XY SCANNING MECHANISM: A DYNAMIC APPROACH · drive stages, (2) clamp release inchworm-screw stages and (3) elastic deformation stages. Assumed Modes Method: Double Parallelogram Flexure

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Int. J. Mech. Eng. & Rob. Res. 2014 Sollapur Shrishail B and Deshmukh Suhas P, 2014

XY SCANNING MECHANISM:A DYNAMIC APPROACH

Sollapur Shrishail B1* and Deshmukh Suhas P1

*Corresponding Author: Sollapur Shrishail B,[email protected]

Flexure mechanisms have immense scope in their use for applications involving high precisionmotion. There are many concepts to build high speed or high precision manipulators, but only afew of them can serve to obtain high speed together with high precision. Mathematical modelingof simple XY manipulator is carried out XY manipulator uses typical Double Flexuralconfigurations. Static and dynamic analysis is carried out using MATLAB. Static analysis iscarried out to determine static deflection of motion stage with force. It is observed that forcedeflection curve is linear. Dynamic analysis is carried out to determine frequencies and modeshapes of flexural manipulator. Further, Finite Element software ANSYS is used to carry outstatic and dynamic analysis of basic DFM configuration and few XY mechanisms. It is observedthat close matching of FEM results with model developed.

Keywords: Flexural mechanism, FEA analysis

INTRODUCTIONSeveral publications talk about the pros andcons of flexure mechanisms and highlight thesignificance of their use in technology requiredto provide energy efficient, wear free, higherresolution and high speed devices. Flexureshave been used as bearings to providesmooth and guided motion, for example inprecision motion stages; as springs to providepreload, for example in the brushes of a DCmotor or a camera lens cap; to avoid over-constraint, as in the case of bellows or helical

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Int. J. Mech. Eng. & Rob. Res. 2014

1 Department of Mechanical Engineering, Sinhgad Academy of Engineering, Pune, India.

coupling; as clamping devices, for example,the collet of a lathe, for elastic averaging as ina windshield wiper, and for energy storage,such as, in a bow or a catapult. This listencompasses applications related to thetransmission of force, displacement as well asenergy, thereby making the versatility offlexures quite evident.

A historical background of flexures ispresented in several texts. Flexure design istraditionally based on creative thinking andengineering intuition, analytical tools can aid

Research Paper

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the design conception, evaluation andoptimization process. Consequently, asystematic study and modeling of thesedevices has been an active area of research.Some of the existing literature deals withprecision mechanisms that use flexures asreplacements for conventional hinges, thuseliminating friction and backlash. Analysis andsynthesis of these mechanisms is simply anextension of the theory that has already beendeveloped for rigid link mechanisms, exceptthat in this case the range of motion is typicallysmall. The key aspect of these mechanismsis flexure hinge design. Unlike these caseswhere compliance in the system is limited tothe hinges, other flexure mechanisms exist inwhich compliance is distributed over a largerpart of the entire topology. Both these kinds ofmechanisms offer a rich mine of innovative andelegant design solutions for a wide range ofapplications.

Any systematic flexure design exercise hasto be based on performance measures. Whiledetailed performance measures can be laidout depending on specific applications, ageneral set of measures are highlighted here.These measures are based on the deviationof flexures from ideal constraints.

One of the primary applications of flexuresis in the design of motion stages. This thesisstrives to bridge the gap between intuition andmathematical analysis in flexure mechanismdesign. Accordingly, the following list highlightsthe specific contributions of this thesis.

1. Dynamic Modeling of double flexuralmanipulator is carried to determine itsnatural modes and mode shapes usingassumed modes method.

2. Modal analysis and frequency response isdetermined for DFM considering actuatordynamics.

3. Static analysis is carried for DFM.

4. FEM analysis of DFM is carried out andcomparison of results of assumed modesmethod and ANSYS results is done.

High precision measuring technologiessuch as the Scanning Probe Microscope(SPM), Atomic Force Microscope (AFM),electron microscope, X-ray microscope, andconfocal scanning microscope are rapidlyadvancing via the development ofsemiconductor processing technologies andbiomedical technologies. Most of thesemicroscopes require a precise multi-axisscanner for sample scanning, and manyscanners with nanometer-level resolution havebeen developed.

