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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