Integrated Design of Mechanism and Control for High-Precision Stagesby the Interaction Index in the Direct Nyquist Array Method
Wataru Ohnishi1 , Hiroshi Fujimoto1 , Koichi Sakata2 , Kazuhiro Suzuki2 and Kazuaki Saiki2
Abstract— High-precision stages are widely used in thesemiconductor and flat panel industry. Because these stages
have six degrees of freedom to control, coupling forces candeteriorate its control performance and stability. If heightsof the center of gravity (CoG), the center of rotation (CoR),
the actuation point, and the measurement point are not thesame, couping between the x and θy motions occurs. In thispaper, integrated design method of mechanism and control is
proposed utilizing the Direct Nyquist Array (DNA) method
and a changeable actuation height stage by means of multiple
actuator arrangement in the x direction. Due to the model
analysis and the optimal actuation height, the coupling canbe reduced with a simple precompensator. By drawing theGeneralized Gershgorin Band (GGB), the effectiveness of the
proposed method is verified through experiments.

I. I NTRODUCTION
High-precision scan stages play an important role in the
manufacturing processes for semiconductors and liquid crystal displays [1]. In these applications, high-precision stages
have six degrees of freedom (x, y, z, θx , θy , θz ) to control [2].
In this paper, a high-precision stage which has multiple
actuators in the x direction shown in Fig. 1–3 is considered.
Additional actuator approach is provided to deal with flexible
dynamics in e.g. [3]. Here, this paper focuses on a decoupling
control between x and θy motions by changing the heightof the actuation point using multiple actuators based on adetailed model. Decoupling control by data-based approachis provided in e.g. [4].
For 6-DOF motion systems, coupling forces degrade
control performance. In our previous paper [5], a detailed
model considering the misalignment among the center ofgravity (CoG) of the stage, the center of rotation (CoR)of the stage, the actuation point of the translational force
fx , and the measurement point is proposed. According to
the model, by changing the height of the actuation point,
coupling characteristic from the translational force fx to therotational angle θy can be varied. On the other hand, coupling
characteristic from the torque τy to the translational position
x is defined by the height of the measurement position which
is difficult to be changed.
In this paper, integrated design of mechanism and control
is proposed. The coupling from fx to θy is suppressed by
the optimal actuation height (mechanism). Next, the couplingfrom τy to x is suppressed by the precompensator used in
the Direct Nyquist Array (DNA) method [6]–[8] (control).
1 The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 2778561, Japan, ohnishi@hflab.k.u-tokyo.ac.jp, fujimoto@k.u-tokyo.ac.jp
2 Nikon Corporation, 47-1, Nagaodaityou, Sakae, Yokohama, 2448533, Japan, koichi.sakata@nikon.com, kazuhiro.suzuki@nikon.com,
kazuaki.saiki@nikon.com

Linear motor f X1

Fine stage

Relative position
sensor x r1
Relative position
sensor x r2
Linear motor f X2

Fig. 1.

Picture of the 6-DOF high-precision stage.

z
x

Coarse stage

y

z

y

VCM (f y1 )x
VCM (f x2 )VCM (f z4 )VCM (f z1 )VCM (f z3 )Gravity Canceller (f gc)

VCM (f y2 )Encoder (z4 )Encoder (z3 )VCM (f x1 )Encoder (x 1 )Encoder (z1 )VCM (f z2 )Encoder (x 2 )Encoder (z2 )Encoder (y)(a) Actuator arrangement of the(b) Sensor arrangement offine stage. This stage has two
the fine stage.
VCMs in the x direction.
Fig. 2. Structure of the fine stage.

Fig. 3. Side view of the fine stage. By changing the thrust distribution
ratio a (see (1) and (2)), the height of the virtual actuation point can beplaced arbitrarily.