The majority of practical nano-positionersutilize flexure-based structures, such ascompliant mechanisms and notch-flexure-based mechanisms, due to their smooth andfriction-free motion and high durability withoutwear and deterioration. Piezoelectric actuatorsand high resolution displacement sensors arewidely used with flexure-based mechanismsto obtain displacement with nanometer levelresolution. Among the flexure-basedmechanisms, parallelogram mechanismsrestrict all rotational degrees-of-freedom of theconnector (end-effector) and keep theconnector parallel to the base, because of theequal lengths for its crank and followers (themembers connected to the base). In spite ofrotation of the crank and the follower, theconnector undergoes pure translation along acircular path. Due to this feature, parallelogram

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mechanisms can be directly used as singledegree-of-freedom nano-positioning modules.Such nano-positioning modules can be easilyconfigured as building blocks to build a multi-degree-of-freedom nano-positioner.Parallelogram mechanisms have also beenused in many other applications, such asdelta-robots and other low degree-of-freedomparallel kinematics nano-positioning stages.

Compliant mechanisms transmit motionand force by deflection of their flexiblemembers. They are usually made of amonolithic piece of material and thus involveno wear, backlash, noise, and lubrication. Topredict more accurately their deflected shapein larger working range, the analysis ofcompliant mechanisms has usually based onnonlinear numerical techniques such as thefinite element method. To increase the workingrange of a compliant mechanism, its membersusually undergo large displacement androtation. Unlike structural members whosedeflection is small so that the linear beamtheory holds, compliant members capable oflarge displacement and rotation require anonlinear analysis to accurately predict theirdeformed configuration. Some commonly usednonlinear analytical methods include theelliptical integrals, the chain algorithm, and theFinite Element Method (FEM).Sincepositioning systems play a crucial role in manynanotechnology applications, they haveattracted considerable interest from manyscholars and research bodies in recent years.A review of the available literature reveals thatpositioning devices can broadly be dividedinto three categories in terms of the particularmethod utilized to drive the stage. Thesecategories are: (1) stick-slip induced friction

drive stages, (2) clamp release inchworm-screw stages and (3) elastic deformationstages.

Assumed Modes Method: DoubleParallelogram Flexure Mechanism:Using the assumed modes technique, wewould first find the natural frequencies andmode shapes of the X-stage and later usethese to arrive at the dynamic equations of thissystem.

From the Euler-Bernoulli beam theory, eachbeam satisfies

2

2

2

2

2

2

tym

xyEI

x

...(1)

By separation of variables:

tFxYtxy , ...(2)

Assuming a harmonic form (conserving thetotal energy of a conservative system) for thetime dependent component:

xmYxEIYdxd

4

4

...(3)

Since ‘EI’ is independent of x and t

EImbYb

dxYd 2

444

4

;0 ...(4)

Equation gives us the beam mode shapesY corresponding to each natural frequency .The general solution (for the ith beam) to thisproblem is of the form:

bxCbxCxY ii cossin 21

hChbxC ii cossin 43 ...(5)

By symmetry, the two beams in the upperhalf will have the same mode shapes as theircorresponding counterparts in the lower half.We therefore analyze the boundary conditions

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for beam 1 and beam 2 as depicted in fig forthe case of free vibrations.

Boundary Conditions (subscript representsthe beam number):

02

01

0201 ||,|| xxxx dxdY

dxdYYY ...(6)

0|,0|,0| 211 LxLxLx dx

dYdxdYY

LxLx Ybm

Mpdx

Yd |

2| 2

432

3

02141

032

3

031

3

|2

|| xorxx Ybm

Mdx

Yddx

Yd ...(7)

The above boundary conditions give useight equations in the nine unknowns andsolving these boundary condition we get:

hbLb

mM

bLbm

MbLbLc pp sin

2sin

2coshcos21

bLb

mM

bLc p cos2

sin22

0cos2

sin24

hbLb

mM

bLc p...(8)

022 232114

11312

121 ccbc

mMcbc

mMc

...(9)

Using matrix notation, they take the form’s(b) = 0. These equations were then solvedusing an optimization algorithm to find thevalues of the unknowns.

