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Comparison of Two Kinds of Large DisplacementPrecision Parallel Mechanisms for Micro/nanoPositioning ApplicationsYuan Yun and Yangmin LiDepartment of Electromechanical Engineering, Faculty of Science and TechnologyUniversity of MacauAv. Padre Tomás Pereira, Taipa, Macau, [email protected], [email protected]— This paper presents kinematic analysis of two kindsof large displacement parallel platforms for micro/nano positioning applications. The kinematics model of the dual parallelmechanism systems is established via the stiffness model ofindividual wide-range flexure hinge. The displacements of theend platform and the input parameters of prismatic actuatorsare discussed and the corrected values of input motions areproposed on some checking points in workspace referring to thereal parameters of two kinds of In-Plane parallel mechanisms.The FEA models are established by ANSYS software, both thetheoretical analysis and FEA simulation results are presentedand compared. The investigations of this paper will providesuggestions to improve the structure optimization for a class ofparallel mechanism in order to achieve such features as largerworkspace and higher motion precision.Index Terms— In-plane parallel mechanism, Flexure hinge,Stiffness matrix, FEA model.I. I NTRODUCTIONParallel manipulators offer the advantages of high stiffness,low inertia, and high speed capability at the expense of smallerworkspace, more complex mechanical design, difficult directkinematics and complicated control algorithms. The errors ofparallel manipulators are distributed averagely in the serialchains while the errors of serial manipulator are accumulated,therefore parallel manipulators have been intensively investigated by both industry and academic fields in recent years.Micro/nano positioning devices are increasingly being madeof parallel manipulators due to their characteristics of highprecision. Some designers adopt the flexure hinges instead ofconventional mechanism joints since the backlash and frictionsin the conventional joints affect the performances of high precision parallel mechanisms remarkably. Although the adoptionof flexure hinges in the parallel mechanism systems increasesthe high precision, short stroke actuators with nanometer scalelevel precision result in a very small workspace. A spatialcompliant parallel robot with pseudo-elastic flexure hinges ispresented in [1]. The material of the flexure hinges is a kindof shape memory alloy (SMA) which offers elastic strains upto 17%. The robot has 3-DOF and a workspace larger than200 200 60 mm3 . Due to the lack of backlash, friction andstick-slip effects within the mechanism, the repeatability andc 2008 IEEE978–1–4244–1676–9/08/ 25.00 resolution should be better than 1µm and 0.1µm, respectively.A parallel structure for macro-micro systems is proposed in [2].In this new design, the macro-motion (DC motor) and micromotion (PZT actuator) are connected by a parallel structure,the two motions are coupled under one compliant mechanismframework. This kind of parallel structure eliminates the interface between the macro motion mechanism and the micromotion mechanism. At the same time, a kind of dual parallelmechanism is developed [3], called a 6-PSS parallel mechanism and a 6-SPS one, which is integrated with wide-rangeflexure hinges as passive joints to ensure the large workspace ofthe whole system and high precision motion. The experimentalprototype platform actuated by piezoelectric motors which canrealize a centimeter-scale stroke with positioning precisionbetter than 100nm and piezoelectric ceramics actuators witha high resolution about 1nm and non-backlash design ensuresnanometer scale precision over the cubic centimeter workspace[4], [5]. A XYZ-flexure parallel mechanism (FPM) is proposedwith large displacement and decoupled kinematics structure[6]. The large-displacement FPM has a large motion rangemore than 1mm. Moreover, the FPM can achieve decoupled X-,Y- and Z-axes translational motions with small cross-axis errorless than 1.9% and small parasitic rotation less than 1.5mrad.A variant