1732556 Teleoperated 3-DOF Micromanipulation System

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Owing to its ability to move, mobile micromanipulator offersapplications exibility together with macroscopic workspaceoptions. Unfortunately, their outcomes were mostly on thehardware design, but only few explanations were provided forsystem modelling, performance evaluations and controllerdesign.

In view of all these efforts, the objective of this study is to

provide the details of the development of micromanipulationsystem (MMS) with teleoperation ability using forcefeedback. In order to meet some standards of the above-mentioned works, our teleoperated MMS has to be simpleand easy to be implemented. Moreover, it has to be stable andable to track the human operator command with reasonabletracking error. Furthermore, the functional aspects of thesystem needed for comfortable use such as sensorredundancy, is proposed as well. In this paper, however,effects of the environmental inuence such as van der Waals,electrostatic, and capillary forces, are not considered. Themain problems considered in this paper are the positionaltracking when the MMS moves in free space and forcetracking when the MMS has a contact with the environment.

The present paper is organized as follows. Following thisintroduction, the current MMS in laboratory which consistsof micromanipulation subsystem including force sensing andhaptic feedback subsystem will be described in detail. Nextpresented is the preliminary design of a simple control schemefor stable teleoperation system. In this section passivity-basedcontroller is discussed. As a matter of fact there must be somekind of delay in the communication channel, i.e. internet.However, because the micro teleoperation is mostlyperformed on site, in this preliminary design, this issue isnot our main concern. In the next session, some experimentalresults which include remote operations as well as locallyconnected operations will be shown in order to verify theeffectiveness of the design. Finally, concluding remarks andpossible future work will be given.

2. Micromanipulation systemThe close view of the MMS is shown in Figure 1. Each of itsaxes is constructed from piezoelectric actuator attached inexure hinge mechanism. A nite element analysis usingMATLABs Partial Differential Equation Toolbox isconducted to predict the behavior of the exure hingestructure. The strain analysis is used with the parameters thatare given in Appendix 1. In extreme case, the results areshown in Figure 2. It can be seen that the displacement fromthe piezoelectric actuator is amplied in perpendiculardirection after the action of two series of lever mechanism.By stacking three of this structure one after another in X -Y - Z directions, a 3 degrees-of-freedom (DOF) Cartesianmanipulation system can be constructed. From Figures 1and 2, it is seen that the shape of the exure hinge structure iswell matched and easy to construct by using this arrangementfor achieving a compact integrated structure. The X -Y - Z direction is chosen because three axes are orthogonal anddecoupled, i.e. the motion of one axis is independent of theother axis displacement. By using this conguration, itskinematics is simplied and so is the dynamics modelling.Moreover, considering the shape of the exure hingemechanism, using X -Y - Z congurat ion themicromanipulator is easy to design and fabricate. Once it isassembled, by assuming that the motion is quasi-static or

quite slow and also smooth, the decoupling property isvirtually maintained.The MMS is obtained by composing three almost similarcongurations of this structure-namely left manipulator, rightmanipulator and moving table as also shown in Figure 1. Theleft and right manipulators have the same construction in amirror symmetric manner, except the way to sense contactforces. The left manipulator is equipped with a set of foil straingauges while the right manipulator with a set of semiconductorstrain gauges which provide much higher sensitivity than foilstrain gauges. The left manipulator has less rigidity than theright one to assure a sensing resolution. Major specications of the manipulators and the specimen table are summarized inTable I. Each manipulator has a 3-DOF ne positioning

mechanism and a 3-DOF coarse positioning mechanism inwhich the former utilizes piezoelectric elements with exurehinges to magnify displacements while the latter uses a manualfeeding mechanism. Travel distance of the ne positioningmechanism is 200 mm in each direction. The specimen table ismounted atop a 3-DOF ne motion mechanism, which has thesimilar structure to those for the manipulators. Furthermore, arotary DOF is added to it for angular positioning of thespecimen used. All of the manipulators are driven using voltagesteering scheme using voltage ampliers. The commandsignals are supplied from the computer via digital to analogconverter PCI3346A 16 bit from Interface Co.

