IOP PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING

J. Micromech. Microeng. 22 (2012) 125013 (18pp) doi:10.1088/0960-1317/22/12/125013

Automated boundary interaction forcecontrol of micromanipulators with in situapplications to microsurgerySohrab Eslami1 and Nader Jalili2,3

1 Engineering Research Center–Computer Integrated Surgical Systems and Technology,Johns Hopkins University, Baltimore, MD 21218, USA2 Piezoactive Systems Laboratory, Department of Mechanical and Industrial Engineering,Northeastern University, Boston, MA 02115, USA

E-mail: s.eslami@jhu.edu and n.jalili@neu.edu

Received 12 June 2012, in final form 17 August 2012Published 6 November 2012Online at stacks.iop.org/JMM/22/125013

AbstractMost recent works on miniature tasks are concentrated on developing tools to take advantageof the visual servoing feedback to control the ultra-small interaction forces. This paper spansan extensive platform for automatic controlling of boundary interaction forces with highprecision in the level of micro/nano-Newton with extensive micro/nanoengineeringapplications such as the microsurgery. To this end, a comprehensive piezoresistivemicrocantilever (PMC) model considering the shear deformation and rotary inertia effectstreating as the distributed-parameters model along with the Hertzian contact force is presented.The purpose of considering the Hertzian contact force model is to investigate the dynamicresponse of the interaction force between the microcantilever’s tip and the specimen.Afterward, a control platform is introduced to automatically manipulate the PMC to follow anideal micro/nano-interaction force. By using the integrated PMC with the micromanipulatorand a digital signal processor, an intuitive programming code is written to incorporate themicromanipulator and the controller in a real-time framework. To calibrate and verify theinduced voltage in the PMC, a self-sensing experiment on the piezoelectric microcantilever iscarried out to warrant the calibration procedure. Some experiments are established to affirm thevalidity of the proposed control for the autonomous real-time tasks on the boundary interactionforce control. Unlike the conventional research studies, the measured force here contributes asthe feedback source in contrast to the vision feedback while force sensors possess moreprecision, productivity and small size. This technique has several potential applications listedbut not limited to the micro/nanomanipulation, developing artificial biological systems (e.g.,fabricating hydrogel for the scaffold), and medicine such as microsurgery. As a result, usingthe proposed platform, we are able to manipulate and control the boundary interaction forcesprecisely at the level of few μN and the presented theoretical and experimental results to studythe mechanical responses of the interaction force are in a good agreement.

(Some figures may appear in colour only in the online journal)

1. Introduction

Industrial automation has been extensively used in nowadaysapparatuses and systems in an immense range from aerospace

3 Author to whom any correspondence should be addressed.

to nanotechnology and medical instrumentations. Due tophysical and cognitive limitations of human in operatingsystems in hard situations and dangerous environments,automation for the purpose of enhancing human’s dexterityis, however, replaced. In the category of medical systemsand robotics, the automation subject has a very crucial place

0960-1317/12/125013+18$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA

J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

in acquiring high precision and long-time operation withthe least mistakes and tiredness. This could become moresignificant when instruments are downsized. Robot-assistedsystems with applications in the surgical instruments haveattracted concentration of the research in the medical robotics,such as the in situ eye microsurgery. Accuracy assessmentof neurosurgical robots [1], robot-assisted surgery in skulldrilling [2, 3] and automated implantation of medical gadgetsin the patient’s body [4] are some of the recent researchworks. Robotic surgery has been very beneficial to enhancethe human capabilities to assist the surgeon during theoperation [5–7]. By recent advancements in different aspects ofengineering, more surgical operations today can be performedat the sub-millimeter and sub-Newton accuracy, such as therobotic microsurgery for corneal transplantation [8] and microneurosurgery [9, 10].

In any automation tasks, the feedback source is one of themost essential factors in assessing real-time and fast operation,reliability and more accuracy. Many robotic systems benefitfrom imaging feedback through different sources, such as themagnetic resistance imaging (MRI), computed tomography(CT), or a CCD camera. Despite significant advantages of thesetechniques, lack of the true real-time medical imaging andperhaps low resolution, or difficulty in finding exact positionsof the objects in the image makes the automated closed-loop control tasks difficult to tackle when fast operation isessential. Innovative force sensor instruments are alternativelyimplemented in which the precision and repeatability areimportant [11–13]. Durability and small sizes of force sensorshave made them more promising to be operated in the clinicaluse, while in contrast, due to the large size of image-guidedtools and their difficulty/risk in implementation such as theMRI and CT, they have confined their usage in the real-timesurgery [14, 15].

Recent automated microinjection of biological cellcultures achievements have opened a widespread windowto scientists to carry out high resolution and delicate taskswith absolute precision to penetrate into the cell and injectmicrofluidic of genetic materials. The applied force forthe microinjection is essential to be accurately sensed andcontrolled not to exceed a threshold and destroy the cell;therefore, the force control paradigm is necessary [16, 17].Several BioMEMS devices are correspondingly developed tofabricate the scaffolds and artificial extracellular matrices withapplications in biological structures and cultivated cells [18].

In this paper, we introduce a novel fully automatedframework integrating a micromanipulator with thepiezoresistive microcantilever (serving as a force sensor) alongwith the designed closed-loop control scheme to autonomouslycontrol the boundary interaction forces with promisingapplications in microsurgery. In contrast to other recentdeclared works, by employing a piezoresistive microcantileveras a force sensor, the precision in this technique is very highand very close to few μN along with other aspects suchas high-speed operation and repeatability. First, a detailedmodel of the piezoresistive microcantilever with the Hertziancontact force model is considered and the model-basedcontrol of such system is designed for the control purposes

Figure 1. MM3A R©micro/nanomanipulator [19].

of the boundary interaction forces. The comprehensivemodel of the microcantilever consists of the rotary inertiaand shear deformation effects treating as the distributedparameters’ model (DPM). In the experimental aspect, usingthe MM3A R© micromanipulator and integrating with a digitalsignal processor and a computer casting as the controller tomake sensible decisions, a platform for the automated controlof the interaction force is designed and discussed. An intuitiveprogramming code is provided to integrate the controllerwith the NanoControl and the micromanipulator and, hence,to transform the outgoing signals in the form of the stringcommands into the electrical voltages. It is concluded that thepresented experimental and numerical results for the case ofconsidering a comprehensive microcantilever with a Hertziancontact force are in a good agreement.

2. System description—MM3A R©: driving principle,operation and force measurement system

Most of the recent works have concentrated on the designand development of novel devices and techniques to imageand manipulate ultra-small objects under different conditionsand environments. There are a numerous number of studiesusing the online specimen image as the feedback source forthe purpose of controlling a micro/nanoelectromechanicalsystem. Having reliability as one of the most advantagesof this technique, however, lack of the precision and thereal-time tracking position of the manipulator’s tip stillremains an essential problem. The micromanipulator usedhere, namely MM3A R© (Kleindiek R© Nanotechnik) shown infigure 1 is aimed for the micro/nanomanipulation tasks. Thismicromanipulator is comprised of three joints capable ofperforming tasks with high precision and adaptability undervarious operating conditions.

