxiliary rotation matricesQi1= Rotz(i1) =cosi1sini10sini1cosi10001(4)and (j = 2,3)Qij= Rotx?(ij) =1000 cosijsinij0 sinijcosij.(5)The z- and x?- axes are shown in Figure 3. In the following,vector si= sizsiysiz is the vector connecting point Pi0the attachment point of leg i to the baseto point Pi5theattachmentpointoflegi totheplatform;thatis,si= pi5pi0.This equation can be rewritten in terms of angles i1, i2, andi3in the fixed frame R such thatsi= Qi1(Qi3gi1+ Qi2gi5)=cosi1sini10sini1cosi10001?i10cosi3sini3+ ?i50cosi2sini2.(6)We also define the vector f?i5= f?i5xf?i5yf?i5zTthat denotesthe force that the mobile platform exerts on leg i at the pointPi5expressed in the frame R?. Then, the total virtual work ofthe ith leg can be described byW = Ci1i1+ Ci2i2+ Ci3i3+ f?i5 s?i5(7)wheres?i5= QTi1sidenotesthevirtualdisplacementofpointPi5in frame R?such thats?i5= QTi1dQi1di1(Qi3gi1+Qi2gi5)i1+QTi1Qi1?dQi3di3gi1i3+dQi2di2gi5i2?(8)Given that QTi1Qi1is the 33 identity matrix, we can rewritethe expression of vector s?i5in the forms?i5=0 1 0100000?i10cosi3sini3+ ?i50cosi2sini2i1+?i10sini3cosi3i3+ ?i50sini2cosi2i2.(9) at Tsinghua University on January 8, 2012Downloaded from 316THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2001According to the principle of virtual work, the total virtualwork of the system must vanish; that is, W = 0. This allowsus to rewrite eq. (7) in the compact form(?i1cosi3+ ?i5cosi2)f?i5x+ Ci1?i5sini2f?i5y+ ?i5cosi2f?i5z+ Ci2?i1sini3f?i5y+ ?i1cosi3f?i5z+ Ci3Ti1i2i3= 0.(10)Solving eq. (10) for f?i5x, f?i5y, and f?i5z, one can find theexpression of vector f?i5as a function of the joint torques Ci1,Ci2, and Ci3with respect to frame R?:f?i5=1?i1?i5sin?i3?i1Ci20cosi3sini3+ ?i5Ci30cosi2sini2+1?i1cosi3+ ?i5cosi2Ci1100.(11)Let fi5be the corresponding expression of vector f?i5in thefixed frame. We can writefi5= Qi1f?i5= Ci1vi1+ Ci2vi2+ Ci3vi3.(12)From eq. (11), vector vi1is given byvi1=1?i1cosi3+ ?i5cosi2cosi1sini10=1|ui|2ui,(13)where vector ui= siy six0Tis normal to the (y?,z?)-plane. Moreover, vectors vi2and vi3are obtained in the fol-lowing compact form:vi2=1?i5sin?i3sini1cosi3cosi1cosi3sini3=1?i1?i5sin?i3Qi1Qi3gi1(14)andvi3=1?i1sin?i3sini1cosi2cosi1cosi2sini2=1?i1?i5sin?i3Qi1Qi2gi5.(15)Then, we can express the generalized force vector as a func-tion of the joint torques:F =D?i=1fi5=D?i=13?j=1Cijvij(16)M =D?i=1(Qbi) fi5=D?i=13?j=1Cij(Qbi) vij).(17)Furthermore, it is well-known thatf = JT? ? ?,(18)whereJistheJacobianmatrixforthemanipulator. Therefore,eachinputangleijgeneratesonerowoftheJacobianmatrix,which is of the formJij=?vTij(Qbi) vij)T?(19)and vectors vijare given by eqs. (13), (14), or (15). For thecaseinwhichangle?i3isactuated,thecorrespondingvectorwould simply be given by vi?3= vi3 vi2. Consideringeqs. (14) and (15), one thus obtainsvi?3=1?i1?i5sin?i3(Qi1Qi2gi5+ Qi1Qi3gi1). (20)Consideringeq.(6), thisvectorcanberewritteninthefollow-ing compact form:vi?3=1?i1?i5sin?i3si.(21)Note that for any case of actuation, the Jacobian matrix doesnot depend on the geometric parameter ?i2.3.2. Computation of the Jacobian Matrices for theManipulator with Three LegsLet xbethevectorcontainingthegeneralizedvelocitiesofthemechanism; thatis, x=? pT T?T. Usingeqs.(2),(14),(15),and (19), we can directly obtain an expression of the vectorrelation between the vector of actuated joint velocities andthe generalized velocity vector x of the mechanism for theparticular case in which for each leg i, the input angles are i2and i3. This vector relation is obtained in a particular formwhere all denominators have been eliminated; that is,A x + B = 0,(22)where =?122232132333?T.