Sensor-less Vibration Suppression and Scan Compensation for Piezoelectric Tube Nanopositioners

Transcription

1 Proceeing of the 44th IEEE Conference on Deciion an Control, an the European Control Conference 25 Seville, Spain, December 2-5, 25 MoB5. Senor-le Vibration Suppreion an Scan Compenation for Piezoelectric Tube Nanopoitioner (Invite Paper) Anre J. Fleming an S. O. Reza Moheimani Abtract Piezoelectric tube canner are employe in highreolution poitioning application uch a canning probe microcopy an nano-fabrication. Much reearch ha proceee ith the aim of reucing hyterei an vibration, the foremot problem aociate ith piezoelectric tube canner. In thi paper, to imple techniue are propoe for imultaneouly reucing hyterei an vibration. Experimental reult emontrate ignificant reuction in hyterei ue to the ue of a charge amplifier. Previou problem involve ith the implementation of uch amplifier are reolve to provie DC accurate performance ith zero voltage rift. Seconly, piezoelectric hunt amping, a techniue previouly reient in the fiel of mart tructure, i applie to amp tube vibration. By attaching an LCR impeance to a ingle tube electroe, the firt mechanical moe i reuce in magnitue by more than 2 B. I. INTRODUCTION Piezoelectric tube canner ere firt reporte in [] for ue in canning tunneling microcope. They ere foun to provie a higher poitioning reolution an greater banith than traitional tripo poitioner hilt being imple to manufacture an eaier to integrate into a microcope. Piezoelectric tube canner are no ue extenively in canning probe microcope an many other application reuiring preciion poitioning e.g. nanomachining [2], [3] etc. A piezoelectric canner comprie a tube of raially pole piezoelectric material, four external electroe, an a groune internal electroe. Other configuration may inclue: a circumferential electroe for inepenent vertical extenion or iameter contraction, an/or ectore internal electroe. Small-eflection expreion for the lateral tip tranlation, erive from the IEEE Piezoelectricity Stanar [4], can be foun in [5]. The foremot ifficultie aociate ith piezoelectric tube canner are the lo mechanical reonance freuency, an hyterei [6]. Techniue aime at areing both mechanical ynamic an hyterei can be groupe generally into to broa categorie, feeforar an feeback. Feeforar techniue, o not inclue a enor but reuire accurate knolege of the uneirable ynamic. Feeback ytem, although more robut to moeling error, are limite by the noie performance an banith of the enor. In many cae it i alo ifficult an/or prohibitively expenive to integrate iplacement enor into the canning apparatu. Coniering the breath of reearch aime at improving can performance, it i urpriing to fin that commercial microcope manufacturer have been reluctant to aopt any ophoticate techniue. The majority of commercial canning ytem operate in much the ame fahion a they i in the early 9. Regarle of the potential benefit, the reuirement for ata acuiition, ophiticate moeling experiment, an aitional enor have everely limite the application of feeforar an feeback can compenation. With thi in min, the reearch preente in thi paper Both author are ith the School of Electrical Engineering an Computer Sciece, Univerity of Necatle, Callaghan 238, Autralia anre.fleming@necatle.eu.au Fig.. Charge riven tube canner ith piezoelectric hunt amping circuit. Fig. 2. (a) v (a) Charge riven tube canner. (b) Voltage euivalent circuit. introuce to imple non-moel bae techniue for the reuction of hyterei an vibration. Since the late 8, it ha been knon that riving piezoelectric tranucer ith current or charge rather than voltage ignificantly reuce hyterei [7]. Simply by regulating the current or charge, a five-fol reuction in the hyterei can be achieve [8]. A uote from a recent paper [9] typifie the entiment toar thi techniue, While hyterei in a piezoelectric actuator i reuce if the charge i regulate intea of the voltage [7], the implementation complexity of thi techniue prevent a ie acceptance []. The