Figure 1 Micromanipulation system

Left manipulator Right manipulator

Specimen Table

y

z

x

PZT

(a) Manipulators and specimen stage

(b) Manipulators and specimen stage

Teleoperated 3-DOF micromanipulation system

Adha Imam Cahyadi and Yoshio Yamamoto

Industrial Robot: An International Journal

Volume 35 Number 4 2008 337346

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2.1 Mechanical model of the MMS and hysteresis

compensationIn order to get a closed-form model equation, sets of exurehinges and springs are used to predict the behavior of the exure hinge structure for each axis. For convenience, thenotation that will be used is as follows. The index i [ x; y; zdenotes the axis and the dot above the variable means itsvelocity. The approximate model is shown in Figure 3. Itshould be noted that a slightly abused notation, i.e. droppingthe index i, is used in this gure. Static analysis on theapproximate exure hinge mechanism yields the followingresults:

u 1i C M 1 i M 1 i C F 1 i F 1i 2 C M 1 i k2i u 1 i l 21i C F 1i k1 i x pi

C F 1 i k1i

1 C M 1 i k2i l 21i x pi

1

u 2i C M 2 i M 2 i C F 2 i F 2i

2 C M 2 i k2i 2

xsi l 2 i C F 2i k2i u 1 i l 1i 2

After some manipulations for each axis we get:

xsi C F 1i C F 2 i k1i k2 i l 1 i

1 C M 1i k2 i l 21i 1=l 2 i C M 2i k3 i =2l 2 i x pi 3

which shows that the exure hinge structure acts as distanceamplier from the piezoelectric displacement x pi to xsi . In the

above equations all angles are assumed to be very small suchthat sin u i < u i and cos u i < 1. Moreover, k3 i k4 i issupposed in order to get u 2 i l 2 i xsi . All parameters are listedin Appendix 2. The approximate dynamics of the abovestructure can be found by applying Euler-Lagrange method.The Lagrangian function L and the dissipated energy E d aregiven as follow:

L i 12

m1 i _x2 pi

12

m2i _

u 1 i l 1 i 2 12

m3 i m4 i _x2si

212

k1 i x2 pi 12

k2 i u 1 i l 1 i 2 12

k3 i k4i x2si

12 M i

_

x2si

12 K i x

2si

4

E di 12

d 1i _

x2 pi 12

d 2 i _

u 1i l 1 i 2 12

d 3 i d 4 i _

x2si

12

D i _

x2si

5

where M i ; K i and D i are constants due to equation (3) and.Using Euler-Lagranges equation:

dd t

L i

_xsi 2 L i xsi E di _xsi F si v i F hsi 2 F ei 6

Figure 2 Flexure hinge mechanism and its working principle

(a) Relaxed state

24.5

19.5

14.5

9.5

4.5

29.5

0.50 5 10

PZT

15 20Length (mm)

L e n g

t h ( m m

)

25 30 35 40 45

(b) Activated state

PZT

29.5

24.5

19.5

14.5

9.5

4.5

0.5

L e n g

t h ( m m

)

5 10 15 20Length (mm)

25 30 35 400 45

Table I Specication of MMS

Table Manipulator

Fine motion

Translation X -Y -Z X -Y -Z Driving PZT PZTTravel distance ( m m) 200 200Resolution (nm) 20 20Coarse motion

Translation X -Y -Z X -Y -Z Driving Manual ManualTravel distance (mm) ^ 2 ^ 2Resolution ( m m) 10 10Rotation u z NoneRotation angle 3608 endless

Figure 3 Approximation by sets of exure hinges and springs

xs

x p

q 1

q 2k3

k4k1

PZT

k2

FH 1

FH 2





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then the dynamics equation is obtained as:

M i

xsi D i _

xsi K i xsi F si v i F hsi 2 F ei 7

where M i ; K i and D i are the MMS equivalent mass, dampingand stiffness, respectively, x pi ; xsi ;

_

xsi ;

xsi are piezoelectricdisplacement, exure hinge displacement, velocity andacceleration while F si v i ; F hsi and F ei stand for the forceterms coming from the piezoelectric actuator itself, thehysteresis term and the external force that is coming fromenvironment. It is assumed that F si v i is a linear function of input voltage.

