For Peer ReviewScaled Bilateral Teleoperation Using Discrete-Time

Sliding Mode Controller

Journal: Transactions on Industrial Electronics

Manuscript ID: TIE-01239-2007.R1

Manuscript Type: SS on Sliding Mode Control in Industrial Applications

Manuscript Subject: Robotics and Mechatronics

Keywords: Actuators, Digital control, Piezoresistive devices

Are any of authors IEEE Member?:

Yes

Are any of authors IES Member?:

No

Transactions on Industrial Electronics

For Peer Review

1

Scaled Bilateral Teleoperation Using Discrete-TimeSliding Mode Controller

Abstract— In this article design of a discrete time sliding modecontroller based on Lyapunov theory is presented along withtherobust disturbance observer and applied to a piezo-stage for high-precision motion. Linear model of a piezo-stage was used withnominal parameters to compensate the disturbance acting onthesystem in order to achieve nanometer accuracy. The effectivenessof the controller and disturbance observer was verified in term ofclosed loop position performance for nanometer references. Thecontrol structure has been applied in bilateral structure for thecustom built tele-micromanipulation setup. Piezoresistive AFMcantilever with inbuilt Wheatstone bridge is utilized to achievenano-newton level interaction forces between piezoresistive probetip and the environment. Experimental results are provided forthe nano-newton range force sensing and good agreement amongthe experimental data and the theoretical estimates has beendemonstrated. Force/Position tracking and transparency betweenmaster and slave has been clearly demonstrated after necessaryscaling.

Index Terms— Discrete Sliding Mode Control, Disturbance Ob-server, High Precision Motion Control, Force Sensing, BilateralControl, Tele-Micromanipulation.

I. I NTRODUCTION

I N recent years, the demand for microsystems technologyhas grown rapidly especially due to the development of

MEMS (Microelectromechanical Systems) products such asaccelerometers, inkjet printer heads, optical MEMS etc. Com-plex micro/nano systems in general contain much distinctfunctionality in a single product and consists of differenttypesof materials. Use of monolithic (uniform) process to producecomplex micro/nano systems is desirable, but unfortunately isnot always feasible. The current state of art to incorporatemultiple incompatible components into a single product isto assemble simple parts one by one [1], [2], [3], [4]. Thefirst and foremost requirement for the assembly process is to“precisely manipulate” objects. Manipulation includes cutting,pushing, pulling, indenting, or any type of interaction whichchanges the relative position and relation of an entities. Theprocess of manipulation cannot be done autonomously withrobotic systems and/or directly by human operators. Bilateralcontrol, which is typically used for teleoperation, offersasolution to these tasks since it enables the operator to worksomewhere without actually being there. That is, if actualpresence of an operator is not possible, inclusion of a bilateralcontrol system between the operator and the task wouldsimply give a possibility to the so called “telepresence” ofthe operator.

Bilateral control is defined as the control of two systemsworking together on an actual or virtual task. Typically, itis used for teleoperation, in which one system is called the“master” side and the other is called the “slave” side ofbilateral action. Slave subsystem is tracking the positions of

the master subsystem and master side provides the forcesencountered by the slave side to the operator and hence, tele-operation is achieved [5], [6]. Nowadays, many researchershave come up with the notion of multilateral control [7], [8]consisting of more than two systems working with propercoordination to achieve a desired task. In order to perform tele-micromanipulation it is indispensable to achieve robust andtransparent bilateral controllers for human interventionso thathigh fidelity position/force interaction between the operatorand the remote micro/nano environment can be achieved[9], [10]. As bilateral control enables skilled teleoperationon several tasks, it offers better safety, low cost and highaccuracy, on the other hand it also suffers from time delayproblem [11], [12] which does effect the transparency of thesystems.

For high precision motion control problems, robustnessof the control algorithm is the most crucial element evenif the system model is linear. Furthermore, when the plantto be controlled has high nonlinearities such as internalhysteresis, which is the case for the PZT (Lead-Zirconate-Titanate also known as piezo actuator) the advantage of arobust controller, which is designed according to nominalplant parameters and which rejects parameter uncertainties,would be simply less effort on modeling the system andcompensation methods [13], [14]. Moreover, it is a fact thatusing more complicated models may not always lead to bettercompensation results than just using a simple model (e.g.the model of Coulomb friction), since the quality of thecompensation depends not only on the model, but also on theimplementation constraints.