firt contribution of thi paper i to preent a ne cla of groune-loa charge amplifier free from DC an lo-freuency voltage rift. The econ contribution i a ne techniue for the reuction of can inuce an exogenou vibration. Dran from the fiel of Smart Structure, e propoe the connection of an electrical impeance to the terminal of one x an y electroe. Uually referre to a piezoelectric hunt amping, thi techniue reult in a ampe electrical reonance capable of ignificantly reucing the magnitue of one or more tructural moe. Figure illutrate an inuctor an reitor connecte to the terminal of a charge riven piezoelectric tube. In thi configuration, the inuctor an reitor are tune to amp the firt x axi cantilever moe. Uneire reonance excitation ue to canning an external iturbance i attenuate. Piezoelectric hunt amping reuire no feeback enor (b) /5/$2. 25 IEEE 62

2 2 G 2 v z 2 2 Z Fig. 3. The electrical euivalent of a charge riven piezoelectric tube ith attache hunt circuit. an i thu immune to the uual problem of lo-banith an meaurement noie aociate ith optical an capacitive enor. Furthermore, a illutrate in Figure 2, e emontrate that the hunt impeance, can be applie to the ame electroe a the riving charge or voltage ource. Thi allo the reunant electroe to be ue for increaing the can range or a a piezoelectric train enor. In Section 2, e icu the moeling of a piezoelectric tube an analyze the effect of a connecte hunt impeance. Implementation iue are then icue in Section 3, folloe by experimental reult an concluion in Section 4 an 5. II. SHUNT CIRCUIT MODELING Although firt appearing in [], the concept of piezoelectric hunt amping i mainly attribute to Hagoo an von Floto [2]. A erie inuctor-reitor netork, a hon in Figure, a emontrate to ignificantly reuce the magnitue of a ingle tructural moe. Together ith the inherent piezoelectric capacitance, the netork i tune to the reonance freuency of a ingle tructural moe. Analogou to a tune mechanical aborber, aitional ynamic introuce by the hunt circuit act to increae the effective tructural amping [2]. A. To Electroe Cae The euivalent electrical moel of a hunte piezoelectric tube (a hon in Figure ) i illutrate in Figure 3. To fin the tranfer function relating iplacement to the riving charge 2 e begin by riting Kirchoff Voltage La aroun the impeance loop an ubtituting v z =, () ()+ () =. () When the oppoing tube electroe are eual in imenion, the charge an 2 have an eual but oppoite influence on the tube eflection an. Furthermore = 2. We efine the tranfer function relating the applie charge to the reulting piezoelectric voltage an tip iplacement a G v () an G (). Due to the ytem ymmetry, G v () can be ue to relate the folloing ignal: () () = 2() 2 () = () = G v () (2) 2 () The principle of uperpoition can be applie to fin an expreion for. () =G v ()() G v () 2 (). (3) Fig. 4. The euivalent feeback iagram repreenting an electrical impeance connecte to the terminal of one tube electroe. The other electroe i riven ith charge. 2 v z K Fig. 5. The electrical euivalent of a piezoelectric tube ith charge rive an hunt circuit connecte to the ame electroe Rearranging (3) in term of 2 an ubtituting into () yiel () 2 () = G v () (4) +G v ()K() here K() = +. (5) The hunte iplacement tranfer function can be erive in a imilar manner, () 2 () = G () (6) +G v ()K() From Euation (6) it i conclue that the preence of an electrical hunt impeance can be viee euivalently a a train-voltage feeback control ytem. A iagrammatic repreentation of euation (6) i hon in Figure 4. Further interpretation an analyi can be foun in [3]. B. Hybri Operation A mentione in the introuction, it i avantageou to connect the hunt impeance an charge ource to the ame electroe. In thi ubection, the electrical filtering effect of on 2 i erive. If uch a filtering effect can be inverte, the charge ource 2 can be ue for canning, analogou to the cae here a hunt impeance i attache to an inepenent electroe. Writing Kirchoff Voltage La aroun the loop, + v z =, (7) 63

