Md. Emdadul Hoque
e-mail: [email protected]
Takeshi Mizuno
Daisuke Kishita
Masaya Takasaki
Yuji Ishino
Department of Mechanical Engineering,
Saitama University,
255 Shimo-Okubo, Sakura-Ku,
Saitama 338-8570, Japan
Development of an Active
Vibration Isolation System Using
Linearized Zero-Power Control
With Weight Support Springs
This paper presents a hybrid vibration isolation system using linearized zero-power control with weight support springs. The isolation system, fundamentally, is developed by
linking a mechanical spring in series with a negative stiffness spring realized by zeropower control in order to insulate ground vibration as well as to reject the effect of
on-board-generated direct disturbances. In the original system, the table is suspended
from the middle table solely by the attractive force produced by the magnets and therefore, the maximum supporting force on the table is limited by the capacity of the permanent magnets used for zero-power control. To meet the growing demand to support heavy
payload on the table, the physical model is extended by introducing an additional
mechanism-weight support springs, in parallel with the above system. However, the nonlinearity of the zero-power control instigates a nonlinear vibration isolation system,
which leads to a deviation from zero compliance to direct disturbance. Therefore, a
nonlinear compensator for the zero-power control is employed furthermore to the system
to meet the ever-increasing precise disturbance rejection requirements in the hitechnology systems. The fundamental characteristics of the system are explained analytically and the improved control performances are demonstrated experimentally.
关DOI: 10.1115/1.4000968兴
Keywords: active vibration control, vibration isolation, magnetic levitation, nonlinear
zero-power control, weight support springs
1
Introduction
Many mechanical systems, especially those that deal with
micro- and nanoscales in high-technology manufacturing or precision metrology, are subject to vibration disturbances containing
effect from ground vibration and on-board-generated direct disturbances. Researchers in this field typically tackle significant challenges to meet both demands. Passive control techniques, as it is
popular for low cost, uses softer suspension for isolating ground
vibration, whereas harder suspension are used to reject the disturbance forces on the table. A trade-off between them is necessary
to reject these two types of disturbances 关1兴. There is still a challenge for passive control because of its limited performance as
well. Therefore, active vibration isolation technology is receiving
satisfactory momentum as a means to insulate precision machines
from ground vibration and at the same time suppress the effect of
direct disturbance force by introducing different types of active
actuators into these systems to improve performances 关2–6兴. However, power consumption and use of high performance sensors
such as servo-type accelerometers in the active control system
lead to costlier systems and become a bottleneck to implement it.
To overcome the above limitations, the authors have recently
proposed a unique vibration isolation systems using zero-power
magnetic suspension 关7–9兴. The control current in the zero-power
control system converges to zero during its stable operation and
hence save the energy significantly 关10,11兴. The proposed vibration isolation system uses only relative displacement sensors,
which is much cheaper than servo-type accelerometers. A hybrid
magnet consisting of permanent magnet and electromagnet is used
Contributed by the Technical Committee on Vibration and Sound of ASME for
publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received
December 17, 2008; final manuscript received December 9, 2009; published online
May 25, 2010. Assoc. Editor: Alan Palazzolo.
Journal of Vibration and Acoustics
to achieve zero-power control. Since a zero-power system behaves as if it has negative stiffness, infinite stiffness against disturbances on the isolation table can be achieved by combining it
with a mechanical spring. It enables the system to have good
characteristics both in reducing vibration transmitted from the
floor and in suppressing direct disturbances.
However, the isolation table is merely levitated by the permanent magnets in the basic system. Therefore, the system has limitations for supporting heavy systems and suppressing the effect of
large payloads. In order to solve such problems, an additional
mechanism, weight support spring, is added between the isolation
table to the base in parallel with the basic system to improve the
above performances. A single-axis model was proposed for further
investigation 关12兴. Another difficulty becomes apparent due to the
nonlinearity of the zero-power control system 关13,14兴, which also
leads to performance degradation for large payloads. Therefore, a
nonlinear compensator for zero-power control can also be introduced to the system. In this paper, the development of a novel
six-degree-of-freedom 共6DOF兲 vibration isolation system and the
implementation of a mode-based controller are presented for promoting industrialization that can overcome the shortcomings of
the existing systems. Apart from this, emphasis is given to the
suppression of direct disturbing forces by two steps: 共a兲 development of a vibration isolation system with weight support springs
that allows simple design of vibration isolation system and 共b兲
improvement of performance furthermore by using nonlinear
compensation of the zero-power control.