Next, the interaction index in the DNA method is calculated.
Finally, the actuation height and the feedback controllers
are designed to have a low interaction index around thesensitivity function peak. The effectiveness of the proposed
method is demonstrated by drawing the Generalized Gershgorin Band (GGB).
II. E XPERIMENTAL SETUP
A. Overview
Our research group designed the dual stage shown in Fig.
1. The actuator and sensor arrangement of the fine stage isillustrated in Fig. 2. As shown in Fig. 2(a) and Fig. 3, the finestage has two Voice Coil Motors (VCMs) in the x direction.
The fine stage is supported by a 6-DOF air bearing“gravity canceller” [9]. The picture and schematic of the

z

x

y

Fine stage
Center of rotation

Air gyro (òx; òy ; òz)Center of rotation(measured)(measured)y z

Air gyro (òx; òy ; òz)Planar air bearing (x; y)xAir(unmeasured)

Planar air bearing (x; y)Center of rotation
Air bearing
actuator (f gc; z)

Air bearing
actuator (f gc; z)

Air

Fine stage

Coarse stage(a) Side view.

(b) Schematic.
(unmeasured)Fig. 4. Structure of the gravity canceller. The radius of the curvature of
the air gyro decides the height of the center of rotation (CoR) of the finestage.

Gravity canceller

Fig. 5. Fine stage model of the x and θy motion.
(positive directions of Lg2 , Lf x and Lm are defined as illustrated)

gravity canceller are shown in Fig. 4. The gravity canceller
compensates for the gravitational force experienced by thefine stage and supports its 6-DOF without friction. The
gravity canceller is composed of three parts: the air gyro,
the planar air bearing and the air bearing actuator which
support the (θx , θy , θz ), (x, y) and (z)-directional motion,
respectively. As shown in Fig. 4, the shape of the air gyro is
like a hemisphere. The fine stage slides on the hemisphericsurface of the air gyro with an air gap of a few µm. In thispaper, the center of the hemisphere is called the center ofrotation (CoR). In other words, the radius of the curvatureof the air gyro decides the height of the CoR.

TABLE IM ODEL PARAMETERS .
Symbol
xm
xg1
xg2
θy
fx
τy
Mx1
Cx1
Kx1
Mx2
Jθy
Cθy
Kθy
Lm
Lg2
Lf x

Meaning
Measured position of the fine stage
Position of the CoG of the planar air bearing and the air gyro
Position of the CoG of the fine stage
Measured attitude angle of the fine stage
Input force of the fine stage in the x direction
Input torque of the fine stage in the θy direction
Mass of the planar air bearing and the air gyro
Viscosity coefficient in the xg1 motion
Spring coefficient in the xg1 motion
Mass of the fine stage
Moment of inertia of the fine stage
Viscosity coefficient of the fine stage in the θy motion
Spring coefficient of the fine stage in the θy motion
Distance between the measurement point of xm and the CoRDistance between the CoR and the CoG of the fine stage
Distance between the CoR of the fine stage and the actuation point

Value
–
–
–
–
–
–
0.077 kg430 N/(m/s)
11000 N/m
5.3 kg0.10 kgm2
1.6 Nm/(rad/s)
1200 Nm/rad−0.028 m−0.051 mchangeable

B. Changeable actuation point stage by multiple actuators
Due to the spatial limitation of the fine stage, a VCM
for the x direction cannot be placed at the desired actuation
point. In this section, a new fine stage structure which
can shift the virtual actuation point by means of thrust
distribution of multiple actuators is introduced. This is basedon a simple idea shown in Fig. 3. The two VCMs generate
force fx1 and fx2 in the x direction by a thrust distributionlaw
fx1 = afx , fx2 = (1 − a)fx ,(1)

where a denotes the thrust distribution ratio. Here, the heightof the virtual actuation point Lf x is defined as
Lf x = aLf x1 + (1 − a)Lf x2 ,(2)

where Lf x1 and Lf x2 denote the heights of the actuation
point of fx1 and fx2 from the CoR, respectively.
Although this structure doubles the amount of wiring, andthe mass of the fine stage becomes slightly heavier, thisstructure has advantages as described below. By changing thethrust distribution ratio a, the height of the virtual actuation
point Lf x can be placed at desired point such as the positionof the CoG and the CoR. Moreover, during operation, the
CoG shifts caused by load mass variation. Because thetransfer functions depend on the height of the CoG Lg2(formulated in (15)–(19)), the coupling characteristics change
dynamically. Even in this case, this structure can reduce
coupling forces by means of placing the virtual actuation
point at the desired position.