To avoid the difficulties mentioned above and concentrateon the main issues of the control problem, one needs to finda methodology that produces a robust controller designed ac-cording to the nominal parameters and has fine disturbance re-jection to realize high precision motion control with minimumeffort. As performance requirements become more stringent,classical controllers such as the PID (Proportional-Integral-Derivative) controller, which has been the most preferredcontroller and widely used in industry for generations, canno longer provide acceptable results. The theory of variablestructure systems (VSS) has opened up a wide new area ofdevelopment for control designers. Variable structure control(VSC) which is frequently known as sliding mode control(SMC), and is characterized by a discontinuous control actionwhich changes structure upon reaching a set of predeterminedswitching surfaces. Some of the concepts and theoreticaladvances of VSS are covered by Young [15], Utkin [16] andHung [17]. These kinds of control may result in a very robustsystem and thus provides a possibility for achieving the goalsof high-precision motion. Some promising features of SMC

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are listed below:• The order of the motion can be reduced.• The motion equation of the sliding mode can be designed

linear and homogenous, despite the original system maybe governed by non-linear equations.

• The sliding mode does not depend on the system dynam-ics, but is determined by parameters selected by the user.

• Once the sliding motion occurs, the system developsinvariant properties which make the motion independentof certain system parameter variations and disturbances.Thus the system performance can be completely estab-lished by the dynamics of the sliding manifold.

SMC theory was initially developed from a continuous timeperspective. It has been realized that directly applying thecontinuous-time SMC algorithms to discrete time systemswill lead to some indomitable problems, such as the limitedsampling frequency, sample/hold effects and discretizationerrors. Since the switching frequency in sampled-data systemscannot exceed the sampling frequency, a discontinuous controlwill not enable generation of motion in a random manifold indiscrete-time systems. This leads to chattering at the samplingfrequency along the designed sliding surface, or even instabil-ity in case of a too large switching gain. In order to cope upwith above mentioned issues, discrete time SMC structure hasbeen derived.

In order to achieve force transparency between master andslave, it is necessary to sense the force in nano-newton rangewith high accuracy. Many researchers has used different waysfor sensing or estimating force using PZT actuator [18], Ca-pacitive sensors, optical deflection as in AFM (Atomic ForceMicroscope) scheme, tunneling as in STM (Scanning Tunnel-ing Microscope)etc. Piezoresistive AFM cantilever with inbuiltWheatstone bridge is utilized as a force sensor. Piezoresistivesensors have been used for many other MEMS applications,including accelerometers, gyroscopes and AFM cantilevers.

In this paper, design and implementation of discrete slidingmode controller along with disturbance observer based onSMC is presented to eliminate nonlinear disturbances actingon PZT in order to achieve high position accuracy in nano-scale. Moreover force sensing with nano-newton range usingpiezoresistive AFM cantilever is presented and finally theforce/position tracking of master and slave for the microma-nipulation setup is achieved.

This paper is organized as follows. Section II explains thetele-micromanipulation setup, Section III discuss the design ofdiscrete sliding mode controller, Section IV focus on modelingof PZT actuator along with the simulation and experimentalresults for the position control of PZT using SMC. SectionV discuss scaled bilateral teleoperation using SMC includingthe experimental validation of force sensing with nano-newtonaccuracy and Section VI shows the experimental results offorce/position tracking of master and slave. Section VII de-scribes the conclusion.

II. T ELE-M ICROMANIPULATION SETUP

The system is composed of two parts, namely a mastermechanism operated by the human operator and a slave mech-anism interacting with the micro environment. For the master

mechanism a DC servo is utilized, while a piezoresistivemicrocantilever attached on PZT stacks is used for the slave.A bilateral man-machine interface is implemented for controlas shown in schematics figure 1.

Fig. 1. Schematic of tele-micromanipulation system.

The position data from the master side is scaled and transferredto slave side, while simultaneously, the force measured at theslave side is scaled and transferred back to master. XYZ basestages are manually operated for proper alignment of microobject under the workspace. A graphical display is also madeavailable to the operator. Figure 2 shows the experimentalsetup.

1

2

3

4

5

Fig. 2. Experimental setup for micromanipulation, 1-Nanocube, 2-Piezoresistive cantilever, 3-Glass slide, 4-Open loop PZT, 5-Nikon MM-40Microscope.