2
Configuration of Vibration Isolation Systems
2.1 Original Configuration. The vibration isolation system is
developed to generate infinite stiffness for direct disturbing forces
and to maintain low stiffness for floor vibration. Infinite stiffness
can be realized by linking a mechanical spring with a magnetic
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f
d
k2
k3
N
S
N
S
N
S
k1
Fig. 2 Basic model of zero-power magnetic suspension with
weight support spring
Fig. 1 Zero-compliance system using zero-power magnetic
suspension
spring that has negative stiffness 共see Sec. 3.3兲 in series. When
two springs with spring constants of k1 and k2 are connected in
series, the total stiffness kc is given by
kc =
k 1k 2
k1 + k2
共1兲
This equation shows that the total stiffness becomes lower than
that of each spring when normal springs are connected. However,
if one of the springs has negative stiffness that satisfies
k1 = − k2
共2兲
the resultant stiffness becomes infinite, that is,
兩kc兩 = ⬁
共3兲
However, the compliance of the isolation table, ␬, can be written
as
␬ = 1/kc = 0
共4兲
This research applies this principle of generating zero compliance
of the isolation table 共high stiffness兲 against direct disturbance to
vibration isolation systems. On the other hand, if low stiffness of
mechanical spring is used with the base, it can maintain ground
vibration isolation performance as well.
A schematic drawing of the basic vibration isolation system is
shown in Fig. 1. A middle table m1 is connected to the base
through a spring k1 and a damper c1 that work as a conventional
vibration isolator. An electromagnet for zero-power magnetic suspension is fixed to the middle table. The permanent magnets are
attached to the opposite side of the electromagnet 共to the isolation
table兲 placing the north and south poles alternately. The isolation
table m2 is made of soft iron material.
This system can reduce vibration transmitted from ground by
setting k1 small and at the same time have infinite stiffness against
direct disturbance by setting the amplitude of negative stiffness
equal to k1.
2.2 Introduction of Weight Support Springs. The above basic system has some limitations relating to design and cope with
heavy payloads. The whole weight of the isolation table is supported by zero-power magnetic suspension, as described in the
system shown in Fig. 1. When the isolation table is large, therefore, a lot of permanent magnets are necessary for suspending the
weight, which will increase the cost. Another problem expected in
putting the proposed system to practical use is that the reaction
part must be installed under the middle table, as shown in Fig. 1.
This is because hybrid magnet can produce attractive force when
041006-2 / Vol. 132, AUGUST 2010
electromagnet and permanent magnet are attached together, and
zero-power control system suspends the isolation table. It makes
the structure of the vibration isolation system rather complex. It
can be noted that if the permanent magnets are attached to the
target of the electromagnet and the north and south poles are
arranged properly such as Fig. 1, hybrid magnet can generate
repulsive force.
These problems can be overcome by introducing a secondary
suspension for supporting the weight. The concept is explained in
Fig. 2. A spring k3 is added in parallel with the serial connection
of positive and negative springs. The total stiffness k̃c is given by
k̃c =
k 1k 2
+ k3
k1 + k2
共5兲
When Eq. 共2兲 is satisfied, the resultant stiffness becomes infinite
for any finite value of k3, that is,
兩k̃c兩 = ⬁
共6兲
and the compliance of the isolation table, ␬ becomes
␬ = 1/k̃c = 0
共7兲
Figure 3 shows the configuration of the proposed vibration isolation system. A spring k3 together with a damper c3 as a weight
support springs is inserted between the isolation table and the
base. The spring is set to produce upward force in the equilibrium
state. It reduces the static load force that the zero-power control
magnetic suspension system normally sustains. Moreover, when
the upward force is greater than the gravitational force, the zeropower magnetic suspension must produce downward force. Since
the reaction part is installed above the middle table, the structure
fd
x2
Isolation Table m2
Permanent
magnets
k3
x1
c3
Middle Table m1
k1
c1
x0
Fig. 3 Basic structure of the vibration isolation system using
weight support spring
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Electromagnet
i
Permanent
magnet
D0
N
S
S
N
F=K
N
S
x
m
balances the weight of the suspended object.
It is assumed that the permanent magnet is modeled as a
constant-current 共bias current兲 and a constant-gap electromagnet
in the magnetic circuit 关18兴 for simplification in the following
analysis. Attractive force of the electromagnet, F, can be written
as
Fig. 4 Basic model of zero-power magnetic suspension
system
3
I20
D20
冉 冊冉 冊
1−
F=K
I
共8兲
where K is the attractive force coefficient for electromagnet, I is
the coil current, and ␦ is the mean gap between electromagnet and
the suspended object.