Using Values Specific to Our SetupMass of primary motion stage (Mp) =

0.307 kg

Mass of secondary/intermediate motionstage (Mi) = 0.070 kg

Beam material density: 7860 kg per cubicmetre

Beam thickness = 0.05 cm

Mass/ unit length (m) = 0.09825 kg/m

Beam Length (L) = 12.5 cm

Width = 2.5 cm

Area of cross section of beam = 1.25 x10–5 m2

Young’s Modulus (E) = 1.131 x 1011 pa

Results show that the final transfer functiondepends primarily on the first two modes ofthe system. The contribution from the remainingmodes is not very significant. Hence, using onlythe first two modes:

Mode Shapes

Frequency: 3.93 Hz

bxbxbxxY sinh237.0cos108.0sin237.01

bxcosh086.0


bxcosh095.0

Frequency: 16.13 Hz


bxcosh086.0


bxcosh03.0

The value Yi(x) can be scaled using anormalizing constant. It can be easily verifiedthat this constant has no effect on the finalsystem transfer function obtained using thesemodeshapes. The modeshapes are shown inFigures 2 and 4.

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Figure 1: Frequency 3.93 Hz via AMM

Distance from Clamp (x/L)

Def

lect

ion

in y

Dire

ctio

n Beam 2

Beam 1

Figure 2: Mode Shape 1: Frequency 4.03 Hz via FEM (Ansys)

Beam 1

Beam 2

Figure 3: Frequecy 16.13 Hz via AMM

Distance from Clamp (x/L)

Def

lect

ion

in y

Dire

ctio

n Beam 1

Beam 2

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Since the two beams will move together inthe presence of any frequency excitation, thegeneralized time coordinate qi(t) for these willbe the same; the mode shapes will differ. Fromthe assumed modes method theory and usingonly mode 1 and 2, displacement ‘y’ for eachbeam in its own frame can be expressed as:

n

i ii ntqxy1

2,

For the case above, the energy equationsfor the systemare:

1

0

2,212

beam

Ldx

ttxymtK

2

0

2,212

beam

Ldx

ttxym

22 ,021,

21

t

tyMt

tLyM lp ...(10)

10

2

2

2 ,212

beam

Ldx

xtxyEltV

20

2

2

2 ,212

beam

Ldx

xtxyEl

...(11)

The kinetic energy can be written as:

n

i

n

j jij tqtqmtK1 1

...(12)

The potential energy can be written as:

n

i

n

j jij tqtqktV1 1

...(13)

Using Lagrange equation with L = K – V;

0

qL

dtd

qL

This gives the equation of motion for freevibration:

0 qKqM ...(14)

For our case

4

4

10100106.2M

0068.00375.0

,0278.1001475.0

FK

State Space Form

Choosing 2211 ,,, qqqq as the state variables,the equations can be written in the form

where

2

2

1

1

qqqq

x

and A, B,C are matrices given by:

Figure 4: Mode Shape 2: Frequency 16.63 Hz via FEM (Ansys)

Beam 1

Beam 2

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010000100001108.50107.00 64

A

51055.5067.580,

0001

CB

Coupled Equations: Flexural Mechanismand Linear Voice Coil Actuator

Force sensitivity AmpNK f /78.5

Back EMF constant msecVKb /77.5

Resistance OhmR 4.4

Inductance HenryeL 34.1

Mass of the coil assembly KgMca 0249.0

The value of inductance is very low and assuch it affects the transfer function of thesystem only at high frequencies (>800 rad/s).Hence, neglecting inductance and coupling theabove equation with the double parallelogramequation, we get:

txCtytVBtxAtx vcvcvc ,

where

CCBRKBB

RKKAA vc

fvc

bfvc ,,0

010000100001108.50107.00 64

A

51055.5067.580,

000

3136.1

CB

Coil assembly

Field assembly

Primary StageIntermediate

Stage

Ground

MotionSensor

Figure 5: Collocated Actuationand Sensing

The mechanism is actuated using a LinearVoice Coil Actuator (shown above). In the finalassembly, the coil assembly mass of the voicecoil actuator is attached to and moves with theprimary motion stage. Thus, in theequationsthe effective mass of the primarystage changes to mass (primary stage) +mass (coil assembly).

