in an Atomic Force Microscope
RalphC. Smith
CenterforResearchinScientic Computation
Department ofMathematics
North Carolina StateUniversity
Raleigh, NC 27695-8205
MurtiSalapaka
ElectricalEngineeringDepartment
Iowa StateUniversity
Ames, IA 50011
Abstract
This paper addresses the development of distributed models for the piezoceramic positioning
mechanismsemployed incurrent atomic force microscope designs. Two common congurations for
positioning mechanisms employstacked actuators utilizing d
33
motion and piezoceramic shells
uti-lizingad
31
motion. In bothcases, therelationbetweeninputvoltages andgenerated displacements
exhibits hysteresis and constitutive nonlinearities due to the ferroelectric nature of piezoceramic
materials. Model development for these nanopositioners is considered in two steps. In the rst,
nonlinear constitutive relations that incorporate the hysteresis are developed throughenergy
prin-ciples. In the second step, system models based on these constitutive equations are developed for
thestacked actuator and piezoceramicshell. An abstract formulation encompassingboth modelsis
developedandusedto establishwell-posednesscriteria. Numericalapproximationtechniquesforthe
twomodelsaresummarizedandtheaccuracyofthestackedactuatormodelisdemonstratedthrough
Atomicforcemicroscopes(AFM) providethecapabilityforobtaining angstrom-resolution
measure-ments of both organic and inorganic samples by monitoring forces between the tip of a
micro-cantilever employed in the microscope and atoms in the sample. To illustrate the principles and
componentsof a prototypical AFM, considerthedesigndepicted inFigure 1. The sampleis moved
both laterally and vertically beneath a highly exible micro-cantilever by either a stage forced by
a stacked piezoceramic(PZT) actuator ordirectly by a PZT shell. To ascertain the 3-D structure
of the compound, the sample is moved along an x-y grid usingthe lateralpositioning mechanisms.
As the sample moves, displacements in the cantilever tip are monitored using the photodiode and
correspondingforcesaredeterminedviaHooke's law. Thesampleisthendisplacedinthez-direction
to maintain constant forceswith the displacement determined by a feedback law. A complete scan
in this manner provides a surface image of the compound. Details regarding the construction and
applications utilizing atomic force microscopes and scanning tunnelingmicroscopes (STM) can be
foundin[10 ].
ThedegreeofaccuracytowhichthePZTelementscanlaterallyandverticallypositionthesample
iscrucialtotheresolutionofthenalimages. Twocommonpositioningmechanismsemploystacked
actuatorsutilizingd
33
motionandpiezoceramicshellsutilizingd
31
motioninresponsetoinputelds
or voltages (see Figure 2). The stacked actuators oer the advantage of simple construction while
generatingtheforcesnecessaryforlargescanpositioning. Theshell congurationsarebetter suited
for isolation from exogenous vibrations and small stage construction. While both congurations
providehighlyrepeatableandaccuratesetpointplacement,therelationsbetweeninputvoltagesand
generatedstrainsordisplacementsexhibithysteresis andconstitutivenonlinearitiesasillustratedin
Figure3withdatacollectedfromanAFMemployingastackedactuator. Atlowfrequencies,feedback
mechanismscan adequately attenuate these eects thus leading to the success of the technologies.
However, at the higher frequencies required for current and future applications, two phenomena
degradetheaccuracy achieved bypresentcontroltechniques: (i)Thepiezoceramicmaterialsexhibit
increasedhysteresisand(ii)Thenoisetodataratioincreasestothepointwherefeedbackmechanisms
areamplifyingnoiseratherthan attenuatingunmodeled dynamics.
Piezoceramic
Positioner
(x,y,z)
Sample
Micro-cantilever
Laser
Photodiode
z
y
x
Law
Feedback
z
x
Micro-cantilever
(a) (b)
Figure 1: (a) Conguration of a prototypical AFM; (b) Surface image determined after one lateral
(a)
(b)
z actuator
x-y actuator
Figure2: Actuatorcongurationsemployed forpositioninginanatomicforcemicroscope. (a)
Rect-angularstacked actuator; (b)Cylindricaltransducer.
Oneapproachbeingconsideredto addresstheseproblemsandextendthetechnologyto high
fre-quencyand broadbandregimesistheincorporationoffeedforwardcontrollers utilizingmodel-based
inverses to compensate fornonlinearities and hysteresis inherent to the positioning mechanisms. A
crucialaspect ofthese controldesigns is thedevelopment of modelsforthepositioning mechanisms
which incorporate the nonlinear dynamics butare suÆciently low-order to permit real-time
imple-mentation. To achieve thiseÆciency,bothlow-orderconstitutive modelsand reduced-order system
modelsarerequired.
