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MODELLING AND IDENTIFICATION OF A HIGH-PRECISION PLANARPOSITIONING SYSTEM

Kai Treichel, Kai Wulff, Johann Reger

Ilmenau University of TechnologyControl Engineering Group

P.O. Box 10 05 65D-98684, Ilmenau, Germany

Stefan Amthor

TETRA Gesellschaft für Sensorik,Robotik und Automation mbH

Gewerbepark Am Wald 4D-98693 Ilmenau, Germany

ABSTRACT

This contribution is dedicated to developing a thorough phys-ical model of a high-precision planar positioning system. Sub-ject of this investigation is Tetra’s state-of-the-art high-pre-cision planar positioning system “PPS200” using air bear-ings and electromagnetic coupling as a principle of propul-sion. We develop physical models of all components and in-vestigate the influence of various physical effects. The modelcharacteristics and parameters are identified using experimen-tal data taken from the dedicated test rig. The resulting sys-tem is implemented as a simulation model and verified bymeasurements of several experiments conducted at the testrig.

Index Terms— planar positioning system, air bearing,electro magnetic propulsion, physical modelling

1. INTRODUCTION

In order to meet the increasing requirements in high perfor-mance and high dimension accuracy for positioning systems,there has been active research on hovering, high-precisionplanar positioning systems in the past few years [1], [2], [3],[4], [5]. While latest technology provides precise position-ing capabilities down to the Nanometer scale, further im-provements in high-precision dynamic positioning will onlybe possible by applying modern nonlinear control methods.This contribution constitutes the first step towards this am-bitious goal by providing a detailed mathematical model ofTetra’s state-of-the-art planar positioning system “PPS200”.

The “PPS200” system consists of a stator base and a sliderwhich is levitated over the stator plate by means of air bear-ings. The slider is suitable to carry a maximal pay load of14kg which is to be positioned in the plane with a specifiedyaw-angle. The travel range of the slider is 200x200mm2

and the maximal yaw angle is ±0.25◦. A number of perma-nent magnets are mounted at the bottom of the slider. Thosemagnets are hovering above rigidly fixed current-carrying con-ductors embedded into the stator plate. Currents through thoseconductors together with the permanent magnets generate aforce between the stator plate and the slider resulting in anacceleration of the slider (see [6] for a discussion on the con-cept of construction).

Controlling the position of the slider (or even follow-ing predefined trajectories) to the highest degree of precisionis challenged by serval nonlinear dynamics of the system’scomponents. Moreover, at a certain degree of precision char-acteristics such as ripples caused by PWM amplifier or para-

Contact: {kai.treichel, kai.wulff, johann.reger}@tu-ilmenau.de

sitic capacities may be relevant for the dynamical behaviourof the system. In order to take the precision of the controlonto such a level, a detailed physical model of the dynamicsis needed. The goal of this contribution is to develop sucha model and to provide some insights into possible relevanteffects.

The paper is structured as follows. In the next section wegive an overview of the structure and introduce the main com-ponents of the positioning system. A detailed model of theactuator is developed in Section 3 followed by a discussionon the electromechanical coupling in Section 4. In Section 5we present the mechanical model of the slider considering 6degrees of freedom. The simulation results of the developedmodel are verified and compared with the measurements fromthe test rig in Section 6.

2. THE PPS200 STRUCTURE

The “PPS200” positioning system is an electromechanicaldrive composed of two main elements: an aerostatic guidedslider carrying the load to be positioned and a stator platewith imbedded coils (c.f. Fig. 1). The slider is hovering onair-bearings above the stator plate providing nearly friction-less motion. The coils functional complements are permanentmagnets mounted at the bottom of the slider such that theirmagnetic field interacts with the current through the coils ofthe stator plate resulting in a force (Lorentz-force) which con-sequently propels the slider. The position driven is measuredby an optical, incremental measurement system consisting oftwo parts – a measurement grid which is directly located atthe bottom of the slider at its center of gravity and a sen-sor system composed of three optical sensors located at thecenter of the stator plate [5]. One sensor is used to measuremotion in the direction of x and the rest of them being y1 andy2 are used in order to compensate or even actively controlthe yaw φz along the vertical z axis as well as detecting they-position driven. Unsurprisingly other positioning systemsshare similar assemblies [4].

In order to capture the slider’s motion an earth-fixed co-ordinate system (x, y, z) located at the center of the statorplate is used. The coordinates x and y describe the positionof the slider’s center of gravity, while φz denotes the slider’sorientation or yaw.