of Stewart-Gogh Platform made of PSS-chains formicro-positioning applications is presented in [7], which givesthe stiffness analysis by comparing two platforms in terms ofPSS-platform and SPS-platform.Based on our extensive studies on various kinds of parallelmechanisms [8]- [17], this paper compares the precision analysis of two kinds of In-plane parallel platform for micro/nanopositioning applications. The nonlinearity of the wide-rangeflexure hinges cannot be ignored for a precision positioningsystem and must be compensated in the kinematics model. Sothe kinematics model of the dual parallel mechanism systemis established via the stiffness model of individual wide-rangeflexure hinge. Referring to the real parameters of these In-Planeparallel mechanisms, FEA model is established by ANSYSsoftware, both the theoretical analysis and FEA simulationresults are presented, which prove that the theoretical modelis correct. The constraint orientation workspaces of the twoRAM 2008

mechanisms are plotted and compared. Finally, the errors ofthe input motions of prismatic actuators between the two kindsof In-plane parallel platforms are discussed on some checkingpoints in workspace.II. S YSTEM D ESCRIPTIONThe parallel platform is a dual parallel mechanism combining a 6-PSS parallel mechanism with a 6-SPS one. In the 6-PSSparallel structure, the prismatic actuators provide macro motionwith micron level accuracy and cubic centimeter workspace. Atthe same time, the micro motion is provided by 6-SPS structurewhich can increase the accuracy of the whole system to thenanometer level. The wide-range flexure hinge is a slendershaft configuration which is adopted as passive joint to ensurethe large workspace of the whole system with high precisionmotion. The schematic of the parallel platform with arbitraryprismatic joint orientation is shown in Fig. 1. A referenceframe o-xyz is attached to the fixed platform at the center o.And local coordinate system o’-x’y’z’ is attached to the movingplatform at the center o’. The length of each limb is Li . Let thevector of the geometry center point of moving platform in theo-xyz coordinate system be rc . is the transformation matrixfrom o’-x’y’z’ coordinate system to o-xyz. riL is the vector ofpoint Mi (i 1,2,· · · ,6) in the local coordinate system. ri is thevector of point Mi (i 1,2,· · · ,6) in o-xyz coordinate system.rbi is the vector of point Bi (i 1,2,· · · ,6) in o-xyz coordinatesystem.Let Tri be the transformation matrix of the ith limb from oi xi yi zi coordinate system to O-XYZ. Kbi , Kmi and Ksi are thestiffness matrices of the flexure hinge connecting to the fixedplatform, the flexure hinge connecting to the moving platformand the PZT strut in oi -xi yi zi coordinate system respectively.pbi , p1i , p2i and pmi are nodal load vectors of these nodesshown in Fig. 6 in O-XYZ coordinate system. dbi , d1i , d2iand dmi are nodal displacement vector in O-XYZ coordinatesystem. Divide the stiffness matrices into 2 2 sub-matrices: Kbi11 Kbi12Kbi (1)Kbi21 Kbi22The stiffness matrix of the whole single limb in oi -xi yi ziFig. 2.Single limb of the parallel mechanism.coordinate system can be achieved by assembling the nodalstiffness matrices: Kbi11Kbi1200 Kbi21 Kbi22 Ksi11 Ksi120 K i Ksi22 Kmi11 Kmi12 Ksi2100Kmi21Kmi22The Lagrange equation of one singleas: pbidbi p1i d1iT p2i Tri · Ki · Tri · d2ipmidmiFig. 1.The parallel platform with arbitrary prismatic joint orientation.III. K INEMATICS M ODELA. 6-PSS Kinematics Model Based on Stiffness EquationSince the large deformation of the wide-range flexure hingescannot be ignored for a micro/nano positioning applications,the kinematics model is established according to [3]. The widerange flexure hinges and the PZT struts are analyzed as beamelements based on FEM theory. The coordinate systems areestablished on a single limb of the parallel mechanism asshown in Fig. 2. The single limb is divided into four parts.Letlimb can be formulated d̈bi M · d̈1i d̈2i d̈miT Ki T ri · Ki · T ripbidbi p1i d1i pi p2i , di d2ipmidmi(2) (3)Eq. (2) can be written as:pi Ki · di M · d̈i(4)In Eq. (4), the acceleration term can be ignored sincethe system always works in a low acceleration and uniformmotion environment. In addition, let P be the external load