III. M ODELINGIn this section, the model considering the misalignment
among the CoG, CoR, the actuation point, and the measurement point is introduced [5].
A. Lagrange’s equations
In this section, the Lagrange’s equations are formulated
based on the model shown in Fig. 5 and Tab. I. First, therelationship of xg1 , xg2 , and θy is expressed byxg2= xg1 + Lg2 sin(θy ),

(1)

ẋg2

= ẋg1 + Lg2 cos(θy )θ̇y .

(2)

The kinetic energy T , the potential energy U , the dissipation
function B, and the work W are defined as follows:
1
1
1
T =
Mx1 ẋ2g1 + Mx2 ẋ2g2 + Jθy θ̇y2 ,
(3)
2
2
2
1
1
U =
Kx1 x2g1 + Kθy θy2 + Lg2 Mx2 g cos(θy ), (4)
2
2
1
1
2
B =
Cx1 ẋg1 + Cθy θ̇y2 ,(5)2
2
W = fx [xg1 + Lf x sin(θy )] + τy θy .
(6)
According to (3) and (4), the Lagrangian L = T −U is given
by
[
]2 1
1
1
L = Mx1 ẋ2g1 + Mx2 ẋg1 + Lg2 cos(θy )θ̇y + Jθy θ̇y2
2
2
2
1
1
2
2
− Kx1 xg1 − Kθy θy − Lg2 Mx2 g cos(θy ).
(7)22

xm (s)[Jθy + Lf x Lm Mx1 − (Lf x − Lg2 )(Lg2 − Lm )Mx2 ]s2 + (Cθy + Lf x Lm Cx1 )s + Kθy + Lf x Lm Kx1 − Lg2 Mx2 g=
fx (s)D(s)θy (s)[Lf x Mx1 + (Lf x − Lg2 )Mx2 ]s2 + Lf x Cx1 s + Lf x Kx1=
fx (s)
D(s)[Lm Mx1 + (Lm − Lg2 )Mx2 ]s2 + Lm Cx1 s + Lm Kx1
xm (s)=
τy (s)D(s)θy (s)(Mx1 + Mx2 )s2 + Cx1 s + Kx1=
τy (s)
D(s)D(s) = [(Mx1 + Mx2 )Jθy + Mx1 Mx2 L2g2 ]s4 + [(Mx1 + Mx2 )Cθy + (Jθy + Mx2 L2g2 )Cx1 ]s3 +

(15)
(16)
(17)
(18)

[(Jθy + Mx2 L2g2 )Kx1 + (Mx1 + Mx2 )(Kθy − Mx2 Lg2 g) + Cθy Cx1 ]s2 +[Cθy Kx1 + Cx1 (Kθy − Lg2 Mx2 g)]s + Kx1 (Kθy − Lg2 Mx2 g)(8)where qi denotes generalized coordinates, q1 = xg1 and q2 =
θy . Finally, according to (1)–(8), nonlinear equations (9) and(10) are obtained:ẍg1 (Mx1 + Mx2 ) + Cx1 ẋg1 + Kx1 xg1[]+Mx2 Lg2 cos(θy )θ̈y − sin(θy )θ̇y2 = fx ,(9)[
Mx2 Lg2 ẍg1 cos(θy ) − g sin(θy ) + Lg2 θ̈y cos2 (θy )]−Lg2 θ̇y2 sin(θy ) cos(θy ) + Jθy θ̈y + Cθy θ̇y + Kθy θy= τy + fx Lf x cos(θy ).

(10)

B. Linearization
Assuming cos(θy ) ≃ 1, sin(θy ) ≃ θy , θ̇y2 ≃ 0, (9) and (10)are linearized as follows:(Mx1 + Mx2 )ẍg1 + Cx1 ẋg1 + Kx1 xg1 + Mx2 Lg2 θ̈y = fx , (11)(Mx2 L2g2 + Jθy )θ̈y + Cθy θ̇y + Kθy θy + Mx2 Lg2 (ẍg1 − gθy )(12)= τy + fx Lf x .