Each variable is given by the sum of a fixed component, which
determines its operating point and a variable component, such as
I = I0 + i
共9兲
␦ = D0 − x
共10兲
1+
i
I0
2
共12兲
Using the Taylor principle, Eq. 共12兲 can be expanded as
F=K
I20
D20
冉
1+2
x
x2
x3
+3 2 +4 3 +¯
D0
D0
D0
冊冉
1+2
i i2
+
I0 I20
冊
共13兲
For zero-power control system, control current is very small, especially in the phase approaches to steady-state condition and
therefore, the higher-order terms are not considered. Equation 共13兲
can then be written as
F = Fe + kii + ks共x + p2x2 + p3x3 + ¯兲
共14兲
where
Fe = K
I20
D20
ki = 2K
ks = 2K
p2 =
p3 =
2
␦2
−2
x
D0
Zero-Power Controlled Magnetic Suspension System
3.1 Basic Magnetic Suspension System. Active magnetic
suspension has been a viable choice for many industrial machines
and devices as a noncontact, lubrication-free support 关15,16兴. It
has become an essential machine element from high-speed rotating machines to the development of precision vibration isolation
system. Magnetic suspension can be achieved by using electromagnet and/or permanent magnet. Electromagnet or permanent
magnet in the magnetic suspension system causes flux to circulate
in a magnetic circuit, and magnetic fields can be generated by
moving charges or current. The attractive force of an electromagnet, F, can be expressed approximately as 关15兴
共11兲
where the bias current, I0, is modified to equivalent current in the
steady-state condition provided by the permanent magnet and
nominal gap, D0 is modified to the nominal air gap in the steadystate condition including the height of the permanent magnet.
Equation 共11兲 can be transformed as
F=K
is simpler than the original shown in Fig. 1. It is to be noted that
the performance of isolation from ground vibration is maintained
by using a soft spring as k3.
The behavior of the basic system for direct disturbance acting
on the isolation table is explained as follows. Assuming that the
table is subject to a downward force, the gap between the electromagnet and the table becomes smaller due to zero-power control;
in other words, the table would move upward if the middle table
were fixed. Meanwhile, the middle table moves downward because of the increase in the electromagnetic force. When the decrease in the gap is canceled by the downward displacement of the
middle table, the isolation table is maintained at the same position
as before the downward force is active.
共I0 + i兲2
共D0 − x兲2
I0
D20
I20
D30
共15兲
共16兲
共17兲
3
2D0
共18兲
4
共19兲
2D20
For zero-power control system, the control current of the electromagnet is converged to zero to satisfy the following equilibrium
condition:
Fe = mg
共20兲
and the equation of motion of the suspension system can be written as
mẍ = F − mg
共21兲
From Eqs. 共14兲, 共20兲, and 共21兲,
mẍ = kii + ks共x + p2x2 + p3x3 + ¯兲
共22兲
where I0 is the bias current, i is the coil current in the electromagnet, D0 is the nominal gap, and x is the displacement of the suspended object.
This is the fundamental equation for describing the motion of the
suspended object.
3.2 Zero-Power Control Magnetic Suspension System. In
order to reduce power consumption and continuous power supply,
permanent magnets are employed in the suspension system to
avoid providing bias current. The suspension system by using hybrid magnet, which consisted of electromagnet and permanent
magnet, is shown in Fig. 4. The permanent magnet is used for the
purpose of providing bias flux 关17兴. This control realizes the
steady states in which the electromagnet coil current converges to
zero and the attractive force produced by the permanent magnet
3.3 Design of Zero-Power Controller. Negative stiffness is
generated by actively controlled zero-power magnetic suspension.
The basic model, controller, and the characteristic of the zeropower control system are described below.
Model. A basic zero-power controller is designed for simplicity
based on linearized equation of motions. It is assumed that the
displacement of the suspended mass is very small and the nonlinear terms are neglected. Hence the linearized motion equation
from Eq. 共22兲 can be written as
Journal of Vibration and Acoustics
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mẍ = kii + ksx
共23兲
x
The suspended object with mass of m is assumed to move only in
the vertical translational direction, as shown in Fig. 4. The equation of motion is given by
mẍ = ksx + kii + f d
where x is the displacement of the suspended object, ks is the
gap-force coefficient of the magnet, ki is the current-force coefficient of the magnet, i is the control current, and f d is the disturbance acting on the suspended object.
Controller. The zero-power control operates to accomplish
lim i共t兲 = 0
for stepwise disturbances
t→⬁
sh̃共s兲
X共s兲
g共s兲
共26兲
where g共s兲 and h̃共s兲 are coprime polynomials in s and selected for
the closed-loop system to be stable, that is, all the roots of the
characteristic equation
冉 冊
ks
ki
g共s兲 + h共s兲 = 0
tc共s兲 = s2 −
m
m
共27兲
are in the left-half plane.