(23)Let us define the two auxiliary unit vectors ni2and ni3suchthatni2= (?i5sin?i3)vi2= Qi1Qi3j(24)andni3= (?i1sin?i3)vi3= Qi1Qi2j,(25) at Tsinghua University on January 8, 2012Downloaded from Monsarrat and Gosselin / Parallel Platform Mechanism317with j = 0 1 0T. Vectors ni2and ni3are collinear withvectors vi2and vi3, respectively. Using a notation identicalto the one that was first introduced by Gosselin and Ange-les (1990), A and B are the 6 6 Jacobian matrices that, forthe mechanism under study, can be expressed byA =nT12(Qb1) n12)TnT22(Qb2) n22)TnT32(Qb3) n32)TnT13(Qb1) n13)TnT23(Qb2) n23)TnT33(Qb3) n33)T(26)B = diag(b12,b22,b32,b13,b23,b33),(27)where the coefficients bi2and bi3are given, for i = 1,2,3,bybi2= ?i5sin?i3andbi3= ?i1sin?i3.(28)3.3. Singularities of Types I and IIType I. The configurations corresponding to singularities oftype I are the ones that occur whendet(B) =3?i=1(?i5sin?i3)3?i=1(?i1sin?i3) = 0.(29)This condition leads to?i3= i3i2= n, n Z.(30)This defines, for each leg i, the minimum and maximumspheres of radii (?i5?i2) and (?i5+?i1), respectively, withcenter at point of coordinates (pi0 Qbi), that constitutethe boundary of the constant-orientation workspace of themechanism.Type II. Singularities of type II occur when the matrix A issingular, that is, whendet(A) = 0.(31)Then, the equations of the inverse kinematic problem (Ebert-Uphoff and Gosselin 1998) are employed to rewrite eq. (31)in terms only of the generalized coordinates. However, theresulting closed-form expression of this determinant is verycomplex. Therefore, the degeneracies of the matrix A willbe studied here through a geometric approach developed inSection 4.3.4. Plcker Coordinates of LinesAsapreliminaryremark, letusrememberthatalineD canbedefined by its Plcker vector. Let two points on the line D beM1and M2. We also introduce an arbitrary reference framewhose origin is O. Let a be the vector connecting the pointM1to point M2, and let b be the vector connecting the originO to point M1. The corresponding six-dimensional Plckervector PDis then defined byPD=?aT(b a)T?T.(32)Let Uijbe the Plcker vector associated with the jth actuatedangle of leg i. Then, considering eq. (26), we can directlyobtain an expression for the column vector Uij:Uij=?nTij(Qbi) nij)T?T.(33)ThisallowsustoobtainanexpressionfortheJacobianmatrixA in the formA = U12U22U32U13U23U33T.(34)Thus, each row Uijof matrix A can be associated with agiven line that is completely characterized in the Cartesianspace. Subsequently, the singularities of the Jacobian matrixA are obtained when one of the Plcker vectors associatedwith a link is linearly dependent on the other Plcker vectors.Therefore, the manipulator will be in a singular configurationof type II if and only if there is a subset spanned by n of itslines that has a rank less than n (Merlet 1989).In the following section, we use an approach based onGrassmann line geometry to determine the generalized co-ordinates of the mobile platform for which there is a lineardependency between the six Plcker vectors.4. Linear Dependencies of the CorrespondingSet of LinesVarieties of lines were studied by H. Grassmann, and thecharacterization of each variety was established by Dan-durand (1984). Furthermore, the constraints for a set of nlines were discussed by Merlet (1989). Therefore, to findthe loci of singularities of the mechanism, we have to find theconfigurationsofthemobileplatforminwhichtheGrassmanngeometric constraints are fulfilled. As mentioned earlier, weanalyze the three-leg six-degree-of-freedom parallel mecha-nism for which, for each leg, the angles corresponding to i2and i3are actuated.Let di2and di3be the lines defined by the Plcker vectorsUi2andUi3,respectively. Letdi23bethelinepassingthroughthe points Pi2and Pi3. Considering the expressions of unitvectors ni2and ni3given by eqs. (24) and (25), we deducethat (i) the line di2is parallel to the link