C. Transformation to measurable coordinate
The generalized coordinate xg1 cannot be measured. Thus,
xg1 is converted to the measurement position xm by
xm (s)
fx (s)
xm (s)
τy (s)==

xg1 (s)θy (s)+ Lm,
fx (s)
fx (s)
xg1 (s)θy (s)+ Lm,
τy (s)
τy (s)(13)(14)

where Lm denotes the distance between the measurement
point and the center of rotation.
D. Transfer functions
According to (11)–(14), transfer functions (15)–(19) are
obtained. In this paper, (15)–(19) are also expressed as
xm (s) xm (s)[]g11 (s) g12 (s) fx (s)τy (s) G(s) =(20)= θy (s)θy (s) 
g21 (s) g22 (s)fx (s)
τy (s)

Magnitude [dB]

−50
−100
−150

Lfx = 0 (CoR−driven case)
Lfx = Lg2 (CoG−driven case)−200 0
10

1

10

10

2

3

10

0
Phase [deg]The Lagrange’s equations are given by
(
)
∂L
∂W
d ∂L
∂B
−
=
+
(i = 1, 2),dt ∂ q̇i
∂qi
∂ q̇i
∂qi

(19)

−90
−180
−270
−360 0
10

1

10

10

2

3

10

Frequency [Hz]Fig. 6. Frequency characteristics of g21 (s) (model parameters are shownin Tab. I). There is a trade-off between CoR-driven case (Lf x = 0) and
CoG-driven case (Lf x = Lg2 ) in the high and low frequencies.

E. Observations
Equation (16) and (17) show that the numerators of thecoupling transfer functions are defined by the height of theactuation point Lf x , the measurement point Lm , and thecenter of gravity Lg2 .
θ (s)Let us consider the relationship between fxy (s) and theposition of the actuation point Lf x . If Lf x equals to 0, meaning the stage with driven at CoR, Lf x (Mx1 s2 + Mx2 s2 +Cx1 s + Kx1 ) becomes 0. Now, the coupling term is only
−Lg2 Mx2 s2 /D(s). Then, the coupling gain of g21 (s) in
the low frequency is suppressed. On the other hand, if
Lf x equals to Lg2 , meaning the stage with driven at theθ (s)CoG, the numerator of fxy (s) becomes 1st order (assuming
Mx1 ≪ Mx2 ). Then, the coupling gain of g21 (s) in the highfrequency is suppressed. A comparison between the CoRdriven and the CoG-driven cases of g21 (s) is shown in Fig.
6. Fig. 6 shows the coupling gain has a trade-off between
the high and low frequencies.
IV. D IRECT N YQUIST A RRAY METHOD AND
INTERACTION INDEX

If the coupling of the plant G(s) is not sufficiently
weak, the closed loop characteristic becomes undesired one.
Then, the Direct Nyquist Array (DNA) method [6]–[8]was proposed for multivariable control. The DNA method
consists of four steps: 1) design and implementation of aprecompensator K(s) to reduce coupling of the plant G(s),2) design of SISO feedback controllers, 3) calculation of
the interaction index λ(s|Q), 4) drawing the Generalized
Gershgorin Band (GGB). The control object Q(s) and theprecompensator are defined by[][]
q11 (s) q12 (s)k11 (s) k12 (s)Q(s) =, K(s) =. (21)
q21 (s) q22 (s)k21 (s) k22 (s)A. Design of the precompensator K(s)In the DNA method, the precompensator K(s) is implemented to reduce the coupling of the control object Q(s)shown in Fig. 7. Here, the control object Q(s) is expressedasQ(s) = G(s)K(s).

Control object

Precompensator

Plant

Fig. 7. Feedback system with precompensator K(s) for the plant G(s)formulated in (20).