However, as a matter of fact rather than nding theparameters of the exure hinge that is quite complex andcumbersome, it will be advantageous to just nd M i ; K i andD i from experiments. By dividing equation (7) with M i thenthe left hand side of equation (7) can be equated with

xsi 2z i v i _

xsi v 2i xsi , the natural frequency v i and thedamping factor z i can be found from the open-loopresponses. As the hysteresis enters the system in the form of force equivalent, we can treat it as disturbance entering to thesystem thus it is possible to develop disturbance observer to

cancel its effect.As the voltage amplier is used, the stroke responsegenerated by the piezoelectric actuator will be nonlinear dueto hysteresis, creep and drift. Indeed, the response of inputvoltage to the MMS displacement is nonlinear as seen inFigures 4 and 5. Qualitatively they show that hysteresis is themain source of nonlinearity while another kind of it, forexample, creep does not appear in Figure 4. There is a smalldrift appearing in Figure 4, but it should be able to becompensated by the main controller that is developed in thenext section. In order to achieve precise movement, however,such nonlinearities, especially the hysteresis, have to betreated carefully.

Related to the hysteresis issues, there have been tremendousefforts for either characterizing,-developing its model orcompensating its effect. For example, Richter characterizedseveral nonlinearities in piezoelectric positioning deviceincluding the transient response, drift and hysteresis byexperiments. However, it did not reach the analytical model(Ritcher et al. , 1997). Several attempts were made utilizing

the famous Preisach model or its derivation for compensatingthe hysteresis (Ge and Jouaneh, 1996; Yokokohji et al. , 1994;Tanikawa, 2001). Despite the nice properties of this modelfamily, a lot of parameters have to be identied in order toachieve a sufciently good model. Another recent workdirectly related to hysteresis compensation for piezoelectricmicromanipulator has been done by Rakotondrabe et al.(2007). In this work, they proposed an H 1 controller tocompensate the hysteresis in their piezo-actuated micro-gripper. Even though the resulted controller is linear, it is stillin a differential equation of order ve that is quite complex,hence approximated to be third order to simplify theimplementation.

In order to avoid the complexity in this part, thehysteresis compensation scheme has to be simple and easyto be built. For this reason, the compensation scheme usedin this work is based on the extension of the Dahl modelfrom Helmick and Messner (2003) that is originally aimedfor predicting the hysteresis in disk drive actuators.Although not well-known, the Dahl model is simpler andmuch easier to be implemented compared with theaforementioned works.

For each axis the hysteresis force term that is in the form of high-order Dahl model can be summarized as follows:

_y Ahi y _

xsi B hi u pi _

xsi

F hsi C hi y 8

For second order model, y qi _

qi T :

Ahi 0 1

2 a 2 i 2 a 1 i sgn _

xsi " #; B hi 01" #; C hi b1 i b0 i sgn _xsi where the initial values of the hysteresis parameters a 1i ; a 2 i ; b0 i and b1 i have to be obtained from the experimental data andu pi is set to 1.

For the full three axes the more compact representationusing:

j xsx_

xsx xsy_

xsy xsz_

xszT

Figure 4 Poor tracking response due to voltage steering

0 2 4 6 8 10 125

0

5

V o

l t a g e

( v o l

t )

Time (second)

Time (second)

output from dispsensor

0 2 4 6 8 10 125

0

5

V o

l t a g e

( v o l

t )tracking error

control signal

Figure 5 Hysteresis due to voltage steering

5 4 3 2 1 0 1 2 3 4 5100

50

0

50

100

150

Control signal (volt)

D i s p l a c e m e n

t ( m i c r o m e t e r

)





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and considering the hysteresis term in equations (7) and (8)the total system can be represented as:

_

j

A sx 0 0

0 A sy 0

0 0 A sz

2664

3775

j

B 1x

B 1 y

B 1z

2664

3775

col{ F si vsi }

B 2x

B 2 y

B 2z

2664

3775

col{ F hsi 2 F ei }

9

ys

C sx 0 0

0 C sy 0

0 0 C sz

26643775

j 10

where:

A i 0 1

2 K i = M i 2 D i = M i " #; B 1 i B 2i 01= M i " #; C si 1 0 and col{ } is a column vector. The output ysi is the measured

variables of the MMS, i.e. the displacement of each axis. Asthe velocity sensor is not provided a reduced orderLuenberger observer for each axis is built as follows:

_z i 2 K i M i

ysi 2D i

M i z i Ly si U i 2 Lz i Ly si 11

^

x2 i z i Ly si 12

where:

U i 1

M i F si v i F hsi 2 F ei

The observer pole is 2 D i = M i 2 L where the observer gain Lcan be set to any positive number. The function of the velocityobserver is two-fold. First it can be used to compute the

hysteresis term that is further employed as hysteresiscompensator. Second, it will be used as stabilizing controllerfor teleoperation as will be mentioned soon in the nextsection.