Pi2Pi3and (ii) theline di3is parallel to the link Pi1Pi5. Moreover, we knowfrom eq. (33), j = 2 and 3, that the lines di2and di3bothpass through the point Pi5, Qbibeing the vector connectingthe centroid C to the point Pi5expressed in the fixed frame at Tsinghua University on January 8, 2012Downloaded from 318THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2001Pi5bpPPi2Pi1Pi3iOxyzi2i3i1i0Cpi0 di2 di3Fig. 4. Grassmann lines di2and di3associated with the actu-ated angles i2and i3of leg i.R. Therefore, the line di2passes through the point Pi5and isparalleltothelinedi23. Thelinedi3passesthroughthepointsPi1and Pi5. Figure 4 shows, for the ith leg, the lines definedby the Plcker vectors Ui2and Ui3.Let ?ibe the plane defined by the two lines di2and di3.The particular topology of the kinematic chain constitutingthe ith leg imposes that for any configuration of the mobileplatform, the plane ?icontain all the links of the ith leg andbe normal to the plane of the base (i.e., the plane containingthe three points Pi0, i = 1,2,3).We implement here the same methodology that was firstpresentedbyMerlet(1989). Thereadershouldalsorefertothelatter reference for a detailed classification of the Grassmannvarieties. The notation used in that reference will be used inthis paper. Then, to study the singularities of the three-legparallel mechanism, we have to consider the linear varietiesof rank 1 to 5. The resulting linear dependencies of the set oflines that correspond to singular configurations of type II willbe described here using simple geometric conditions.4.1. Subset of Two LinesCONDITION1.Let us begin with the linear variety of rank1. In this case, we obtain a singular configuration when twolines dijform a line in the three-dimensional space. For themanipulator under study, the set of two lines can be of twotypes.1. In the first case, the two lines are associated with thesame leg i. A singularity occurs when the two linksof the ith leg are aligned; that is, di2 di3. In theseconfigurations, a change of branch set occurs and theleg loses one degree of freedom. Such a singularityis typically encountered by four-bar linkage structures.Note, correspondingly, that the closed-form expressionof the resulting singularity locus is the same as the onedescribing the occurrence of singularities of type I.2. In the second case, the two lines belong to differentplanes ?i. Without loss of generality, let us considerthe three lines d12, d13, and d2j, j = 2 or 3. We havethe possibilities d12 d2jor d13 d2j. Such singularconfigurations can occur only if the planes ?1and ?2are coincident.4.2. Subset of Three LinesCONDITION2.The lines belong to a flat pencil of lines: thethreelinesbelongtoaplaneandallintersectatthesamepoint.The sets of three lines dijcan be divided into two families.1. First, we analyze the case in which two of the threelines belong to the same plane ?i(e.g., d12, d13, andd2j,j = 2 or 3). A singular configuration is obtainedwhen the line d2jbelongs to the plane ?1and passesthrough the point P15. Therefore, the line d2jmust becollinear with the edge P15P25. Such a case can occuronly if the two planes ?1and ?2are coincident.2. Inthesecondfamily,eachofthethreelinesisassociatedwith one leg i. We therefore consider the lines d1j,d2j, and d3j, j = 2 or 3. Subsequently, a singularconfiguration occurs when the three lines dijbelong tothe plane of the mobile platform and have one point incommon. Otherwise, the three planes ?i, i = 1,2,3,should be coincident, which is clearly impossible.4.3. Subset of Four LinesCONDITION3a.A set of four lines dijbelong to a regulus.Let us consider three skew lines in space. The family of linesthat intersect these lines generates a hyperboloid of one sheetandiscalledaregulus(MoulyandMerlet1992). Amongasetof