vector acting on the moving platform, the force and momentequilibrium equation can be written as:K(n) τ6pmi 0P (5)i 1dbi is zero vector in terms of solving the deformable errorof the moving platform. d1i and d2i are unknown vectors,and dmi (i 1,· · · ,6) have six unknown elements related to thedisplacement of the moving platform. p1i and p2i are zerovectors because of no load acting on node 1 and 2, Eq. (4)can be written as: dbi d1i K1ip1i (6) · d2i p2iK2idmi Kbi K1i Ki K2i KmiwhereB. Geometric Nonlinearity of the Wide-range Flexure HingesSince the wide-range flexure hinges are adopted in thisparallel mechanism, the system can not be expressed by alinear equation exactly. The stiffness matrix and nodal loadis a function of nodal position. Eq. (4) can be written as:pi (di ) Ki (di ) · di M · d̈i φ(di )(8)where φ(di ) is the imbalance load vector. If d i is an accuratesolution of Eq. (8), we can obtain: pi (d i ) Ki (d i ) · d i M · d̈i φ(d i ) 0(n) di(n) (n)(n) pi (di ) (n) di(n) pi (di(n)where Fi (di ) Ki ·di , andsystem. Eq. (8) can be written as:)(n) di(n) Fi (di )(n) di 0 in conservative(n)K(n) φ(di )τ · di(n 1)(n)(n) di didi(11)K(n)is the tangential stiffness matrix of each limb. It isτobviously that the tangential stiffness matrices of each limbhave to be updated in order to converge faster to a solution ineach sub-step of the iterated algorithm. It is a very huge amountof calculation for a multi-body parallel system. Therefore, akind of Modified Newton-Rephson method which uses theoriginal tangential stiffness matrix instead of renewal one isadopted to save some time for computations.φ(di )(n) φ(di)After the 6-PSS macro motion, the 6-SPS motion is designed to adjust the moving platform in a micro space. The 6SPS can be analyzed by the pseudo-rigid-body model (PRBM).In order to simplify the analysis and design, the PRB modelis always proposed for modeling the kinematics input-outputbehavior of flexure mechanism by expressing a methodologyof treating each flexure hinge as a revolute joint with a torsionspring. The model makes analysis of mechanisms with flexurehinges easier and faster.IV. S IMULATION R ESULTS AND A NALYSISA FEA model is established with the macro prismaticactuators self-locked as shown in Fig. 3 according to theparameters of the parallel mechanism shown in Table I.The wide-range flexure hinges and struts are meshed withBEAM188 and the material is beryllium bronze and hardaluminum. The moving platform is meshed with SOLID92and the material is hard aluminum. The external load is 27Nalong -z axes on the moving platform.The prismatic actuators directions of the two kinds ofTABLE IG EOMETRIC AND M ATERIAL PARAMETERS(9)Therefore, the Newton-Raphson method is utilized, which usesan iterative process to approach one root of a function. Thespecific root that the process locates depends on the initial,arbitrarily chosen di -value.(n 1)(n) di(n) C. 6-SPS Kinematics Model(7)From Eq. (6) for each limb and Eq. (5) that have 78 equationsand unknown elements, we can solve the deformation of themoving platform.When we solve the correcting values of drivers, each dbihas one unknown element generated by actuator. d1i and d2iare unknown vectors too, and dmi can be solved by thedisplacement of the moving platform. And we can solve thecorrected values of actuators from Eq. (6) for each limb andEq. (5) too.di(n)(n) φ(di )(10)(n) diItemRadius of moving platform rRadius of fixed platform RAngle of short side ratio αRadius of flexure hinge rfLength of flexure hinge lfRadius of strut rsLength of strut lsModulus of elasticity of flexure hinge EfModulus of elasticity of strut EsThe acceleration term can be ignored in static analysis. Assume(n) di(n) φ(di )(n) φ(di(n) di)parallel mechanisms are set to:a1 a(1)2 3a(1)20TValue20mm90mm50 0.9mm12mm4mm76mm130GPa70GPa