The MM3A R© considering here has three degrees-of-freedom (DOFs) with revolute–revolute–prismatic (RRP)joints. An important feature of the MM3A R© is having nobacklash or reversal shortfall in the joints while driving, inaddition to the fact that it can operate with a high velocity up

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

Amplifier

PiezoresistiveMicrocantilever(Force Sensor)

NanoControlController

Command signals(string command)

Command signals(electrical voltage)Reference Force

+-

Adjustable LCD

A/D Channel ofdSPACE®

Output BNC

Bridge supply voltage (Default = 2.5 V)

Force Measurement System

Figure 2. Closed-loop control block diagram with the force feedback.

to 10 mm s−1. Interested readers may refer to [19] for moredetailed information about the MM3A R© specifications.

The detailed concept of the automatic boundary forcecontrol is presented in figure 2. Having a PMC casting asa force sensor, some electrical voltages corresponding tothe interaction forces are induced in the piezoresistive layer.The force measurement system (FMS) is composed of aninternal amplifier operating to amplify the received signalfrom the PMC and sent it to the analogue-to-digital (A/D)channel of a digital signal processor (dSPACE R©). Based onthe desired value of the interaction force, the controller makesan appropriate decision to manipulate the micromanipulatorin any desired direction until the PMC senses the favoriteinteraction force.

This introduced force sensor herein is capable ofmeasuring ultra-small interaction forces, and hence, somemathematical representation of the corresponding system withthe precise controller is required in this regard.

3. Micromanipulator mathematical modeling

MM3A R© could be substituted by a 3-DOF manipulator withrevolute–revolute–prismatic joints is assumed with a fixedbase. This manipulator is able to manipulate in a 3D spacesuch that its end-effector is able to follow a determined pathfor some specific duties such as manipulating nanoparticles orcontrolling small forces in some biological cells. A schematicview of this manipulator is shown in figure 3. By implementingthe Denavit–Hartenberg convention [20] and computing thekinematics equations, the position of the end-effector can beobtained as follows:⎡

⎣Px

Py

Pz

⎤⎦ =

⎡⎣c1c2x + c1c2l1+c1l2

s1c2x + s1c2l1+s1l2s2x + s2l1

⎤⎦ , (1)

where c and s denote the cosine and sine, Px, Py, Pz are thepositions of the end-effector in the three principal directions,respectively.

Furthermore, the velocity and acceleration of the end-effector may be calculated by taking the first and second timederivatives from both sides of (1). Therefore, by having thecurrent position, velocity and acceleration of the end-effectorand applying the Jacobian matrix, the Cartesian space couldbe converted into the Joint space.

Figure 3. A scheme of the RRP micromanipulator.

The Jacobian matrix is defined as follows:

θ = J−1(θ )v. (2)

Using the Newton–Euler formulation and calculating theinternal and external iterative relations [20], the dynamicequation of the MM3A R© can be expressed as follows:

τ = M (θ1) θ + V(θ, θ

) + G (θ ) , (3)

where θ = [θ1 θ2 · · · θn]T ∈ Rn is the joint position vector,

M(θ ) is the n × n inertia matrix of the manipulator, v(θ, θ ) isan n × 1 vector of the centrifugal and coriolis terms, and G(θ )

is an n × 1 vector of gravity terms. Equation (3) is derived forthe torques and force of the joint actuators in such a way thatτ1 is defined as the torque in the first revolute joint, τ2 is thetorque in the second revolute joint, and f is the force in theprismatic joint. All these terms are considered as the inputs tothe system, while θ1, θ2 and x are the system outputs. For thispurpose, the velocity and acceleration of each joint must becalculated.

3.1. Comprehensive beam model assumption with theHertzian contact force model

There are various models introduced to demonstrate thedynamic behavior of the atomic force microscope (AFM)

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

microcantilever which can be categorized as the lumped-parameters’ models (LPMs) and the DPMs. It is apparent thatthe LPMs are aimed only for the sake of the simplicity and,hence, are not comprehensive enough to consider all aspects ofthe mechanical properties of the probe. The best representativeof the LPMs could be the mass–spring–damper systems havingsome definite numbers of DOFs. In contrast, DPMs are morecomprehensive but require more effort to determine the closedform solution of the dynamic response for which the mosttwo popular models are the Euler–Bernoulli and Timoshenkomodels. In the more comprehensive model (i.e. Timoshenkobeam) the rotary inertia and the shear deformation effectswhen the aspect ratio of the microcantilever is relativelysmall are included. In the design and manufacturing processof microcantilevers, there are still needs for having suchmodels to represent the realistic behavior of the probes, andby emphasizing on this point here, we present an impressionof the Timoshenko beam to provide a basis to investigate theeffects of the contact AFM with the Hertzian force model.The novelty of this work here is to consider the piezoresistivemicrocantilever representation as the most comprehensivemodel for the mechanical probe implemented in an AFMsystem. Consequently, the interaction of the contact forceis considered and its influence on the dynamic behavior ofthe probe is followed by applying an advanced model-basedautonomous controller framework.

Having the framework of the continuum mechanics andapplying the extended Hamilton’s principle for a continuoussystem with an infinite number of DOFs, the partial differentialequation (PDE) of the microcantilever based on the energymethod may be represented by implementing the procedurediscussed here. The kinetic energy of the piezoresistivemicrocantilever system can be represented as

T = 1

2

∫ L

0ρ(x)(h(t) + w(x, t))2 dx

+1

2

∫ L

0

ρ(x)I(x)

Aϕ2(x, t) dx + 1

2me(h(t) + w(L, t))2

+1

2mbh2(t), (4)

where ρ is the volumetric density, A is the beam cross-sectionalarea, h(t) is the base oscillation, w(x, t) and ϕ(x, t) are thetransverse displacement and the orientation of the beam crosssection, respectively, me is the tip mass, mb is the base massand h(t) is the driving of the microcantilever’s base.