Equations: Voice Coil Actuator

The voltage drop across the actuator can bewritten a inductancebackemfresistance tVtVtVtV

The values specific to our actuator modelare:

Figure 6: Circuit Diagram for the VoiceCoil Actuator

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The voice coil back emf acts as a resistanceand reduces bode plot amplitude in the lowerfrequency zone. The inductance has a similareffect over the higher frequency range.

Static and Dynamic Analysis of DFMStatic Analysis of FlexuralMechanismThis flexure unit is also referred to as acompound parallelogram flexure, folded beamflexure or crab-leg flexure. Analysis shows thatthis flexure allows relative Y translationbetween bodies A and B, but is stiff in relativeX displacement and rotation. The parasiticerror along X direction, is considerably smallerbecause any length contraction due to beamdeformation is absorbed by a secondarymotion stage. There does exist a rotationalparasitic motion, which may be eliminated byappropriate location of the Y direction force.

Hence, body A exhibits perfect Y-translationwith respect to body B on the application of aY direction force. These statements are trueonly in the absence of X direction forces.

The double parallelogram may be employedto construct XY mechanisms as shown inFigure. In these cases, cross axis coupling and

Figure 7: Bode Plots of Theoretically Derived Models: 1) Double Parallelogram FlexuralMechanism, and 2) The Overall Assembly, Including the Voice Coil

Frequency (rad/s)

Phas

e (d

eg)

Mag

nitu

de (d

b)

Figure 8: Double Flexure Mechanism

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motion stage yaw should be small and actuatorisolation should also be better than previousdesigns.

Beam material density = 7860 kg per cubicmetre

Beam thickness = 0.05 cm

M a s s / u n i t l e n g t h ( m) = 0.09825 kg/m

0,11,12 2

221

23

Lbbt

ElFL

Beam length (L) = 12.5 cm

Width = 2.5 cm

Area of cross section of beam = 1.23 x10–5 m2

Young’s modulus (E) = 1.31 x 1011 pa

Figure 9: Comparison of Analytical andFEM-ANSYS Results for Static Analysis

Fig 9: Comparison of Analytical andFEM-ANSYS results for Static

Analysis

Figure 10: Static Analysis: Finite ElementResults by Application of 1 N Load

at Motion Stage

Figure 11: Static Analysis: Finite ElementResults by Application of 2 N Load

at Motion Stage

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Deflection, angle of rotation and parasiticmotion for double parallelogram flexural unitis given by following relations,

Mechanism AnalysisThis chapter looks at the analytic developmentof the dynamic equations of the system usingthe assumed modes method. The results arethen compared with those obtained from FEM(Ansys) and via the open loop experiments onthe setup.

X-Y positioning is achieved using twodouble parallelogram mechanisms arrangedas shown below:

By symmetry, we can say that the load isequally shared by the upper and lower halvesof the structure. This is depicted in Figure 13.

Dynamic Analysis: Double FlexureMechanismUse modal analysis to determine the vibrationcharacteristics (natural frequencies and modeshapes) of a structure or a machinecomponent while it is being designed. It canalso serve as a starting point for another, more

Figure 11 (Cont.)

Each stage operates independently of theother. Based on this premise, we can modelthe X-stage (inside the enclosure) as:

Figure 12: X-Y Positioning Stage withTwo Double Parallelogram Flexure

Mechanisms

Figure 13: a) X-Stage b) SymmetricDistribution of the Load

SecondaryMotionStage

PrimaryMotionStage

F

F/X Y

Beam 2Beam 1

Mode Frequency [Hz]

1 14.3

2 170.19

3 174.75

4 324.53

5 468.33

6 474.22

7 479.39

8 560.71

9 590.08

10 918.42

Table 1: Modal Analysis Result

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detailed, dynamic analysis, such as a transientdynamic analysis, a harmonic responseanalysis, or a spectrum analysis. Summarizingthe modal analysis method of analyzing linearmechanical systems and the benefits derived:

• Solve the undamped eigenvalue problem,which identifies the resonant frequenciesand mode shapes (eigenvalues andeigenvectors), useful in themselves forunderstanding basic motions of the system.

