Modeling and Control of a High-thrust Direct-driveSpiral Motor

Yasutaka Fujimoto, Issam A. Smadi, Hiroko Omori, Koichiro Suzuki, and Hiroshi HamadaDepartment of ECE, Yokohama National University, 79-5 Tokiwadai, Yokohama 240-8501, JAPAN

Abstract— In this paper, modeling and control of spiral motoris proposed. The voltage equation and motion equation of thespiral motor is proposed. Based on this modeling, control systemfor the spiral motor is proposed. The proposed controller consistsof three parts; the first part is PI current controller with backEMF compensation specialized for spiral motor, the second partis disturbance observer based PD controller for linear and gapmotion interacting each other, and the third part is zero-powercontroller for equilibrium fluctuation of gap displacement. Itis confirmed that the proposed controller achieves independentlinear position and gap control simultaneously.

Index Terms— Axial flux machine, Linear actuator, Motioncontrol, Robotics.

I. INTRODUCTION

Efficiency and power density of the electric motors arevery high compared to other actuators, but its torque is notenough for several applications. Therefore high-ratio gears arecombined with electric motors in many applications. Mainlosses in such actuator systems with the geared servo motorsare mechanical transmission loss in the gears, iron loss inthe iron core, copper loss in the windings, and switchingloss in the power converter. From a control viewpoint, themechanical loss, i. e., friction loss reduces adaptability, safety,and backdrivability in the motion systems. Various motionmechanisms and controls that recover the backdrivability werereported in the past works[1]–[10].

The authors have proposed a novel helical mechanism ofactuator that realizes direct-drive motion without mechanicalgears[11]–[13]. The structure and developed prototype areshown in Fig. 1. Permanent magnets are attached on thesurfaces of the mover. Slots are provided for winding onthe surface of the stator. Three-phase winding in the slotsgenerates flux in the axial direction. Thus, the spiral motoris a helical motion permanent magnet axial flux machine. Dueto its large air-gap area, the motor can generate relatively highthrust force.

In this paper, circuit equation and motion equation of thespiral motor are introduced. In order to realize direct-drivemotion, we need to keep the air-gap constant. A magneticlevitation control is proposed for this purpose. A variation ofthe zero power controller is also proposed.

This work was supported by KAKENHI 19676003.

phase W coil phase W’ coil

phase V coilphase U coil

phase V’ coilphase U’ coil

(a) A stator with three-phase windings.

magnet

(b) A mover with permanent magnets.

(c) A motor combined with the stator and mover.

Slide Rotary BushSlide Rotary Bush

StatorMotor Frame Mover Magnet

Air Gap Windings

(d) A cross-sectional view.

Fig. 1. A preliminary structure of a spiral motor.

Fig. 2. A helical-shape stator yoke.

II. MODELING OF SPIRAL MOTOR

A. Mechanism of Spiral Motor

Figure 1 (a)-(c) shows a preliminary structure of the spiralmotor. Permanent magnets are attached on the surfaces ofthe mover. Slots are provided for windings on the surfaceof the stator. The spiral motor is a helical motion axial fluxpermanent magnet motor. Three-phase windings in the slotsgenerates flux in the axial direction. Due to its large air-gaparea, the motor can generate relatively high thrust force.

The cross sectional view of the spiral motor is shown inFig. 1 (d). The radial load applied to the output shaft issupported by two slide rotary bushes and the thrust loadis directly controlled by the electromagnetic force. Thus amagnetic levitation control is required to keep the air-gapconstant. For this levitation control, two independent three-phase inverters are required.

B. Prototypes

Four types of prototype of spiral motors have been devel-oped as shown in Table III. A helical-shape stator yoke ismade of soft magnetic composite (SMC) as shown in Fig. 2.A helical-shape magnet is made of Nd-Fe-B as shown in Fig.3. Internal structure of the stator is shown in Fig. 4. The Nd-Fe-B magnets are attached on the mover yoke which is madeof silicon steel, as shown in Fig. 5. Precise helical shapes ofthe stator and mover enable uniform short length of air-gap andavoid concentration of stress when the mover touches downto the stator.

Exterior of the spiral motor is shown in Fig. 6. A rotaryencoder and linear encoder attached at the mover measure thelinear position x and rotation angle θ for control. Displacementof the air gap xg is computed by using these measurement asfollows.

xg = x − �p

2πθ (1)

C. Permeance Model

In order to derive analytical voltage equation, and thrust-force and torque equations, a permeance model of the motorfor 360 degree electric angular displacement in polar coordi-nates is presented.