An open architecture micromanipulation system that satis-fies the requirements has been developed and used as the slavemechanism. Nano scale positioning of the microcantilever hasbeen provided using three axes piezo stages (P-611 by PhysikInstrumente) which are driven by a power amplifier (E-664)in closed loop external control mode. Strain gauge sensorsintegrated in the amplifier, are utilized for position measure-ment of the closed loop stages which possess a travel rangeof 100µm per axis with one nanometer theoretical resolution.An open loop piezoelectric micrometer drive (PiezoMike PI-854 from Physik Instrumente) has been utilized as the basestage, which is equipped with integrated high resolution piezolinear drives [19]. Manually operable linear drives are capable

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of 1 µm resolution and the automatic movement range of themicrometer tip with respect to the position can be set50 µm

(25 µm in/out). As for the force feedback, a piezoresistiveAFM cantilever with inbuilt Wheatstone bridge from AppNanoInc. has been utilized. A real time capable control card(dSPACE DS1103) is used as control platform and an opticalmicroscope (Nikon MM-40) is used for visual feedback.

III. SLIDING MODE IN VARIABLE STRUCTURE SYSTEMS

Sliding mode control (SMC), which is sometimes knownas variable structure control (VSC), is characterized by adiscontinuous control action which changes structure uponreaching a set of predetermined switching surfaces. Thiscontrol structure may result in a very robust system and thusprovides a possibility for achieving the goals of high-precisionmotion. Consider the system below

x = f(x, t) + B(x, t)u(x, t) x ⊂ Rn, u ⊂ Rm (1)

where all the elements of vectorf(x, t) and matrixB(x, t)are continuous and bounded having continuous and boundedfirst order time derivatives;rank(B(x, t)) = m, ∀x, t > 0The discontinuous control is given by

u =

{u+(x, t), σ(x) > 0u−(x, t), σ(x) < 0

(2)

σ(x)T = {σ1(x), σ2(x), ...., σm(x)}, σ(x) = G(xr − x) (3)

hereu+(x, t) , u−(x, t) and σ(x) are continuous functions.G is a positive integer chosen for the error converging re-sponse time andxr is the reference position. u(x,t) undergoesdiscontinuity on the manifoldσ = 0.

Let S = X |σ(x)=0 be a switching manifold that includes theorigin x = 0. If, for any x0 in S, x(t) is in S for all t > t0,then x(t) is a sliding mode of the system and the manifoldS is called a sliding manifold. A sliding mode exists, if inthe vicinity of the switching surfaceS, the tangent or thevelocity vectors of the state trajectory always point towardsthe switching surface.

A. Design of Discrete Sliding Mode Controller

Drakunov and Utkin [20] introduced a continuous approachto SMC for an arbitrary finite dimensional discrete-time sys-tem. This approach implies that for a sampled-data controller,as the system becomes discrete, the controller should be con-tinuous to overcome the sampling frequency limitations of thediscontinuous approach. For such continuous implementationof SMC, plant motion is proven to reach the sliding manifoldof predefined state trajectory in finite time.

The derivation of the controller structure can be achievedusing proper selection of the Lyapunov functionV (σ), and anappropriate form of the derivates of the Lyapunov function,V (σ).Suitable candidate of the Lyapunov function can be taken as

V (σ) =σ2

2(4)

Hence, the derivative of the Lyapunov function is

V (σ) = σσ (5)

In order to guarantee the asymptotic stability of the solutionσ(x, xr) = 0, the derivates of the Lyapunov function may beselected to be

V (σ) = −Dσ2 − µσ2

|σ|(6)

Here D and µ are positive constants. Hence, if the controlcan be determined from Eqn.(5) and Eqn.(6), the asymptoticstability will be guaranteed sinceV (σ) > 0, V (0) = 0 andV (σ) < 0. By combining Eqn.(5) and Eqn.(6) the followingequation can be deducedσ(σ+Dσ+µ σ

|σ| ) = 0 which means,

σ + Dσ + µσ

|σ|= 0 (7)

The derivative of the sliding function can be written asσ =G(xr − x) = Gxr − Gx. Here G = {λ 1} with λ being apositive constant. After some simplification,σ = Gxr −Gf −GBu(t) = GB(ueq − u(t)) and solving we get,

u(t) = ueq + (GB)−1(Dσ + µσ

|σ|) (8)

It can be seen from above equation thatueq are difficult tocalculate. Using the fact thatueq is a continuous function, itcan be rewritten in discrete form using Euler’s approximationas σ((k+1)Ts)−σ(kTs)

Ts= GB(ueq(kTs) − u(kTs)).

