vibration isolation using six degree-of-freedom quasi...
TRANSCRIPT
1
Vibration isolation using six degree-of-freedom quasi-zero stiffness
magnetic levitation
Authors:
Name: Dr Tao Zhu1 (corresponding author)
Address: School of Mechanical Engineering, The University of Adelaide, North Terrace
Campus, South Australia, Australia. 5005
Institution: The University of Adelaide; The University of Nottingham UNNC
Fax: (61) 8 83134367
Email: [email protected]
Name: Associate Professor Benjamin Cazzolato
Address: School of Mechanical Engineering, The University of Adelaide, North Terrace
Campus, South Australia, Australia. 5005
Institution: The University of Adelaide;
Tel: (61) 8 83135449
Fax: (61) 8 83134367
Email: [email protected]
Name: Dr William S P Robertson
Address: School of Mechanical Engineering, The University of Adelaide, North Terrace
Campus, South Australia, Australia. 5005
Institution: The University of Adelaide;
Fax: (61) 8 83134367
Email: [email protected]
Name: Associate Professor Anthony Zander
Address: School of Mechanical Engineering, The University of Adelaide, North Terrace
Campus, South Australia, Australia. 5005
Institution: The University of Adelaide;
Fax: (61) 8 83134367
1 Present address: The University of Nottingham UNNC
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Abstract
In laboratories and many high-tech manufacturing applications, passive vibration isolators are
often used to isolate vibration sensitive equipment from ground-borne vibrations. However,
in traditional passive isolation devices, where the payload weight is supported by elastic
structures with finite stiffness, a design trade-off between the load capacity and the vibration
isolation performance is unavoidable. Low stiffness springs are often required to achieve
vibration isolation, whilst high stiffness is desired for supporting payload weight. In this
paper, a novel design of a six-DOF (six degree of freedom) vibration isolator is presented, as
well as the control algorithms necessary for stabilising the passively unstable maglev system.
The system applies magnetic levitation as the payload support mechanism, which realizes
inherent quasi-zero stiffness levitation in the vertical direction, and zero stiffness in the other
five DOFs. While providing near zero stiffness in multiple DOFs, the design is also able to
generate static magnetic forces to support the payload weight. This negates the trade-off
between load capacity and vibration isolation that often exists in traditional isolator designs.
The paper firstly presents the novel design concept of the isolator and associated theories,
followed by the mechanical and control system designs. Experimental results are then
presented to demonstrate the vibration isolation performance of the proposed system in all six
directions.
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1. Introduction and background
In laboratories and many high-tech manufacturing applications, passive vibration isolators are
often used to isolate vibration sensitive equipment from ground-borne vibration. However, in
traditional passive isolation devices, where the payload weight is supported by a resilient
structure, a design trade-off between the load capacity and the isolation performance is
unavoidable [1, 2]. This is due to the necessity of using a high stiffness support to carry the
payload weight to avoid excessive static isolator deflection, which is contrary to the
requirements for vibration isolation. In order to isolate ground vibration, low stiffness support
is normally preferred for reducing the sensitivity of the plant to external disturbances.
Another disadvantage of many spring-damper type designs is that they normally only provide
effective isolation in one direction. However, vibrations potentially exist in all six degrees of
freedom (DOF). Cross coupling between axes often occurs in single-DOF isolator designs,
such that vibration is transmitted from other DOFs into the primary working direction of the
isolator, which significantly limits the performance of the single DOF designs. In some
advanced applications, such as in semi-conductor manufacturing, multi-DOF vibration
isolators are often essential to fulfil the stringent vibration isolation requirements.
Whilst passive designs are commonly used in most applications, actively controlled isolators
are finding increased applications due to their enhanced performance over the passive designs.
Some examples of commercially available active isolators are the optical tables produced by
Newport [3] and Herzan [4], which are designed for laboratory uses to isolate vibration
sensitive equipment. In gravitational wave monitoring projects (LIGO [5] and VIRGO [6]),
multi-DOF active vibration isolators [7] are also used to isolate the instruments from ground-
borne vibrations in multiple directions. The advantages of active vibration control have been
well demonstrated by the literature. However, the application of active vibration control is
still limited by the drawbacks of the active designs, such as high power consumption and
significantly increased system complexity and cost over traditional passive designs.
The demand for high performance vibration isolation systems has drawn significant research
attention over the past two decades, and magnetic levitation (maglev) has demonstrated its
potential in vibration isolation applications. Previously the technology was mainly applied to
maintain a constant levitation gap, such as in maglev trains and maglev bearings [8-10]. The
non-linear magnetic force-displacement relationship was also found to benefit vibration
isolation as this unique feature of maglev enables creation of low stiffness supporting
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structures. Robertson et al. [11, 12] proposed a quasi-zero (infinitely close to zero around the
operating point) stiffness magnetic spring for vibration isolation. In their model, permanent
magnets were used to establish maglev system which had quasi-zero stiffness around the
nominal operating position. A tuneable high-static-low-dynamic stiffness vibration isolator
was developed by Zhou and Liu [13], in which electromagnets were used to statically
manipulate the magnetic field to generate a quasi-zero stiffness zone. An alternative approach
to quasi-zero stiffness is quasi-infinite stiffness, whereby the plant is tied to a high impedance
element (such as a plinth), thus isolating the plant from direct on-board excitation. For
example, Mizuno et al. [14] proposed a single-DOF maglev vibration isolator, which used a
number of electromagnets to create a negative stiffness member and counteract the stiffness
of the mechanical springs so that the resultant system stiffness is infinitely large. The infinite
stiffness design was later expanded to a three DOF [15, 16], and a full six-DOF system [17,
18] by applying the same design principle in multiple directions. More developments on
quasi-zero stiffness isolators and maglev isolators are presented in [19-24].