2.2 Contact force sensingAs mentioned earlier, the both manipulators are equippedwith force sensors. The right manipulator employssemiconductor strain gauges while the left one uses foilstrain gauges. The semiconductor strain gauges have higherresolution than the foil strain gauges. However, some driftsalways occur in spite of the absence of environment contact.Moreover, the right manipulator serves as the primarymanipulator thus the valid information has to be kept inorder to achieve a stable teleoperation performance. Hence,the right manipulator, shown in Figure 6, is provided with anextra force sensing means to realize a so-called fault-tolerancing function. That is, the contact force experiencedby the right manipulator can be measured by two differentways, i.e. a set of semiconductor strain gauges and acapacitive displacement sensor.

In this setting, the primary force sensor, i.e. strain gaugetype force sensor, continues to provide a force feedback signalunder normal circumstances. However, if something goeswrong with the primary sensor during its operation, e.g.mechanical or electrical failure, it may ruin or damage thesystem as well as the work piece. Under such a situation, a

secondary force sensing instrument, i.e. capacitive forcesensor in Figure 6, takes over the primary sensor to continueproviding the force feedback signal without a pause. Use of this kind of back-up sensing enables the operator to keepexecuting a required task without even noticing the failure of the primary sensor. To remove the noisy signal generated bythe capacitive force sensor, a digital low pass lter isemployed. It is required that the signal transition betweenthe two force sensor should be smooth. Therefore, anotherdigital low pass lter is used for smoothing the signaltransition. The switching condition is as follows:

f

0 if abs f p . abs f s; abs f s , l 0

f p if abs f p . l 0 ; no error f s if error8>>>:

13

where f is the outputted force, f p is force reading from primarysensor, f s is force reading from fault-tolerance sensor, l 0 is aband for zero force, while error and no error conditionsare specied from a set of rules that indicates error that oftenoccurs in experiments. An example of such rule that infers theerror condition in this work is shown in Figure 7.

The rst part of equation (13) is used to compensate thedrift in the primary sensor, while the second and the last partresponsible for force output decision during fault. Forverication purposes, a fault occurrence is demonstrated

Figure 6 Force sensors of the right manipulator

Semiconductor strain gauge

Capacitive displacement sensor

Figure 7 Condition for error

get two last samples of fp and fs: p(k) , fp(k1) , fs(k) and fs(k1)

is abs(fp(k) - fs(k)) big

is abs((fp(k) - fp(k1)) -(fs(k) - fs(k1)) big

no error condition is set

no

yeserror condition is set

error condition is setyesno





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under a local connection between MMS and haptic devicesystem (HDS). The results are shown in Figure 8. In thisexperiment, a specimen table approaches the probe andmakes a contact with the probe after approximately 4 s. Afterthe specimen table approaches the manipulator probe it startsexerting a force against the probe. An articially generatederror is applied to the primary force sensor at 7.5 s.

Figure 8(b) shows the force feedback information from thesemiconductor strain gauge force sensor. It is seen that thesignal is drifted away from the origin and uctuates in verylow frequency in spite of no contact with the environment.Such condition is undesirable in bilateral teleoperationsystem. On the other hand as seen in Figure 8(c), regardlessits lower resolution and noise, the response of capacitivedisplacement sensor is almost constant when there is nocontact with the environment. Moreover, due to its non-contact nature, it is more durable than the primary sensor.From this gure it is also shown that as soon as its abnormalsignal is detected, the signal from the primary force sensor iscut off and the secondary force sensor will take over. Thereare two regions in the middle gure where a sensor failure is

assumed to occur. During the two regions, the force feedbacksignal is generated based on the sensor reading of thesecondary sensor. As can be seen, there are minor spikescaused by the switching of control, but it mostly succeeds toback up the failure of the primary sensor.