Figure 7 shows the polar coordinates expression of themotor. From this model, a simple magnetic circuit as shownin Fig. 8 is obtained. For simplicity magnetic resistance of

Fig. 3. A helical-shape Nd-Fe-B magnet.

TABLE I

SPECIFICATIONS OF SPIRAL MOTORS.

#1 #2 #3 #4Number of stator layers 6 8.5 12 12Number of mover layers 2 4 7 5Length of flame*[mm] 132 182 252 252Diameter of flame*[mm] 55 55 55 55Stroke[mm] 60 70 80 120Lead length of screw[mm] 20 20 20 20Nominal length of air gap[mm] 1 1 1 1*excluding projecting parts

Fig. 4. Internal structure of a stator. (model #1)

the iron core is ignored. Rg = 3p(�g − xg)/Sμ0 is magneticresistance of forward side air gap for an area of each phasewindings, where S is a gap area of the cross section ofthe cylinder. R′

g = 3p(�g + xg)/Sμ0 is magnetic resistanceof backward side air gap. Rm = 3p�m/Sμm is magneticresistance of a permanent magnet. c(·)If is a spatial functionof magnetomotive force of the field magnet by the permanentmagnet and approximated by a cosine function as c(θ) =k cos(θ) where k = 6

√3

π2 sin(pα/2) is the fundamental compo-nent of c(θ). If = Br�m/μm is an equivalent magnetizationcurrent, i. e., magnetomotive force of the permanent magnet.These parameters are shown in Table II. For simplicity, weassume that the permeability of the permanent magnet μm isequivalent to μ0. Also the edge effect is ignored.

In the part (A) of the magnetic circuit, interlinkage fluxΦ = [Φu, Φv, Φw, Φf ]T for each current is obtained as Φ =LI where I = [Iu, Iv, Iw, If ]T is current vector and L is aninductance matrix:

L = P

⎡⎢⎢⎣

n2 − 12n2 − 1

2n2 32nkc0

− 12n2 n2 − 1

3n2 32nkc1

− 12n2 − 1

2n2 n2 32nkc2

32nkc0

32nkc1

32nkc2 (3k

2 )2

⎤⎥⎥⎦ (2)

Fig. 5. A mover. (model #2)

Fig. 6. Exterior of a spiral motor. (model #1)

S

N

N

S

S

S N

N S

S N

N

N S

S

N

N

N

S

S

S

N

S

S

S

N

N

N S

Iw

Iv

Iu

Iu

I’w

I’v

I’u

I’u

S

N

N

N

S

S

S

N

S

S

S

N

N

N S N

r2

r1

(A) (B)lp

lm lg+xglg-xg

a (i)

(ii)

(iii)Iw

Iv

Iu

Iu

I’w

I’v

I’u

I’u

x

θ

N S

θ

rb

ra

lpmlps

Fig. 7. A spiral motor in polar coordinates.

where P = 23(Rg+Rm) is permeance for one-phase current and

ci = cos(pθ − 2iπ3 ).

In general, a salient-mover machine has a mutual induc-tance between ni-turns windings at position θi and nj-turnswindings at position θj represented by

Lij = Pdninj cos(θi − pθ) cos(θj − pθ)+ Pqninj sin(θi − pθ) sin(θj − pθ) (3)

where Pd ∝ 1/(� − xg) is d-axis permeance and Pq ∝1/(� − xg) is q-axis permeance. � = �g + �m is a nominalgap between the mover iron and the stator. Instead of (2), we

(A) (B)

(i)

(ii)

(iii)Rg Rm

c(θ-4π/3)If

nIw

Rg Rm

c(θ-2π/3)If

nIv

Rg Rm

c(θ)If

nIu

c(θ-4π/3)If’

c(θ-2π/3)If’

c(θ)If’

Rm

Rm

Rm

Rg’

Rg’

Rg’

nIw’

nIv’

nIu’

Fig. 8. A magnetic circuit.

