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1、IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,VOL.60,NO.10,OCTOBER 20134613Optimal Design,Fabrication,and Controlof anXYMicropositioning Stage Drivenby Electromagnetic ActuatorsShunli Xiao and Yangmin Li,Senior Member,IEEEAbstractThis paper presents the optimal design,fabrication,and control of a nove
2、l compliant flexure-based totally decoupledXY micropositioning stage driven by electromagnetic actuators.The stage is constructed with a simple structure by employingdouble four-bar parallelogram flexures and four noncontact typesof electromagnetic actuators to realize the kinematic decouplingand fo
3、rce decoupling,respectively.The kinematics and dynam-ics modeling of the stage are conducted by resorting to com-pliance and stiffness analysis based on matrix method,and theparameters are obtained by multiobjective genetic algorithm(GA)optimization method.The analytical models for electromagneticfo
4、rces are also established,and both mechanical structure andelectromagnetic models are validated by finite-element analysisvia ANSYS software.It is found that the system is with hysteresisand nonlinear characteristics when a preliminary open-loop test isconducted;thereafter,a simple PID controller is
5、 applied.There-fore,an inverse Preisach model-based feedforward sliding-modecontroller is exploited to control the micromanipulator system.Experiments show that the moving range can achieve 1 mm 1 mm and the resolution can reach 0.4 m.Moreover,thedesigned micromanipulator can bear a heavy load becau
6、se of itsoptimal mechanical structure.Index TermsElectromagnetic actuators,hysteresis,micro-/nanopositioning,sliding-mode control.I.INTRODUCTIONCOMPLIANT positioning stages with ultrahigh precisionplay more and more important roles in such a situationwhere a high-resolution motion over a micro-/nano
7、range isexpected in the cases of microelectromechanical systems,opti-cal fiber alignment,biological cell manipulation,and scanningprobe microscope.Compared with conventional mechanicaljoints,flexure hinges can provide more ideal high-precisionManuscript received October 30,2011;revised March 2,2012
8、and May 6,2012;accepted June 18,2012.Date of publication July 19,2012;date ofcurrent version May 16,2013.This work was supported in part by the NationalNatural Science Foundation of China under Grant 61128008,by the MacaoScience and Technology Development Fund under Grant 016/2008/A1,bythe Research
9、Committee of the University of Macau under Grant MYRG203(Y1-L4)-FST11-LYM,and by the State Key Laboratory of Robotics ofShenyang Institute of Automation under Grant O8A120S.S.Xiao is with the Department of Electromechanical Engineering,Facultyof Science and Technology,University of Macau,Taipa,Macau
10、(e-mail:yb07408umac.mo).Y.Li is with the Department of Electromechanical Engineering,Faculty ofScience and Technology,University of Macau,Taipa,Macau,and also withTianjin Key Laboratory for Control Theory and Applications in ComplicatedSystems,Tianjin University of Technology,Tianjin 300384,China(e-
11、mail:ymliumac.mo).Color versions of one or more of the figures in this paper are available onlineat http:/ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIE.2012.2209613motions since they have advantages,including no backlash,nofriction losses,no need for lubrication,vacuum compatibility,and
12、ease of fabrication and assembly.Hence,many compliantmechanisms adopt flexural hinges to realize precise micro-/nanopositioning tasks.Due to the advantages of the compliant parallel stages withhigh bearing load capacity,high accuracy,low inertia,and com-pact size,many parallel stages are proposed fo
13、r the pertinentapplications 15.In these structures,the closed-loop natureof the parallel mechanism limits the motion of the platform andcreates complex kinematic singularities inside the work space.Because of the limited work space along with the singularitiesand voids inside it,trajectory planning
14、and control are difficultto realize.With respect to simplifying the control strategies,there are many studies concentrated on decoupled parallelmicro/nano operational stages 6,in which the stages aremostly driven by piezoelectric actuators(PZT).Consideringthat the PZT cannot bear shearing force and
15、usually can onlyoffertensofmicrometersofstrokes,aforcedecoupleddisplace-ment amplifier is required for those micro-/nanopositioningstages.However,these motion decoupled structures and ad-ditional force decoupled displacement amplifiers really com-plicate the structure because of the very limited wor
16、k spaceavailable 7.Aside from PZT actuators,there are many otherdriving technologies,such as shape memory alloy actuator,thermal actuator,and magnetostrictive actuator 8,which alsoencounter the similar problems as when using PZT.For such practical applications such as IC assembly andbiological cell
17、manipulation,a large motion range of the stageis needed for effective operation.In order to make a balance onlarge motion range,high resolution in positioning,and simplecontroller design,considering the small payload of the maglevsystem in the normal gravity environment,we need not onlyto construct