The total potential energy of the system may bewritten as

U = 1

2

∫ L

0[(EI(x))eff ϕ

′2(x, t)

+ K(AG(x))eff(w′(x, t) − ϕ(x, t))2] dx, (5)

with EI(x) being the beam rigidity, K the shear deformation,G(x) shear modulus, fb(t) is the applied force to the base ofthe microcantilever, the interaction force fts(t) (assuming to bethe Hertzian contact force model here) can be upper bounded| fts(t)| � C1, and viscous and structural damping forces withdamping coefficients B and C, respectively, can be expressed

in the virtual work as follows:

δW nc = fbδh(t) + fts(t)δw(L, t) − B∫ L

0w(x, t)δw(x, t) dx

−C∫ L

0w′(x, t)δw(x, t) dx. (6)

In equations (4)–(6), the term ρ(x) as the volumetric densitycan be defined as

ρ(x) = (S(x)ρptp + ρbtb

)b (7)

where ρp is the piezoresistive layer volumetric density and bis the width of the microcantilever which is equal to the widthof the piezoresistive layer bp, tb and tp are the thicknesses ofthe microcantilever and piezoresistive layer, respectively [21].

The function S(x) in (7) is calculated as

S(x) = H(x − x1) − H(x − x2) (8)

where H(x) is the Heaviside function, and

EI(x) = EbIb + EpIpS(x), (9)

EbIb = 1

12bEbt3

b and

EpIp = bEptp

[t2b

4+ tbtp

2+ t2

p

3− zn

2(tb + tp)

]

for zn = Eptp(tb + tp)

(Ebtb + Eptp), (10)

where EbIb and EpIp are the rigidity, Eb and Ep are Young’smoduli of the microcantilever and piezoresistive layers,respectively, and zn is the neutral surface [21]; all theparameters presented in figure 4.

The equations of motion can be acquired by using theextended Hamilton’s principle:∫ t2

t1

(δT − δU + δW nc) dt = 0. (11)

After substituting the kinetic and potential energies andvirtual work into Hamilton’s principle and performing somemanipulation and mathematical arrangements, the PDE of apiezoresistive microcantilever subjected to an interaction forcecan be summarized as⎧⎪⎪⎨⎪⎪⎩

−K(AG(x))eff(w′′(x, t) − ϕ′(x, t)) + ρ(x)(h(t) + w(x, t))

+ Bw(x, t) + Cw′(x, t) = 0, (a)

((EI(x))eff ϕ′(x, t))′ + K(AG(x))eff(w

′(x, t) − ϕ(x, t))− ρ(x)I(x)

A ϕ(x, t) = 0, (b)(mb + me +

∫ L

0ρ(x) dx

)h(t) +

∫ L

0ρ(x)w(x, t) dx

+ mew(L, t) = fb(t) + fts(t). (c)

(12)

Subject to the boundary conditions (BCs) as⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

K(AG(x))eff(w′(L, t) − ϕ(L, t))

+ me(h(t) + w(L, t)) − fts(t) = 0,

(EI(x))eff ϕ′(L, t) = 0,

w(0, t) = 0,

ϕ(0, t) = 0.

(13)

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

Figure 4. Schematic view of the AFM contact mode with the Hertzian force model.

In order to facilitate solving of PDE (12), ϕ(x, t) from (12a) iscalculated and substituted into (12b) to provide a fourth-orderPDE for w(x, t) integrating a piezoresistive layer as follows:

(EI(x))eff w′′′′(x, t) −

(ρ(x) (EI(x))eff

K (AG(x))eff+ ρ(x)I(x)

A

)w′′(x, t)

− B (EI(x))eff

K (AG(x))effw′′(x, t) − C (EI(x))eff

K (AG(x))effw′′′(x, t)

+ ρ(x)(h(t) + w(x, t)) + Bw(x, t) + Cw′(x, t)

+ ρ2(x)I(x)

K(A2G(x)

)eff

˙h(t) + ρ2(x)I(x)

K(A2G(x)

)eff

˙w(x, t)

+ Bρ(x)I(x)

K(A2G(x)

)eff

w(x, t) + Cρ(x)I(x)

K(A2G(x)

)eff

w′(x, t) = 0.

(14)

In (14), the corresponding PDE of the piezoresistivemicrocantilever is a comprehensive description of thedynamics of the system considering the shear deformationand rotary inertia effects as well as the viscous and structuraldamping forces.

3.2. Hertzian contact force model

In the literature, there are several models proposed andconsidered to visualize and identify the effects of theinteraction forces between small particles in which the linearforce, Hertzian contact force, Derjaguin–Muller–Toporov(DMT) [23], non-contact force model [22] could be listed assome of these models. In this research work, we are interestedin considering the Hertzian force owing to the contact regionbetween the microcantilever’s tip and the sample’s surface andthe challenge to sense and control the boundary interactionforce. Having a microcantilever with a sphere tip with an elastichalf-space of the tip–sample distance as δ, the Hertzian contactforce may be represented as follows [23]:

Fts(δ) ={

43 E∗√R(−δ)

32 δ � 0

0 δ > 0.(15)

Here, E∗ is the effective elastic modulus between the probe’stip and the sample and is equal to E∗ = [ 1−ν2

tEt

+ 1−ν2s

Es

]−1,

where Et,s, νt,s are the elastic moduli and Poisson’s ratiosand subscripts t and s correspond to the tip and the sample,respectively, and R is the tip radius [23]. (Note that Et = Eb,here).

3.3. Control objective

The mathematical model introduced for the piezoresistivemicrocantilever is a comprehensive model having thiscapability of representing the precise dynamic behavior of theAFM probe under the interaction forces mainly arising fromthe van der Waals or contact Hertzian interactions. In order tohave a precise control for the interaction force tracking tasksa Lyapunov-based controller is contributed herein. Possessinga model of the probe which can mimic the realistic dynamicresponse of the probe, it provides a better foundation for theadvanced controller still under various uncertainties to drivethe microcantilever until its tip approaches a desired force.

In this section, a more suitable form of the dynamic PDE(12) is carried out to determine the appropriate form of theboundary input force fb(t). To this end, we use the followingchange of variable [24]:

u(x, t) = w(x, t) + h(t), (16)

which can be easily calculated for the following terms:

u(x, t) = w(x, t) + h(t), u′(x, t) = w′(x, t),u(x, t) = w(x, t) + h(t), u′′(x, t) = w′′(x, t),u(x, t) = w(x, t) + h(t), u′′′(x, t) = w′′′(x, t),u(x, t) = w(x, t) + h(t), u′′′′(x, t) = w′′′′(x, t),

u′(x, t) = w′(x, t), u′′′(x, t) = w′′′(x, t),u′′(x, t) = w′′(x, t), u′(x, t) = w′(x, t),u′′(x, t) = w′′(x, t).