TABLE II

NOMENCLATURE

α angle of a sector-type permanent magnetBr residual flux density of a permanent magnetf thrust force of the moverIf magnetomotive force of a field magnetIi i-axis current on forward side windings, i ∈ {d, q}I ′

i i-axis current on backward side windings, i ∈ {d, q}J moment of inertia of the mover around the axisk a fundamental Fourier component of c(θ)� a nominal gap between the mover iron and the stator�g nominal length of air gap�m thickness of magnet�p lead length of screwμ0 the permeability in vacuumμm permeability of a permanent magnetM mass of the movern ampere-turnp number of pole pairs per 360 degree mechanical

angular displacementq number of layersθ mechanical rotational angle of the moverS a gap area per 360 degree mechanical angle

displacementτ torque of the moverx a linear position of the moverxg displacement of air gap

adopt a inductance matrix as follows

L =

⎡⎢⎢⎣

L00 L01 L02 L03

L10 L11 L12 L13

L20 L21 L22 L23

L30 L31 L32 L33

⎤⎥⎥⎦ (4)

where n0 = n1 = n2 = n, n3 = nf = 3k/2, θ0 = 0, θ1 =2π/3, θ2 = 4π/3, and θ3 = pθ. The model (4) is equivalentto (2) if the machine is not salient, i. e., Pd = Pq = P holds.

The voltage equation, thrust-force equation, and torqueequation of the spiral motor are derived by using the induc-tance matrix (4) as follows.

V = RI + LdI

dt+ θ

∂L

∂θI + x

∂L

∂xI (5)

f =12IT ∂L

∂xI (6)

τ =12IT ∂L

∂θI (7)

where V = [Vu, Vv, Vw, Vf ]T is voltage of each windings,

R = diag(Rs, Rs, Rs, Rf ) is resistance of windings, and fand τ are thrust-force and torque of the mover, respectively.

D. Voltage Equation on dq-axis

In order to apply field oriented control to the spiral motor,dq-axis model of the spiral motor is derived as follows. Letthe transformation matrix C be

C =

⎡⎢⎢⎢⎢⎣

√23c0

√23c1

√23c2 0

−√

23s0 −

√23s1 −

√23s2 0

1√3

1√3

1√3

00 0 0 1

⎤⎥⎥⎥⎥⎦ (8)

where ci = cos(pθ− 2iπ3 ), si = sin(pθ− 2iπ

3 ). Then the dq-axiscurrent and voltage are represented by Idq = CI and Vdq =CV where Idq = [Id, Iq, I0, If ]T , Vdq = [Vd, Vq, V0, Vf ]T .And Id, Iq , I0 are d-axis, q-axis, and zero-phase current, Vd,Vq , V0 are d-axis, q-axis, and zero-phase voltage, respectively.

The voltage equation on dq-axis is

Vdq = CRCT Idq + CLCT Idq + θC∂L

∂θCT Idq

+ θCL∂CT

∂θIdq + xC

∂L

∂xCT Idq (9)

In (9), note that

∂L

∂θ=

∂L(xg, θ)∂θ

∣∣∣∣xg=x− �p

2π θ

=∂L(xg, θ)

∂θ− �p

2π

∂L(xg, θ)∂xg

(10)

Then, the voltage equation (9) is expanded as follows.

Vd = RsId + LdId − pθLqIq + Ldf If

+xg

� − xg(LdId + LdfIf ) (11)

Vq = RsIq + Lq Iq + pθ(LdId + LdfIf )

+xg

� − xgLqIq (12)

V0 = RsI0 (13)

Vf = Lf If + Ldf Id +xg

� − xg(LdIf + LdfId) (14)

where Ld = 32n2Pd ∝ 1/(�− xg) and Lq = 3

2n2Pq ∝ 1/(�−xg) are d-axis and q-axis inductances, respectively. Ldf =√

32nnfPd ∝ 1/(� − xg) is equivalent mutual inductance

between d-axis stator windings and field magnet windings.Lf = n2

fPd ∝ 1/(� − xg) is equivalent self-inductance offield magnet windings.

Finally, following dq-axis voltage equation is obtained bytaking first two equations (11), (12) and substituting If = 0.[

Vd

Vq

]=[

Rs + Ld( ddt ) 0

0 Rs + Lq( ddt )

] [Id

Iq

]+ Ψf

[1

�−xgxg

pθ

]

+xg

� − xg

[Ld 00 Lq

] [Id

Iq

]+ pθ

[0 −Lq

Ld 0

] [Id

Iq

](15)

where Ψf = LdfIf corresponds to field flux by the permanentmagnet. Fig. 9 shows block diagram of this dq-axis circuitmodel.