four lines dij, at least two belong to the same plane ?iandtherefore pass through the same point Pi5. Thus, we cannotfind a set of four lines dijthat belong to the same regulus.CONDITION3b.The lines belong to two flat pencils havinga line in common but lying in two distinct planes and withdistinct centers. We also have to consider two different setsof four lines.1. First,twopairsoflinesdijhaveacommonpoint,whichis their common point Pi5on the mobile platform. Let at Tsinghua University on January 8, 2012Downloaded from Monsarrat and Gosselin / Parallel Platform Mechanism319us first consider the lines d12, d13, d22, and d23. Thetwo flat pencils necessarily lie in the planes ?1and?2, respectively. Thus, the common line d betweenthe two flat pencils is the line going through the pointsP15and P25. We therefore obtain a singular configura-tion if and only if the line d is the intersection line ofthe planes ?1and ?2; that is, d ?1 ?2, the lined being perpendicular to the plane of the base. More-over, the structure of the three-leg parallel manipulatorimplies that such a singular configuration also occurswhen considering the quadruplets (d12,d13,d32,d33)and (d22,d23,d32,d33).Thus, with an appropriatemodification of the base and mobile frames, such a sin-gular configuration can be described for each of theabove-mentioned quadruplets by the equation = 2.(35)Equation(35)isvalidwhenconsideringthesetofEulerangles as defined in Section 2.2. Second, only one pair of the four lines dijhas a com-mon point (e.g., d12, d13, d2j, and d3j). Condition 3bimposes that the lines d2jand d3jbe coplanar. Let I23be their intersection point. The common line betweenthe two flat pencils is the line going through the pointsP15and I23. In that case, we obtain a singularity whenthe point I23belongs to the plane ?1and the pointP15belongs to the plane spanned by the two lines d2jand d3j. We point out that such a configuration of themobile platform can only occur if the three planes ?i,i = 1,2,3, all intersect in a line perpendicular to theplane of the base.CONDITION3c.Asetoffourlinesdijpassthroughthesamepoint but are not coplanar. Let us examine the possible cases.1. First,twopairsoflinesdijhaveacommonpoint,whichis their common point Pi5on the mobile platform. Letus consider the lines d12, d13, d22, and d23. The pointP15is the intersection point of lines d12and d13. Thepoint P25is the intersection point of lines d22and d23.Thus, thepointsP15andP25mustbecoincident, whichis clearly impossible.2. The second subcase is obtained when only one pair ofthe four lines dijhas a common point (e.g., d12, d13,d2j, and d3j). The common intersection point is P15.We therefore obtain a singular configuration when thelines d2jand d3jpass through the point P15.CONDITION3d.The four lines belong to the same plane butdo not belong to a flat pencil of lines. We have to consider aset of four lines dijsuch that each pair of lines belongs to aplane ?i(e.g., d12, d13, d22, and d23). Such a case is obtainedwhen the planes ?1and ?2are coincident.4.4. Subset of Five LinesBecausethetopologyofthearchitectureimposesthatwehaveat most three skew lines dijin space, we do not have to con-siderthedegeneracyofthesubsetoffivelinesinconfiguration4a (elliptic congruence).CONDITION4b.The five lines are concurrent with two skewlines in space (hyperbolic congruence).Without loss ofgenerality, let us consider the lines d12, d13, d22, d23, andd3j,j = 2 or 3. We first determine a set of two skew linesd?and d?that intersect four lines dij. We analyze the twopossible cases.1. First, d? ?1and crosses the point P25, and d? ?2and crosses the point P15. These conditions imposethat P25 ?1and P15 ?2. This implies that thetwo planes ?1and ?2must be coincident, thus con-taining the edge