In thispaper,wedevelopsystemmodelsforthestackedactuator andPZTshelldepictedin
Fig-ure2. Low-orderconstitutiverelationsaredevelopedthroughthequanticationofdomaindynamics
usingenergy principles. Thesenonlinearconstitutive relationsarethenincorporatedinclassicalrod
and shelltheory toobtainPDEmodelswithnonlinearinputs. Anabstract formwhich encompasses
bothmodelsis developedand criteria requiredformodelwell-posedness areestablished. Numerical
methods for discretizing the two modelsare summarized and theaccuracy of the stacked actuator
model isillustratedthrougha comparisonwith theexperimental dataplottedinFigure 3.
−2000
0
2000
4000
6000
8000
10000
12000
14000
−4
−2
0
2
4
6
8
x 10
−6
Electric Field (V/m)
Displacement (m)
Figure 3: Relation between the inputeld E and displacements generated bythe PZT positioning
ToestablishconstitutiverelationsforthePZTmechanisms,wemaketheassumptionthatthe
stress-strain behavior of the material is linear whereas the relation between the applied voltage V or
eld E and generated strains e exhibits hysteresis and saturation nonlinearities as illustrated in
Figure 3. Furthermore, we make the assumption that the relation between the polarization P and
strain is linear so that the hysteresis and nonlinear eects occur in the map F between the eld
and polarization. The assumptionof linear relations betweenP and eis motivated by a number of
classicalreferences(e.g., [5 ,11]) and experimentsarebeingdesignedto establishthevalidityofthis
assumption in the regimes under consideration. The nonlinearities and hysteresis in F have been
experimentallyvalidatedforanumberofPZTcompounds[19 ]andtheseexperimentshavebeenused
to evaluatetheaccuracy of various modelsforquantifyingthenonlinearhysteresismap F.
2.1 1-D Constitutive Relations
In the absence of structural damping, constitutive nonlinearities, or hysteresis, it is illustrated in
[11 ] that consideration of the elastic Gibbs free energy for piezoelectric materials yields the linear
constitutiverelations
e=s P
+P
E = b+
P
orequivalently
=c P
e c P
P
P ="
0
E+:
(1)
Here e;;P and E respectivelydenote the strain, stress, polarizationand eld while s P
and c P
=
1=s P
are thecompliance and Young's modulusat constant polarization values. Moreover, b; and
=1=("
0
) arepiezoelectricandelectriccouplingcoeÆcients,denotestheelectricsusceptibility,
and "
0
is the permittivity of free space. To express the actuator relation in (1) in terms of input
voltages V, thelinear polarizabilityrelationP ="
0
E can be invoked along withthe approximate
formulationV =Eh, whereh isthematerial thickness, to obtain
=c P
e c P
dV=h
P ="
0
E+:
Two congurations for the piezoelectric coeÆcient d = "
0
have been previously mentioned in
the context of the stacked and cylindrical actuators. The stacked actuator utilizes a d
33
eect in
which elds in the 3 direction generate displacements in the 3 direction. Alternatively, transverse
displacements in the cylindrical transducer are generated by the d
31
eect in which eldsthrough
theshell thickness generatelongitudinal(1-direction) displacements.
To incorporate internal damping, saturation nonlinearities, and hysteresis, we generalize (1) to
obtain
=c P
e+c
D _ e c
P
P(E;)
P =F(E;)
(2)
where c
D
is the Kelvin-Voigt damping parameter and F quanties the hysteresis and constitutive
nonlinearities inherent to the materials. A number of techniquesexist fordetermining appropriate
maps F includingPreisach characterizations [3 , 9 ,14 , 22 ,23 ] and domainmodelsbased on
poorlyunderstoodordiÆculttoquantify. However, thelackof anunderlyingenergyprincipleleads
tomodelsrequiringalargenumberofnonphysicalparametersfortheasymmetricresponsesarisingin
atomicforcemicroscopy. Itisalso diÆcultinthisapproachtoincorporatethefrequency-dependence
inherentto thematerials. Forthese reasons,we employdomainwall modelsto specifythemap F.
Asdetailedin[18 ,19 ],therelationbetweentheinputeldandpolarizationP isquantiedintwo
steps. In therst, Boltzmannprinciplesare employed to formulate theprobability(E) =e E=k
B T
of attainingan electrostatic energystate E = Ep wherek
B
T is theBoltzmann orkineticenergy
of the dipoles p. Under the assumption that dipoles can align onlyin thedirection of the eective
eld
E
e
(t)=E(t)+P(t)+E
(t); (3)
oroppositeto it, summation ofdipolecontributionsyieldstheIsingspinrelation
e
P
an
(E(t);P(t);(t))=P
s
tanh(E
e
(t)=a) (4)
for the anhysteretic polarization. Here P
s
denotes the saturation polarization, a is a
temperature-dependentnormalization constant, quantiescoupling eects dueto neighboringdipoles,and E
quantiestheeldcontributionsduetostress-induceddipolerotation. Theexpression(4)isderived
intheabsenceofbiaseldsandhenceyieldsasymmetriccharacterizationofcertaindomainswitching
mechanismswhich producecertainsaturation eectsand hysteresis.