As the same manner, voltage equation of part (B) in Fig. 7and 8 is obtained as follows.[V ′

d

V ′q

]=[

Rs + L′d(

ddt ) 0

0 Rs + L′q(

ddt)

] [I ′dI ′q

]+ Ψ′

f

[− 1

�+xgxg

pθ

]

− xg

� + xg

[L′

d 00 L′

q

] [I ′dI ′q

]+ pθ

[0 −L′

q

L′d 0

] [I ′dI ′q

](16)

where L′d = 3

2n2P ′d ∝ 1/(� + xg) and L′

q = 32n2P ′

q ∝1/(� + xg) are d-axis and q-axis inductances in part (B),respectively. Ψ′

f = L′dfI ′f corresponds to field flux by the

permanent magnet in part (B), where L′df =

√32nnfP ′

d ∝1/(� + xg).

Two independent three-phase inverters are required fordriving the spiral motor. One inverter controls the circuit (15)and the other controls the circuit (16).

E. Thrust-force/Torque Equation

Thrust-force equation (6) and torque equation (7) are rewrit-ten by using Idq as follows.

f =12ITdqC

∂L

∂xCT Idq (17)

=1

� − xg

(ΨfId +

LdI2d + LqI

2q + LfI2

f

2

)(18)

τ =12ITdqC

∂L

∂θCT Idq (19)

= p(ΨfIq + (Ld − Lq)IdIq)

− �p

2π

1� − xg

(ΨfId +

LdI2d + LqI

2q + LfI2

f

2

)(20)

As the same manner, thrust-force equation and torque equa-tion in part (B) are obtained as follows.

f ′ = − 1� + xg

(Ψ′

fI ′d +L′

dI′2d + L′

qI′2q + L′

fI ′2f2

)(21)

τ ′ = p(Ψ′fI ′q + (L′

d − L′q)I

′dI

′q)

+�p

2π

1� + xg

(Ψ′

fI ′d +L′

dI′2d + L′

qI′2q + L′

fI ′2f2

)(22)

Finally, total thrust-force and torque per 360 degree electri-cal angle are obtained as ftotal = f + f ′ and τtotal = τ + τ ′.Hense, the equation of motion is obtained as follows.

Mx = pq(f + f ′) − Dx − d (23)

Jθ = pq(τ + τ ′) − Dτ θ − dτ (24)

where M , J are mass and inertia of the mover, respectively.D, Dτ are friction coefficient for linear and rotaty motion,respectively. d, dτ are disturbance force and torque applied tothe mover, respectively.

Vd

Vq

Id

Iq

1

Lds+Rs

1

Lqs+Rs

Ld

Lq

xg

θ

1

�−xg

p

Ψf

Ψf

××

××

+

+ + + +

+++

+

−

Fig. 9. dq-axis circuit model of the spiral motor in part A.

F. Simplified Thrust-Force/Torque Model

For control design, a simplified model is obtained bylinearlizing (18), (20), (21), and (22) around xg = 0.

ftotal � 2Lf0I2f

�2xg +

Ψf0

�(Id − I ′d)

+Ld0

2�(I2

d − I ′2d ) +Lq0

2�(I2

q − I ′2q ) (25)

τtotal � p(Ψf0(Iq + I ′q) + (Ld0 − Lq0)(IdIq + I ′dI

′q))

− �p

2π

(2Lf0I2f

�2xg +

Ψf0

�(Id − I ′d)

+Ld0

2�(I2

d − I ′2d ) +Lq0

2�(I2

q − I ′2q ))

(26)

where Ld0 = Ld|xg=0 = L′d|xg=0, Lq0 = Lq|xg=0 = L′

q|xg=0,Ψf0 = Ψf |xg=0 = Ψ′

f |xg=0, and Lf0 = Lf |xg=0 = L′f |xg=0.

From (25) and (26), linear motion is mainly controlledby d-axis current and rotational motion is mainly controlledby q-axis current. The first term in (25) corresponds to aunstable drift force which is a positive feedback in terms ofgap displacement.

When the field oriented control is applied to (15) and (16)and their current hold conditions Id = −I ′d and Iq = I ′q bythe current control, then we have

ftotal � Kgxg + KfId (27)

τtotal � KτIq − �p

2π(Kgxg + KfId) (28)

where Kg =2Lf0I2

f

�2 , Kf = 2Ψf0� , and Kτ = 2pΨf0. From

(23), (24), (27), and (28), it turns out that the motion systemhas a spring whose coefficient is negative. Figure 10 shows theblock diagram of the simplified model of the motion system.

x

xg

θ

x

θ

d

dτ

Id

Iq

1

Ms+D

1

Js+Dr

�p

2π

�p

2π

1

s

1

s

Kg

Kf

Kτ

+

+

+

+

++

−

−

−

−

Fig. 10. Simplified motion model of the spiral motor.