(17)

After implementing (17) to (14), the resulting equation for˙u (x, t) can be

˙u (x, t) = K(A2G(x))eff

ρ2(x)I(x)

[−(EI(x))eff u′′′′(x, t)

+(

ρ(x)(EI(x))eff

K(AG(x))eff+ ρ(x)I(x)

A)u′′(x, t)

+ B (EI(x))eff

K(AG(x))effu′′(x, t) + C(EI(x))eff

K(AG(x))effu′′′(x, t)

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

− ρ(x)u(x, t) − B(u(x, t) − h(t)) − Cu′(x, t)

− ρ2(x)I(x)

K(A2G(x)

)eff

˙h(t) − Bρ(x)I(x)

K(A2G(x)

)eff

˙u (x, t)

− Cρ(x)I(x)

K(A2G(x)

)eff

u′(x, t)

], (18)

and substituting (18) into (12c) after applying (17) yields(mb + me +

∫ L

0ρ(x) dx

)˙h(t) +

∫ L

0ρ(x)(u (x, t) −˙h(t)) dx

+ me(L, t) = fb(t) + fts(t). (19)

Utilizing (13), we can define error signal as

mee(t) = fts(t) − f dts(t), (20)

where f dts(t) is the desired interaction force.

Extracting term ϕ′(x, t) from (12a), implementingauxiliary variable u(x, t) (i.e. (17)) and integrating over x from0 to L, after simplifying, gives

ϕ(L, t) = u′(L, t) − u′(0, t) −∫ L

0

ρ(x)

K (AG(x))effu(x, t) dx

−∫ L

0

B

K (AG(x))eff

(u(x, t) − h(t)

)dx

−∫ L

0

C

K(AG(x))effu′(x, t) dx + C2. (21)

Plugging (21) into BCs (13),

K(AG(x))eff

(u(0, t) +

∫ L

0

ρ(x)

K(AG(x))effu(x, t) dx

+∫ L

0

B

K (AG(x))eff

(u(x, t) − h(t)

)dx

+∫ L

0

C

K (AG(x))effu′(x, t) dx

)+ meu(L, t)) = fts(t).

(22)

After using (18), (19) and (22), the resulting equation formbh(t) becomes

mbh(t) =∫ L

0Bu (x, t) dx +

∫ L

0Cu′(x, t) dx

+∫ L

0

K(A2G(x)

)eff (EI(x))eff

ρ2(x)I(x)u′′′′(x, t) dx

−∫ L

0

(A (EI(x))eff

ρ(x)I(x)+ K (AG(x))eff

ρ(x)

)u′′(x, t) dx

−∫ L

0

BA (EI(x))eff

ρ2(x)I(x)u′′(x, t) dx

−∫ L

0

AC (EI(x))eff

ρ2(x)I(x)u′′′(x, t) dx

+∫ L

0

K(A2G(x)

)eff

ρ(x)I(x)u(x, t) dx

+∫ L

0

BK(A2G(x)

)ρ2(x)I(x)

(u(x, t) − h(t)

)dx

+∫ L

0

CK(A2G(x)

)ρ2(x)I(x)

u′(x, t)dx + fb(t)

+ K (AG(x))eff

(h(t) +

∫ L

0

ρ(x)

K (AG(x))eff˙u (x, t) dx

+∫ L

0

C

K(AG(x))e f fu′(x, t) dx

), (23)

where u(0, t) = h(t) is considered.For stabilizing the controller, we use the error signals r0(t)

and rL(t) in the following forms [24]:

r0(t) = u(0, t) + γ u(0, t), (24)

rL(t) = u(L, t) + γ u(L, t). (25)

By taking time derivative of (24) and multiplying it by anonzero value mb, the term mb

...r 0 (t) can be expressed as

mb...r 0 (t) =

∫ L

0B u(x, t) dx +

∫ L

0Cu′(x, t) dx

+∫ L

0

K(A2G(x))eff(EI(x))eff

ρ2(x)I(x)u′′′′(x, t) dx

−∫ L

0

(A(EI(x))eff

ρ(x)I(x)+ K(AG(x))eff

ρ(x)

)u′′(x, t) dx

−∫ L

0

BA(EI(x))eff

ρ2(x)I(x)u′′(x, t) dx

−∫ L

0

AC(EI(x))eff

ρ2(x)I(x)u′′′(x, t) dx

+∫ L

0

K(A2G(x)

)eff

ρ(x)I(x)u(x, t) dx + γ h(t)

+∫ L

0

BK(A2G(x))eff

ρ2(x)I(x)(u(x, t) − h(t)) dx

+∫ L

0

CK(A2G(x))eff

ρ2(x)I(x)u′(x, t) dx + fb(t)

+ K(AG(x))eff

(h(t) +

∫ L

0

ρ(x)

K(AG(x))eff˙u (x, t) dx

+∫ L

0

C

K(AG(x))effu′(x, t) dx

+∫ L

0

B

K(AG(x))eff( u(x, t) − h(t)) dx

). (26)

By taking into account the above equation, the control forceinput fb(t) can be established as

fb(t) =∫ t

0

∫ t

0

(−mbγ +

∫ L

0

(A(EI(x))eff

ρ(x)I(x)+ K(AG(x))eff

ρ(x)

)

× u′′(x, t) dx +∫ L

0

BA(EI(x))eff

ρ2(x)I(x)

× u′′(x, t) dx +∫ L

0

AC (EI(x))eff

ρ2(x)I(x)u′′′(x, t) dx

+ K(AG(x))eff u(0, t) + Cu(0, t)

)dt dt

−β1r0(t) − β2u(0, t) + faux, (27)

where β1, β2 >∈ �1 > 0 and auxiliary force | faux| � C2, forC2 denoting a positive constant.

In the similar approach, the value of me...rL (t) by

taking time derivative of (25) and multiplying by me may becalculated as

merL(t) = −K(AG(x))eff

[h(t) +

∫ L

0

ρ(x)

K(AG(x))effu(x, t) dx

+∫ L

0

C

K (AG(x))effu(x, t) dx

]+ fts(t) + meγ u(L, t).

(28)

6

J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

3.4. Stability analysis

Select the Lyapunov candidate in the following form [24]:

V (t) = V1(t) + γV2(t) + V3(t) (29)

such that V1(t), V2(t) and V3(t) are defined in the followingfashion:

V1(t) = 1

2

∫ L

0ρ(x)u2(x, t) dx + 1

2

∫ L

0

ρ(x)I(x)

A(x)ϕ2(x, t) dx

+1

2

∫ L

0[(EI(x))eff ϕ

′2(x, t) dx

+ K(AG(x))eff(u′(x, t) − ϕ(x, t))2] dx,

V2(t) =∫ L

0ρ(x)u(x, t)u(x, t) dx + 1

2B

∫ L

0u2(x, t) dx

+1

2C

∫ L

0u(x, t)u′(x, t) dx,

V3(t) = 1

2k0u2(0, t) + 1

2mbr2

0(t) + 1

2mer2

L(t) + 1

2mee2(t),

(30)

where it is assumed that function V (t) is bounded in 0 �[λ1, λ2] � C3 for a constant positive C3.