P15P25. This implies that d?is notskew to d?. We therefore cannot obtain such a singularconfiguration.2. In the second case, we study the configuration in whichd? ?1?2and d?is the line that passes through thepoints P15and P25. The line d?is therefore perpendic-ular to the plane of the base containing the three pointsPi0, i = 1,2,3. Then, the line d3jintersects both d?and d?if and only if the line d3jbelongs to the planeof the mobile platform and d? ?1?2?3. There-fore,alocusofsingularitycorrespondstothisparticularconfiguration.CONDITION4c.The five lines define three flat pencils lyingin distinct planes with distinct centers having a line in com-mon (parabolic congruence). Let us consider the singularconfiguration obtained for case 1 satisfying condition 3b. Insuch a configuration, the lines d12, d13, d22, and d23belong totwo flat pencils having the line d ?1?2in common withcenters at points P15and P25. We consider here the set of fivelines d12, d13, d22, d23, and d3j, j = 2 or 3. Then, we pointout that a linear variety of type 4c is obtained if and only ifthe line d3jintersects the line d. In such a configuration, thethree planes ?i, i = 1,2,3, all intersect the line d, which isperpendicular to the plane of the base.CONDITION4d.All the lines belong to the same plane orpass through one point in that plane (degenerate congruence).We have to consider two cases.1. Three lines, for example d13, d23, and d33, belong tothe plane of the mobile platform. However, the tworemaining lines d12and d22do not have a commonpoint that belongs to that plane. Therefore, we cannotfind such a configuration that satisfies condition 4d.2. Three lines belong to a plane ?i(e.g., d12, d13, andd2j, j = 2 or 3). Such a case can occur only if the two at Tsinghua University on January 8, 2012Downloaded from 320THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2001planes?1and?i, i = 2 or 3, arecoincident. Then, theintersection point P35of the two remaining lines, d32and d33, must belong to that common plane ?1 ?2.4.5. Subset of Six LinesCONDITION5a.The variety spanned by the six lines dijis ageneral linear complex. In such a configuration, all coplanarlinesthatbelongtothatcomplexintersectthesamepoint(i.e.,all coplanar lines of that complex define a flat pencil of lines).Let us consider the three lines di, i = 1,2,3, belonging tothe three flat pencils di2 di3, i = 1,2,3, and lying in the(x,y)-plane of the base. The three lines di, i = 1,2,3, arethe three intersection lines between the planes ?iand theplane of the base. If the six lines dijbelong to the complex,it implies that the lines di, i = 1,2,3, also belong to thatcomplex. Therefore, we obtain a general complex if and onlyifthethreelinesdi,i = 1,2,3,intersectthesamepoint. Sucha configuration occurs if and only if the geometric conditiond? ?1?2?3is satisfied, the intersection line d?beingperpendicular to the plane of the base.CONDITION5b.All the six lines dijcross the same line (sin-gular linear complex). Such a configuration can occur if andonly if we have d? ?1 ?2 ?3. As mentioned previ-ously, the three planes ?iare normal to the plane of the basecontaining the three points Pi0, i = 1,2,3, and the line d?istherefore also perpendicular to that plane.Thus, it appears that the singular linear complex (condi-tion 5b) is a subcase of the general linear complex (condition5a). However, for the mechanism under study, the geometricconditioncorrespondingtothesingularcomplexisequivalentto the one corresponding to the general complex, and therebythe resulting singularity loci coincide.Aninterestingpropertyofthesingularitylociofpointssat-isfying conditions 5a and 5b is that they can be determinedthrough a two-dimensional analysis in the (x,y)-plane par-allel to the one containing the base of the parallel mecha-nism. Once the closed-form equation of the correspondingcross section of the singularity surface is determined (i.e.,FQ(x,y) = 0), the equation describing the complete singu-larity loci in the Cartesian space will directly be given by?FQ(x,y) = 0z = z.