In laboratory, the semiconductor strain gauge sensor (KSP-6-350-E4, Kyowa Co.) is arranged in bridge connection suchthat theoretically has 50 mN/V resolution. The capacitivedisplacement sensor (D-050.00, Physik Instrumente)theoretically should have 15 mN/V resolution; however, dueto noises its resolution degrades signicantly. The signalsfrom the force sensors are inputted to the slave-side

controller, i.e. computer, through 12 bit A/D converter(PCI3177C, Interface Co.). Moreover, a positional signalsensed with a laser displacement sensor (LC-2400A, KeyenceInc.) is also inputted with the same A/D converter.The command voltage is fed to 9-CH voltage ampliersfrom the 16 bit D/A converter (PCI3346, Interface Co.). Thevoltage amplier used to drive the piezoelectric actuators in

each part of MMS is assumed to linearly map the inputvoltage ranged from 0-10 to 0-150 V. As it is sometimesdesirable to locally operate the MMS on site, a joystickcontroller is also provided. The signal from the nine-channelMMS plus one rotary table share the same A/D converterwith those mentioned above. The joystick operation uses asimple position control with proper scaling. It will bementioned later in the next section for convenience. Theslave-side conguration can be seen in Figure 9.

2.3 Haptic device systemAs commonly used in teleoperation system, a HDS serves asmaster device for sending a command signal as well as ahaptic presentation device to the operator. On the other handMMS undergoes actual micromanipulation tasks according tothe command given. Either 3-DOF PHANToM Desktop w orPHANToM Omni w can be used as HDS which ismanufactured by SensAble Technologies Inc. The dynamicsof the master device is given in the following equations:

M mxm xm C m xm ;

_xm_xm N mxm ;

_xm F hum u m 14

where M m xm is an inertia matrix, C m xm ;_xm is a Coriolis

and centrifugal matrix force, N mxm ;_

xm is a gravity term,while F hum and u m are, respectively, external forces from thehuman operator and input force. The details of the abovematrices are omitted in order to get a good continuationbetween the pages. As PHANToM w only has three actuators,

i.e. DC motors, in the rst three links and for sake of simplicity, it should be noted that only these rst three linksare considered in the modelling. The last three links aretreated as a modelling error or disturbance regardless of theincomplete modelling of the rst three links. Manysimplication has been done in order to get the equation of motion of the HDS. For instance, the material of PHANToM

Figure 8 Demonstration of fault-tolerancing mechanism: (a) outputtedforce signal; (b) signal from the primary sensor; (c) signal measured bythe secondary sensor

F o r c e s s e n s e

d a t m a s

t e r s i

d e

( N ) 0 2 4 6 8 10 12

101234

(a)

0 2 4 6 8 10 121

01234

(b)

0 2 4 6 8 10 121

01234

Time (second)

(c)

Figure 9 Slave side conguration

Local operation

(joystick control)

Slave controller

AD

DA

Voltage amplifier(9 channels)

MMS

Laser position sensor

xs

x jc

f p

f s

vsus

Note: Reproduced from the only available original





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master side, i.e. equation (21) is re-stated as follows:um um R

_

xm 28

um 2 K e f s 2 F hum 29

where K e is a constant error gain. In order to get passivetelecommunication channel irrespective of time delay thefamous wave variables or scattering variables transformationcould be used. However, we are not going to implement thisway as the experiments is done in virtually no delaycommunication channel. Moreover, the performance couldbe degraded because of the wave reection. In summary, theproposed teleoperation system for the MMS is shown inFigure 10.

4. Experiments4.1 Hysteresis compensationAnalysis from the open loop responses of the MMS using fastFourier transform reveals that the natural frequency of theMMS is approximately 6 Hz. As the open loop response isstable and under damped then the damping factor should be

in the range of 0 , z i , 1. Hence, z i is approximated to be0.01. The hysteresis parameters a 1 i ; a 2i ; b0 i and b1i are,respectively, tuned to 0.082, 0.95, 0.80 and 2.32. The velocityobserver is implemented using Runge-Kutta method and itspole is placed at 2 10. For the joystick operation, the scalingfactor is set to 25 103.

It is seen in Figure 5 in Section 2 that the open loopresponse between positive and negative displacement is notsymmetric due to the behavior of the exure hinge. Usingjoystick operation, the response of the z-axis piezoelectricinput voltage to the z -axis of the MMS displacement when thehysteresis compensation is applied are shown in Figures 11and 12. From Figure 11 it is seen that the MMS has bettertracking capabilities than the uncompensated one. Moreover,it is noticed from Figure 12 that the hysteresis area could be

minimized despite the chattering occur due to non-smoothSignum function. It should be noted that the control signal isproportional to the joystick displacement.