Assuming that h共s兲 / g共s兲 is restricted to be proper, the minimal
order compensator achieving zero-power control and assigning
the closed-loop poles arbitrarily can be represented as
s共h̃2s + h̃1兲
X共s兲
I共s兲 = − 2
s + g 1s + g 0
共28兲
Negative stiffness. When a constant force F0 is applied to the
suspended object, m, the suspended object is maintained at a position satisfying
0 = ksx共⬁兲 + kii共⬁兲 + F0
共29兲
in the steady states. In the zero-power control system, the coil
current converges to zero, that is,
i共⬁兲 = 0
共30兲
Fig. 5 Block
arrangement
F0
ks
共31兲
ks 1
.
)x2
k i D02
diagram
of
i = iZP − d2
the
冉
nonlinear
compensator
冊
ks 1 2
·
x
ki D20
共32兲
where d2 is the nonlinear control gain, and ks, ki, and D0 are
constants for the system. The square of the displacement 共x2兲 is
fed back to the normal zero-power controller. The block diagram
of the nonlinear controller arrangement is shown in Fig. 5. The air
gap between the permanent magnet and the suspended object can
be changed in order to choose a suitable operating point.
4
Developed Vibration Isolation System
4.1 Basic Model. In this section, the high-order terms will be
neglected for simplicity. The equations of motion of the model
shown in Fig. 3 about the vertical translational motion can be
written as
m1ẍ1 = − m1g + k1⌬x1 − k1共x1 − x0兲 − c1共ẋ1 − ẋ0兲 + f e
共33兲
m2ẍ2 = − m2g + k3⌬x3 − k3共x2 − x0兲 − c3共ẋ2 − ẋ0兲 − f e + f d
共34兲
where x0, x1, and x2 are the displacements of the floor, the middle
mass, and the isolation table, f e is the attractive force of the hybrid
magnet, f d is the direct disturbance acting on the isolation table,
⌬x1 is the initial compressed length of spring k1, and ⌬x3 is the
initial compressed length of spring k3.
The attractive force of the hybrid magnet is approximately represented by
f e = f̄ e + ks共x1 − x2兲 + kii
共35兲
where f̄ e is the attractive force in the equilibrium states. The following equations are satisfied in the equilibrium states:
Therefore,
x共⬁兲 = −
i
Nonlinear compensator
共25兲
The controller achieving the control objective 共25兲 is generally
represented by 关17兴
I共s兲 = −
d2 (
共24兲
+
_
Zero-power controller
k1⌬x1 − m1g + f̄ e = 0
共36兲
k3⌬x3 − m2g − f̄ e = 0
共37兲
f̄ e = k3⌬x3 − m2g
共38兲
From Eq. 共37兲,
The negative sign appearing in the right-hand side verifies that the
new equilibrium position is in the direction opposite to the applied
force. It indicates that the zero-power control system reveals a
unique characteristic and behaves as if it has negative stiffness.
Therefore, the condition that allows the configuration shown in
Fig. 3 is given by
3.4 Nonlinear Compensation of Zero-Power Controller. It
is shown that the zero-power control can generate negative stiffness. The control current of the zero-power controlled magnetic
suspension system is converged to zero for any added mass. To
counterbalance the added force due to the mass, the stable position of the suspended object is changed. Due to the air gap change
between permanent magnet and the object, the magnetic force is
also changed and hence, the negative stiffness generated by this
system is also varied according to the gap 共see Eq. 共17兲兲. To
compensate the nonlinearity of the basic zero-power control system, the first nonlinear term of Eq. 共22兲 is considered and added to
the basic system. From Eq. 共22兲, the control current can be expressed as
It also indicates that the steady-state force to be generated by the
hybrid magnet can be reduced by setting the upward force of the
spring k3 appropriately. There is a wide range of flexibility to
change the weight support force. This allows the simple design of
vibration isolation system than that of without weight support
springs even in the case of MDOF systems. In Fig. 1, the design
of isolation table seems rather difficult 共C-shaped structure兲 because one part of it must be placed under the middle table. Therefore, the design of MDOF systems will be more complicated and
costly. This problem is solved by using weight support springs.