To quantify asymmetric minor loops due to biases in the eld and polarization, it is necessary
to modifytheanhysteretic relationinthemannerdescribedin[17 ]. Forabias level(E
0 ;P
0
), dueto
eldand polarizationreversalat thepoint (E
1 ;P
1
), we employthemodiedanhysteretic relation
P(E;P;)= [ e
P
an (2E
0
E;2P
0
P;) (1 )ÆP
s ]+2P
0
=
P
1 +P
s
e
P
an (E
1 ;P
1
;)+P
s :
(5)
Here Æ=sign dE
dt
=1 and e
P
an
isgiven by(4).
Secondly,thereversibleandirreversibleenergiesassociatedwithdomainwallbendingand
trans-lation are quantied by determining the energy required to reorient dipoles in the presence of the
eective eld E
e
. As detailed in [18 , 19 ], for low frequencydrive regimes the resulting irreversible
and reversiblepolarizations arerespectivelygiven by
dP
irr
dt =
dE
dt
e
Æ(P
an P
irr )
kÆ (P
an P
irr )
(6)
and
P
rev
(t)=c(P
an (t) P
irr )(t):
The total polarizationis thenquantiedbythesum
P(t)=P
irr (t)+P
rev
(t): (7)
Theparametersc;k;P
s
;andaaredeterminedthrougheitherasymptoticrelationsoraleastsquares
t to data[19]whilethe parameter e
Æ =0 or1 is determined directly from eld measurements. We
note that for broadband orhigh frequency drive regimes, frequency-dependent models of the type
=c P
e+c
D _ e c
P
P(E;)
P =cP
an
(E;P;)+(1 c) Z
P
irr
0 e
Æ[P
an
(E;P;) P
irr
(E;P;)]
kÆ [P
an
(E;P;) P
irr
(E;P;)] dP
irr
(8)
which,inthefullycoupledform,areobservedtobenonlineardueto thedependenceofthe
polariza-tion on stress as quantiedbythe termE
in the eective eld. Forapplications (e.g., self-sensing
actuators) which utilize both the direct and converse piezoelectric eects, the coupling which
pro-duces the nonlinearity is important and must be retained. In purely actuator applications, such
as forthe AFMnanopositioner,one can obtainreasonable accuracy withE
=0. In thiscase, the
systemmodelresultingfrom(8)willbelinearinthestatewiththehysteresisandothernonlinearities
appearingonlyintheinputs. Finally,wenotethatwhiletheintegralformulation(8)isadvantageous
for certaintheoretical issues, theformulation (7), withP
irr
specied throughnumerical solution of
thedierentialequation (6), istypicallyemployed formaterial characterization andcontrol design.
As a prelude to establishing model well-posedness for the atomic force microscope positioning
mechanisms, we establish conditions which guarantee that the map between input elds and
gen-erated polarizations is well-dened. It is rst necessary to restrict the admissible parameter set to
guaranteethat
jkÆ (P
an P
irr )j"
0
(9)
for Æ = 1. This places a bound on the polarization curve which is reasonable from both
mathe-maticaland physicalperspectives. Secondly,wedene theset ofturningpointsofE inthemanner
speciedinDenition2.1. TheexistenceofauniquesolutiontotheconstitutivemodelforE
0is
establishedinProposition2.3. The more general case E
6=0 is lessrelevant fortheinitial analysis
of thisactuatorapplication andis notaddressed here.
Denition 2.1 Let T = ft
0 ;t
1 ;;t
n
g denote the set of critical points of E at which dE
dt
changes
sign within the interval [0;T].
Proposition 2.2 Assume that (9) is satised, E 2 C[0;T], and the partition T is specied. The
dierential equation (6) then has a unique solution on [0;T].
Proof. Equation (6)has theform
@P
irr
@t
=G(P
irr ;t)
where
G(P
irr ;t)=
dE
dt
e
Æ(P
an (t) P
irr )(t)
kÆ [P
an (t) P
irr (t)]
:
SinceÆ =1and e
Æ=0or1,itfollowsthatoneachsubinterval(t
j ;t
j+1
),Gand @G
@P
irr
arecontinuous
and boundedwith
@G
@P
irr
k
" 2
0 :
From the theoryof ordinarydierential equations(e.g., see [12 , Chapter 3]), it follows that(6) has
a unique, continuously dierentiable solution on each subinterval with well dened values at the
endpointst
j
. Hence (6)hasa uniquesolutionP
irr
P 2C[0;T]:
Proof. The result followsfrom thedenitionsof P
irr ;P
an
and P.
2.2 2-D Constitutive Relations
The1-Dconstitutiverelations(8)areappropriateformodelingthestacked actuator. However,they
must be extendedto 2-D to quantifythebehavior ofthecylindricalactuator.