18、a kinematic decoupled structure with compliantflexure hinges but also to eliminate force contact by usingnoncontactorafrictionlessforcegeneratedbetweenthemovingplatform and the stator.Several kinds of actuators can real-ize noncontact or frictionless forces,such as electromagneticforce and air beari
19、ngs,but air bearings are not suitable for avacuum environment.There are many advantages associatedwith electromagnetic actuators in terms of no contamination,nofriction,fast response,large travel range,and low cost 9,10.Moreover,comparing with the maglev micropositioning sys-tems 11,12,the electroma
20、gnetic driven micromanipulatorwith flexure-based mechanism can achieve a large load capa-bility and a good positioning precision.0278-0046/$31.00 2012 IEEEAuthorized licensed use limited to:Shanghai University of Engineering Science.Downloaded on April 04,2022 at 11:27:36 UTC from IEEE Xplore.Restri
21、ctions apply.4614IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,VOL.60,NO.10,OCTOBER 2013Fig.1.Schematic diagrams of decoupled XY stages.Fig.2.Typical applications of parallelogram flexures.II.DESIGN OF ADECOUPLEDXY COMPLIANTSTAGEIn order to construct a decoupled XY compliant stage,themechanical scheme
22、 is designed as shown in Fig.1(a)and(b),and a simple mechanism with sliding pairs is employed dueto its simple structure,where two configurations are producedaccording to the assembled sequence of the sliding-pair ele-ments.It is noticeable that the sliding pairs in a common mech-anism are not suita
23、ble for application in micro-/nanodevices.A compliant mechanism with flexure hinges can be used toreplace the rigid-body mechanism with sliding pairs.Similarto the design of a rigid-body manipulator based on four slidingpairs,a compliant mechanism with four parallelogram flexurescan be designed as s
24、hown in Fig.2(a)and(b),respectively.Inview of a heavier capability in payload and smaller total size,a symmetrical structure of the stage is designed as shown inFig.3(a)and(b).In Fig.3(a),the mobile platform is arrangedinside the stage,while in Fig.3(b),the mobile platform isconstructed as an outsid
25、e square frame.If a force is exerted onthe mobile platform in the x-or y-direction,it will move alongone direction without any interference from the other direction.A decoupled electromagnetic force actuator assembled witha perpendicular motion is used to drive the mobile platform.To achieve a symme
26、trical structure and a simple controller,single-coil electromagnet is adopted;meanwhile,two single-coil electromagnets are arranged opposite each other on the twosides of the stage in the x-or y-direction,respectively,as shownin Fig.4.The armatures are fixed onto the mobile platform,andthe electroma
27、gnetic actuators are fixed onto the fixed platform;then,a noncontact force can be exerted on the mobile platformto form a force decoupled structure.Moreover,the noncontactlaser sensors are used to measure the displacements,whichcan also avoid any contacts between the mobile platform andfixed platfor
28、m.The electromagnets have very simple structures,which can be purchased or fabricated easily.After all the struc-ture scheme is carefully designed,a 3-D assembly schematicof the stage is expressed in Fig.5.The mechanical structurecan be fabricated monolithically by wire electrodischarge ma-chining.A
29、luminum alloy material(AL7075-T651)with a highy/E ratio is selected as the material to build up the stageFig.3.Conceptual design.Fig.4.Symmetrical XY decoupled stage driven by electromagnets.so that a large deflection is allowed.If proper dimension isselected,the motion range can reach 1 mm 1 mm wit
30、hultrahigh precision,which will have wide applications in manymicro-/nanofields.Authorized licensed use limited to:Shanghai University of Engineering Science.Downloaded on April 04,2022 at 11:27:36 UTC from IEEE Xplore.Restrictions apply.XIAO AND LI:OPTIMAL DESIGN,FABRICATION,AND CONTROL OF XY MICRO
31、POSITIONING STAGE4615Fig.5.Three-dimensional mechanical design of the manipulator.III.COMPLIANCE ANDSTIFFNESS OF THEXY STAGEVarious methods can be adopted in modeling the compliantmechanisms 7,13.One of those typical methods is thepseudo-rigid-body method,which is based upon simplificationanalysis,a
32、nd only the deformation around one axis is consid-ered.However,other members of the mechanism except thehinges are assumed as rigid bodies,and their deformations areneglected.The compliance CD1is defined as the compliance ofpoint O1with respect to the ground point D.All the limbs andhinges are conne
33、cted in serial or parallel,as shown in Fig.3(b).For the purpose of calculating the compliance of the stage,thecompliance of every flexure hinge with respect to the groundCDican be derived byCDi=TDiCi?TDi?T(1)where the transformation matrix takes on the following form:Tji=?RjiS?rji?Rji0Rji?.(2)where