The block-diagram in Fig. 2 shows the basic structure ofthe closed loop system. The aim is to control the slider coor-dinates (x, y, φz) measured by the sensor system described,via the current through the coils considered as control input.Therefore the variables (xd, yd, φzd) define the reference tra-jectory for the slider to travel.

430

Φzx

y

z

driving coilcouple

air bearing

slider

measurementsystem

i.e. positionsensor

Btm2 Btm1

Rgt2

Rgt1

Top2 Top1

Lft2

Lft2

x

z

air gapTcommutation period

k

row ofpermanentmagnets

granitestatorplate

12

3 4

1, 42, 3y

Fig. 1. Basic assembly of the positioning system

computer withcontrol algorithm& commutation

pwmcurrentamplifier

coils,chokes

slidercoupling offorces

(coil/magnet)

positionx, y, Φz

x , y , Φd d zd

zx, y, ΦFj

jM

ijidj

Fig. 2. Block-structure of the PPS200 positioning system

These reference values are fed into the controller whichgenerates an actuator signal being the desired current vectorid = (idx idy idφz )

T . id is then fed into a commutationfunction which separates the currents position dependent onthe actuating element.

The actuating element is composed of j PWM-amplifiersconnected to the j driving coils, where j ∈ {Btm1, Btm2,Rgt1, Rgt2, T op1, T op2, Lft1, Lft2} represents the coildenotation. Chokes in series before and behind each drivingcoil are used to ensure EMC compatibility. Accordingly thecurrents ij through the coils are the actuator signals or con-trol inputs. These currents are transformed into the Lorentz-forces Fj via the coupling of forces between coils and perma-nent magnets thus actuating the slider (plant) and its position(x, y, φz). The forces applied between the permanent mag-nets and the bottom and top coil couple will move the sliderin the direction of x, while by interaction between the mag-nets and the left and right coil couple the slider will movein the direction of y. The slider’s acceleration and thus thetotal force acting on the slider is proportional to the currentsapplied. Since the slider is levitated it will also experience asuspension z, roll θx and pitch ψy . Hence, the slider’s statesare x, y, z, θx, ψy and φz .

3. ELECTRICAL PWM CURRENT AMPLIFIER ANDDRIVING COIL MODEL

The slider is driven by eight actuators each consisting of aPWM amplifier and a driving coil. In this section we inves-tigate the dynamics of these actuators. For simplicity of no-tation we shall omit the index j denoting the location of therespective actuator in this section.

The PWM-amplifier is especially designed by the IMMS1

to match the requirements of the “PPS200”. In order to modelthe amplifier, we extracted its basic structure from its electri-cal circuit which can be seen in Fig. 3. It basically consists of

inputfilter

PI-controller

comp

oscillatordriving coiland chokes

desiredcurrent

switchingstage

u a uR

uΔ

eueL u aL

uaL

10

f

RM

R S L S= ueid f

KRK

Fig. 3. Basic structure of one of the eight PWM-current-amplifiers

an input filter determining the cut-off frequency of the inputsignal, a controller to control the current i through the loadto the desired set point specified by the input signal id, a tri-angle wave form oscillator in conjunction with a comparatorto modulate the actuator signal of the controller to a PWM-signal and a switching stage composed of power transistorsin switching mode amplifying the PWM-signal. The load isconnected in an H-bridge, while the current through the loadis measured through the voltage across the measuring resis-tance RM and fed back to the controller in order to track thedesired value.

The desired current through the coils id is representedby the voltage uef across the input filter which is a simpleRC-low-pass filter. It can be described by

u̇af (t) =1

Tf(uef (t) − uaf (t)), (1)

where Tf = RC is the lag time constant. After filtering, thedesired set point for the current through the coil is denoted bythe voltage uaf being the input for the PI-controller whichgenerates the actuator signal uR, also measured in volts. Thecontrol law for the PI-controller is given by

uR(t) = KR

(e(t) +

1

TnR

t∫0

e(τ)dτ

). (2)