(22)

Clearly, to make Q(s) diagonal, K(s) is calculated as
g12 (s) k21 (s)g21 (s)k12 (s)
=−
,
=−
.
(23)
k22 (s)g11 (s) k11 (s)g22 (s)In the DNA method, a reduced order of precompensator
K(s) is used.
B. Calculation of the interaction index λ(s|Q)
Here, the matrix M (s|Q) is defined by
q12 (s)
0
q
(s)
22
.
M (s|Q) =  q (s)210
q11 (s)Fig. 8. Control system seen from x∗m and xm . Control object Q(s)is defined as Q(s) = G(s)K(s). Closed loop characteristic of feedback
controller c1 (s) is affected by the coupling transfer function and the otherloop’s feedback controller c2 (s).

V. P ROPOSAL OF INTEGRATED DESIGN OF MECHANISM(24)

The interaction index λ(s|Q) is defined as a maximum eigenvalue of M (s|Q). The interaction index λ(s|Q) represents
the strength of the coupling of the control object Q(s).
C. Drawing the Generalized Gershgorin Band (GGB)
Fig. 7 can be converted into Fig. 8, where h11 (s|c2 )denotes the control object of the SISO feedback controller
c1 (s). h11 (s|c2 ) is calculated byc2 (s)q12 (s)q21 (s).
(25)
1 + c2 (s)q22 (s)As shown in (25), the control stability and performance areaffected by the other loop’s SISO feedback controller c2 (s).
The same can be said for:c1 (s)q12 (s)q21 (s)h22 (s|c1 ) = q22 (s) −,(26)
1 + c1 (s)q11 (s)h11 (s|c2 ) = q11 (s) −where h22 (s|c1 ) denotes the control object of c2 (s).
In the conclusion of DNA method, (27) and (28) areproven:|c1 (jω)h11 (jω|c2 ) − c1 (jω)q11 (jω)|< λ(jω|Q), (27)|c1 (jω)q11 (jω)||c2 (jω)h22 (jω|c1 ) − c2 (jω)q22 (jω)|< λ(jω|Q). (28)|c2 (jω)q22 (jω)|
Equation (27) means that the Nyquist diagram ofc1 (jω)h11 (jω|c2 ) is within the circle, the center of which isc1 (jω)q11 (jω), and the radius of which is c1 (jω)q11 (jω) ·λ(jω|Q), whatever c2 (s) is.

AND CONTROLIn this section, integrated design of mechanism and control
method is proposed for the decoupling of the x and θymotions utilizing the changeable actuation point stage andthe DNA method.
A. Simple precompensator design by the changeable actuation point stage
First in this paper, k11 = k22 = 1 is considered in (23) not
to change diagonal characteristics. Here, k12 (s) and k21 (s)compensate the coupling of the plant g12 (s) and g21 (s),respectively.
As shown in Sec. II, this stage can change the height of theactuation point Lf x by the thrust distribution of the multiple
actuators. Next, according to (16), the coupling transfer
function g21 (s) can be varied by Lf x . In this proposed
method, the coupling transfer function g21 (s) is suppressed
without using the precompensator k21 (s). On the other hand,the coupling transfer function of g12 (s) is defined by theheight of the measurement point Lm which is difficult to bechanged. Here, the coupling of g12 (s) is suppressed by theprecompensator k12 (s).
From the above, the precompensator K(s) is set as][(s)
1 − gg12
(s)
11
.
(29)
K(s) =
01
B. Feedback controller design and the interaction index
The nominal plants for the feedback controllers c1 (s)and c2 (s) are defined by the 2nd order model considering

−80

2

10

−160

0

0

−90

−90

−180
−270
−360 110

210Frequency [Hz]0−0.033−0.069−180 110310Phase [deg]Phase [deg]−180 1
10−140

2Magnitude [dB]

Magnitude [dB]640−0.033−0.069
2

10

2

0 1
10

3

10
Frequency [Hz]

10

2

10
Frequency [Hz]

3

−130

0
−0.033−0.069
2

3

10

10

0
Phase [deg]

−90

−270
−360 110

210Frequency [Hz](b) Enhanced figure of Fig.
11(a).