Foranappliedeldwhichproducesd
31
motion,theextensionofHooke'slawto a2-Dcylindrical
domainyieldstherelations
e
x =
1
c P
(
x
)+d
31 E
e
=
1
c P
(
x )+d
31 E:
Heree
x ;
x ande
;
denotethestrainsandstressesinthelongitudinalandcircumferentialdirections
and is the Poisson ratio. The inclusion of Kelvin-Voigt damping and formulation interms of the
voltage yieldsthe linearconstitutiverelations
x =
c P
1
2 (e
x +e
)+
c
D
1
2 (e_
x +e_
)
c P
d
31
1
V
=
c P
1
2 (e
+e
x )+
c
D
1
2 (e_
+e_
x )
c P
d
31
1
V
P =d
31 +
V
h :
(10)
The model(10) isappropriate ifhysteresis andnonlinearitiesare negligible.
Forregimesinwhichit isnecessary to incorporatetheinherent hysteresis and constitutive
non-linearities, the polarization-based relation (2) is generalized to obtain the nonlinear constitutive
relations
x =
c P
1
2 (e
x +e
)+
c
D
1
2 (e_
x +e_
)
c P
1
P(E;)
=
c P
1
2 (e
+e
x )+
c
D
1
2 (e_
+e_
x )
c P
1
P(E;)
P(E;)=cP
an
(E;P;)+(1 c) Z
P
irr
0 e
Æ P
an P
irr
kÆ (P
an P
irr )
dP
irr :
(11)
As in the 1-D case, state nonlinearitiescan be eliminated through the assumption that E
= 0 in
(3) which indicates that stresses have negligible eect on the polarization. For low drive regimes
in which sensor (direct)piezoelectric eects are ignored, thisassumption is reasonable and will be
Two common actuator congurations for nanopositioning employ the stacked and cylindrical
actu-ators depicted in Figure 2. We develop here system models for these congurations based on the
nonlinearconstitutive relations summarizedinSection2.
3.1 Stacked Actuator
WhenmodelingthestackedactuatordepictedinFigure2a,itisassumedthatthecross-sectionalarea
issmallascompared tothelengthsothatonlylongitudinalmotionis considered. Thisisconsistent
withthe observationthat thematerial isdrivenina d
33
motion. For initialmodel development,we
assume that theendof theactuator bondedto the AFMbase is xed and theopposite endisfree.
The modelcan bedirectly modiedto accommodatemassorelasticloading at thefreeend.
Therodisassumedtohavecross-sectionalareaAandlength`, anddisplacementsinthe
longitu-dinal(x-axis)aredenotedbyu. Thedensity,Young'smodulusandKelvin-Voigtdampingparameters
arerespectivelydenoted by;c P
and c
D .
Force balancing and the enforcement of boundaryconditions thenyields thestrong form of the
model
A @
2
u
@t 2
= @N
@x
u(t;0)=N(t;`)=0:
(12)
By employing the constitutive relation (8), along with the linear strain relation e = @u
@x
, the force
resultant N = R
A
dAforthe rodis givenby
N =c P
A @u
@x +c
D A
@ 2
u
@x@t c
P
AP(E)
where thepolarizationis speciedby(7)or(8). Ifthecoupling termE
is retained intheeective
eld relation (3), it is observed that the polarization is stress-dependent and hence (12) is fully
nonlinear. However, undertheassumptionthatE
=0 ,thenonlinearityoccurssolelyintherelation
betweentheinputeldE and thepolarizationP,and itis thisregime thatwe consider.
Todeneaweakorvariationalformofthemodelwhichisappropriateforwell-posednessanalysis,
approximation,orcontrol design,the state u is consideredinthe state space X=L 2
(0;`) withthe
innerproduct
h; i
X =
Z
`
0
A dx: (13)
The space of test functions is taken to be V =H 1
`
(0;`) =f 2H 1
(0;`)j (0) =0g with theinner
product
(c P
);
V =
Z
`
0 c
P
A 0 0
dx: (14)
Multiplicationby testfunctions followed byintegration thenyieldsthe weak form
Z
`
0 A
@ 2
u
@t 2
dx+ Z
`
0
c P
A @u
@x +c
D A
@ 2
u
@x@t
@
@x =
Z
`
0 c
P
AP(E) dx (15)
which must be satised forall 2V. Asdetailed inSection 3.3, thisprovides a framework which
A second class of nanopositioning mechanisms are comprised of piezoceramic shells. We focus on
the actuator employed for z-displacements since real-time control of this component is required to
maintain constant forcesbetween thesample and micro-cantilever. The massof theshell employed
for the x-y translationis combined with the mass of the sample to provide an inertial force acting
on thefreeendof thez-actuator.