(36)4.6. Summary of the Geometric Conditions Leadingto SingularitiesSubsequently, the geometric analysis of the singularities ofthe mechanism allows us to make the following observations:i. Cases1-2(case2satisfyingcondition1),cases2-1,and4d-2aresubcasesofthemoregeneralcase3dforwhichtwo of the three planes ?imust be coincident.ii. Cases 2-2, 3b-2, 3c-2, 4b-2, 4c, and 5b are differentsubcases of case 5a, which implies a configuration ofthe mobile platform of the manipulator such that d?1 ?2 ?3.iii. Underthenonrestrictiveassumptionthatthegeometriesof the base and mobile platform are such that b ?= r,the three lines dialso define a flat pencil of lines inthe (x,y)-plane in the configuration for which two ofthe three planes ?iare coincident. This implies thatthe variety spanned by the six lines dijalso forms ageneral linear complex when condition 3d is satisfied.It follows that all the cases 1-2, 2-1, 2-2, 3b-2, 3c-2,3d, 4b-2, 4c, 4d-2, and 5b are subcases of the generallinear complex (condition 5a).Except case 3b-1, all the singular configurations of the three-leg six-degree-of-freedom parallel manipulator can be re-duced to the generation of a general linear complex. In prac-tice, we must determine the general equations of the singular-ity loci corresponding to condition 5a in terms of the gener-alized coordinates (x,y,z,). Then, the curves corre-spondingtothedifferentsubcaseswillbeimplicitlytakenintoaccount by the resulting equations. As the closed-form equa-tions corresponding to the singularities of type I and thosecorresponding to subcases 1-1 and 3b-1 have been obtainedat this stage of the procedure, we still have to determine theequations, as a function of the position p and orientation Q ofthe mobile platform, for case 5a.5. Closed-Form Equations of the SingularityLociWe develop in this section the general closed-form expres-sions of the different loci of singularity of the three-legsix-degree-of-freedom parallel mechanism, which were de-scribed geometrically in the previous section.5.1. Locus of Points Satisfying Condition 5aLet us begin with the determination of the three-dimensionallocus that contains all the points satisfying condition 5a. Asmentioned previously, we can first derive the correspondingtwo-dimensionalequationofthecurve, inthe(x,y)-plane, asa function of the coordinates (x,y,). Then, we canextrudethatcurveinthe(+z)-directiontoobtainthecompletesingularity surface in the Cartesian space.(a) General Case: Arbitrary Orientation of the PlatformLet di, i = 1,2,3, be the three intersection lines between theplanes ?iand the (x,y)-plane of the base. We can define thepointP?suchthatP? d1d2d3. WealsodefinethepointsP?i5and C?as the projections of the points Pi5and C onto the(x,y)-plane. The position of the points P?i5, denoted by thevector p?i5= pi5xpi5y0T, depends on the position vector at Tsinghua University on January 8, 2012Downloaded from Monsarrat and Gosselin / Parallel Platform Mechanism321p and orientation matrix Q of the mobile platform. Then, wedetermine the equations of the three lines dithat pass throughthe points Pi0and P?i5. These equations can be expressed inthe following vector form, i = 1,2,3:?uv?=?pi0xpi0y?+ ?pi0x pi5xpi0y pi5y?, R,(37)where u vTdescribes the position of an arbitrary point thatbelongs to the line diand is located in the (x,y)-plane andpi0xand pi0ydenote the x and y coordinates of point Pi0,i = 1,2,3, respectively. Eliminating parameter in eq. (37),one obtains, i = 1,2,3,?pi0ypi5y?u + (pi5xpi0x)v+?pi0xpi5ypi0ypi5x?