4.2 Teleoperation experimentsThe parameters for the bilateral teleoperation of the MMS arechosen as follows. The scaling factors a x and a f using

equations (17) and (18) are, respectively, 1.2

103 and1 103. R_xm is set to 0 :1 diag{ _

xmx_

xmy_

xmz }, l is set to0.1 and K s; K e are chosen to be 10. The master and the slavecontroller are placed in near distance and connected via high-bandwidth internet, therefore, the communication delayvirtually could be assumed to be zero. The responses of thissystem in free space are shown in Figure 13. All of thequantities are plotted from the slave point of view.Displacement of the master side is scaled by an appropriatefactor in order to display in the same gure for comparisonpurpose. It is seen that the responses are almost perfect in thesense of tracking and force feedback capability. It is noticedthat the operator sensed a very small force that could be arisenfrom noise measurements due to interference with thePHANToM w movement. Figure 14 shows the responses of positions and force of both sides when there is a contact withthe environment. It is seen that the force feedback sent to themaster device can be tracked well. However, some smalldisturbance occurs when it returns to the free space as theswitching routine is invoked as a result from the primarysensor failure.

5. Conclusion and future worksThis paper presented design, modelling and experiments of teleoperation using a MMS under force feedback. Thedeveloped MMS has simple structure and is easy to construct.The MMS modelling has been done by considering hysteresisphenomenon in its piezo-actuator. It has been shown that thehysteresis curve due to voltage steering of the MMS can be

Figure 10 Scheme of the teleoperated MMS using haptic device

HDS

Communication channel

Master controller Slave controller

MMS

xs , f sum us

us

um

xm ,xm ,F hum.

xm.

r sr md r m

r m = l xm + xm.

xm

F hum

f sd f s

1 /a x

a f

K

K e+

++l

Note: Reproduced from the only available original





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minimized. In addition, the efcacy of a redundant sensing forimplementing the fault-tolerancing concept for comfortableuse of the MMS was demonstrated through the experiments.The teleoperated MMS via a commercially availablePHANToM

w

has been done and some experimental resultshave also been shown under ineligible communicationchannel delay.

In this preliminary design, the exact model of the slavedevice, i.e. MMS, needs to be known and the responses mightbe poor or even unstable otherwise. Therefore, in the futurethe teleoperation scheme has to be extended to overcome thiscontroller difculty. The possibility of operating the MMSfrom remote side can give good prospect for future use.However, the delay has to be taken into account in designingthe controller as well. The real use of the MMS, for examplefor scanning probe microscopy in STM/AFM, can also

be enlarged by improving currently developed MMS such asthe development of end effectors tools like micro-gripper oradhesive tools. Moreover, an additional extra DOF to movesample in vertical direction needs to be implemented for SEMwith energy dispersive spectrometer facility.

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Figure 14 Master and slave positions and forces under contact fromslave point of view

0 2 4 6 8 10 12100

50

0

50

100

D i s p

l a c e m e n

t ( m i c r o m e t e r

)

masterslave

0 2 4 6 8 10 121

0123456

10 4

F o r c e

( N )

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Control signal (volt)

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l a c e m

e n t ( m i c r o m e t e r

)

Figure 11 Tracking response of the compensated system

0 2 4 6 8 10 125

0

5

Time (second)

(a)

V o

l t a g e

( v o

l t )

control signaloutput from dispsensor

0 2 4 6 8 10 12

Time (second)(b)

5

0

5

V o

l t a g e

( v o

l t )

tracking error





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Appendix 1

The nite element analysis dataThe exure hinge mechanism is made from copper alloy withparameters given as follows:. Young modulus 655N/m 2 .. Density 8,940 kg/m 3 .. Poisson ratio 0.37.

Appendix 2

Variables of exure hinge mechanism. u 1 ; u 2 rotations due to exure hinge deformation. C M 1 i ; C M 2 i compliances with respect to acting moments. C F 1 i ; C F 2 i compliances with respect to acting force. M 1 i ; M 2i acting torques. F 1i ; F 2 i acting forces. l 1i ; l 2i lengths of exure hinge mechanism in vertical and

horizontal directions. k1 i ; k2 i ; k3 i ; k4 i spring constants. m1i ; m2 i ; m3 i ; m4 i equivalent mass of the springs. d 1 i ; d 2i ; d 3 i ; d 4i equivalent damping of the springs.

Corresponding authorYoshio Yamamoto can be contacted at: [email protected]





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