After taking time derivative of (30), relations (12a), (12b)and (17) are used to derive terms u(x, t) and ϕ(x, t) andtheir first and second time derivatives in which the Holderinequality [25] is applied. It is attempted to find the upperbound for V1 in this manner. There is the same scenario forV2 which by substituting for value u(x, t) and its first spatialand time derivatives, the upper bound for V2 can be obtained.Similarly, taking time derivative of V3 and using closed-loopdynamic (26) and (28), the upper bound for V3 is followed.After combining the resultant expressions for the upper boundsof V1, V2 and V3, the upper bound for V will be resulted.The described procedures here are straightforward but needextensive mathematical calculation and arrangements. Afterfollowing the preceding procedure, the upper bound for V (t)can be eventually obtained as follows:

V (t) � −α

βV (t) + C4, (31)

where α, β and C4 are some positive and bounded constantsand C4 � |δ| such that δ is the tip–sample distance.

Remark 1. In acquiring (31), the following inequality is used[26]:∫ L

0u2(x, t) dx � L2

∫ L

0u′2(x, t) dx � L4

∫ L

0u′′2(x, t) dx

(32)

and we have

u2(x, t) � L∫ L

0u′2(x, t) dx � L3

∫ L

0u′′2(x, t) dx,

u′2(x, t) � L∫ L

0u′′2(x, t) dx ∀x ∈ [0, L],

f 2(t)

ξi+ ξig

2(t) � 2| f (t)g(t)| � | f (t)g(t)| ∀ f , g ∈ �1,

∀ξi > 0,

2[ f 2(t) + g2(t)] � [ f (t) + g(t)]2 ∀ f , g ∈ �1. (33)

4. Simulation results

The purpose of designing the controller is developing amethodology to identify the required force that must be appliedto the base of the piezoresistive microcantilever to dynamicallycontrol the boundary interaction force. Here, a numericalexample is carried out to demonstrate the performance andstability of the controller.

To solve PDE (14) and truncate it to a reduced-ordermodel, the assumed mode model (AMM) is utilized:

w(x, t) =n∑

i=1

Wi(x)qi(t). (34)

In (34), Wi(x) are the microcantilever mode functions, qi(t)are the generalized coordinates for the elastic deflection and nis the number of modes.

The eigenfrequency equation for a clamped-freeTimoshenko beam with a tip mass can be obtained bysubstituting the solution of (12) into (13) as follows [27–30]:(

χ2i + s2

χi

)(R3R′

4 − R′3R4 + R4R′

1 − R1R′4)

+(

κ2i − s2

κi

)(R2R′

3 − R′2R3 + R1R′

2 − R2R′1) = 0, (35)

where{χi

κi

}=

√2

2

{∓(r2 + s2) +

[(r2 − s2)2 + 4

b2i

]1/2}1/2

,

R1 =(

bi

χi

)sinh(biχi) + Mb2

i cosh(biαi),

R2 =(

bi

χi

)cosh(biχi) + Mb2

i sinh(biχi),

R3 =(

bi

κi

)sin(biκi) + Mb2

i cos(biκi),

R4 = −(

bi

κi

)cos(biκi) + Mb2

i sin(biκi),

R′1 =

[(χ2

i + s2)

χi

](biαi) cosh(biαi),

R′2 =

[(χ2

i + s2)

χi

](biχi) sinh(biχi),

R′3 = −

[(κ2

i − s2)

κi

](biκi) cos(biκi),

R′4 = −

[(κ2

i − s2)

κi

](biκi) sin(biκi), (36)

with the shape functions Zi(ξ ) and �i(ξ ) defined as

Zi(ξ ) = ϒ1

[cosh(biχiξ ) − (R1 − R3)

(R2 − RR4)sinh(biχiξ )

− cos(biκiξ ) + R(R1 − R3)

(R2 − RR4)sin(biκiξ )

],

�i(ξ ) = ϒ2

[sinh(biχiξ ) − (RR1 − R3)

R(R2 − R4)cosh(biχiξ )

+ 1

Rsin(biκiξ ) + (RR1 − R3)

R(R2 − R4)cos(biκiξ )

]. i = 1, 2, . . . ,

(37)

7

J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

0 0.02 0.04 0.06 0.08 0.1 0.12-3

-2

-1

0

1

2

3

4

5

Time (Sec)

Bas

e D

isp

lace

men

t (n

m)

Figure 5. Time history of the base displacement (h(t)).

0 0.02 0.04 0.06 0.08 0.1 0.12-200

-150

-100

-50

0

50

100

150

200

250

Time (Sec)

Inp

ut

Fo

rce

(mic

ro-N

)

Control Input

Figure 6. Control input as the input force to the base of the microcantilever ( fb(t)).

with any arbitrary constants ϒ1 and ϒ2 and variables as

R = (χ2i + s2)

χi

κi

(κ2i − s2)

, R1 = χis2

bi(χ2i + s2)

R1,

R2 = χis2

bi(χ2i + s2)

R2,

R3 = κis2

bi(κ2i − s2)

R3, R4 = κis2

bi(κ2i − s2)

R4, (38)

and

r =√

I

AL2, s =

√EbIb

KAGL2, K = 10(1 + νt )

12 + 11 νt, (39)

where νt is Poisson’s ratio expressed by the followingrelationship:

Eb = 2G(1 + νt ), 0 � νt � 0.5. (40)

In figure 5, the base displacement of the microcantilever h(t)is shown while applying the input force to the base. In figure 6,the evolution of the control action fb(t) is depicted so that it

specifies the required force that must be applied to the base tomanipulate the piezoresistive microcantilever and control theboundary interaction force.

Figure 7 demonstrates the time history of themicrocantilever’s tip displacement and figure 8 shows thecomparison between the ideal and the actual interaction forcesand their respective error signal. The performance of thecontroller is validated by comparing these curves and impliesthat the applied force to the base of the microcantilever and thecontroller is able to manipulate the microcantilever to followthe ideal force.

The numerical parameters and control gains in table 1 areused in establishing the numerical simulations.

5. Force measurement system

The piezoresistive layer mounted on the microcantilever istogether called the force measurement tip (FMT). This FMThas the capability of cooperating with the FMS unit through the

8

J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

0 0.02 0.04 0.06 0.08 0.1 0.12-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (Sec)

Tip

Dis

pla

cem

ent

(nm

)

Figure 7. Evolution of the tip displacement (w(L, t)).

0 0.02 0.04 0.06 0.08 0.1 0.12-2

-1

0

1

2

3

4

5

Time (Sec)

Inte

ract

ion

Fo

rce

(mic

ro-N

)

ftsd (t)

fts(t)

e(t)

Figure 8. Comparison between the ideal and the actual interaction force and their corresponding error signal ( f dts(t), fts(t), e(t)).

Figure 9. Front and rear view of the force measurement system(FMS), FMS plug-in holder (bottom right) and FMS adaptor(bottom left).