= 0.(38)Then, we can mathematically express the fact that these threelines all intersect a common point. The occurrence of sucha configuration can be expressed by forming the followingdeterminant whose elements are the coefficients of eq. (38).This determinant must vanish; that is,?p10y p15yp20y p25yp15x p10xp25x p20x(p10xp15y p10yp15x)(p20xp25y p20yp25x)p30y p35yp35x p30x(p30xp35y p30yp35x)?= 0.(39)This equation describes the locus of the configurations of themanipulator in which condition 5a is satisfied. After substi-tuting the elements of vectors pi0and pi5in eq. (39), oneobtains the closed-form equation describing the singularitylocus of the manipulator located in the (x,y)-plane. We ob-tain in fact an equation that contains terms up to the secondpower in x and y. After simplification, this equation reducesto the following form:E1x2+ E2y2+ E3xy + E4x + E5y + E6= 0,(40)where the coefficients Ei, i = 1,. ,6, are given in theappendix.Thelocusofsingularitiescorrespondingtothegenerallin-ear complex (condition 5a) is therefore represented in theCartesian space by the set of equations?E1x2+ E2y2+ E3xy + E4x + E5y + E6= 0z = z.(41)This represents, for a given orientation of the mobile plat-form, the equation of a quadratic surface whose coefficientsEidepend on the geometric parameters b and r of the ma-nipulator and on the orientation Q of the mobile platform.Figure 5 represents a top view of the three-leg mechanismFig. 5. Top view of the three-leg parallel platform mech-anism in a configuration satisfying condition 5a, (b,r) =(0.207m,0.300m), (,) = (10,30,30).in a configuration such that the curve given by eq. (40) is anellipse.Then, it is well-known that the nature of that surface de-pends on the following quantity: = E1E2E234.(42)If = 1, the surface is a circular cylinder.If 0, the surface is an elliptic cylinder.If = 0, the surface is a parabolic cylinder.If 0(48)?i5 ?i2 Mi ?i5+ ?i1,(49)where the Jacobian matrix J is determined in Section 3 andthe necessary leg lengths Miare given, i = 1,2,3, byMi= ?pi5 pi0? = ?p + Qbi pi0?.(50)The resulting constant-orientation workspaces located in-side the region that is free of platform singularities are rep-resented in Figures 6, 7, and 8 for different orientations at Tsinghua University on January 8, 2012Downloaded from Monsarrat and Gosselin / Parallel Platform Mechanism323Fig.6. (a)Perspectiveand(b)topviewsoftheregionoftheconstant-orientationworkspacelocatedinsidethesurfacesatisfyingcondition 5a, = 2.382, (b,r) = (0.207m,0.300m), (,) = (10,30,30). at Tsinghua University on January 8, 2012Downloaded from 324THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2001Fig.7. (a)Perspectiveand(b)topviewsoftheregionoftheconstant-orientationworkspacelocatedinsidethesurfacesatisfyingcondition 5a, = 0.048, (b,r) = (0.207m,0.400m), (,) = (10,0,50). at Tsinghua University on January 8, 2012Downloaded from Monsarrat and Gosselin / Parallel Platform Mechanism325Fig.8. (a)Perspectiveand(b)topviewsoftheregionoftheconstant-orientationworkspacelocatedinsidethesurfacesatisfyingcondition 5a, = 1.000, (b,r) = (0.207m,0.300m), (,) = (10,0,0). at Tsinghua University on January 8, 2012Downloaded from 326THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2001(,) and values of parameter r. Moreover, the curvecorresponding to eq. (40) is superimposed on the resultingworkspace. For visualization purposes, only the correspond-ing curve located in the (x,y)-plane was represented. Note,correspondingly, that there is a complete correlation betweenthe results obtained with the geometric method based on theGrassmann line analysis and those obtained with the dis-cretization algorithm.Finally, a few cross sections of the constant-orientationworkspace superimposed on the