Formodeling purposes, we assume that the shell has length`, thickness h, and radius R . The
axial direction is specied along the z-axis and the longitudinal, circumferential and transverse
displacements are respectively denoted by u;v and w. The density, Young's modulus and
Kelvin-Voigtdampingparametersaredesignatedby;c P
;c
D
,andtheregionoccupiedbythemiddlesurface
oftheshellisspeciedby
0
=[0;`][0;2]. Finally,weconsiderthecaseinwhichthebottomedge
of theshell(x=0)is clamped and theoppositeend(x=`) isactedupon onlybytheinertial force
associatedwiththecombinedmassofthex-yactuatorand thesample. Forinitialanalysisinvolving
longitudinalinputsand motion,thecombined masscan be represented bya point massm.
As detailedin [4 ,13 ], force and moment balancingyield theDonnell-Mushtarishellequations
R h @ 2 u @t 2 R @N x @x @N x @ =0 R h @ 2 v @t 2 @N @ R @N x @x =0 R h @ 2 w @t 2 R @ 2 M x @x 2 1 R @ 2 M @ 2 2 M x @x@ +N =0 where N x ;N and N x
are general force resultants and M
x ;M
and M
x
are moment resultants.
The boundaryconditions at thexededge x=0are taken to be
u=v=w= @w
@x =0;
and theconditions
N x = m @ 2 u @t 2 ; N x + M x R =0 Q x + 1 R @M x @
=0 ; M
x =0
are enforcedat x =`. The rst resultant condition incorporatesthe inertial force due to the mass
of thepiezoceramiccylinderemployed fory-z translationalong withthemassof thesample.
The force and moment resultants are specied by integrating the stress relations (11), or the
product of the stress and moment arm, throughthe thickness of theshell. In theabsence of shear
stresses,thisyields
N x = c P h 1 2 (e x +e )+ c D h 1 2 (e_
x +e_
) c P h 1 P(E) N = c P h 1 2 (e +e
x )+ c D h 1 2 (e_
+e_
x ) c P h 1 P(E) N x = c P h
2(1+) e x + c D h
2(1+) _ e
M x = c P h 3 12(1 2 ) ( x + )+ c D h 3 12(1 2 ) (_ x +_ ) c P h 3 12(1 ) P(E)
M = c P h 3 12(1 2 ) ( + x )+ c D h 3 12(1 2 ) (_ +_ x ) c P h 3
12(1 ) P(E)
M x = c P h 3
24(1+) +
c
D h
3
24(1+) _ :
The midsurfacestrainsand changesincurvature aregiven by
e x = @u @x ; e = 1 R @v @ + w R ; e x = @v @x + 1 R @u @ x = @ 2 w @x 2 ; = 1 R 2 @ 2 w @ 2 ; = 2 R @ 2 w @x@ : (16)
To reduce smoothness requirements on approximating solutions, it is again advantageous to
consideraweakorvariationalform ofthe problem. To accomplishthis, we considerthe statespace
and space oftest functions
X=L 2
(
0 )L
2
(
0 )L
2
(
0 )
V =H 1
` (
0 )H
1
` (
0 )H
2 ` ( 0 ) where H 1 ` ( 0
) =f 2H 1
(
0
)j(0;) =0g and H 2
` (
0
) =f 2H 2
(
0
)j(0;)=
x
(0;)=0g: For
=(u;v;w) and =(
1 ;
2 ;
3
),theinnerproducts are
h; i
X = Z 0 hu 1 d+ Z 0 hv 2 d+ Z 0 hw 3
d+mu
1 (`) (17) and (c P ); V = Z 0 c P h 1 2 (e x +e ) @ 1 @x + 1 2R (1 )e x @ 1 @ d + Z 0 c P h 1 2 (e +e x ) @ 2 @ + 1 2R (1 )e x @ 2 @x d + Z 0 c P h 1 2 " 1 R (e +e
Z 0 R h @ 2 u @t 2 1 +R N
x @ 1 @x +N x @ 1 @ R c P h 1 P(E) @ 1 @x d R m @ 2 u @t 2 1
(`)=0
Z 0 R h @ 2 v @t 2 2 +N @ 2 @
+R N
x @ 2 @x c P h 1 P(E) @ 2 @
d =0
Z 0 R h @ 2 w @t 2 3 +N 3 R M x @ 2 3 @x 2 1 R M @ 2 3 @ 2 2M x @ 2 3 @x@ + c P h 1
P(E)
3 + c P h 3 R 12(1 ) P(E)r 2 3
d=0
(19)
for all = (
1 ;
2 ;
3
) 2 V. The internal resultants N
x =N x c P h 1
P(E); N
= N c P h 1 P(E); N x =N x ;M x =M x c P h 3 12(1 )
P(E);M
=M c P h 3 12(1 )
P(E) and M
x =M
x
are comprised of
the material contributions present in the absence of applied elds E. It is the model (19) that we
considerwhen establishing criteria formodel well-posedness and developing fulland reduced-order
approximation techniques.