According to the discussion in Sec. 3.3, the control current
achieving the zero-power control is generally represented by
041006-4 / Vol. 132, AUGUST 2010
k3⌬x3 ⬎ m2g
共39兲
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Weight Support Springs
Table
Hybrid Magnet
Middle Table
Coil Spring
Base
(a)
Isolation Table
Weight Support
Spring
Hybrid Magnet
Middle Table
Coil Spring
Electromagnet
(b)
Fig. 6 Developed vibration isolation system: „a… photograph and „b… schematic diagram „front view…
I共s兲 = − q共s兲s共X1共s兲 − X2共s兲兲
共40兲
where
t̃c共s兲 = t1共s兲t̃2共s兲 + k̃2共s兲共t1共s兲 + t̃2共s兲兲
共49兲
To estimate the stiffness for direct disturbance, the direct disturbance Fd is assumed to be stepwise on the isolation table, that is,
q共s兲 =
h̃共s兲
g共s兲
共41兲
4.2 Response to Direct Disturbance. In the following analysis, each Laplace-transformed variable is denoted by its capital
and it is assumed for simplicity that the initial values of the variables are zero. Therefore, the transfer function representation of
the dynamics described in Eqs. 共33兲–共37兲 and 共40兲 becomes
X1共s兲 =
k̃1共s兲共t̃2共s兲 + k̃2共s兲兲 + k̃2共s兲k̃3共s兲
t̃c共s兲
X0共s兲 +
k̃2共s兲
t̃c共s兲
X2共s兲 =
共F0:const兲
共50兲
k1 − ks
t1共s兲 + k̃2共s兲
x2共⬁兲
=
= lim
F0
k3共k1 − ks兲 − k1ks
s→0
t̃c共s兲
共51兲
Therefore, if
X0共s兲
k1 = ks
共52兲
x2共⬁兲
=0
F0
共53兲
then
Fd共s兲
共43兲
t1共s兲 = m1s2 + c1s + k1
共44兲
for any finite value of k3. Equation 共53兲 shows that the suspension
system between the isolation table and the floor has infinite stiffness statically because there is no steady-state deflection even in
the presence of stepwise disturbance acting on the table. It should
be noted that if ks is not constant against load increment, as explained in Eq. 共17兲, Eq. 共53兲 will not be realized in the end. In
such a case, nonlinear compensator presented in Eq. 共32兲 can
solve such problem.
t̃2共s兲 = m2s2 + c3s + k3
共45兲
5
k̃1共s兲 = c1s + k1
共46兲
k̃2共s兲 = kiq共s兲s − ks
共47兲
k̃3共s兲 = c3s + k3
共48兲
t̃c共s兲
+
F0
s
When the vibration of the floor is neglected 共x0 = 0兲, the steadystate displacement of the table is obtained as
Fd共s兲
共42兲
k̃1共s兲k̃2共s兲 + k̃2共s兲k̃3共s兲 + k̃3共s兲t1共s兲
Fd共s兲 =
t1共s兲 + k̃2共s兲
t̃c共s兲
where
Journal of Vibration and Acoustics
Experiments
5.1 Experimental Apparatus. The developed vibration isolation system with weight support spring is depicted in Fig. 6共a兲 and
simplified structure of the system 共front view兲 is shown in Fig.
6共b兲. It consisted of a rectangular isolation table, middle table, and
base. The height, length, width, and mass of the apparatus were
300 mm, 740 mm, 590 mm, and 400 kg, respectively. The isolation and middle tables weighed 88 kg and 158 kg, respectively.
AUGUST 2010, Vol. 132 / 041006-5
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x
3
Pitch
Sensors
z
Yaw
4
Actuators
y
300
Roll
2
1
410
Fig. 7 Sensors and actuators position „top view…. Dimensions
are in millimeters.
The middle table was supported by four pairs of coil springs and
dampers, and the isolation table was supported by additional four
coil springs, as weight support springs, in addition to the zeropower control system by four sets of hybrid magnets. The sensor
and hybrid magnet positions are shown in Fig. 7. These actuators
were used for table levitation as well as for controlling the threedegree-of-freedom motions 共z, roll, and pitch兲 of the table in the
vertical direction. Each set of hybrid magnet for zero-power suspension consisted of five square-shaped permanent magnets 共20
⫻ 20⫻ 2 mm3兲 and five 585-turn electromagnets. The permanent
magnet is made of NdFeB materials. The stiffness of each normal
spring was 12.1 N/mm and that of weight support spring was 25.5
N/mm. The stiffnesses of the springs were statically measured by
load-displacement characteristics of the springs. There was flexibility to change the position of the weight support springs to
make it compatible for designing stable zero-power controlled
magnetic suspension system.
The relative displacements of the isolation table to the middle
table and those of the middle table to the base were detected by
eight eddy-current gap sensors. The displacements of the isolation
table from base were measured by additional four gap sensors.