−270

12
2

3

10
Frequency [Hz]10(d) g22 .

Fig. 9. Measured frequency responses. The height of the actuation point
Lf x is changed as 0, −0.033, −0.069. g11 , g12 , and g22 are not largely
affected by the variation of the Lf x . According to Fig. 9(c), there is a trade
off between Lf x = 0 and Lf x = −0.069 in a high and low frequencies.
The validity of the model shown in (15)–(19) and Fig. 6 is demonstrated.

Magnitude [dB]

0
−20

0.8
0
−0.033−0.069

8
6
4

−40
−60
10

2

0
−0.033−0.069

0.6
0.4
0.2

2
0 1
10

2

10
Frequency [Hz]

3

10

0 1
10

2

10
Frequency [Hz](b) Enhanced figure of Fig.
12(a).

Fig. 12. Interaction index with the precompensator k12 (s). The height of
the actuation point Lf x is changed as 0, −0.033, −0.069. The interaction
index is suppressed compared to Fig. 11.

0−0.033−0.069HPF
3

10off between Lf x = 0 and Lf x = −0.069 in a high and low
frequency as shown in the model (see Fig. 6).

180
Phase [deg]10(a) From 10 Hz to 1000 Hz.

20

90
0

B. Design of the precompensator k12 (s) and feedback controllers−90−180 1
10

Fig. 11. Interaction index without the precompensator k12 (s). The heightof the actuation point Lf x is changed as 0, −0.033, −0.069. Clearly, there
is a trade-off between Lf x = 0 and Lf x = −0.069 in a high and low
frequency.

−90(c) g21 .

−80 1
10

2

10

−180

−360 1
10

3

10

0.2

Frequency [Hz](a) From 10 Hz to 1000 Hz.

−90
−110

0

−180

0.4

0 1
10

3

10

−70

−150 110

100−0.033−0.0690.62(b) g12 .

−100−160 110Phase [deg]

8

−50

−80

−140

0.8
0
−0.033−0.069−270(a) g11 .
−60

−120

10

−180

−360 1
10

3

10

3

10

10

Interaction index λ−160

0−0.033−0.069−120

Interaction index λ

−140

12

−100

Interaction index λ−120Interaction index λMagnitude [dB]

Magnitude [dB]

−80
−100

2

3

10
Frequency [Hz]

10

g

(jω)

Measured frequency responses of g12 (jω) (Lf x is set as
Fig. 10.
110, −0.033, −0.069). These frequency responses are fitted by 1st orderhigh-pass filter with 16 Hz cut-off frequency.

(jω)
are
According to the frequency responses, the gg12
11 (jω)calculated and fitted by the 1st order high-pass filter asshown in Fig. 10. This 1st order high-pass filter is used as
the precompensator k12 (s). In this experiment, the feedback
bandwidth is designed as 20 Hz by Sec. VB approach.

C. Interaction index and GGB
diagonal part only. The feedback controllers c1 (s) and c2 (s)are PID compensators designed by pole assignment [10].
In this step, the interaction index λ(s|Q) is calculatedby the measured frequency responses of the plant and theprecompensator. Here, the optimal actuation height Lf x
should be selected to have a low interaction index aroundthe frequency of the sensitivity function peak.
VI. E XPERIMENT
A. Frequency responses of the plant
In the experiment, the height of the actuation point ischanged as Lf x = 0, −0.033, −0.069 by the thrust distribution using the multiple actuators described in Sec. II.
The measured frequency responses are shown in Fig. 9. Fig.
9 shows that g11 , g12 , and g22 are not largely affected by thevariation of the Lf x . According to Fig. 9(c), there is a tradeFig. 11 shows the interaction index without the precompensator k12 (s). The GGBs are illustrated in Fig. 13, 14,
and 15 from Fig. 11. c1 q11 and c2 q22 denote the desired
feedback characteristics considering the diagonal part only.
c1 h1 and c2 h2 denote the actual feedback characteristics
considering the coupling of the plant and the other loop’s
feedback controller. In Lf x = 0 case shown in Fig. 13,
the actual feedback characteristics c1 h1 and c2 h2 have lessphase margin compared to c1 q11 and c2 q22 because the GGBs
around the sensitivity function peak are wide. In Lf x =−0.069 case shown in Fig. 15, the GGBs become narrow
and the phase margin improves because the interaction index
is small. In this case, however, the GGBs still overlap thepoint of (−1 + j0).
Fig. 12, 16, 17, and 18 depict the interaction index and the
GGBs with the simple precompensator k12 (s) shown in Fig.
10. The GGBs illustrated in Fig. 18 are away from the point