3 an ubtituting the folloing, () = v z() + 2(), (8) reult in the loop euation () ()+()+ 2 () =. (9) Given that = G v, e can ubtitute = /G v into (9). After implification, the tranfer function from 2 to can be foun: () 2 () = K()G v() +G v ()K(), here K i a given in (5). Similarly, v z ~ 2 - () 2 () = K()G () +G v ()K(). () Unlike the to-electroe cae, the impeance itort the tube tranfer function from the riving charge 2 to the eflection. Rather than imply aing a train feeback controller to the mechanical ytem, the tranfer function from 2 to no contain a filter F () =K(). An obviou techniue for recovering the natural tube ynamic i to pre-filter the riving charge ith F (). Fortunately thi pre-filtering an inverion i traight-forar to implement in practice. Thi olution i icue in Section III. C. Shunt Impeance Deign The Smart Structure an Vibration Control literature contain a multitue of paive, active, linear, an non-linear piezoelectric hunt impeance eign (reviee in [4], [5]). Only a mall ubet of techniue are uitable for piezoelectric tube amping. The o-calle reonant linear hunt meet all of the reuiite criteria, primarily, they are eay to eign, implement an tune, they offer excellent amping performance (epecially for ingle moe of vibration), they are trictly paive an inject no harmonic, an finally, their preence influence the mechanical ynamic only over a mall freuency range. Reonant linear hunt have been hon to emulate the effect of a tunema mechanical aborber [2]. After examining variou impeance eign, the LCR circuit epicte in Figure 2 a foun to offer goo performance. The preence of a erie capacitance i neceitate by the reuirement for DC tracking. If the impeance of the netork a not infinity at DC, contant tube eflection oul reuire a ramp ignal in charge (eventually aturating the amplifier), thi i reflecte in the can filter F () an it invere F () To amp a ingle moe of tructural vibration, the circuit inuctance L, capacitance C, an piezoelectric capacitance are tune to reonate at the target mechanical freuency ω. Although the capacitance value C i eentially arbitrary, value of to time the piezoelectric capacitance have been foun uitable. To euate the freuencie of electrical an mechanical reonance, the inuctor i tune a follo: L = C +. () C ω The reitance value, epenent on the inherent ytem amping, i mot eaily foun experimentally. For uch ytem, reitance in the orer of kω are typical. Fig. 6. Simplifie implementation of a charge amplifier an piezoelectric hunt impeance. III. IMPLEMENTATION Reonant piezoelectric hunt amping circuit reuire impractically large value of inuctance, typically in the ten of Henry. For thi reaon the hunt amping circuit ill be yntheize artificially uing the charge amplifier. Conier the chematic hon in Figure 6. Neglecting the input 2, the charge applie to the piezoelectric tube i eual to = v z. (2) The impeance (or amittance) experience by the piezoelectric tranucer can be calculate by examining the ratio of current to voltage at it terminal. A the current i eual to, an i efine by (2), the impeance preente to the terminal i imply (a efine by the filter in Figure 6). By implementing the filter any arbitrary impeance can be preente to the terminal of the tranucer. Simple techniue for eigning analog an igital filter that repreent can be foun in [6]. In thi ork a Space DSP ytem i ue to implement an tune the filter. In aition to the charge reuire for hunt impeance ynthei, the aitive charge 2 i ue for tube canning. A mentione in Section II-B, the aitive charge 2 reuire a filter F () to compenate for the electrical ynamic of the hunt impeance hen attache to the ame electroe. A ubtantial implification can be mae by tuying the tructure of the filter F (), F () = K() = + = +. (3) Coniering that the tranfer function ha alreay been implemente, F () can be replace a hon in Figure 6. A. DC Accurate Charge Amplifier One of the key component utilize ithout reference in the previou ection i the charge amplifier. A mentione in the introuction, ubtantial ifficultie have been reveale in attempt to contruct uch a evice for capacitive loa. A olution to the problem of voltage rift in charge amplifier ith capacitive loa a firt preente in [7]. An auxiliary voltage feeback loop a inclue to correct 64

