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Int 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 PointsErrorZ Axis Displacement Error (mm)Roll Angle Error (Deg)Pitch Angle Error (Deg)Fig. 18 Displacement androtational error analysis199the calibration results, the errors of the rotational angle andposition increase when the platform has reached themaximum rotational angle, which is the boundary of thecalculated workspace.9 ConclusionsThis paper addresses the kinematics analysis of the hybridPKM that has been designed and developed in this researchto perform translation along the zaxis and rotation aboutthe x and yaxes. Modular configurations of the variousPKMs are successfully assembled using the same modularunits. An algorithm for the optimal design of the hybridPKM has been presented. In particular, the proposed for-mulation represents an integration of the relevant aspects ofthe dimension
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