The isolation table was also supported by several normal
springs and hybrid magnets for controlling other three-degree-offreedom motions 共x, y, and yaw兲 in the horizontal directions. Two
pairs of hybrid magnets were used in the y-direction and one pair
in the x-direction between isolation table to middle table. Similarly six pairs of normal springs and actively controlled electromagnets 共two pairs in the x-direction and four pairs in the
y-direction兲 were used between base and middle table to adjust the
positive stiffness. The isolation table was also supported by four
S1
+
1/4
1/4
_
+
_
+
_
Zero-Power
Controller (Z)
+
1/4
Zero-Power
Controller (Roll)
Zero-Power
Controller (Pitch)
+
+
_
1/4
Nonlinear
Compensator
1/4
i1
+
Nonlinear
Compensator
+
+
_
S4
5.3 Determination of Controller Parameters. The measurements of controller parameters ks and ki are important for designing zero-power controller and its nonlinear compensator. The parameters can be determined from a suspension system levitated by
proportional-derivative 共PD兲 control or integral-proportionalderivative 共I-PD兲 control. In this research, a basic suspension system presented in Fig. 4 with PD control was used because PD
control is the most stable and fundamental control system. It can
be noted that if the middle table is fixed in Figs. 1, 3, and 6, and
the isolation table is levitated by PD control, those systems are
+
+
S3
5.2 Design of Controller. Design of controller in the vertical
direction is a challenging task because it includes stable levitation
as well as motion control. The developed system was a fourchannel multiple-input and multiple-output 共MIMO兲 system.
There were four sensors and four actuators for controlling 3DOF
motions 共three modes兲 in the vertical direction, as shown in Fig. 7.
For the developed system, there was redundancy in actuator, as
four actuators were used to control the three modes. Therefore, a
mode-based centralized controller was designed to control the
three modes 共Z, ⌰y, and ⌰x兲 based on each mode, as shown in
Fig. 8. This controller can deal with motion control of any mode
effectively. The control current of each mode was distributed to
the four actuators according to their positions. The designed control algorithms were implemented with a DSP-based digital controller DS1103, manufactured by dSPACE, Germany. The sampling rate was 10 kHz.
+
+
S2
pairs of normal springs from base, as weight support spring for the
horizontal directions. Hence the isolation table was also capable to
control other three modes in the horizontal directions.
One pair of displacement sensors was used in the x-direction
and two pairs in the y-direction to measure the relative displacement between isolation table and middle table for horizontal displacements. Similarly six pairs of sensors were used to measure
the relative displacement between middle table and base.
This paper will focus on the control of three-degree-of-freedom
motions in vertical direction. The isolation table was levitated and
negative stiffness was realized by the four sets of hybrid magnets,
which was fixed to the middle table. The stiffness of the positive
springs can be adjusted by the actively controlled electromagnets.
Hence, the isolation table was levitated by four pairs of normal
and negative springs, as well as weight support springs so that the
three modes 共three-degree-of-freedom motions兲 of the table can be
controlled by the proposed mechanism. They are one translational
motion in the vertical direction 共Z兲 and two rotational motions,
roll 共⌰y兲 and pitch 共⌰x兲.
+
_
i2
i3
+
1/4
+
_
i4
_
Nonlinear
Compensator
Fig. 8 Block diagram of the mode-based controller
041006-6 / Vol. 132, AUGUST 2010
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1
共b0I共s兲 + d0Fd共s兲兲
s − a0
2
共54兲
in which each Laplace-transformed variable is denoted by its capital, and
ks
a0 = ,
m
ki
b0 = ,
m
1
d0 =
m
1
X共s兲
=
Fd共s兲 s2 + b0 pvs + 共b0 pd − a0兲
共56兲
共57兲
where X共s兲 is the displacement output and Fd共s兲 is the disturbance
input.
Again general form of transfer function of a PD control system
can be represented as
G⬘共s兲 =
A
s + 2␨␻ns + ␻2n
2
共58兲
where A is constant, ␨ is the damping ratio, and ␻n is the natural
frequency of the system.
Comparing Eqs. 共57兲 and 共58兲, it is found that
␻2n = b0 pd − a0
共59兲
The natural frequency of the system, ␻n, can be measured from
the frequency response of the PD control system for a particular
pd. If pd gain is changed, natural frequency ␻n also changes.
When ␻2n is plotted against pd, the parameters a0 and b0 are automatically extracted from the intercept and slope of the line, respectively. Finally ks and ki are determined from Eq. 共55兲 as the
mass of the isolation table or floater is known. Moreover, even
weight support mechanism is included in the system, as shown in
Figs. 2, 3, and 6, ks can be determined from a0⬘ = 共ks − k3兲 / m, where
ki remains unchanged.
5.4
Experimental Results
5.4.1 Static Characteristics
5.4.1.1 Nonlinearity of the zero-power control. Several experiments have been conducted for the developed vibration isolation
system with weight support springs both for stabilization and performance enhancement. First of all, zero-power control was realized between isolation table and middle table for stable levitation.
The positive and negative stiffness springs were, then, adjusted to
satisfy Eq. 共52兲. The stiffness could either be adjusted in the positive or negative stiffness part. In the former, PD control could be
used in the electromagnets that were employed in parallel with the
coil springs. In the latter, a PD control could be used in parallel
with the zero-power control. For better performance, the latter
was adopted in this experiment.