4 vref R C v ref C L v z LC C L L R L LC Fig. 8. Tet for voltage / charge ominance. L Fig. 7. DC accurate charge ource for groune capacitive loa lo-freuency behavior an allo for contant charge offet. The circuit implementation reuire the eign of eparate voltage an charge feeback controller. A implifie eign relying on the intrinic voltage control offere by the paraitic reitance a later preente in [8]. Neither of the amplifier icue have been capable of riving groune loa. A piezoelectric tube have multiple external electroe an a common (often groune) internal electroe, the reuirement for a groune-loa i a neceity. Folloing e preent the eign of a DC accurate groune-loa charge amplifier. Shon in Figure 7, the amplifier incorporate a high common-moe rejection, high common-moe range ifferential tage coniting of the loer opamp, voltage brige, an intrumentation amplifier. The amplifier ork to euate the voltage meaure acro the ening impeance to the reference voltage v ref. To unertan the operation of the amplifier e tuy the tranfer function from the reference voltage v ref to the loa charge LC. L () v ref () = C + C R (4) The tranfer function from reference to actual loa charge can be foun by combining euation (4) ith an expreion relating L to LC, LC () v ref () = L() LC () (5) v ref () L () = C + C R By etting C L R L = C R, i.e. R L R + R LC L = C C L (6) the amplifier ha no lo freuency ynamic an contant gain C Columb/V olt. Effectively the voltage amplifier, comprie of the to reitance R L an R, yntheize the operation of an ieal charge amplifier at lo freuencie. If the amplifier can be viee a the concatenation of a voltage an charge amplifier, an important uetion i: in hat region of operation oe the amplifier operate a a pure charge amplifier?, likeie for voltage operation. Conier the chematic hon in Figure 8. During perfect charge operation i.e., hen LC i correctly regulate to zero, the voltage Fig Piezoelectric tube imenion (in mm). v z ill be eual to. During voltage ominant behavior, v z ill be regulate to zero. Such characteritic can eaily be meaure experimentally. Although the voltage ynamic have been eigne to perfectly yntheize the operation of an ieal charge amplifier, uring voltage ominant operation, if the loa i not purely capacitive, error in LC ill occur. When v ref =, hich implie L =, the tranfer function from to v z reveal the voltage or charge ominance of the amplifier. At freuencie here v z, the amplifier i charge ominant, an voltage ominant hen v z. For the hybri amplifier hon in Figure 7, hen v ref =, v z () () = + (7) R LC L i.e. at freuencie above RC the amplifier i charge ominant, an voltage ominant belo. Obviouly, given euation (7), the objective ill be to elect a loa reitor R L a large a poible. Thi may be limite by other factor uch a opamp current noie attenuation, bia-current bae offet voltage, an the common-moe an ifferential leakage of the opamp. In practice vz() () i bet meaure by imply applying a voltage to another electroe an uing that a a reference. A the freuencie uner conieration are ell belo the tube firt mechanical reonance, the applie voltage ill be relate by a contant. IV. EXPERIMENTAL RESULTS In thi ection, the prototype hunt circuit an charge amplifier are employe to rive a piezoelectric tube poitioner in one imenion. Phyical imenion of the tube can be foun in Figure 9. An ADE Tech capacitive enor a ue to meaure the iplacement ith enitivity V/µm an banith khz. An aluminium cube ( cm cm 65