III. CONTROL OF THE SPIRAL MOTOR

A. Current Controller

From (15) and (16), current controller is designed as fol-lows.

V refd =

(Kpd +

Kid

s

)(Iref

d − Id) + Ed (29)

V refq =

(Kpq +

Kiq

s

)(Iref

q − Iq) + Eq (30)

V ′refd =

(Kpd +

Kid

s

)(−Iref

d − I ′d) + E′d (31)

V ′refq =

(Kpq +

Kiq

s

)(Iref

q − I ′q) + E′q (32)

where the controller gains are chosen so that Kpd = ωidLd0,Kpq = ωiqLq0, Kid = ωidRs, and Kiq = ωiqRs. Also Ed,Eq , E′

d, and E′q are back EMF compensation given by

Ed =xg

�(Ψf0 + Ld0Id) − pθLq0Iq (33)

Eq = pθ(Ψf0 + Ld0Id) +xg

�Lq0Iq (34)

E′d = − xg

�(Ψf0 + Ld0I

′d) − pθLq0I

′q (35)

E′q = pθ(Ψf0 + Ld0I

′d) −

xg

�Lq0I

′q. (36)

B. Position/Gap Controller with Disturbance Observer

From (27) and (28), current references are obtained asfollows.

Irefd =

1Kf

(f ref − Kgxg) (37)

Irefq =

1Kτ

(�p

2πf ref + τref

)(38)

where f ref and τref correspond to thrust-force and torque ref-erences. The modeling error and disturbance are compensatedby using disturbance observer[2] as follows.

f ref = Mnxref + d (39)

τref = Jnθref + dτ (40)

where the terms d and dτ are estimated disturbandes.

d =ωd

s + ωd(f ref − Mnsx) (41)

dτ =ωd

s + ωd(τref − Jnsθ) (42)

The acceleration references in (39) and (40) are computedby trajectory tracking feedback controller for position x andregulator for gap displacement xg as follows.

xref = Kpx(xcmd − x) + Kdx(xcmd − x) + xcmd(43)

xrefg = Kpg(xcmd

g − xg) + Kdg(xcmdg − xg) (44)

θref =2π

�p(xref − xref

g ) (45)

where xcmd represents position command of the mover, xcmdg

represents gap displacement command, respectively. Ideally,the unstable equilibrium point corresponds to being the gapdisplacement xg = 0. However, there is a offset of theequilibrium point according to the manufacturing accuracy ofthe parts of the stator and mover. Thrust load also affects theoffset. In such a case, there remains constant current in orderto achieve xg = 0, which causes copper loss in the windings.xcmd

g should be set to be the equilibrium point by using thezero power controller.

C. Zero Power Controller

Zero-power control[14] was proposed for electromagneticsuspension systems using permanent magnets, which achievesautomatic gap adjustment so that the input current convergesto zero. In this paper, we propose a variation of zero-powercontroller suitable for our control system.

The equilibrium gap displacement xcmdg in (44) is obtained

by integrating d-axis current reference Irefd as follows. xcmd

g

approaches to the equilibrium point and finally Id = 0 isachieved if xcmd

g corresponds to the equilibrium.

xcmdg =

Kz

sIrefd (46)

xcmdg = KzI

refd (47)

where Kz represents a controller gain.

D. Overall Control System

Finally, the overall control system of the spiral motor isdescribed as shown in Fig. 11.

IV. SIMULATION

A. Plant and Control Parameters

Table III shows plant and control parameters. The electricalplant parameters in this table are for 360 degree electricalangle.