• Use the eigenvectors to uncouple ordiagonalizable the original set of coupledequations, allowing the solution of n-uncoupled s dof problems instead of solvinga set of n-coupled equations.

• Calculate the contribution of each mode tothe overall response. This also allows oneto reduce the size of the problem byeliminating modes that cannot be excitedand/or modes that have no outputs at thedesired dof's. Also, high frequency modesthat have little contribution to the system atlower frequencies can be eliminated orapproximately accounted for, furtherreducing the size of the system to beanalyzed.

• Write the system matrix, A, by inspection.Assemble the input and output matrices, Band C, using appropriate eigenvector terms.Frequency domain and forced transientresponse problems can be solved at thispoint. If complete eigenvectors are available,initial condition transient problems can alsobe solved. For lightly damped systems,proportional damping can be added, whilestill allowing the equations to be uncoupled.

ANSYS V11 Workbench is used here tocarry out modal analysis of flexural unit used

as basic building blocks of XY planar flexuralmechanisms

Modal Analysis of ParallelogramSingle Flexure

Figure 14: Analysis of Single FlexureMechanism

Modal Analysis of Double FlexuralUnitIt can be analytically shown that parallelogramflexure offers small resistance to relative

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motion in the Y direction but is stiff with respectto relative motion in X and rotation. Hence, itis a much better approximation for single DOFflexure as compared to the simple beamflexure used in the previous case. wo rigidmoving platforms are referred to as the primaryand secondary platforms. Loads f, m and pare applied at the primary platform. The twoparallelograms are treated as identical, exceptfor the beam spacing, w1 and w2. Assumptionssimilar to the ones stated in the previoussection hold here as well. x1, y1 and 1 are theabsolute displacement coordinates of thesecondary stage, and x2, y2 and 2 are thoseof the primary stage, along the directions

indicated in Figure 15. Force displacementresults obtained in the previous section areapplied to the constituent parallelograms in thepresent case. Force equilibrium conditions areobtained by drawing FBDs for each of theplatforms.

Figure 15: Double Flexure Mechanism

Figure 16: Modal AnalysisDouble FlexureMechanism

Figure 17: Modal Analysis of FlexureMechanism 1

FEM Analysis of FlexuralMechanisms 1 (Static)The parallelogram flexure unit is a classicdesign that has been employed in manyflexural mechanisms. Due to their finitestiffness, the additional flexure units do notover-constrain the mechanism. This symmetricarrangement should result in severalperformance improvements. The motion stageyaw should be further reduced to due to the

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additional rotational constraints arising fromthe parallelogram flexures. On the applicationof an X actuation force, the two sides of themechanism tend to produce displacements inY direction that counter each other, andtherefore reduce the cross-axis couplingerrors. Out of plane stiffness also improves dueto better support of the motion stage.

Figure 18: Modal Analysis of FlexuralMechanism 1 (Dynamic)

Figure 19: Plot Force via Deflection

Mode Frequency [Hz]

1 12.334

2 33.002

3 169.79

4 170.32

5 173.79

6 194.58

7 348.42

8 451.83

9 467.8

10 469.21



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CONCLUSIONThere exist many two-axes planer flexuralmechanisms that allow for small translationswithin the plane of the flexure. Most of thesedesigns incorporate a stacked assemblywhere one linear stage in mountedperpendicular on a second linear stageresulting in a relatively bulky design.Nevertheless, in this arrangement the two axesare entirely decoupled and the actuation of oneaxis has no effect on the other. Such anassembly is commonly referred to as a ‘serialdesign’ in robotics terminology. In some cleverserial designs, the above-mentioned stackingis achieved within a plane.