KR and TnR denote the controller gain and the time constantof the integrator, respectively. The control error is

e(t) = uaf (t) − KRK uaL(t). (3)In the comparator unit (denoted “comp” in Fig. 3) the sig-nal uR is compared to a triangle wave form uΔ with a basefrequency of roughly 22 kHz. Thus the output of the com-parator switches to high, if uR > uΔ or to low, if uR < uΔgenerating the pulse width modulated signal with respect tothe desired input signal. This signal is amplified by powerMOSFETs operating in switching mode (see switching stagein Fig. 3). The amplifiers output is given by the piecewiseconstant function

ueL(t) =

{+ueLmax for uR(t) > uΔ(t)−ueLmax for uR(t) < uΔ(t)

, (4)

where ueLmax = 150V .The second part of the actuator consists of the driving

coils and two further coils in series acting as chokes for im-proving EMC. The resistance RS in Fig. 3 represents the sum

1Institut für Mikroelektronik- und Mechatronik-SystemegGmbH, Ilmenau

431

of the coil winding resistances of those three coils and LS de-notes their overall inductance. Finally RM is a resistance toground used to measure the current through the coils. Theamplified pulse width modulated signal ueL acts at the inputof this network. The dynamics of this circuit is given by:

di(t)

dt= − 1

LS

((RM + RS)i(t) − ueL(t)

). (5)

The output voltage uaL = RM i is fed back via the gain KRKin the feedback loop.

-1 0 1 2 3x 10-4

0

0.1

0.2

0.3

0.4

0.5

0.6

time [s]

Am

plitu

de [V

]

measured outputoutput model

(a)

-5 0 5 10 15 20x 10-4

0

0.5

1

1.5

time [s]

Am

plitu

de [V

]

measured outputoutput model

(b)

Fig. 4. Comparison of measurements and simulationof the actuator model using the nonlinear inductance (6)(100mV =̂1A). (a) Detail of the slope. (b) Step responsesfor input signals of 1-4V .

For verification of the derived model we conducted sev-eral experiments at the test rig. At the input of the PWM-amplifier we applied rectangular pulse signals with ampli-tudes varying between 1 and 4V while the slider was re-moved. The current i through the coils was measured witha clamp-on ammeter.

Fig. 4 (b) shows the measured data for different ampli-tudes, where 100mV =̂1A. Besides the ripples that are char-acteristic for a PWM-amplifier at constant currents, we canobserve a significant increase of the slope when the currentexceeds 2A, c.f. the detail in Fig 4 (a). This slope increasecannot be reproduced using the linear model of the coils inequation (5). In fact, this nonlinear behaviour is due to mag-netic saturation of the metal core of the driving coil. At acertain current isat, called saturation current, the permeabil-ity of the core decreases. This leads to a loss of inductanceof the coils until eventually the coil behaves like an air-corecoil. Thus, the inductance of the coils LS is to be modelledas a nonlinear function of the current i(t). To approximate

this nonlinear behaviour of the inductance we choose the fol-lowing approach:

LS(i(t)) = K1 arctan(K2 |i(t)| + K3) + K4 . (6)

The Parameters K1, . . . , K4 were obtained by standard non-linear least-squares optimization.

In Fig. 4 the measured data is compared to the simulationusing the nonlinear inductance model (6). The overall perfor-mance of the model is very satisfying and even that charac-teristic increase of the slope due to core saturation (c.f. Fig 4(a)) is very well matched.

4. ELECTROMECHANICAL COUPLING

row ofpermanentmagnets

commutation-periodTK

airgap

airbearings1, 4

Top2,Btm2

Top1,Btm1

Rgt1, Rgt2Lft1, Lft2

stator platex

z

y

airbearings2, 3

Fig. 5. Shape of the magnetic field of the permanent magnetsalong the x axis

The forces acting on the slider are generated by the in-teraction between the electromagnetic field induced by thecurrent through the driving coils and the magnetic field of thepermanent magnets at the bottom of the slider. According toLorentz’ law the electromagnetic force is given by

FL = il × B, (7)

where i is the current through a wire with constant length land B is the flux density of the homogeneous magnetic fieldof a permanent magnet [7].

In order to derive a sufficiently precise description of theelectromagnetic coupling we make the following (mild) as-sumptions:

1. The vertical distance between the magnets and thecoil-surface is sufficiently small so that the magneticfield is homogeneous (in the vertical direction) andperpendicular with respect to the coil surface.

2. The sag (subsidence) of the air bearings (roughly 5μm) is coupled with the quantities z, θx and ψy of theslider. The influence of these quantities on the abso-lute value as well as the direction of the Lorentz-forceis negligibly small. Hence, we assume that the mag-netic field is always perpendicular to every single coil-winding such that the forces for the x- and y-directionare decoupled from each other. The forces act alwaysperpendicular with respect to the corresponding driv-ing coil regardless of the orientation of the slider.