HereTs is the sampling time andk = Z+. It is also necessaryto write u(t) in the discrete form which results in

u(kTs) = ueq(kTs) + (GB)−1(Dσ(kTs) + µσ(kTs)

|σ(kTs)|) (9)

ueq(kTs) can be written as

ueq(kTs) = u(kTs) + (GB)−1(σ((k + 1)Ts) − σ(kTs)

Ts

)

(10)

Since the system is causal, and it is required to avoid thecalculation of the predicted value forσ as control cannot bedependent on future value ofσ. Since the equivalent controlis a continuous function, the current value of the equivalentcontrol can be approximated with the single-step backwardvalue calculated forueq(kTs) as

ueqk−1= uk−1 + (GB)−1(

σk − σk−1

Ts

) (11)

Hereueqk(or ueq(kTs)) is the estimate of the current value of

the equivalent control. After some simplification the resultingcontrol structure as be written as

uk = uk−1 + (GBTs)−1((DTs + 1)σk − σk−1 + Tsµ

σ(k)

|σ(k)|)

(12)

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The control structure Eqn.(12) is suitable for implemen-tation, since it requires measurement of the sliding modefunction and the value of the control applied in the precedingstep. For a discrete-time system, the discrete sliding modecanbe interpreted as the only states that are required to be kepton the sliding surface at each sampling instant. Between thesamples, the states are allowed to diverge from the surfacewithin a boundary layer. Note that the control defined byEqn.(12) is continuous in nature unlike the case for continuoustime. Estimation of boundary layer is explained in next section.

B. Estimation of Boundary layer in Discrete Sliding-ModeControl

During the course of designing a controller, it is crucialto analyze the robustness of the controller or, in other words,whether it satisfies the condition defined by Eqn.(7). Moreoverthe estimation of boundary layer of the sliding manifold issignificant in relation with the robustness of the controller.The analysis shown are concerned with a general system asin Eqn.(1).

wheref(x, t) andB(x, t) are assumed to be continuous andbounded. The derivative of the sliding surface is given by

dσ(t)

dt= G(xr − x) +

∂σ(t)

∂t= Gxr(t) − GBu(t) +

∂σ(t)

∂t(13)

If instead of u(t) the control defined by Eqn.(12) is usedassumingσ = 0, the following results is obtained

dσ(t)dt

= Gxr(t) − Gf(t) − GB{u(t−)+

(GB)−1(Dσ + σ)|−t + ∂σ(t)∂t

(14)

heret− = t− Ts for discrete time applications withTs is thesampling time. Further simplification of Eqn.(14) lead to

dσ(t)

dt= Gxr(t)−Gf(t)−GBu(t−)+

∂σ(t)

∂t−(Dσ+

dσ

dt)|t−

(15)

Finally Eqn.(15) can be written as

dσ(t)dt

= dσ(t−)dt

− dσ(t−)dt

− Dσ(t)+

G∆xr − G∆f + ∆(∂σ

∂t) − D∆σ

︸ ︷︷ ︸

ζ(Ts)

(16)

Here

∆xr = xt − xr(t−);∆f = f(t) − f(t−);∆σ = σ(t) − σ(t−);

∆(∂σ∂t

) = ∂σ(t)∂t

− ∂σ(t−)∂t

;

(17)

Hence,

dσ(t)

dt+ Dσ(t) = ζ(Ts) (18)

Sincef(t), xr(t) and σ(t) are smooth functions, thenζ(Ts)has orderO(Ts) which implies order of one sample period.

Hence, the states will remain within anO(Ts2) boundary layer

of the sliding surface.

IV. H IGH PRECISION MOTION USING PIEZOACTUATOR

A. Modeling PZT Actuator

Since piezoceramic is a known dielectric, one would expecta PZT stack actuator to exhibit capacitive behavior alongwith rate-independent hysteresis exhibited which effectsthenet electrical charge delivered to the actuator. Additionally,dynamic observation indicates that endpoint displacementasa function of electrical charge is well approximated by second-order linear dynamics.