5 Charge Gain 77.8 nc/v Voltage Meaurement Gain. V/V L 2.9 H C 5 nf R 3.3 kω TABLE I PARAMETERS OF THE CHARGE AMPLIFIER AND SHUNT IMPEDANCE mag (B) cm) i glue onto the tube tip an groune to provie a return for the capacitive enor. Parameter of the hunt impeance an amplifier are hon in Table I. The nominal firt reonance freuency an DC charge enitivity of the tube ere meaure to be 88 Hz an 5.7 m/c (= 5.7 µm/µc). A. Amplifier Performance A icue in Section III-A, the banith of charge ominance a acertaine by zeroing the charge reference an introucing an internal loa voltage. We oberve a charge ominance banith of.8 Hz. Freuencie above thi banith ill experience the full linearity benefit of charge actuation. 3 2 θ f (hz) Fig. 2. Experimental Repone. Natural ( ) an hunt-ampe (- -) tube ynamic meaure from the aitive charge input 2 (C) to the tip iplacement (m). range of ±3 µm i aroun 2% of the full cale eflection, it i often aume that hyterei i negligible at uch lo rive. Similar plot for the ame apparatu ith a ±8 µm rive can be foun in [7], a greater hyterei i exhibite, an heavily reuce through the ue of a imilar charge rive. (nm) r (V) Fig.. Relationhip beteen an applie voltage an the reulting tube iplacement. ( Hz rampe inuoial input). (nm) r (V) Fig.. Relationhip beteen an applie charge reference an the reulting tube iplacement. ( Hz rampe inuoial input). To jutify the ue of charge actuation e emontrate the benefit in Figure an. Hyterei i reuce by approximately 89% imply through the ue of a charge amplifier. Percentage reuction i calculate by meauring the maximum excurion in the minor axi of each plot, then taking the ratio voltage charge. It houl be note that a can B. Shunt Damping Performance Whilt canning at high freuencie, the greatet caue of tracking error i excitation of the mechanical reonance. Such high freuency component can be reuce by filtering, or ignal optimization in cae of perioic canning. Neverthele, ome reiual excitation of the mechanical reonance i inevitable. To illutrate the improvement in triangular canning fielity, an unfiltere 46 Hz triangular aveform a applie to the ytem. The freuency an lack of filtering a choen to illutrate the ort-cae inuce ripple. In practice, the triangle oul be filtere or pae through a feeforar controller to reuce vibration. Regarle of the ripple magnitue, the preence of a hunt circuit provie the ame ecreae in ettling time. At high pee, ignificant increae in fat-axi reolution can be expecte. In the cae here feeforar vibration control [9] i applie, the ampe mechanical ytem oul allo a le evere pre-filter an provie greater immunity to moeling error. The influence of the hunt impeance can be oberve to ignificantly increae the effective amping in Figure 2. Time omain improvement for a triangular canning ignal can be oberve in Figure 3. Another ignificant ource of tracking error i external mechanical noie. Due to the highly reonant nature of the tube, high freuency noie component can excite the mechanical reonance an lea to large erroneou excurion. By applying a voltage to an oppoite electroe, e can imulate the effect of a train iturbance. A ignificant amping of the mechanical reonance by greater than 2 B a oberve. The time omain reuction of reonant vibration can be een in Figure 4. The final tet of uch an apparatu i the ability to track DC charge offet. In Figure 5 a lo freuency triangle ignal a applie to the charge amplifier, at time 3 ec a DC offet euivalent to aroun µm a applie. Aie from the faithful reprouction of a. Hz triangle ave, the charge amplifier reprouce the offet ithout rift. 66

6 (a) r (V) (b) t () Fig. 3. Tube eflection (in nm) reulting from a 46 Hz triangle ave excitation. Uncontrolle (b), an ith L C R hunt impeance (a). (nm) t () Fig. 5. Lo freuency canning reference an reultant tube iplacement ith aitive DC offet. 