The disadvantage of serial designs is thatthe actuator for the second stage has to bemounted on the moving member of the firststage. This not only makes the designunnecessarily complex but also limits thesystem dynamic performance, for example,speed of response. Ideally, it is desirable tomount the actuators for both the axes onground, i.e., the fixed base.

Furthermore, if one tries to increase thedegrees of freedom of a flexural mechanismusing the serial approach, the design becomesincreasingly cumbersome and bulky. Therefore,instead of taking this path designers usuallydevelop assemblies that are based on closed-chain parallel designs (as opposed to serialdesigns). In this kind of designs however,parasitic coupling between the degrees offreedom and cross-sensitivity of actuatorsbecome performance limiting factors. These twofactors are explained in the next section. Thereare situations where such parallel designs areused, but accuracy is compromised for economyin size. There are other situations where any

degree of errors is unacceptable either at themotion stage itself or at the points of actuatorforce application. It would be desirable to havemechanisms that have the compactness ofparallel designs while axes decoupling andactuator isolation of serial designs.

In this disclosure we shall present a groupof flexural mechanisms that are based onparallel elasto-kinematics. It is worthwhile tomention here that the motion of compliantmechanisms is not completely characterizedby kinematics; it is strongly dependent onelastic deformations as well. Hence, the studyof motion of flexural mechanisms is commonlyreferred to as elasto-kinematics. Mechanismspresented here make unique use of knownflexural units and novel geometric symmetryto minimize or even completely eliminateactuator cross-sensitivity, and parasiticcoupling between the two axes.

FUTURE SCOPEThe designs presented in this document arevery fundamental and can be used over a widerange of macro, meso or micro scale precisionmachines where decoupled multiple degreesof freedom are required. Potential applicationscan be found in optical instruments, Micro andNano Electro Mechanical Systems, precisionmetrology, etc. A few specific applications arementioned here.

High Precision Two or More Axis MotionStage: In certain high precision microscopethat is used for observing the interactionbetween protein and DNA molecules, the stagethat holds the specimen needs to be pannedwith sub-micron precision. The 2 DOF planerdesigns presented in this document are idealcandidates for such applications.

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Micro-Electro Mechanical (MEM) MotionStages for Actuators and Bearings: Thesedesigns are of very significant consequenceto MEMS technology where structures needto be etched on Silicon wafers. Planer designsthat provide multiple DOF can have anunprecedented impact on MEMS actuators,bearings and guides.

REFERENCES1. Awtar S and Slocum A H (2004),

“Apparatus Having Motion with Pre-Determined Degree of Freedom”, USPatent 6,688,183 B2.

2. Davies P A (2001), “PositioningMechanism”, US Patent 6,193,226.

3. Howell L L (2001), CompliantMechanisms, John Wiley & Sons.

4. Hwa Soo Kim and Young Man Cho(2009), “Design and Modeling of a Novel3-DOF Precision Micro-Stage”,Mechatronics, Vol. 19, pp. 598-608.

5. Lobontiu N (2003), CompliantMechanisms: Design of Flexure Hinges,CRC Press.

6. Qing Yao J and Dong P M Ferreira (2007),“Design, Analysis, Fabrication and Testingof a Parallel-Kinematic MicropositioningXY Stage”, International Journal ofMachine Tools & Manufacture, Vol. 47,pp. 946-961.

7. Tian Y, Shirinzadeh B, Zhang D, Liu Xand Chetwynd D (2009), “Design andForward Kinematics of the CompliantMicro-Manipulator with LeaverMechanisms”, Precision Engineering,Vol. 33, pp. 466-475.

8. Yangmin Li and QingsongXu (2009),“Modeling and Performance Evalutrion ofa Flexure-Based XY ParallelMicromanipulator”, Mechanism andMachine Theory, Vol. 44, pp. 2127-2152.

Documents

XY SCANNING MECHANISM: A DYNAMIC APPROACH · drive stages, (2) clamp release inchworm-screw stages and (3) elastic deformation stages. Assumed Modes Method: Double Parallelogram Flexure