Static characteristic of the isolation table was measured, as
shown in Fig. 9, when the payloads were increased to produce
static direct disturbances on the isolation table in the vertical direction. It is seen from the figure that zero compliance to direct
disturbance of the isolation table was realized up to 50 N, while
the position of middle table and the gap between table and middle
table was changed according to the increment of payloads. A high
stiffness was realized between base and table in this range which
practically satisfied Eq. 共53兲. However, the figure also reveals that
Journal of Vibration and Acoustics
0
-1
20
40
60
80
100
Load [N]
where pd is the proportional feedback gain and pv is the derivative
feedback gain.
From Eqs. 共54兲 and 共56兲, transfer function G共s兲 can be written
as
G共s兲 =
1
0
共55兲
For PD control, the control current can be written as
I共s兲 = − 共pd + spv兲X共s兲
Middle table
Gap
Table
2
-2
Fig. 9 Static response of the isolation table to direct disturbance in the vertical direction „Z…
the zero-compliance characteristic has been lost after this range
and the gap, which is generated by zero-power control, was also
nonlinear. That performance degradation might occur due to the
nonlinearity of the zero-power control system.
Subsequently, the insight of zero-power control is investigated
by measuring its characteristics. In this case, the middle table was
fixed and the table was levitated by zero-power control. Figure 10
presents the load-stiffness characteristic of the zero-power control
system. This figure indicates that there was a wide variation in
stiffness when the downward load force was changed. For the
uniform load increment, the change in gap was not equal due to
the nonlinear magnetic force. Therefore, the negative stiffness
generated from zero-power control was also nonlinear, which severely affected the vibration isolation system, as exposed in Fig.
9.
5.4.1.2 Nonlinear compensation. To overcome the above
problem, the nonlinear compensator was introduced in parallel
with the zero-power control system. The nonlinear control gain
共d2兲 was chosen by trial and error method. The gap 共D0兲 between
the table and the electromagnet was 5.1 mm after stable levitation
by zero-power control. The values of ks and ki were determined
from the system characteristics. The load-stiffness characteristic
using nonlinear compensation for different control gains is shown
in Fig. 11. It is obvious from the figure that the linearity error was
reduced when control gain 共d2兲 was increased. For d2 = 55, the
linearity error was very low and the stiffness generated from the
system was approximately constant. This result showed the potential to improve the static response performance of the isolation
table to direct disturbance.
Therefore, further experiments were conducted with the linearized zero-power controller. Figure 12 demonstrates the performance improvement of the controller for static response to direct
disturbance. The displacements of the isolation table and middle
table were plotted against disturbing forces produced by payload
in the vertical direction. It is clear that zero compliance to direct
disturbance was realized up to 100 N payloads with nonlinear
300
Stiffness [N/mm]
X共s兲 =
3
Displacement [mm]
equivalent to the model presented in Fig. 4.
Hence, the transfer function representation of the dynamics described in Eq. 共24兲 becomes
250
200
150
100
0
20
40
60
80
Load [N]
Fig. 10 Load-stiffness characteristic of the zero-power control
system
AUGUST 2010, Vol. 132 / 041006-7
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d2 = 0
250
d2 = 45
d2 = 35
200
d2 = 55
150
0
10
Table (with nonlinear control)
Middle table
Gap
Table (without nonlinear control)
0.006
Roll Angle [deg]
Stiffness [N/mm]
300
20
30
40
50
60
70
0.004
0.002
0
-0.002
-0.004
-0.006
0
Load [N]
controller 共d2 = 55兲. The stiffness of the isolation system was increased to 960 N/mm, which was approximately 2.8 times more
than that of without nonlinear control. The figure illustrates significant improvement in rejecting on-board-generated disturbances. However, the zero-power feature has been lost when nonlinear compensator was introduced in the zero-power control, as
shown in Fig. 13. The figure shows that the zero-power characteristic was realized over 90 N and below this range, small current
was necessary. Maximum current necessary for all the four sets of
actuator was 1.2 A. Therefore, the control current for each set of
electromagnet was around 0.3 A, which verified that the nonlinear
controller could, nevertheless, save energy significantly.
Next, the static performances of the isolation table for rotational
motions were also measured. Static characteristics of the isolation
table to direct disturbance for roll and pitch modes are shown in
Figs. 14 and 15, respectively. For roll mode, the zero-compliance
characteristic was realized up to 18 N m, whereas the stiffness of
the isolation table was increased from 33,700 N m/deg to 74,400
N m/deg when nonlinear control was used. Again, for pitch mode
the generated stiffness without nonlinear control was 33,730 N m/
12
15
18
deg, while that of using nonlinear control was 65,200 N m/deg. In
both cases for rotational mode, direct disturbance suppression performance was enhanced by more than twofold when nonlinear
controller was introduced.