A. Constraint Orientation Workspace(a)This kind of 6-PSS parallel mechanism as shown in Fig.4(a) is a special case with all actuators lying on xy plane andeach two of them on one guiding-rail. The errors of movingplatform may be decreased through reducing the amount ofmounting parts which may introduce more installation errors.The other one is shown in Fig. 4(b) with an In-plane orienta tions of the actuators along OBi . This kind of platforms havesuperior stiffness values, whereas has very different workspaceshape. If the strokes of the prismatic actuators are -5mm 5mm, the constraint orientation workspaces are solved bythe nonlinear inverse kinematic model as shown in Fig. 5. Itis obvious that the structure (b) has a larger workspace than(a) at the same strokes.(b) 3x 104(c)2a2 a(2)2a4 a6 a(5)2 a(6)2 sin( α2 β)b(2)b2 sin( β2 )b(3)b3 b4 b5 sin( β2 )b(4)sin( α2 β)b(5)b6 sin( α2 )b(6)3a(2)20a(4) sin( α2 )b(1)b1 a(3)a3 a5 00 0 3a(5)23a(6)2x(m)FEA model developed in ANSYS.T00 2 44 T20 3x 10 T00T 242 3x 10z(m) 3x 105 T0cos( α2 β)b(2) cos( β2 )b(3)0 T00cos( α2 β)b(5)cos( α2 )b(6) 4 40Tcos( α2 )b(1) cos( β2 )b(4) 2y(m)0x(m)Fig. 3. T0 55 3x 10 T0 Ty(m) T500 5 5 3x 10z(m)Fig. 5. Constraint orientation workspaces of the macro motions of these twokinds of parallel mechanisms.B. Simulation Results of Deformation(a)Fig. 4.(b)The prismatic actuators configurations.According to section II, the kinematics models based onthe stiffness matrices are tested by ANSYS. The z-axes deformations of the moving platform are plotted in case that theinitial position height is z 71.03mm as shown in Fig. 6. Oneof the ANSYS results is shown in Fig. 7, which has a 27Nload along -z axes acted on moving platform.The figures indicate that two lines are linear and almost thesame with the errors under 0.0875% between the two models.Therefore, the nonlinear results can be tested in ANSYS withthe large displacement static analysis.

Deformation of the movingDeformation of the movingplatform in theoretical model (m) platform in ANSYS (m)TABLE IIC ORRECTED VALUES AND I NPUT M OTIONS OF ACTUATORS 61.5x 101Displacements(mm )[0 0 2 0 0 0]0.500510152025Load along z axes (N)30 61.5x 10[0 0 -2 0 00]10.500510152025Load along z axes (N)30[0 0 1 0 0 0]Fig. 6. Simulation results of deformation solved by ANSYS and theoreticalmodel.[0 2 0 1 0 0][0 0 0 3 3 3][2 2 0 0 0 0]Fig. 7.Corrected Values and Input Motions(mm)(a)(b)Corrected (-0.2531 -0.2531 (-0.0714 -0.0714values-0.2531 -0.2531 - -0.0714 -0.0714 0.2531 -0.2531)0.0714 -0.0714)Input(-3.7302 -3.7302 (-2.1513 -2.1513motions-3.7302 -3.7302 - -2.1513 -2.1513 3.7302 -3.7302)2.1513 -2.1513)Corrected (-0.1545 -0.1545 (-0.0588 -0.0588values-0.1545 -0.1545 - -0.0588 -0.0588 0.1545 -0.1545)0.0588 -0.0588)Input(2.9764 2.9764 (1.9060 1.9060motions2.97642.9764 1.90601.90602.9764 2.9764)1.9060 1.9060)Corrected (-0.0559 -0.0559 (-0.0178 -0.0178values-0.0559 -0.0559 - -0.0178 -0.0178 0.0559 -0.0559)0.0178 -0.0178)Input(-1.7460 -1.7460 (-1.0427 -1.0427motions-1.7460 -1.7460 - -1.0427 -1.0427 1.7460 -1.7460)1.0427 -1.0427)Corrected (-0.0392 -0.0335 (-0.0022 -0.0285values-0.1460 -0.2011 - -0.0058 -0.0180 0.0523 -0.0213)0.0517 -0.0110)Input(2.8105 -0.3779 - (1.7698 -0.2460 motions2.9653 -2.8078 - 1.6726 -1.5408 0.6667 3.0984)0.4440 1.9080)Corrected (-0.1665 -0.1612 (-0.1019 -0.0512values-0.0165 -0.1082 - -0.0055 -0.0730 0.1542 -0.0660)0.0804 -0.0354)Input(-3.1257 -1.8392 (-1.9134 -1.0484motions0.37712.3686 0.24121.44640.8943 -0.1796)0.5805 -0.1038)Corrected (-0.0152 -0.0254 (-0.0529 -0.0297values-0.1725 -0.1633 - -0.0042 -0.0585 0.2082 -0.2063)0.0403 -0.0091)Input(1.7572 -3.7171 (0.9955 -2.3098motions-4.8795 -0.8703 -2.8157 -0.46722.3653 3.8314)1.6372 2.5943)Simulation result: 27N load along -z axes on moving platform.C. Simulation Results of Drive ParametersIn order to improve the control accuracy, the inputparameters of prismatic actuators have to be corrected bythe inverse solution of the kinematics model based on thestiffness equations.In Table II, the input motions of actuators are given bytheoretical model and the corrected values of the initial rigidmodel are listed comparing with the rigid body control. Itindicates that the correct values of the checking points aremillimeter-scale and gradually increase when the movingplatform departures from initial position that can not