3. The rotation about the z-axis is sufficiently small suchthat the electromagnetically relevant area between themagnets and the coils’ surfaces is independent of theslider’s orientation φz . In fact the maximum rotationis restricted to φz,max = ±0.25◦.

432

With the above assumptions the electromagnetic forcegenerated by the j-th coil and the slider’s magnetic field isgiven by

Fj = ij lB , (8)

where ij is the current through the respective coil, l is theeffective length of the conductor and B is the magnetic fluxdensity of the magnetic field effective for the coil.

The magnetic field generated by the magnets is not ho-mogeneous along the x-axis (bottom and top coils) and they-axis (left and right coils), see Fig. 5 for an illustration. Thusthe flux density B of the magnetic field interacting with thecoils depends on the slider position above the stator plate.Fig. 6 shows the measurements of the magnetic flux over thex-position of the slider. Approximating this function by

Bx = B̂ cos

(2π

TKx

)(9)

where TK is the distance between the centers of adjacentmagnets, yields satisfying results as shown also in Fig. 6.

-60 -40 -20 0 20 40 60 80 100-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

x-position [mm]

flux

dens

ity B

[T]

measured Bk cos(2π(1/Tk) x)

Fig. 6. Magnetic flux density along the x-direction.

In order to compensate for the spacial dependency of themagnetic field we consider the structure depicted in Fig. 7(suggested in [6]). The inputs are the desired forces in x- andy-direction and rotation torque expressed by (idx , idy , idφz ),respectively. The outputs are the resulting forces Fx, Fy andthe torque Mz acting on the slider.

Comm-utation

pwm amp/coils

ijidj coupling offorces(coil/magnet)

idΦzidx idy( , , ) ( F , F , M )x y z

Fig. 7. Feedforward-structure to compensate the spacial de-pendance of the magnetic field.

The resulting force on the slider in x- direction is given by

Fx = FTop1 + FTop2 + FBtm1 + FBtm2 . (10)

By choosing the currents idj as sinusoidal functions of theslider position x with appropriate phase-shift accounting forthe offsets due to the coil position on the stator, the spacial de-pendency can be compensated by the commutation. Assum-ing an ideal compensation yields for the force on the slider:

Fx = 2lB̂ idx , (11)

where l in [m] is the effective length of the conductor and B̂is the magnitude of the magnetic flux density in [T ]. Note,that lB̂ is constant such that the force Fx only depends onthe current idx , but is independent of the slider’s position.Similar considerations yield according results for Fy and Mz .

5. MECHANICAL SLIDER MODEL

x

y

zCG

Mz

Φz

d

leverarm

Top Top2F = F + FTop1

Btm Btm2F = F + FBtm1

ΦzLft

Lft2F=F

+F

Lft1

Rgt

Rgt2

F=F

+F

Rgt1

Mx

x

z

y

ψy

Fc1,4

z1,4F , FBtm Top

My

zcg

g

x

z

y

θxzcg

g

b/2

Fc2,3

z2,3

z1,2z3,4Fc3,4 Fc1,2

F , FRgt Lft

(a)

(b)

(c)

Fig. 8. Planar (a) and vertical (b),(c) forces as well as torquealong the x (c), y (b) and z (a) axes (red:forces and torque,blue:angles, green:distances)

The slider is essentially a mass with six degrees of free-dom suspended by air bearings. Deriving the equations ofmotion we shall distinguish between the planar movements(x, y, φz) and the additional spacial movements (z, θx, ψy)considering the third dimension. In the former case the promi-nent forces on the slider are the propelling electromagneticforces and some viscous friction forces due to the air bear-ings. In the latter case, the slider motion is dominantly in-fluenced by the suspending forces, but we shall also considerthe fact that the propelling forces attack outside the center ofgravity and thus cause for roll and pitch motion.

To begin with we derive the planar equations of motion,see Fig. 8 (a). The black square in the figure denotes thepoints of attack of the electromagnetic forces along the rowsof permanent magnets at the bottom of the slider. Assumingthat the planar forces are decoupled (see Section 4) we obtainthe following balance of forces by applying the principle oflinear momentum (Newton’s second law):

mẍ = FBtm + FTop − FRx (12)mÿ = FRgt + FLft − FRy , (13)

where m is the slider mass and FBtm, FTop, FRgt, FLft arethe resulting forces of each respective pair of coils. The lineardamping terms FRx = krẋ and FRy = kr ẏ denote the so-called drag- or viscous friction forces (Stokes law) evoked bythe air bearings and the viscosity of air (see [2]).