3.3 Abstract Model Formulation
To providea frameworkinwhichto establish modelwell-posedness,weconsideran abstract
formu-lation of the models based upon stiness and damping sesquilinear forms. To this end, we dene
i
:V V !Cl ;i=1;2,by
1 (; )= (c P ); V 2
(; )=h(c
D ); i
V
(20)
where V = H 1
`
(0;`) with the inner product (14) for the rod model and V = H 1
` (
0 )H
1 ` ( 0 ) H 2 ` ( 0
)with theinnerproduct(18)forthecylindricalshell. Inbothcases, (c P ); V diersfrom h(c D ); i
V
onlyin that Young'smoduliare replaced byKelvin-Voigt damping parameters. It can
bedirectly veriedthat thestinessform
1
satises
(H1) j
1
(; )jc
1 jj
V j j
V
; forsome c
1
2lR (Bounded)
(H2) Re
1
(;)c
2 jj
2
V
; forsome c
2
>0 (V-Elliptic)
(H3)
1
(; )=
1
( ;) (Symmetric)
forall ; 2V. Moreover, the dampingterm
2
satises
(H4) j
2
(; )jc
3 jj
V j j
V
; forsome c
3
2lR (Bounded)
(H5) Re
2
(;)c
4 jj
2
V
; forsome c
4
Theinputoperatorsdependuponthespecicmodel. Fortherodmodel(15), B:U !V
isdened
by
h [B(E)](t); i
V
;V
=[P(E)](t) Z
`
0 c
P
A d
dx
dx (21)
whereasfortheshellmodel(19), B isdened by
h [B(E)](t); i
V
;V
= [P(E)](t) Z
0
R c
P
h
1
@
1
@x +
c P
h
1
@
2
@
+ c
P
h
1
3 +
c P
h 3
12(1 )R r
2
3
d:
(22)
Inbothcases,h ;i
V
;V
denotestheusualdualityproduct. ItisobservedthattheoperatorB canbe
expressedas
[B(E)](t)=[b(E)](t)g ; g2V
(23)
where
[b(E)](t)=c P
A[P(E)](t)
g( )= Z
`
0 0
dx
(24)
fortherod and
[b(E)](t) = c
P
h
1
[P(E)](t)
g( )= Z
0
R @
1
@x +
@
1
@ +
3 +
h 2
(1+)12R r
2
3
d
(25)
fortheshell.
The weak form of the rod model (15) or shell model (19) can then be written in the abstract
variationalform
hy(t); i
V
;V +
2
(y(t);_ )+
1
(y(t); )=h [B(E)](t); i
V
;V
for all 2 V. For therod model,y =u, V =H 1
`
(0;`), the duality product is theextension of the
X-inner product dened by (13) and B is dened by (21). For theshell model,y = (u;v;w), B is
denedin(22), V =H 1
` (
0 )H
1
` (
0 )H
2
` (
0
), and thedualityproductistheextension of(17).
Alternatively,one can denetheoperators A
i
2L(V;V
);i=1;2,by
h A
i
1 ;
2 i
V
;V =
i (
1 ;
2 )
and formulatethe modelinoperatorform as
y(t)+A
2 _
y(t)+A
1
y(t)=[B(E)](t) (26)
inthedual spaceV
. Well-posedness criteria areestablishedinthefollowinglemma andtheorem.
Lemma 3.1 Dene the operator B by (21) or (22) and let T denote the partition specied in
Def-inition 2.1. Under the assumption that E 2 C[0;T] and that the parameters satisfy (9), it follows
that
B(E)2L 2
((0;T);V
Proof. From (23), theoperator B can beexpressed as [B(E)](t)=[b(E)](t)g whereg 2V and b
aredened in(24)or(25). It follows fromProposition2.3that
b():C[0;T]!C[0;T]
sothat thenorm
k[B(E)](t)k
V
=sup
v2V
j[b(E)](t)g(v)j
kvk
V
existsforeach t2[0;T]. Since k[B(E)](t)k
V
=j[b(E)](t)jkgk
V ;
itfollows that
kB(E)k 2
L 2
((0;T);V
)
max
t2[0;T]
j[b(E)](t)j 2
T kgk 2
V
which impliesthat
B(E)2L 2
((0;T);V
):
Theorem 3.2 Let
1 and
2
be given by (20) and consider the assumptions of Lemma 3.1. There
then exists a unique solution y to (26) which satises
y2C((0;T);V)
_
y2C((0;T);X):
The resultfollows directly fromTheorem4.1 of[4] orTheorem2.1 and Remark2.1 of[2 ].
4 Numerical Approximation Techniques
To implement the models for either the rectangular stacked actuator or the cylindrical actuator,
it is necessary to develop appropriate approximation techniques to discretize the modeling PDE.
To accomplish this, we consider general Galerkin methods in which basis functions are comprised
of spline or spline-Fourier tensor products. The resulting methods can accommodate a variety of
boundary conditions,are suÆcientlyaccurate to resolve ne-scale dynamics,and can be employed
forconstructing reduced-order PODapproximatesforreal-time implementation.