The piezo-stage consists of a piezo-drive with a flexureguided structure which is designed to possess zero stictionand friction. Moreover the flexure stages exhibit high stiffness,high load capacity and insensitive to shock and vibration.Figure 3 describes the overall electromechanical model [21]of a PZT actuator.

qɺ

H

pqɺ

emT

hu

pu

inu

eC

emT

pq y

pu pF

M

pF

pF extF

extF

y

Fig. 3. Electromechanical model of a PZT actuator

The hysteresis and piezoelectric effects are separated.H

represents the hysteresis effect anduh is the voltage due to thiseffect. The piezoelectric effect is represented byTem whichis an electromechanical transducer with transformer ratio. ThecapacitanceCe represents the sum of the capacitances of theindividual PZT wafers, which are electrically in parallel.Thetotal current flowing through the circuit isq. Furthermore,qmay be seen as the total charge in the PZT actuator. The chargeqp is the transducer charge from the mechanical side. Thevoltageup is due to the piezo effect. The total voltage overthe PZT actuator isuin, Fp is the transducer force from theelectrical side,Fext is the externally applied force, and theresulting elongation of the PZT actuator is denoted byy. Themechanical relation betweenFp andy is denoted byM . Notethat we have equal electrical and mechanical energy at theports of interaction i.e.upqp = Fpy.

The piezoelectric ceramic has elasticity modulusE, vis-cosity η, and mass densityρ. Furthermore, the geometricproperties of the PZT actuator are lengthL and cross-sectionalareaAp. Nominal Massmp, nominal stiffnesskp and dampingco-efficientcp can be calculated asmp = ρApL, kp =

EAp

L

andcp =ηAp

L.

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The complete electromechanical equations can be written as:

mpy + cpy + kpy = Tem(uin(t) − H(y, uin)) − Fext (19)

Here y represents the displacement of the piezo stage andH(y, uin) denotes the non-linear hysteresis which is a functionof y anduin. The properties of utilized piezoelectric actuatoris shown in Table 1. Since modeling hysteresis and usingcompensation is cumbersome process due to its dependenceon many factors, thus disturbance observer which is discussedin the following section will be used for compensation ofhysteretic behavior in the system.

TABLE I

PROPERTIESOF PIEZO-STAGE

Symbol Quantity Value in SImp Nominal Mass 1.5× 10

−3kg

cp Nominal Damping 220Nsm

kp Nominal Stiffness 300000Nm

fr Resonant Frequency 350Hz

Tem Transformation Ratio 0.3NV

B. Disturbance Observer based on SMC

There are several hindrances for high precision motionwhich is highly nonlinear in nature and arises from severalfactors such as hysteresis, dead zone, saturation, backlashetc of the actuators and/or sensing devices, high parametervariations and time delay. It might be possible to combineall the effects of these different kind of disturbances on theplant response (i.e. observe their position) and provide acompensation for them as an addition to the controller outputand use this sum as the plant input. This kind of compensationis called “disturbance compensation” and the observer usediscalled “disturbance observer”.

The observer structure is deduced based on the Eqn.(19) un-der the assumption that all the plant parameters uncertainties,nonlinearities and external disturbances can be represented asa lumped disturbance. It is assumed thaty is the displacementand measurable and similarlyut is the input and also ameasurable quantity.

mpy + cpy + kpy = Temu(t) − Fdis

Fdis = TpH + ∆T (uin + vh) + ∆my + ∆ky(20)

Here mp, cp, kp and Tp are the nominal plant parameterswhile ∆m, ∆c, ∆k and∆T are the uncertainties associatedwith the plant parameters. Sincey and u(t) are measurablequantity, observer structure can be written in following form,

mp¨y + cp

˙y + kpy = Tpu − Tpuc (21)

Here y, ˙y and ¨y are estimated position, velocity and acceler-ation respectively.u is the plant control input anduc is theobserver control input as shown in figure 4.The estimated positiony should be forced to tracky. Thederivation process of SMC structure is also used for deriving