FMS plug-in holder and SMC adaptor (see figure 9). The FMSconsists of a Wheatstone resistive bridge, an amplifier, an LCD

and a loudspeaker. By approaching the sample and touchingit, the piezoresistive layer on the microcantilever bends, andhence, the resistance of the Wheatstone bridge changes whichresults in inducing an appropriate voltage. The induced voltageis enhanced by an amplifier so that its magnitude is displayedon the LCD of the FMS, while its corresponding sound issimultaneously heard from the loudspeaker. Another way toread the output voltage is the output BNC port on the rearside of the FMS unit which is outlined by a yellow circle infigure 9. This read-out voltage from this port can be furthersent to the digital signal processor device for data acquisitionand it can be considered as the feedback signal for the onlinecontrol purposes.

To carry out the experiment, first of all, the FMS isnecessary to be calibrated. Without calibrating the FMS,there is a chance of bringing about ±40% error into themeasurement. To calibrate the FMS, a small ball is attached tothe end of a strip of metal and functions based on the see-sawprinciple (figure 10). The mass of the ball is about 1.67 mgand it is expected to observe its appropriate value on the LCD

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

Force Measurement Tip

Calibration Weight = 1.67 mg

Calibrated Force = 16.4 micro-N

Figure 10. Calibration of the force sensor.

Table 1. Geometrical and mechanical parameters of thepiezoresistive microcantilever and control gains.

Symbol Properties Value (unit SI)

me Tip mass 1.16 × 10−11 Kgmb Base mass 0.001 KgEb = Et Microcantilever Young’s modulus 131 × 109 PaL Microcantilever beam length 393 × 10−6 mb Microcantilever beam width 25 × 10−6 mtb Microcantilever beam thickness 1 × 10−6 mρ Microcantilever beam density 2330 Kg m−3

νt Microcantilever Poisson’s ratio 0.3Es Sample Young’s modulus 10 × 109 Paνs Sample Poisson’s ratio 0.3Ep Piezoresistive Young’s modulus 160 × 109 PaLp = a2 − a1 Piezoresistive layer length 25 × 10−6 mρp Piezoresistive beam density 7750 Kg m−3

B Microcantilever viscous damping 0.01 Kg ms−1

C Microcantilever structural damping 0.01 Kg s−1

β1 Controller gain 136β2 Controller gain 8γ Gain 1.2

by applying the same force on the opposite side of the strip.For this, we start bringing the piezoresistive microcantilever(PRC-400 self-sensing microcantilever [31]) down until themicrocantilever’s tip touches the strip and the sound is heardfrom the loudspeaker. We then continue pushing the see-sawdownward until the small ball on the other end is suspended.Since the mass of the ball is 1.67 mg, we now turn the adjustingknob on the FMS so that the amount of 16.4 μN is shownon the LCD. Now the force sensor is calibrated and readyfor performing the force sensing or surface imaging tasks ofspecimens by modulating the measured force.

The Wheatstone bridge of the piezoresistive microcan-tilever as well as the reference layer is shown in figure 11.

R3

R4

R1

R2

Vout

+

-

Piezoresistive Microcantilever

Reference Lever

Vin

Figure 11. Piezoresistive microcantilever circuit with theWheatstone bridge.

Figure 12. Joypad (left) and NanoControl (right) for operating theMM3A R©.

There are different ways for operating the micromanip-ulator as shown in figure 12. The NanoControl consists ofthree channels (or knobs) A, B and C to control the individualaxes of the manipulator. The amplitude, speed and frequencyof the three joints can be controlled through the NanoCon-trol panel as well. The fourth channel (channel D) is used asthe gripping task by attaching a microgripper to the end ofthe micromanipulator. The other tool is the Joypad by whichthe respective speed is adjusted, and by using the NanoCon-trol, the manipulator can be moved and guided in the desired

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

Table 2. MM3A R©operational specifications.

Serial configuration Configuration Command Frequency/Amplitude Value

Baud rate 19 200 Coarse First joint 2.7 kHzData bit 8 ‘coarse B +50’ Second joint 2.7 kHzParity None Fine Third joint 2.7 kHzStop bits 1 ‘fine C –20’ First joint 40 VFlow control None Abort Second joint 15 VTimeout 0.1 ‘CR’ Third joint 23 V

Figure 13. Integration of the piezoelectric self-sensing and piezoresistive microcantilever.

path. Based on the movement concept of the Nanomotors R©,there are two types of operation modes: ‘Fine’ and ‘Coarse’modes [19].

The communication bridge between the MM3A R© anda computer is executed through the standard serial port(RS232) by converting the NanoControl language in theform of the string commands (ASCII code char 32). Thepurpose of interfacing the computer with the NanoControlis to automatically manipulate the micromanipulator, andhence, there is no need to use the NanoControl (manipulationknobs on its panel) or Joypad for manually manipulatingthe MM3A R©. This aspect itself has several benefits such aseliminating the operator mistakes that may occur during thereal-time experiment, creating more precision and flexibility,and applying advanced controllers on the system.

Table 2 illustrates the specifications of the MM3A R© dur-ing the operation. Two types of commands are designated forthis system: Coarse and Fine. Each command needs to bespecified in the string command as the ASCII code shown intable 2.

The range of the frequency for each joint is varied from 0.5to 10 kHz and the selected frequency here is 2.7 kHz. However,higher frequencies are not recommended due to wear down ofthe piezomotors.

6. Experimental procedures

6.1. Boundary interaction force self-sensing

We are interested in self-sensing measurements of thepiezoelectric actuators to validate the force-sensing quantitycarried out by the piezoresistive microcantilever. Thegoverning constitutive law for piezoelectric may be calculatedfrom [32]

δWp = −∫ Lb

0Epd31abpVaδ

(∂2u(x, t)

∂x2

)

Cp

v0

vs(t)

V2

C1

Cr C1

V1

vc(t)

Figure 14. Self-sensing circuit schematic.

× [H(x − x1) − H(x − x2)] dx, (41)

where d31 is the piezoelectric constant, bp is the width of thepiezoelectric beam, Lb is the length of the microcantileverbeam, H is the Heaviside function, x1 and x2 are the positionsof the piezoelectric ends mounted on the microcantilever, anda = 1

2 (t ′b + t ′p) for t ′b and t ′p being the thicknesses of the beamand piezoelectric layer, respectively.

The shown Active Probe R© in figure 13 is mainlycomprised of two parts: (1) piezoelectrically drivenmicrocantilever which is covered by a layer of piezoelectricconsisting of 0.25 μm Ti/Au, 3.5 μm ZnO, 0.25 μmTi/Au and (2) a silicon microcantilever so that the wholesystem serves as a biomorph to control the vibration of themicrocantilever [33].