TABLE III

PLANT AND CONTROL PARAMETERS

p number of pole pairs per 360 degree 2q number of layers 2�g nominal length of air gap 1 [mm]�m thickness of magnet 2 [mm]Br residual flux density of magnet 1.2 [T]n ampere-turn 50Rs registance of windings 0.374 [Ω]Ld0 nominal d-axis inductance 0.329 [mH]Lq0 nominal q-axis inductance 0.329 [mH]Ψf0 nominal field flux 0.0195 [Wb]Kf thrust-force constant 13.0 [N/A]Kτ torque constant 0.0781 [N.m/A]J moment of inertia of the mover 715 [g cm2]M mass of the mover 0.229 [kg]Kg equivalent spring coefficient 25800 [N/m]Kpd, Kpq propotional gain of current controller 1.64Kid, Kiq integral gain of current controller 1870ωid, ωiq bandwidth of current control system 5000 [rad/sec]ωd bandwidth of disturbance observer 500 [rad/s]Kpx propotional gain of position controller 10000Kdx derivative gain of position controller 200ωx bandwidth of position control system 100 [rad/sec]Kpg propotional gain of position controller 2500Kdg derivative gain of position controller 100ωg bandwidth of gap control system 50 [rad/sec]Kz gain of zero-power controller 0.002Vdc dc link voltage of inverter 80 [V]Ts control sampling period 50 [μsec]Δx resolution of linear encoder with

quad edge evaluation method 0.25 [μm]Δθ resolution of rotary encoder with

quad edge evaluation method 2π/20000 [rad]

B. Results

The proposed control system is applied to spiral motormodel (5)–(7). Figure 12 shows simulation results of theproposed control. The mover touches down at t = 0[sec],i. e., the gap displacement is set to xg = 0.7[mm] as theinitial condition. Step command xcmd = 1[mm] for the linearposition x is given at t = 0.4[sec]. Figure 12(a) shows theposition and gap response of the mover. In this figure, therotation angle θ is converted to equivalent linear displacementby multiplying �p/2π. We can see that the gap displacementconverges to neutral position and the mover position convergesto the command value without interfaring gap control. Fig.12(b)(c) show armature current and input voltage on dq-axis.

C. Response under Inertia Fluctuation

Figure 13 shows simulation results when the actual mass Mfluctuates 10 times bigger than the nominal value Mn. Veryrobust response is obtained.

D. Response under Equilibrium Fluctuation

Figure 14–15 show simulation results under equilibriumfluctuation. The offset of equilibrium is set at xg = 50[μm].Figure 14 is a case with the proposed zero-power controller.The gap displacement converges to equilibrium point and zerocurrent is realized. On the other hand, in a case withoutthe proposed zero-power controller as shown in Fig. 15, thegap controller manages to keep xg = 0 where continuous

Controlled Plant

Disturbance Observerfor Linear Motion

Disturbance Observerfor Rotational MotionZero Power

Controller

Gap Controller

Position Controller

x

xg

xg

θ

x

θ

d

dτ

Id

Iq

Irefd

Irefq

1

Ms+D

1

Js+Dr

�p

2π

�p

2π

�p

2π

�p

2π

�p

2π

1

s

1

s

Kg

Kf

Kτ

+

+

+

++

+

+

+

++

++

+

+

+

++

++

+ +

+

+

+

+

+

++

−

−

−

−

−

−

−

−

−

−

−

−

−

−

ωid

s+ωid

ωiq

s+ωiq

1

Kfn

1

Kτn

Kgn

fref

τref

xref

θref

xrefg

2π

�p

xcmd

xcmd

xcmd

xcmdg

xcmdg

Kpx

Kdx

Kpg

Kdg

Mns

Jns

ωxs

s+ωx

ωθs

s+ωθ

ωd

s+ωd

ωd

s+ωd

d

dr

Kz

s

Kz

Fig. 11. Overall control system of spiral motor.

current required. Positioning accuracy of both cases are almostsame. Thus, the proposed control has a advantage of energyefficiency.

V. CONCLUSION

In this paper, modeling and control of spiral motor isproposed. The model consists of two parts; one is voltageequation that is an extension of well-known dq-axis model,and the other is motion equation that has a negative spring.The proposed controller consists of three parts; the first part isPI current controller with back EMF compensation specializedfor spiral motor, the second part is disturbance observer basedPD controller for linear and gap motion interacting each other,and the third part is zero-power controller for equilibriumfluctuation of gap displacement. It is comfirmed that theproposed controller achieves independent linear position andgap control simultaneously.

REFERENCES

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Fig. 12. Simulation results.

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Fig. 13. Simulation results with mass fluctuation. (M = 10Mn)

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Fig. 14. Simulation results with equilibrium fluctuation. (offset = 50 [μm])

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Fig. 15. Simulation results with equilibrium fluctuation without zero-powercontroller. (offset = 50 [μm])