Dis

turb

an

ce

Ob

serv

er

Discrete Sliding Mode

ControllerPiezoelectric Actuator

Linear Plant Model

Discrete Sliding Mode

Controller

+

+

+

+

-

y

-

y

y^

e

u

u u inref

c

Fig. 4. Controller and Disturbance Observer for Position Control of the PZTActuator

the observer controller whose sliding manifold is defined asσobs = λobs(y − y) + (y − ˙y).Hereλobs is a positive constant. Ifσobs is forced to becomezero theny should be forced toy. As described in the previoussection, with the same analogy it can be written as(σobs +Dobsσobs) = 0.which guaranteesσobs → 0. After some modification theresulting equation is can be written as(y − ¨y) + (λobs +Dobs)(y − ˙y) + λobsDobs(y − y) = 0.It can be seen that the transient of the closed-loop system aredefined by the roots−λobs and−Dobs . The same structureof the controller will be used in the observer as described inEqn.(12). From structure Eqn.(21) it can be seen that the inputmatrix is given byBobs = [0 −

Tp

mp]T . The matrixG for

this case is defined asG = [λobs 1].Thus, after some simplification the observer structure can bewritten as,

uck= uck−1

−mp

Tp

(Dobsλobsk+

σobsk− σobsk−1

Ts

) (22)

Hereuc is the compensated control input to the system. Thepositive feedback by inputuc forces the system to behaveclosely towards the ideal system having the nominal parame-ters. But in reality there is also some amount of differencebetween the real disturbance and estimated disturbances.

C. Experimental Validation of Position Control

In order to verify the performance of discrete time slidingmode controller along with the disturbance observer, smoothstep inputs are given to one of the piezo stages and responseis drawn in Figure 5 which represents the step response forposition reference of50nm. The rise times and steady stateerror is23 ms and2% respectively. An overshoot behavior isnot observed in any tested cases. Operation with no overshootis the foremost requirement for micromanipulation applica-tions since overshoot may result in damage to the probe orparticles. However, the system suffers from noise coming fromthe measurement devices, which shows up in the steady stateplots.

Figure 6 represents the response for trapezoidal input withaheight of0.5µm and the result shows that it precisely followsthe reference position and tracking error is found to be lessthan 10nm. Figure 7 demonstrates the position response forsinusoidal input with an amplitude of1µm with low frequency

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of 2Hz due to the requirement of slow motion on the slaveside. It can be clearly observed that the actual position tracksthe reference with very accurately and the tracking erroris within ±20nm. These experimental results suggests thatthe proposed controller along with the disturbance observerproduces acceptable results for positioning with very high-precision.

Fig. 5. Position response for a reference of50nm [22]

0 5 10 15 20 25 30 35−0.2

0

0.2

0.4

0.6

Time (sec)Re

fere

nc

e a

nd

Ac

tua

l P

osi

tio

ns

(mic

ron

s)

0 0.5 1 1.5 2 2.5 3 3.5

x 10

−0.02

−0.01

0

0.01

0.02

Time (sec)

Po

siti

on

Err

or

(mic

ron

s)

Reference

Position

Fig. 6. Position response for a trapezoidal reference

V. SCALED BILATERAL TELEOPERATION

In the micromanipulation applications, scaled bilateral con-trol is used for teleoperation where master/human is not ableto access the micro environment on the slave side. Since themaster and the slave are working on macro and micro scalesrespectively, thus it’s indispensable to use general bilateralcontroller to scale the position and forces between two sidesfor extensive capability. In other words, position informationfrom the master is scaled down to slave and force informationfrom the slave side is scaled up to master.

A. Schematic of Tele-Micromanipulation Setup

The complete bilateral structure is shown in figure 1 com-prising of master and slave side. Piezo-stage on the slave

0 5 10 15 20 25−1.5

−1

−0.5

0

0.5

1

1.5

Time (sec)

Re

fere

nc

e a

nd

Ac

tua

l P

osi

tio

ns

(mic

ron

s)

0 0.5 1 1.5 2 2.5

x 10

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time (sec)

Po

siti

on

Err

or

(mic

ron

s)

Reference

Position

Fig. 7. Position response for a sinusoidal reference for1µm amplitude

side is required to track master’s position as dictated byoperator using discrete sliding mode controller structureasdiscussed in previous section. The1D force of interactionwith environment, generated by piezoresistive cantilever, onthe slave side is transferred to the master as a force opposingits motion, therefore causing a “feeling” of the environmentby the operator. The conformity of this feeling with the realforces is called the “transparency”. Transparency is crucialfor micro/nanomanipulation application for stability of theoverall system. Furthermore, for micro system applications,position and forces should be scaled in order to adjust to theoperators requirements. Position of the master manipulator,scaled by a factorα, is used as a position reference for theslave manipulator, while the calculated force due to contactwith environment, scaled by a factorβ, is fed-back to theoperator through the master manipulator.