In the experimental setup shown in figure 13, we attach thepiezoresistive microcantilever at the end of the MM3A R© andplace its tip over the DMASP (the Bruker AFM Active Probe R©)and gradually bring the tip down until the piezoresistivemicrocantilever’s tip touches the tip of the piezoelectric

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

0 0.01 0.02 0.03 0.04 0.05 0.060

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Piezoresistive Microcantilever Voltage Output - Vout

(V)

Sel

f-se

nsin

g V

oltg

ae -

V o (

V)

Figure 15. Self-sensing relation with the piezoresistive microcantilever voltage.

Figure 16. Automated closed-loop boundary force control through the force feedback using MM3A R© and dSPACE R©.

microcantilever. The self-sensing microcantilever diagram is,however, displayed in figure 14:

Vo(t) =[

Cp

C1 + Cp− Cr

C1 + Cr

]Vc(t) + Cp

C1 + CpVs(t), (42)

where Vo(t) is the output voltage, C1 and Cr are the bridgecapacitors, Cp is the equivalent capacitance of the piezoelectricelement and Vs(t) is the piezoelectric self-induced voltage.

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

A capacitance bridge circuit is schemed in figure 14separating the actuation and self-induced voltage.

For the values of C1 = 52 nF, Cr = Cp = 30 pF,

the sensing voltage of the piezoelectric is obtained withan assumption of the linear relation between Vo(t) andpiezoresistive voltage and the constant gain of 13.3 as theslope of curve shown in figure 15. The material presented inthis section can be used in calibrating the PMC and verifythe induced voltage in the PMC in accordance with theActive Probe R©’s excitation. However, in an experiment ona live biological specimen, it may not be probable to verifythe produced voltage in the PMC, and by this way, we cancheck and affirm the induced voltage in the PMC to be inaccordance with the actuated voltage in the Active Probe R©.The self-sensing method could be potentially applied inmicro/nanomanipulation and microassembly tasks. It shouldbe noted here that the nonlinearities of the piezoelectric andthe hysteresis effects are not studied in this research.

6.2. Boundary interaction force control

After ensuring that the PMC is calibrated for sensing ultra-small forces and testing the results through the self-sensingapproach introduced in the preceding section, we present thecontrol framework to evaluate the performance of the proposedcontroller in tracking the ideal interaction force as shown infigure 8 in which the see-saw setup described previously is usedherein. There is a strip of metal carrying an immobile smallball with mass of 1.67 mg on one end while it is expected toexert a desired force to the other end of the see-saw. To avoidexerting maximum load on the piezoresistive microcantileverto lift and suspend the small mass, it is sought to apply an idealconstant force to the strip that is assumed to be 2 μN.

The control aspect of the boundary interaction forces atthe ultra-small scale is one of the most recent challengingproblems demanding high accurate tools and strong expertiseto deal with. For this purpose, a closed-loop boundary forcecontrol platform is established to perform the tip forcecontrol of the flexible beam (microcantilever) mounted ona rigid micromanipulator which could be executed in thedynamic mode. This platform consists of a computer, theNanoControl, the FMS and the dSPACE R©. The advantage ofthis platform is its full automatic feature that discriminatesfrom other recent works in the category of control schemesin micro/nanomanipulation. In this approach, the interactionforce between the probe’s tip and the sample is used asthe feedback signal; rather, many other approaches use thevisual servoing in the closed-loop control. The generatedcommands in the PC are sent to the NanoControl to operateevery joint of the micromanipulator. Having the piezoresistivemicrocantilever as the force sensor can produce correspondingforce signals when the probe’s tip touches the specimens. Forthe FMT-400 force sensor (with tip-force constant of 2–4 Nm−1), the maximum measured force is about 80 μN withthe sensitivity 3.1 × 10−3 mVnm−1. This amount of force issufficient to precisely measure the interaction forces existingbetween the biological tissues and in surgical applicationssuch as the microsurgery [34], cutting the single-walled carbon

Figure 17. Flowchart of the automated algorithm for the boundaryinteraction force control problem.

nanotubes [35, 36], and future resolution for stem cells creationby the nuclear reprogramming.

The force signals are fed back to the A/D channelof the signal processor (dSPACE R©) for the quasi-onlinedata acquisition and compared with the reference signalsuch that the controller decides the next movement of themicromanipulator in which all this procedure is performed inan automatic fashion (figure 16).

Figure 17 shows a flowchart of the automated closed-loopcontrol of the interaction force. Initially, the micromanipulatorshould manually bring the microcantilever’s tip down until thetip touches the sample’s surface. Once the sound is heard, thesystem is set to run automatically. Every force signal sensedby the force sensor in the form of the electrical voltage issent to the A/D channel of the dSPACE R©. In each iteration,a conditional function is allocated to decide if the measuredforce is less than the desired force or not. If the answer isyes, the controller makes the MM3A R© to manipulate the linksand configure them to bring the microcantilever’s tip slightlydownward. To avoid the unsolicited effects of the nonlinearityof the piezoresistive layer, a time delay is intentionally takeninto account. Hence, after the response of the MM3A R©,depending on the amount of the time delay, the system stopsat the new position until it senses a new quantity of the force.This procedure is automatically repeated until the amount ofthe measured force becomes greater than the desired force andthe system stops operating.

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

0 0.5 1 1.5 2 2.5 30.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (Sec)

Am

plit

ud

e

Ramp Signal (Reference)Acquired Data

0 1 2 3 4 5 6 7 8-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (Sec)

Am

plit

ud

e

Pulse Signal (Reference)Acquired Data

0 1 2 3 4 5 6 7 8

-1

-0.5

0

0.5

1

1.5

Time (Sec)

Am

plit

ud

e

Noise Signal (Reference)Acquired Data

(a)

(b)

(c)

Figure 18. Different types of signals for the online quasi-data acquisition of some sample inputs: (a) ramp, (b) pulse and (c) noise.

As stated earlier, some specific string commands (i.e. theASCII code) need be sent to the NanoControl to operate theMM3A R©. On the other hand, those signals received bythe force sensor are in the form of the electrical voltage

sent to the A/D channel of the dSPACE R©. MATLAB R© canbe regarded as a connection chain to correlate the receivingsignals (voltage) and sending out signals (string). In the typicalprocess of working with the dSPACE R©, any type of controller

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

50 100 150 200-0.5

0

0.5

1

1.5

2

2.5

Time (Sec)

Mea

sure

d F

orc

eso

r (m

icro

-N)

Captured Interaction Force by the Digital Signal Processor after Every 1000 Sample (or 1 sec.)