34、Rjiis the rotation matrix of coordinate Oiwith respect toOj,rjiis the position vector of point Oiexpressed in referenceframe Oj,and S(r)represents the skew-symmetric operator fora vector r=rx,ry,rzTwith the notationS(r)=0rzryrz0rxryrx0.(3)Considering the symmetrical structure and arrangement ofthe a
35、ctuators in the stage,we can assume that both the inputforce and output force will pass through the central pointA of the mobile platform so as to simplify the calculation.The virtual conceptual point A and point D will be at thesame place when no forces are applied to the XY stage orthose forces do
36、 not result in any deformations.For calculatingthose compliances of each limb,the stiffness KL1=CL11is defined as the compliance of the limb from O1 O2withrespect to D.Similarly,we can obtain the stiffness KL2,KL3,KL4,KL5,KL6,KL7,and KL8as follows:KL1=CL11=?CD1+CD2?1KL2=CL21=?CD5+CD6?1KL3=CL31=?CD9+
37、CD10?1KL4=CL41=?CD13+CD14?1KL5=CL51=?CD3+CD4?1KL6=CL61=?CD7+CD8?1KL7=CL71=?CD11+CD12?1KL8=CL81=?CD15+CD16?1.(4)It can be observed that the compliance CDAcan be describedas four parallel limbs(CL1234=CL1/CL2/CL3/CL4)connected serially with other four parallel limbs(CL5678=CL5/CL6/CL7/CL8).The complia
38、nce of input point Awith respect to the ground point D can be derived byCDA=CL1234+CL5678(5)whereCL1234=KL1+KL2+KL3+KL41CL5678=KL5+KL6+KL7+KL81.(6)According to the simple structure,the input compliance isapproximately equal to the output compliance.IV.DYNAMICMODELING ANDANALYSISLagranges method is e
39、mployed for the dynamics modelingof the stage 13.Variable d=d1,d2Tdenotes the displace-ment along the x-or y-axis,respectively so that the kineticand potential energies of the compliant mechanism can beexpressed by generalized coordinates.It is assumed that thekinetic energies are generated from the
40、 rigid links connectingwith flexure hinges;hence,the potential energies are introducedby the elastic deformations of flexure hinges.Since the stageis horizontally placed,the gravity potential energy remains thesame during the motion,so it will not be taken into account.The kinetic energy of the stag
41、e is derived byT=12ma?12d1?24+12?112?mal2?d1l?24+12mb(d1)2+12mc(d1)24+12mc?12d2?24+12?112?mcl2?d2l?24+12md?(d1)2+(d2)2?(7)where ma,mb,mc,and mdare the masses of the limb a,the middle platform b,the limb c,and the mobile platform d,respectively.Authorized licensed use limited to:Shanghai University o
42、f Engineering Science.Downloaded on April 04,2022 at 11:27:36 UTC from IEEE Xplore.Restrictions apply.4616IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,VOL.60,NO.10,OCTOBER 2013The potential energy of the whole mechanism can bederived byV=12k1d21+12k2d22(8)where k1or k2is the input stiffness in the x-
43、or y-direction,respectively.SubstitutingthekineticandpotentialenergiesintoLagranges equation leads toddtTdiTdi+Vdi=Fi(9)where Fi(i=1,2);Fi(i=1)is the input actuation force,andFi(i=2)is the output force,respectively.The dynamics equation of the undamped free vibration of thecompliant mechanism system
44、 can be derived byMd+Kd=0(10)whereM=?23ma+12mb+2mc+12md0023mc+12md?(11)is the mass matrix andK=diagk1,k222(12)is the stiffness matrix.Based on the theory of vibrations,the modal equation de-scribing the free vibration of the system can be obtained by?K M2i?i=0(13)where 2iand iare the eigenvalue and
45、eigenvector related tothe ith mode shape of the mechanism,respectively.Solving the following characteristic equation?K M2i?=0(14)leads to generation of the eigenvalue 2i.The fundamental natural frequency can be calculated byfi=i2.(15)V.SAFETYFACTOR OF THEPOSITIONINGSTAGEWith adenoting the allowable
46、stress of the material and yexpressing the yield strength of the material,the safety factorcan be written as na=y/a.It is obvious that the stress ismainly concentrated on the notched circular flexures becausethe rotations(r)and axial loads(t)of the flexure hingesremain within the allowable stress ao
47、f the materiala=yna=r+s.(16)First,concerning a notch hinge bearing a bending momentaround its rotational axis,the maximum angular displacementmaxarises when the maximum stress rmaxat the outermostsurface of the thinnest portion of the hinge reaches the allow-able stress a.Referring to 13,the relatio
48、nship between themaximum stress and maximum rotation of the flexure hinge canbe calculated byrmax=E(1+)9202f()max(17)where =t/2r is a dimensionless geometry factor with theaccuracy range of 0 0.3 and f()is a dimensionlesscompliance factor defined in 13 asf()=12+2?3+4+22(1+)(2+2)+6(1+)(2+2)32tan1?2+?
49、12?.(18)Second,the maximum tensile stress subject to the axialload may occur on the thinnest portions of the flexure hingesconnecting the displacement amplifiers or other links of thestage,which can be determined byt1=FinSmin=KinQwt(19)where Smindenotes the minimum cross-sectional area of thehinge.T
50、hen,the safety factor can be described by a function oft,R,input compliance CA,and input displacement Q,whichgives the relationship between the stiffness/compliance valuesand the architectural parameters of the gripper.VI.MODELING ANDANALYSIS OF THEELECTROMAGNETICNONCONTACTLINEARACTUATORThe electrom