singularity loci correspond-ing to condition 5a for different values of coordinate z are ob-tained as shown in Figure 9. The sections of the workspace,locatedinplanesparalleltothe(x,y)-plane,weredeterminedgeometrically through the multistep algorithm first presentedby Gosselin (1990). The algorithm is based on the geometricdescriptionoftheconstant-orientationworkspaceofthethree-leg six-degree-of-freedom parallel mechanism, which wasshown to be obtained from the intersection of three regions,eachbeingthedifferencebetweentwoconcentricspheres(seeSection 3.3). The two-dimensional graphical representationsaregivenfordifferentorientationsofthemobileplatformandfor different values of the geometric parameter r.In light of the above results, one can visualize how thecorresponding singularity surface has a critical impact on theset of points that are reachable in practice by the end effector.Thus, the prediction of such singular configurations is to betaken into account during the design stage of this type ofparallel architecture.6. ConclusionIn this paper, a complete kinematic analysis of a new classof six-degree-of-freedom parallel platform manipulator withthree legs using five-bar linkages was presented. First, a gen-eral expression of the Jacobian matrix was obtained using theprinciple of virtual work, and the singularities of type I wererecalled. TheresultingPlckervectorsassociatedwiththesixinputangleswerederivedforthecaseinwhichforeachleg,theangles i2and i3are actuated. Subsequently, the linear de-pendenciesbetweenthecorrespondingGrassmannlineswerestudied for this case, and the singular configurations were de-scribed using simple geometric rules. It was shown that mostof the singular configurations of the three-leg six-degree-of-freedomparallelmanipulatorcanbereducedtothegenerationof a general linear complex. Then, closed-form expressionsdescribingthesingularitylocioftypeIIwereobtained. Itwasshown that for a given orientation of the mobile platform, thesingularity locus corresponding to the general linear complexis a quadratic surface (i.e., either a hyperbolic, a parabolic, oran elliptic cylinder) oriented along the z-axis. Additional sin-gularconfigurationsthatoccurwhentheequationsrelatingtheactiveandpassivejointvelocitiesarenotdefinedwerealsode-scribed. The corresponding singularity loci are three verticallinesthatpassthroughthecentersofthespheresthatconstitutethe boundary of the constant-orientation workspace. Finally,the corresponding expressions were used to graphically rep-resent the intersection between the singularity loci and theconstant-orientation workspace of the mechanism.A fewcross sections of the constant-orientation workspace super-imposed on the singularity loci corresponding to the generallinear complex (condition 5a) for different values of coordi-nate z were also given.In the context of singularity analysis of this new classof spatial parallel mechanisms, the geometric approach hasproved to be very efficient, allowing simultaneously (i) thegeometric characterization of all singularities of type II and(ii)thedeterminationoftheinherentclosed-formequationsofthe singularity loci in a very compact form. The closed-formequations and the graphical representations of the singular-ity loci have been shown to be very powerful design toolsthat can be of great help during the design process of parallelarchitectures of this new type.AppendixCoefficients Ei, i = 1,. ,6, of eq. (40):E1= 2rbcss+ 2rbscE2= 2rbscE3= 2rbcc 2rbcc+ 2rbsssE4=rb2s2ssc 2rb2c2ssc+rb2cc2s 2b2ccrsc+ r2bcss+rb2c2css+ r2bsc rb2ss2s2cr2scb + rb2cc2sE5=r2bsss rb2c2c2+ r2bccb2s2crc+ 2rb2sccss+ rb2s2c2rb2s2s2s2+ rb2c2cc r2bcc2b2cssrscE6=rb3s3c2c rb3s2cccss r3bscr3bsc+ r2b2sc2c+ r2b2c2ssc+r2b2c2ssc r2b2s2cssr2b2s2css rb3sc2c2c+2r2b2sccc+ r2b2