In order to eliminate oscillations both on master sidebecause of oscillatory human hand and on the slave sidedue to piezoresistive cantilever dynamics, position of mastermanipulator and force of slave manipulator are filtered by lowpass filters before scaling.

B. Force Sensing Using Piezoresistive AFM Cantilever

A commercially available piezoresistive microcantileverfrom AppliedNanostructureswith an integrated inbuilt Wheat-stone along with lightly-doped strain gauge is utilized as theforce sensor as shown in figure 8. As the force is applied at thefree end of the cantilever, the change of resistance takes placedepending on deflection. The amount of deflection is measuredby a Wheatstone bridge which provides a voltage output andamplified by the amplifier as shown in the figure 1.To match with the initial cantilever resistance value, one ofthe passive resistors in the full bridge is used with parallelto a potentiometer. The amplified voltage is send to the dataacquisition card, and the force is calculated using Hooke’slawF = Kc × z.

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piezoresistive

AFM cantilever

Wheatstone

Full Bridge

Resistor Pads

Fig. 8. Piezoresistive AFM Cantilever with inbuilt Wheatstone bridge.

where Kc is the known spring constant of0.3603Nm

and z

is the amount of cantilever deflection. The spring constant iscalculated by considering a linear beam equation and verifiedvia a natural frequency test using an AFM [23]. Linear beamequation is represented asKc = 3EI

L3 .where E represents the modulus of elasticity (190GPa forsilicon) and I represent the moment of inertia calculated asI = bh3

12 .Whereb andh represents the width and height of the micro-cantilever50 µm and1.6 µm respectively and the value ofIis calculated as17.067× 10−24m4.

The cantilever is mounted on the three axes closed loopstage and the interaction (contact and non-contact) forcesbetween the tip and glass slide are measured. The movementof the cantilever is selected to be perpendicular to the planeof the optical axis in order to achieve better visibility of thedistance between the cantilever and the glass slide. Since thedisplacement range of the x-axis of the closed loop stage is100µm, the glass slide is brought within the range using open-loop manual PZT axes. Finally, the change of the resistanceis converted to change in voltage (millivolt range) using theinbuilt full bridge along with offset potentiometer, whichinturn is converted to±10V ranges using the amplifier.

C. Experimental Validation of Force Sensing

In order to verify the force sensing the piezoresistive AFMcantilever is made to interact with the glass slide and fig-ure 9 presents the attractive forces between the tip and glassslide [24]. Decreasing distance between the tip and glass slidecorresponds to an increase in the position of PZT along thesame axis. It can be observed that as the distance between thetip and glass slide decreases, attractive forces increases. Thefirst part of the graph is dominated by electrostatic forces whilethe remaining part is dominated by van der Waals forces. Thechange in slope of the force measurement plot correspondingto these two regions can be observed in figure 9.

In order to verify force measurement, theoretical values ofpull-off force (breaking load during the withdrawal of tip)between the silicon tip and the glass surface is compared withthe experimental results. In case of the interaction between a

0 10 20 30 40 50 60 70 80 90−20

0

20

40

60

80

100

Fo

rce

(n

N)

0 10 20 30 40 50 60 70 80 90−20

0

20

40

60

80

Time (sec)

Po

siti

on

(µ

)

Fig. 9. Force for smooth step position reference.

spherical tip and a planar surface, the interaction force can beapproximated by Dugdale model [25] as

Fpull−off =

(

7

4−

1

4

4.04λ1

4 − 1

4.04λ1

4 + 1

)

πWR (23)

whereW is the work of adhesion between the two mediums,R

is the radius of the sphere andλ is a coefficient, which can beused to choose the most appropriate contact model for a givencase [26]. Using the interfacial energy the pull-off force canbe calculated forλ = 0.54 according to the Dugdale model as39.43 nN [27], [28]. Figure 10 demonstrates experimentallydetermined the pull-off force is close to40 nN , indicatinga close match between the theoretically and experimentallydetermined values.