0.5

0

0

0

0

c1q11
c1h1
GGB

−1.5
−2
−2

−1.5

−1
−0.5
Real axis

0

−0.5
−1
c2q22
c2h2
GGB

−1.5
−2
−2

0.5

−1.5

(a) x.

0

0

−0.5
−1
c1q11
c1h1
GGB
−1.5

−1
−0.5
Real axis

0

−1
c2q22
c2h2
GGB
−1.5

0

−0.5
−1
c1q11
c1h1
GGB
−1.5

−1
−0.5
Real axis

0

0.5

0

−1
c1q11
c1h1
GGB

−1.5

(a) x.

−1−0.5
Real axis

0

0.5

−1.5

Fig. 17.

−1−0.5
Real axis

0

−2
−2

0.5

c2q22
c2h2
GGB
−1.5

of (−1 + j0). The effectiveness of the proposed integrated
design of mechanism and control is demonstrated throughexperiments.
VII. C ONCLUSION
This paper proposes the integrated design of mechanism
and control by utilizing the precise model of x and θy motions and the changeable actuation point stage with multiple
actuators. Due to proposed integrated design, coupling canbe suppressed by simple precompensator. The effectivenessof the proposed method is verified by experiments drawing
the GGBs.
Although this paper dealt with the x and the θy motions
as an example, it can also be applied to other coupling suchas the y and θx motions. Furthermore, this method could be
used for other applications such as robots, automobiles, andsatellites, all of which have a misalignment of its CoG, its
CoR, its actuation point, and its measurement point.
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00.5

GGB (Lf x = −0.033) with precompensator k12 (s).

0.5
0

−1
c1q11
c1h1
GGB

−1.5
−1.5

−1
−0.5
Real axis(a) x.

(b) θy .

GGB (Lf x = −0.069) without the precompensator k12 (s).

−1−0.5Real axis(b) θy .

−0.5

−2
−2

0.5

−1
−1.5

0

c2q22
c2h2
GGB

0

−0.5

(a) x.

−1−1−0.5
Real axis

0

0.5

−1.5

−1.5

(b) θy .

−0.5

−2
−2

0.5

−0.5

−2
−2

−2
−2

0.5

GGB (Lf x = 0) with precompensator k12 (s).

−1.5

Imaginary axis

0
Imaginary axis

Imaginary axis

0.5

Fig. 15.

−1−0.5
Real axis

c2q22
c2h2
GGB

0.5

(b) θy .

0.5

−2
−2

Fig. 16.

−0.5

−2
−2

0

0

−1.5

0.5

−1
−0.5
Real axis−1(a) x.

GGB (Lf x = −0.033) without the precompensator k12 (s).

−1.5

−1.5

−0.5

−1.5

0.5

(a) x.
Fig. 14.

−2
−2

0.5

Imaginary axis

0.5

−1.5

c1q11
c1h1
GGB

−1.5

GGB (Lf x = 0) without the precompensator k12 (s).

0.5

−2
−2

0

−1

(b) θy .

Imaginary axis

Imaginary axis

Fig. 13.

−1−0.5
Real axis−0.5

Imaginary axis

−1

Fig. 18.

0

0.5

Imaginary axis

−0.5

Imaginary axis

0.5

Imaginary axis

0.5

Imaginary axis

Imaginary axis

0.5

−0.5
−1
c2q22
c2h2
GGB

−1.5
−2
−2

−1.5

−1
−0.5
Real axis

00.5(b) θy .

GGB (Lf x = −0.069) with precompensator k12 (s).

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