2 (a) (b) t () Fig. 4. Tube eflection (in nm) reulting from a.6 khz banlimite uniformly itribute ranom train iturbance. Uncontrolle (b), an ith L C R hunt impeance (a). V. CONCLUSIONS In thi paper, e have preente a ne charge amplifier for reuction of hyterei in piezoelectric tube actuator. Uing the intrinic voltage feeback offere by paraitic reitance, lo-freuency voltage rift ha been eliminate to provie DC accurate charge actuation. The propoe charge amplifier i imple to fabricate, an eaily integrate into exiting open-loop or controlle ytem. In aition to hyterei reuction, a imple techniue ha been propoe for the reuction of vibration. Piezoelectric hunt amping involve the connection of an electrical impeance to the terminal of a piezoelectric tranucer. In experiment coniering can-inuce an externallyinuce vibration, an LCR netork reuce the firt reonance freuency by 2 B in magnitue. No feeback enor are reuire. Although charge riven hunt ampe piezoelectric tube can be integrate into previou eign methoologie, the implicity-of-implementation an achievable performance arrant their ue inepenently. Current ork inclue the eign of an all-analog amplifier incorporating both charge actuation, an hunt impeance implementation. REFERENCES [] G. Binnig an D. P. E. Smith, Single-tube three-imenional canner for canning tunneling microcopy, Revie of Scientific Intrument, vol. 57, no. 8, pp , Augut 986. [2] D. Croft, D. McAlliter, an S. Devaia, High-pee canning of piezo-probe for nano-fabrication, Tranaction of the ASME, Journal of Manufacturing Science an Technology, vol. 2, pp , Augut 998. [3] W. Gao, R. J. Hocken, J. A. Patten, J. Lovingoo, an D. A. Lucca, Contruction an teting of a nanomachining intrument. Preciion Engineering, vol. 24, no. 4, pp , October 2. [4] I. of Electrical an E. E. Inc., IEEE tanar on piezoelectricity, ANSI/IEEE St , 988. [5] C. J. Chen, Electromechanical eflection of piezoelectric tube ith uartere electroe. Applie Phyic Letter, vol. 6, no., pp , January 992. [6] H. J. M. T. A. Ariaen, W. L. e Koning, an R. Banning, Moeling piezoelectric actuator, IEEE/ASME tranaction on mechatronic, vol. 5, no. 4, pp , December 2. [7] C. V. Necomb an I. Flinn, Improving the linearity of piezoelectric ceramic actuator, IEE Electronic Letter, vol. 8, no., pp , May 982. [8] P. Ge an M. Jouaneh, Tracking control of a piezoelectric actuator, IEEE Tranaction on control ytem technology, vol. 4, no. 3, pp , May 996. [9] J. M. Cruz-Hernanez an V. Hayar, Phae control approach to hyterei reuction, IEEE tranaction on control ytem technology, vol. 9, no., pp. 7 26, January 2. [] H. Kaizuka an B. Siu, imple ay to reuce hyterei an creep hen uing piezoelectric actuator, Japan Journal of Applie Phyic, Part 2 - Letter, vol. 27, no. 5, pp , May 988. [] R. L. Forar, Electronic amping of vibration in optical tructure, Applie Optic, vol. 8, no. 5, pp , March 979. [2] N. W. Hagoo an A. Von Floto, Damping of tructural vibration ith piezoelectric material an paive electrical netork, Journal of Soun an Vibration, vol. 46, no. 2, pp , 99. [3] S. O. R. Moheimani, A. J. Fleming, an S. Behren, On the feeback tructure of ieban piezoelectric hunt amping ytem, Smart Material an Structure, vol. 2, no., pp , February 23. [4] A. J. Fleming, Synthei an implementation of enor-le hunt controller for piezoelectric an electromagnetic vibration control, Ph.D. iertation, The Univerity of Necatle, Callaghan 238, Autralia, February 24. [5] S. O. R. Moheimani, A urvey of recent innovation in vibration amping an control uing hunte piezoelectric tranucer. IEEE Tranaction on Control Sytem Technology, vol., no. 4, pp , July 23. [6] A. J. Fleming an S. O. R. Moheimani, Improve current an charge amplifier for riving piezoelectric loa, an iue in ignal proceing eign for ynthei of hunt amping circuit. Intelligent Material Sytem an Structure, vol. 5, no. 2, pp , February 24. [7], hybri DC accurate charge amplifier for linear piezoelectric poitioning, in Proc. 3r IFAC Sympoium on Mechatronic Sytem, Syney, Autralia, September 24. [8] K. A. Yi an R. J. Veillette, A charge controller for linear operation of a piezoelectric tack actuator, IEEE Tranaction on Control Sytem Technology, vol. 3, no. 4, pp , July 25. [9] D. Croft, G. She, an S. Devaia, Creep, hyterei, an vibration compenation for piezoactuator: Atomic force microcopy application, Tranaction of the ASME, Journal of Dynamic Sytem, Meaurement, an Control, vol. 23, pp , March 2. 67