beignored in the design of control algorithm for a micro/nanopositioning applications. Meanwhile, the correct values ofstructure (b) are very small compared with (a) and the resultshowed that the nonlinear effect of structure (a) is gettingmore serious than that of (b).The corrected input motions can be tested in ANSYSwith the large displacement static analysis. The six chains areconstrained via inputting parameters of actuators and fixedall rotation DOF. The wide-range hinges and the movingplatform are connected by CONTA175 and TARGE170elements instead of using CP command to express rigidconstraints as the CP command is invalid in large-deflectioneffects static analysis. One of the ANSYS results is shownin Fig. 8. The max error ratios of the moving platform areshown in Table III to verify the nonlinear inverse kinematicsimulation results.It can be found that the errors of the checking pointsbetween the two kinds of parallel mechanisms are under 10%,when the moving platform departures from initial position,the error ratios are slightly increased which may be producedby the simplified theoretical model. Compared with structure(a), the structure (b) has larger displacement under the sameinput motions and the z-axes motions are with higher accuracybut the xy-plane motions and rotated motions are with lessaccuracy than (a). The configuration of the prismatic actuatorsis necessary to be optimized for both larger workspace andhigher motion precision.

TABLE IIIV ERIFICATION OF I NVERSE K INEMATIC S IMULATION R ESULTSDisplacements(mm )[0 0 2 0 0 0][0 0 -2 0 0 0][0 0 1 0 0 0][0 2 0 1 0 0][0 0 0 3 3 3][2 2 0 0 0 0]Max Error Ratios Simulated by %7.07%9.49%4.01%7.41%improve the structure and control algorithm optimization fora class of parallel mechanism in order to ensure both largerworkspace and higher motion precision. The results will beuseful in modifying the structure of the platform with highdynamic properties.ACKNOWLEDGMENTThis work was supported by the Research Committee of University of Macau under Grant RG065/0607S/08T/LYM/FST and Macao Science and Technology Development Fund under Grant 069/2005/A.R EFERENCESFig. 8.Simulation result of large displacement static analysis by ANSYS.V. C ONCLUSIONSIn this paper, we present precision analysis of two kindsof parallel platform for micro/nano positioning applications.The kinematics model of one class of dual parallel mechanismsystem is established via the stiffness model of individualwide-range flexure hinges. Referring to the real parametersof an In-Plane parallel mechanism, the constraint orientationworkspaces of the macro motions of these two kinds of parallelmechanisms are meshed. FEA model is established in ANSYS,both the theoretical analysis and FEA deformation results arepresented in case that the initial position height is z 71.03mm,which means that the theoretical model is correct. And theinput parameters of prismatic actuators are discussed and thecorrected values are proposed on some checking points inworkspace. We have found that the structure (b) has a largerworkspace than (a) at the same strokes and the correct values ofthe checking points are millimeter-scale and gradually increasewhen the moving platform departures from initial position thatcan extremely influence the control accuracy. Meanwhile, thecorrect values showed that the nonlinear effect of structure (a)is getting more serious than that of (b). The inverse kinematicsimulation results are verified by ANSYS, and the errors of thechecking points between the two kinds of parallel mechanismsare under 10%. Compared with structure (a), the structure (b)has larger displacement under the same input motions andthe z-axes motions are with higher accuracy but the xy-planemotions and rotated motions are with less accuracy than (a).The investigations of this paper will provide suggestions to[1] J. Hesselbach, A. Raatz, J. 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Comparison of Two Kinds of Large Displacement Precision Parallel Mechanisms for Micro/nano . rbi is the vector of point Bi . platform, the ﬂexure hinge connecting to the moving platform and the PZT str