Operating Range of the Proposed Controller

Figure 19. Measured interaction force via dSPACE R© after every 1000 captured samples (equivalent to 1 s).

needs to be designed in the Simulink environment. Then, a real-time interface file is built using the ControlDesktop R©softwareof the dSPACE R©. In the virtual environment of theControlDesktop R©, any gains or parameters in the controllercan be directly adjusted and changed on-the-fly withoutrequiring to go back to the Simulink and change them inthat environment. However, the remaining issue is that wecannot build a real-time interface from our controller since theoutgoing signals are written in the ASCII code and cannot bebrought into the ControlDesktop R©environment. To this end, aMATLAB R© interface incorporated with the dSPACE R©code isintuitively written to perform the online quasi-data acquisition.The detailed circuit of the FMS is represented in figure 2.

Before using the algorithm presented in figure 17, we needto evaluate the error magnitude in data-acquisition procedureby measuring the emitted and received signals in Simulinkenvironment. Figures 18(a)–(c) show representative signalsgenerated and sent out from the D/A channel of the dSPACE R©.

On the other hand, the received signals are sent intothe dSPACE R© through the A/D channel. The produced andmeasured signals (i.e. ramp, pulse and noise) are shown infigures 18(a)–(c). As noticed, the magnitude of the error isrelatively less than 1% in reading the incoming signals whichmay have insignificant influence on sensing and controllingthe boundary interaction force; however, the main source ofthe error is due to the time delay in the real-time measurement.

By using the proposed procedure in figure 16,implementing and incorporating it with the dSPACE R©,programming code and MM3A R©, the second link ofMM3A R© starts moving downward until it touches the strip.Once the microcantilever’s tip touches the sample, themeasured interaction force gradually becomes larger until itreaches to 2 μN presumed as the ideal force. At this point,the controller stops sending out any signals to the MM3A R©’slinks, while the microcantilever is still exerting the same

force to the strip downward. Without loss of generality, itis supposed that there is a linear relationship between theoutput voltage and the measured interaction force. Figure 19depicts the measured interaction force via dSPACE R© afterevery 1000 captured samples which is equal to 1 s. Thesedata are delivered to the software program at each time instantand the controller makes the proper decision for the nextaction on the micromanipulator’s link. However, the process ofsending and receiving signals includes some inevitable delaysin transferring data as noticed.

As observed in figure 19, the second link of themicromanipulator is brought downward to the strip to let theprobe’s tip reach the strip and the controller is then engaged.This mechanism demonstrates two sections: non-measuredforce (from 0–110 s) and applied measured force with aconstant slope (from 111–200 s). In the second segment,the proposed controller is in effect until the force sensorapproaches the ideal force. Another experiment for about 350 sis run and shown in figure 20. This figure depicts the automatedcontrol of the interaction force by implementing the MM3A R©,while it is divided into three segments: non-contact region,contact region and the applied desired force region.

In the non-contact region, since the tip is not in contactwith the sample, no interaction force is expected to beseen. However, after touching the strip of metal by themicrocantilever’s tip, the interaction force is persistentlyincreasing, while the tip is simultaneously pushing the see-saw downward until the ideal force (for instance 2 μN)is attained. After this stage, the system has no motion,while the tip is consistently exerting the ideal force to thesample. It is observed that there are noises due to differentsources mainly arising from the environment. Some otherconcerns could be listed as the point of contact between themicrocantilever’s tip and the strip, the small ratio of the weightof the microcantilever with respect to the weight of the smallball to lift it and the nonlinearity of the piezoresistive layer.

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

0 50 100 150 200 250 300-0.5

0

0.5

1

1.5

2

2.5

Time (Sec)

Forc

e (m

icro

-N)

Applied Desired Interaction Force from the Non-contact to Contact Region

Non-contact Region

Contact Region

Applying the Desired Force Region

Offset ErrorDesired Force = 2 (micro-N)

Figure 20. Controlling the boundary interaction force through the proposed controller.

-100 -50 0 50 100-0.5

0

0.5

1

1.5

2

2.5

Distance (nm)

Forc

e (m

icro

-N)

Experimental DataCurve Fitted DataHertz contact Model

Intermittent Contact

Figure 21. Experimental and theoretical evolution of the force–distance curve.

It is essential to study the behavior of the force–distance diagram under different circumstances. As for theintermolecular forces, there are mainly two types of forces:attraction and repulsion [22] depending on the distancebetween the particles which is well described by the Lennard–Jones potential curve. The experimentally controlled contactforce curve versus the tip–sample distance is shown infigure 21 and is compared with the theoretical model describedearlier including the comprehensive DPM considering theHertzian contact force model. Due to the sophisticated andundetermined nature of the ultra-small forces, it would becrucial to find a model to resemble the realistic behavior of theinteraction forces. Given the experimental data, a polynomialcurve is best fitted to the force–distance diagram shown in

figure 21 and can be further quantitatively used to studyand identify the interfacial contact forces. The best propercurve is fitted to the acquired contact interaction force and itscorresponding differential equation as well as their parameterscan be identified. However, this model is a polynomial equationwith the representative coefficients based on the order of thesystem shown as follows:

F(x) = aixi + ai−1xi−1 + ai−2xi−2 + ai−3xi−3 + · · ·

+ ai−nxi−n for i � n, i = 1, 2, . . . . (43)

When the probe’s tip is in contact with the surface ofthe sample, the comparative study reveals that the theoreticalmodel can realistically represents the interaction force model

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J. Micromech. Microeng. 22 (2012) 125013 S Eslami and N Jalili

in the contact mode. However, there are numerous medicaland biological applications in measuring and controlling theboundary interaction forces so that this force model couldbe implemented in the control of microcantilevers in thetapping/non-contact AFM mode [37].

7. Conclusions

In this paper, a micromanipulator system comprising apiezoresistive microcantilever was introduced along with itsmechanical/electrical characteristics and driving principlesfor the purpose of micro/nanomanipulation tasks. Acomprehensive model for the piezoresistive microcantileverwas obtained while treating the microcantilever as adistributed-parameters system having the shear deformationand rotary inertia effects. Along with that representation, theHertzian contact force model was formulated and a Lyapunov-based control was developed to control the boundaryinteraction force to resemble the realistic dynamic behaviorof the system. Some numerical simulations were presentedto validate the dynamic response of the microcantilever aswell as the performance and stability of the developed model-based controller. A FMS having the capability of sensingforces in the order of micro-Newton was discussed. Bytaking advantage of the force feedback, some ultra-smallinteraction forces could be automatically controlled in whichmore precision was achieved having promising medicineand biological applications. To this end, the communicationinterface between the computer and the NanoControl moduleof the robot was presented and a digital signal processor wasintegrated with the described system for the online quasi-data acquisition. An experimental setup was contributed forthe boundary interaction force self-sensing and force controlwith the closed-loop control using the force feedback; rather,benefiting from the visual servoing feedback. Through theproposed closed-loop framework, we could automaticallycontrol some ultra-small boundary interaction forces as oneof the other contributions in this work. Consequently, thetheoretical and experimental results were compared togetheron validating the dynamic response of the boundary interactionforce in which they were in a good agreement.

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