433

For the motion about the z-axis we apply the principle ofconservation of angular momentum:

JzCG φ̈z = MzBtm + MzRgt − MzT op − MzLft , (14)where JzCG denotes the slider’s moment of inertia about thez-axis and the respective torques are given by

MzBtm =( d

cos(φz)+ x tan(φz)

)FBtm (15)

MzT op =( d

cos(φz)− x tan(φz)

)FTop (16)

MzRgt =( d

cos(φz)+ y tan(φz)

)FRgt (17)

MzLft =( d

cos(φz)− y tan(φz)

)FLft (18)

For the dynamics in z-direction we are investigating theinfluence of the air bearings. The air bearings are vacuumpreloaded meaning that they provide vacuum air as well aspressurisation (see [8]). We approximate their dynamical be-haviour with a simple parallel spring-damper model similarto the approach in [3]. The suspension force Fci of the i-thair bearing is then given by

Fci = cllzi − dllżi (19)where dll is the damping-constant, zi is the distance betweenthe stator surface and the i-th air bearing, and cll is the springconstant which again depends on the air gap zi. Elementarygeometric considerations (c.f. Fig. 8 (b),(c)) yield for the airgaps at each bearing:

z1 = z +b

2

(sin(θx) + sin(ψy)

)(20)

z2 = z +b

2

(sin(θx) − sin(ψy)

)(21)

z3 = z − b2

(sin(θx) − sin(ψy)

)(22)

z4 = z − b2

(sin(θx) + sin(ψy)

)(23)

For identification of the spring constant cll we resort to ex-tensive experiments conducted at the IMMS who establisheda typical load deflection curve of an air bearing depicted inFig. 9. Note however, that for a particular air bearing thiscurve may differ significantly.

4.5 4.6 4.7 4.8 4.9 5 5.1 5.2x 10-6

1

1.2

1.4

1.6

1.8x 107

air gap [m]

sprin

g co

nsta

nt [N

/m]

Fig. 9. Load deflection curve of a single air bearing estab-lished by the IMMS

The dynamics in the z-direction are then given by

mz̈ =

4∑i=1

Fci − mg . (24)

Finally we consider the balances of torques about the x-and y-axis:

JxCG θ̈x = zcg(FRgt + FLft

)+

+b

2

(Fc1 + Fc2 − Fc3 − Fc4

) (25)JyCG ψ̈y = zcg

(FBtm + FTop

)+

+b

2

(Fc1 + Fc4 − Fc2 − Fc3

),

(26)

where zcg denotes the lever arm of the driving forces withrespect to the center of gravity, c.f. Fig. 8 (b),(c).

The moments of inertia in (14),(25),(26) are given by

JxCG =1

12m(a2 + c2) (27)

JyCG =1

12m(a2 + c2) (28)

JzCG =1

6ma2 (29)

where a is the width of the slider, c its hight and m its mass.The complete model of the slider is given by the differen-

tial equations (12), (13), (14), (24), (25) and (26) representingthe dynamics of the slider in an adequate and simple way.

6. EXPERIMENTAL VERIFICATION

The model derived above was implemented in Simulink andverified using measurements of the test rig. For those ex-periments the test rig was run in open loop and square-waveinput signals with various amplitudes were applied demand-ing a movement in x-direction. During those experiments theslider was fixed in the y-direction to avoid measurement er-rors due to disturbances.

Fig. 10 shows the results for idx = 0.1A, 1A and 2A,respectively. Note that only the position (figures (ai)) wasmeasured at the test rig. The velocities (bi) as well as ac-celerations (ci) were obtained by numerical differentiation.Moreover, the air gaps in Fig. 10 (di) are simulative resultsonly as no sensors for the air gaps are available at the test rig.

By examination of the Figures (a2,3), (b2,3), (c2,3) ofthe second and the third measurement one can observe thatfor currents idx > 1A the overall mathematical model ap-proximates the real system quite good. However, Fig. 10(a1)-(c1) depicting the results of the first measurement il-lustrates that there is some kind of damping effect presentat low currents (idx < 1A). The velocity (a2) increasesrapidly at the beginning (0-0.5s) but decreases around 0.5sand settles in a steady-state. Those damping effects might becaused by eddy-current and magnetization effects as observedin [6]. Another possible reason for this behaviour may be dueto errors in the commutation or the geometrical assembly ofthe permanent magnets. Our derived model does not accountfor any of these effects and thus is not capable to reflect thisdamping behaviour. Extending the model in this regard willbe subject of future investigations.