32 33 34 35 36 37 38

−40

−20

0

20

40

60

80

Distance (µ)

Fo

rce

(n

N)

Fig. 10. Forces of Interaction between Silicon Tip and GlassSlide

VI. POSITION/FORCE TRACKING OF MASTER AND SLAVE

In order to attain “full” transparency, it’s inevitable that theslave precisely tracks the master position and simultaneouslyslave also transmits the interaction force with the environmentto the master. In our case, the force transmitted by piezoresis-tive cantilever is in single dimension.

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A. Scaling of Position/Force Information

Since the master and the slave side resides on macro andmicro scales respectively, thus its very vital to appropriatelychoose the scaling factor in order to attain optimum perfor-mance. In the ideal condition, the steady state condition ofthebilateral controller should be

xs = αxm

Fm = βFs(24)

Where α and β represents the position and force scalingrespectively.xm, xs denotes the master and slave positionrespectively andFm, Fs denotes the master and slave forcerespectively. To be able to meaningfully interact with the microenvironment, positions and forces are scaled to match theoperator requirements.

In the first and second experiments, scaling factors ofα =0.027µm

degand β = 0.00366 N

nNare used, that is an angular

displacement of1deg on the master side corresponds to alinear displacement of1µm on the slave side and a force of0.00366nN on the slave side corresponds to a force of1N

on the master side. The objective of these experiments is toprovide very fine motion on the slave side for a relatively largerdisplacement on the master side, henceα is selected accordingto this objective. Then the corresponding forces/torques foreach amount of displacement were compared for the selectionof β, keeping in mind that the DC servo on the master sidehas low torques.

B. Experimental Validation of Position/Force Tracking

Figure 11 illustrates the experimental results for positiontracking between along with the tracking error of the masterand the slave systems. It can be clearly seen that under differ-ent references, the slave tracks the master position with highaccuracy. This position tracking performance is acceptable forprecisely positioning the micro cantilever.

Figure 12 demonstrates the force tracking between themaster and slave along with the tracking error. It can beclearly observed from the figures that the master tracks theslave force precisely and tracking error is found to be within±20nN . Thus it can be clearly concluded that using discreteSMC structure along with disturbance observer yield veryprecise position tracking. Force tracking also confirms thetransparency between the master and the slave.

VII. C ONCLUSION

In this article the design of a discrete time sliding mode con-troller based on Lyapunov theory is presented. A robust distur-bance observer based on sliding mode control is presented andapplied to a piezo-stage by considering all the nonlinearitiespresent in the system as lumped disturbance. Linear model ofa piezo-stage was used with nominal parameters and used tocompensate the disturbance acting on the system in order toachieve nano scale accuracy. The effectiveness of the controllerand disturbance observer was verified in term of closed loopposition performance. The results show the proposed controllerstructure produced good experimental results eliminatingany

Fig. 11. Position tracking of the bilateral controller for zig-zag motion withamplitude20nm

0 5 10 15 20 25−600

−500

−400

−300

−200

−100

0

Time (sec)

Ma

ste

r a

nd

Sla

ve

Fo

rce

s (n

N)

0 0.5 1 1.5 2 2.5

x 10

−16

−14

−12

−10

−8

−6

−4

−2

0

Time (sec)

Fo

rce

Err

or

(nN

)

Master Force

Slave Force

Fig. 12. Force tracking of the bilateral controller and tracking error

chattering motion but the influence of sensing noise, whichbelongs to high frequency range, effect steady state positionof the system and forces an oscillatory behavior.

Above mentioned discrete time sliding mode controllerhas been applied in bilateral structure for the tele-micromanipulation setup. A piezoresistive AFM micocan-tilever with inbuilt Wheatstone bridge to achieve the nano-newton level interaction forces between piezoresistive probetip and a glass surface. Experimental results are compared tothe theoretical estimates of the change in attractive forces asa function of decreasing distance and of the pull off forcebetween a silicon tip and a glass surface, respectively. Goodagreement among the experimental data and the theoreticalestimates has been demonstrated. Force/Position trackingbe-tween master and slave has been clearly demonstrated afternecessary scaling. It is clearly demonstrated that the slaveposition tracks the master position with high precision andthe master feels the interaction forces between the slave and

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environment in one dimension.

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