4.1 Stacked Actuator Model
To approximate theweak form of thestacked actuator model(15), we employa niteelement
dis-cretization in space followed by a nite dierence discretization in time. The semidiscrete system
resultingfromtheniteelementapproximationisappropriatefornitedimensional,continuoustime
controldesignwhereasthefullydiscretesystemisamenable tosimulationsand control
implementa-tion.
To obtain a semidiscrete system, we consider a uniform partition of [0;`] with points x
i = ih;
basisf
i g
i=1
is thencomprisedof oflinear splines
i (x)= 1 h 8 > < > : (x x i 1 ) ; x
i 1
x<x
i
(x
i+1
x); x
i
xx
i+1
0; otherwise
; i=1;;N 1
N (x)= 1 h ( (x x N 1 ) ; x
N 1
xx
N
0 ; otherwise
(see [16 ] fordetailsregardingtheconvergence analysisforthemethod). The solutionu(t;x)to (15)
isthen approximatedbytheexpansion
u N
(t;x)= N X j=1 u j (t) j (x): BecauseV N =spanf i g N i=1
V =H 1
`
(0;`),theapproximatesolutionsatisestheessentialboundary
conditionu N
(t;0)=0 and can attainarbitrary displacementsat x=`.
The projectionof the problem(15) onto the nitedimensionalsubspace V N
yieldsthe
semidis-crete system
_
y(t)=Ay(t)+[F(E)](t)
y(0)=y
0
(27)
wherey(t)=[u
1
(t);;u
N (t);u_
1
(t);;u_
N (t)] T and A= " 0 I Q 1 K Q 1 C #
; [F(E)](t)= " 0 Q 1 [f(E)](t) # : (28)
The mass,stiness, dampingand forcingmatriceshave thecomponents
[Q] ij = Z ` 0 A i j
dx ; [f(E)]
i
(t)=[P(e)](t) Z ` 0 c P A 0 i dx [K] ij = Z ` 0 c P A 0 i 0 j
dx ; [C]
ij = Z ` 0 c D A 0 i 0 j dx:
Thesystem(27)canbeemployedfornite-dimensionalcontroldesign. Forsubsequent
implemen-tation, we considera temporal discretizationof (27)using a modied trapezoid rule. For temporal
stepsizes t,thisyieldsthedierence equations
y j+1 =Ay j +[B(E)](t j ) y 0
=y(0);
(29)
wheret
j
=jt, y
j approximates y(t j ), and A= I t 2 A 1 I+ t 2 A
; [B(E)](t
j )=t
I t 2 A 1 [F(E)](t j ):
This yields an A-stable, single step method requiring moderate storage and providing moderate
Dueto theinherentcouplingbetweenlongitudinal,circumferential,and transverse displacementsin
combination withthe 2-Dsupportofthemiddlesurface, thenumerical approximation of themodel
forthecylindricalactuator issignicantlymore complicatedthantheapproximation ofthestacked
actuator model. Among the issues which must be addressed when constructing nite element or
general Galerkin methods for the shell is the choice of elements which avoid shear and membrane
locking and the maintenance of boundary conditions. We summarize here a spline-based Galerkin
methoddevelopedin[6 ] forthinshellsanddirect thereadertothat sourcefordetailsregardingthe
construction of constituent matrices and convergence properties of the method. Details regarding
theuseof thisapproximationmethod forLQRcontrolof shellsutilizingpiezoceramicactuatorscan
befoundin[7 , 8 ].
The bases forthe u;v and w displacementsare respectivelytaken to beB
u
k
(;x)=e im
B
u
j (x);
B
v
k
(;x) = e im
B
vj
(x); and B
w
k
(;x) = e im
B
wj
(x) where B
uj ;B
vj
and B
wj
are cubic B splines
modied to satisfytheboundaryconditions(e.g., seepage79 of [16]). The approximating subspace
is V N
= spanfB
u
k
gspanfB
v
k
gspanfB
w
k
g and displacements are approximated through the
expansions
u N
(t;;x)= N
u
X
k=1 u
k (t)B
u
k (;x)
v N
(t;;x)= Nv
X
k=1 v
k (t)B
v
k (;x)
w N
(t;;x)= N
w
X
k=1 w
k (t)B
w
k (;x):
The restrictionoftheproblem(19)toV N
and constructionoftheforcingvectorsthenyieldsthe
matrixsystem
_
y(t)=Ay(t)+[F(E)](t)+G(t)
y(0)=y
0
where y(t) = [#(t); _
# (t)] T
2 lR 2N
, #(t) = [u
1
(t);;u
Nu (t);v
1
(t);;v
Nv (t);w
1
(t);;w
Nw (t)]
T
,
N =N
u +N
v +N
w
;denotesthecoeÆcientsand theirderivatives. The matrixAand vector F have
thegeneralform(28)and thevectorG(t)=[0; Q 1
g(t)] T
incorporatestheboundarycontributions
duetotheinertialmassand appliedloadat theendof thecylinder. The readerisreferredto [6 ]for
detailsconcerning theconstruction ofthe mass,stiness and dampingmatrices Q;K and C.