Fig. 10 (d1,2,3) show the simulation results of the airgaps at two of the bearings for an input amplitude of 3A. Thevariations of the air gap due to the acceleration in x-directionare very small (in the range of 0.07μm), which is deemedto have no impact on the electromagnetic coupling. The re-sulting pitch and roll motion are also small such that theirinfluence on the dynamics of the slider is indeed arguable.

434

0 2 4 6 8-0.15

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0.05PPS200modelinput signal

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PPS200modelinput signal

(a1) (a2) (a3)

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(b1) (b2) (b3)

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0 0.5 1 1.5 2 2.5-1.2

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0.8333

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0 0.5 1 1.5-2.2

-1.4667

-0.7333

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0.7333

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0 0.5 1 1.5 -3.5

-2.3333

-1.1667

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1.1667

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(c1) (c2) (c3)

0 0.5 1 1.5 1.813

5.15

5.2

5.25

x 10-6

z2

0 0.5 1 1.5 1.813

5.15

5.2

5.25

x 10-6

z1

0.825 0.83 0.835 0.84 0.845

5.18

5.2

5.22

5.24

5.26

5.28x 10-6

z1

(d1) (d2) (d3)

Fig. 10. Experimental and simulation results for idx = 0.1A (first column) idx = 1A (second), idx = 2A (third). Subfigures(ai): slider position x [m]; Subfigures (bi): velocity ẋ [m/s], Subfigures (ci): acceleration ẍ [m/s2]; Subfigures (d1,2,3): airgaps z2 and z1, [m] for experiment 3 only.

7. CONCLUDING REMARKS

In this paper we present a study on the dynamic behaviour ofTetra’s high-precision planar positioning system “PPS200”.We derived physical models of the main components of thesystem and analysed various nonlinear effects. The resultingsystem description proves suitable to simulate the dynamicsof the positioning system to a large degree of precision. Sig-nificant deviations of the simulation and experimental resultsare only observable at very low input currents. Further in-vestigations are needed to analyse the cause of the dampingeffect observed for this input range. The derived model issuitable to serve as a basis for analysis and application ofnonlinear control algorithms.

8. REFERENCES

[1] Y. Tomita, Y. Koyanagawa, and F. Satoh, “A surfacemotor-driven precise positioning system,” Precision En-gineering, vol. 16, no. 3, pp. 184–191, 1994.

[2] R.S. Fearing, “A planar milli-robot system on an air bear-ing,” in 7th Int. Symp. Robotics Research. 1996, vol. 7,pp. 570–581, Springer.

[3] S.Q. Lee and D.G. Gweon, “A new 3-DOF Z-tilts mi-cropositioning system using electromagnetic actuators

and air bearings,” Precision Engineering, vol. 24, no.1, pp. 24–31, 2000.

[4] W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono,and Y. Tomita, “A surface motor-driven planar motionstage integrated with an XY θZ surface encoder for pre-cision positioning,” Precision Engineering, vol. 28, no.3, pp. 329–337, 2004.

[5] M. Stubenrauch, A. Albrecht, W. Hild, O. Mollenhauer,B. Guddei, S. Spiller, C. Schaeffel, M. Katzschmann, andF. Spiller, “Novel precision positioning system with in-tegrated planar guides,” in 53rd International ScientificColloquium (IWK 2008), Ilmenau, Germany, 2008, pp.251–252.

[6] St. Hesse, H.-U. Mohr, and C. Schaeffel, “Planar motionsystem for a nanopositioning- and nanomeasuring ma-chine (npmm, nmm) with 200x200m operating range,”in euspen 4th International Conference, Glasgow, Scot-land, 2004.

[7] J.N. Chiasson, Modeling and high performance controlof electric machines, Wiley-IEEE Press, 2005.

[8] St. Hesse, C. Schaeffel, M. Katzschmann, T. Maass, andH.-U. Mohr, “Planar motor concept for positioning withnanometer position uncertainty,” in euspen 8th Interna-tional Conference, Zürich, Switzerland, 2008.

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