5.4.2 Dynamic Characteristics. The dynamic performance of
the isolation table was measured in the vertical direction, as
shown in Fig. 16. In this case, the isolation table was excited to
produce sinusoidal disturbance force by two voice coil motors,
which were attached to the base and can generate force in the
Z-direction. The displacement of the table was measured by gap
sensors and the data were captured by a dynamic signal analyzer.
It is found from the figure that high stiffness was realized at low
frequency region 共⫺66 关dB mm/N兴 at 0.015 Hz兲. It also demonstrates that direct disturbance rejection performance was not worsened even nonlinear zero-power control was introduced. Again the
isolation table was excited in one of the typical rotational motion
and dynamic characteristic was measured. Figure 17 illustrates the
Pitch Angle [deg]
Displacement [mm]
9
Fig. 14 Static response to direct disturbance with nonlinear
control „d2 = 40… in the roll mode „⌰y…
Table (with nonlinear control)
Middle table
Gap
Table (without nonlinear control)
0.008
Table (with nonlinear control)
Middle table
Gap
Table (without nonlinear control)
1
6
Torque [Nm]
Fig. 11 Load-stiffness characteristics of the zero-power control with nonlinear compensation
2
3
0
-1
0.006
0.004
0.002
0
-0.002
-0.004
-0.006
-0.008
0
-2
4
8
12
16
Torque [Nm]
0
20
40
60
80
100
Fig. 15 Static response to direct disturbance with nonlinear
control „d2 = 50… in the pitch mode „⌰x…
Load [N]
Fig. 12 Static response to direct disturbance with nonlinear
control in the vertical direction „Z…
Control Current [A]
0.5
0
-0.5
-1
-1.5
0
20
40
60
80
100
120
Load [N]
Fig. 13 Control current of the electromagnets for the linearized zero-power controller
041006-8 / Vol. 132, AUGUST 2010
Fig. 16 Dynamic response of the isolation table to direct disturbance in the vertical translational direction
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direct disturbance in each mode was derived analytically. This
characteristic was verified experimentally with stable levitation
that even weight support springs were used and showed that performance could be well enhanced by using such mechanism and
the linearized zero-power control. The developed system was
statically and dynamically stable and generated very high stiffness
in different modes. The developed system demonstrated better direct disturbance rejection performances over conventional passive
suspension that even the system had similar base vibration isolation capability. The developed system showed the feasibility of its
use for heavy system, and good controller performances were revealed. Direct disturbance rejection, moreover, could be well improved by incorporating a feed forward controller with the feedback control.
Fig. 17 Dynamic response of the isolation table to direct disturbance in the pitch mode
Acknowledgment
The authors gratefully acknowledge the financial support made
available from the Japan Society for the Promotion of Science
共JSPS兲 as a Grant-in-Aid, the Ministry of Education, Culture,
Sports, Science and Technology of Japan, and Electro-Mechanic
Technology Advancing Foundation as a Grant-in-Aid for the Development of Innovative Technology, and a Grant-in-Aid for Scientific Research 共B兲.
References
Fig. 18 Frequency response of the isolation table to dynamic
direct disturbance using zero-compliance control and conventional passive suspension technique
frequency response of the isolation table when the isolation was
excited in the pitch mode. It also shows that the displacement of
the isolation table at low frequency was very small with or without nonlinear control. A slight deviation for both cases between
nonlinear control and without nonlinear control was occurred due
to the imprecise adjustment of the positive and negative stiffnesses to realize zero-compliance system. Finally, a comparative
study of the disturbance suppression performance was conducted
with zero-compliance control and conventional passive suspension technique, as shown in Fig. 18. The experiment was carried
out with same lower suspension for ground vibration isolation.
First, the isolation table was suspended by positive suspension
共conventional spring damper兲 and frequency response to direct
disturbance was measured. It is seen from the figure that the displacement of the isolation table was almost same below 1 Hz
共approximately ⫺46 dB兲. However, when the isolation table was
suspended by zero-compliance control satisfying Eq. 共7兲, the displacement of the table was abruptly reduced at the low frequency
region below 1 Hz 共⫺66 dB at 0.015 Hz兲. It is confirmed from the
figure that the developed zero-compliance system had better direct
disturbance rejection performance over the conventional passive
suspension even both the systems used similar vibration isolation
performances.
6
Conclusions
A six-axis vibration isolation system using zero-power controlled magnetic suspension with weight support springs was developed. The mode-based control scheme was designed to control
the three-degree-of-freedom motions in the vertical direction actively. The condition for the system to have infinite stiffness for
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