5 Model Validation
To illustrate the attributes and capabilities of the models, we consider the characterization of the
stackedactuator depictedinFigure 2a. To accommodate thedimensionsandlongitudinaldynamics
exhibited by theactuator, the rodmodel(15) resulting from thenonlinearconstitutive relation(8)
isemployed. Fornumericalimplementation, we employtheresulting discretesystem(29).
The stacked actuator has a width and thickness of 5 mm and length of 20 mm. The Young's
modulusand densityspeciedbythemanufacturerare c P
=6:510 10
N/m 2
and =7730 kg/m 3
.
Thevaluesc
D
=6:510 7
Ns/m 2
and =3:5210 2
fortheKelvin-Voigtdampingparameter and
piezoelectriccoupling coeÆcient were estimatedthrougha leastsquarest tothe data. Finally,the
−2000
0
2000
4000
6000
8000
10000
12000
14000
−4
−2
0
2
4
6
8
x 10
−6
Electric Field (V/m)
Displacement (m)
Data
Model
Figure 4: Experimentalstacked actuator dataand modelprediction.
to obtainthe parameters =4:510 6
Vm/C, a= 4:610 5
C/m 2
, k = 1:410 5
C/m 2
, c =:81
and P
s
=:49C/m 2
forthe hysteresis component of themodel.
The model prediction obtained with these parameters is compared with the actuator data in
Figure 4. It is observed that the model accurately quanties both the constitutive nonlinearities
and hysteresisinherent tothisnanopositioningmechanism to adegree which issuÆcientlyaccurate
forbothdevicecharacterizationand theconstructionofinversecompensators forsubsequentcontrol
design.
Furthermore, it was observed that because longitudinal piezoelectric inputs are manifested as
stress resultants at the endof the rod, there is little spatialvariation forthe considered
congura-tion. Hence convergence of the niteelement method was attained by N = 8 with only moderate
discrepancies produced with fewer basis functions. The ramications of this are twofold: (i) the
method is very fast which facilitates real-time implementation, and (ii) suÆcient accuracy for
cer-tain drive regimes may be attained with lumped ODE models for the rod. The quantication of
conditions under which ODE rod models may be applicable is under investigation. It is noted,
however, that such simplicationswill not be feasible for the shell model used to characterize the
cylindricalactuatordueto the inherent coupling betweencomponent displacements. Hencethe full
PDE model or appropriate reduced-order models will be required for all congurations utilizing
cylindricalactuators.
6 Concluding Remarks
ThePDEmodelsdevelopedhereprovideameansofcharacterizingtwooftheactuatorcongurations
employed incurrent atomicforcemicroscopesand projected nanopositioningdesigns. Asillustrated
through data from the stacked actuator, hysteresis and constitutive nonlinearities inherent to the
constituent piezoelectricmaterialsaremanifestedeven atthelowdrivelevelsrequiredfor
nanoposi-tioning. It iscrucial that these nonlinearmechanismsbe accommodated in device characterization
and subsequent controldesignto achieve theangstrom-levelresolution requiredfromthe actuators.
At lowdrive frequencies,the deleteriouseects of the hysteresis and nonlinearitiescan be
basedonlow-ordermodelsinconjunctionwithfeedforwardcontroldesignswillberequiredtoachieve
thenecessary accuracy.
Twocrucialattributes of modelsconsideredforuse ininverse compensator designarethe
prop-erties that they can be inverted and are suÆcientlylow-orderto permit real-time implementation.
Asillustrated [15 ],modelsof thetype presentedhere are amenableto either fullinversion,through
the solutionof complementaryODE, or partialinversionthrough consideration of theanhysteretic
componentofthemodel. Thedesignandimplementationoffeedforwardcontroldesignswhichutilize
theresulting inverse compensators is under investigation. Asdiscussed inSection 5,the rod model
exhibits minimal spatial variation in certain drive regimes which indicates that lumped parameter
ODE modelsmay be adequate for certain nanopositioner designs. For regimesin whichsignicant
spatialvariationnecessitatestheretentionofthefullPDE,aswellasthecharacterizationof
nanopo-sitionersemployingcylindricalactuators,thedevelopmentofreduced-ordermodelswillbenecessary
to achieve real-time, broadbandimplementation. We arecurrentlypursuing thedevelopment of
ac-tuator models based on proper orthogonal decomposition (POD) expansions due to recent success
utilizing this reduced-order approach in control design for cylindrical shells with surface mounted
piezoceramicactuators [1 ]. Theresulting reducedmodelsemployinginverse algorithmswillthenbe
consideredforbroadbandcontroldesigninnanopositioningapplications.
Acknowledgements
ThisresearchwassupportedinpartbytheAirForceOÆceofScienticResearchunderthegrant
AFOSR-F49620-01-1-010 7 and the NationalScience Foundationunder thegrant CMS-0099764.
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