According to Earnshaw’s theorem [25], passive magnetic levitation is inherently unstable.
Therefore, active controls are necessary to enable the operation of maglev systems. The
controller for such isolation systems must be designed carefully such that the vibration
induced by the control system from sensor and electronic noise is negligible. Previous
researchers have investigated a number of low noise control approaches for maglev based
isolation systems. A K-filter based design was proposed by Yang et al. [26]. Their system
was designed to achieve accurate maglev positioning where noisy position sensors are used.
However, in order to realize accurate positioning, the proposed K-filter design has high
levitation stiffness, which is counter to the requirements necessary for ground vibration
isolation. Back-stepping based controller designs were also studied by Wai and Lee [27] for
maglev rail systems. The three controller designs presented in their paper also result in high
levitation stiffness since their aim is to maintain a constant levitation gap. A classic PID
control approach was proposed by Zheng et al. [28]. In their system, the levitation stiffness
and damping are directly controlled through the proportional and derivative gains of the PID
controller. A tracking differentiator was also embedded in the PID controller to suppress
noise within the differentiation loop.
The previous active isolation systems have all demonstrated their potential to improve the
isolation performance when compared to traditional passive systems. However, most of the
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designs showed poor performance at low frequencies, and the trade-off between vibration
isolation and payload support still exists. These designs also exhibited higher resonance
frequencies of the fundamental rigid body modes compared to many of the existing high
performance passive vibration isolation systems. In this paper, a novel design of a six-DOF
maglev vibration isolator is presented. The design provides quasi-zero/zero stiffness (quasi-
zero stiffness in the vertical direction and zero stiffness in the other five DOFs) levitation in
multi-DOF, which enables effective vibration isolation in all six DOFs. The novel design also
allows the payload to be supported solely by the permanent magnets with a static magnetic
force, hence the trade-off between load capacity and vibration isolation is avoided. The
vertical supporting force is adjustable by changing the magnet separations so that the
magnitude of the supporting force can match the payload weight to minimize the energy
consumption of the magnetic levitation.
This paper is structured as follows. It starts with the description of the design concept and
theoretical background, followed by the mechanical and control system designs for
implementation of the proposed maglev vibration isolation. Experimental results and
discussions are presented at the end.
2. Theoretical background
The novel isolator design presented in this paper is based on a unique quasi-zero stiffness
maglev system, which is composed of two pairs of rare earth magnets. The system has no
mechanical connections between the frame and the floater, and hence avoids the issues seen
in some of the traditional mechanical spring based isolators, where parasitic high frequency
modes act to degrade the high frequency isolation performance. This section explains the
theoretical background of the isolation system design.
2.1. Maglev system model
A simplified schematic (also discussed in [29]) of the maglev system, which consists of four
identical cylindrical magnets, is shown in Figure 1, Magnets 1 and 4 are fixed to ground and
Magnets 2 and 3 are floating and connected through the rigid linkage. N and S represent the
north and south poles of the magnetisation respectively. The top pair of magnets (1 and 2) has
the same polarisation, which generates an attraction force F12= [F12X, F12Y, F12Z] T, and the
bottom pair has the opposite magnetisation directions, which generates a repulsive force F43=
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[F43X, F43Y, F43Z]T. The assembly of Magnet 2, Magnet 3 and the Linkage collectively form
the floater. It is able to move in six DOFs and is isolated from external vibration.
2.2. Calculation of magnetic forces and stiffness
The analytical method used here for the calculation of the magnetic forces is discussed in
detail in [30-32]. Robertson et al. further investigated and simplified the calculations [33, 34]
for determining magnetic forces and torques. Robertson has also provided a Matlab script [35]
to carry out the force and torque calculations, which has been applied in this research to
perform the required calculations.
Explicit calculations of 3D magnetic forces and torques are often computationally expensive.
In order to reduce the execution time of the calculations required for analysing the maglev
system, the 3D magnetic forces are approximated using a single dimensional method. In the
3D calculations, all the forces in the X, Y and Z directions have to be modelled and
calculated individually. However, with the 1D approximation, the necessary force modelling
is reduced to only the coaxial force between the two cylindrical magnets. Magnetic forces
along the axes of the coordinate system can then be approximated based on the geometric
relationships between the three directions. Figure 2 shows a schematic of this approximation
α
β
γ
N
S
N S
S N
Lin
kag
e
N
S
F12 (Attraction)
F43 (Repulsion)
Magnet 1 (Fixed)
Magnet 2 (Floating)
Magnet 3 (Floating)
Magnet 4 (Fixed)
Z
Y
X
Floater
Top magnet pair
Bottom magnet pair
Figure 1: Schematic of the magnetic levitation system.
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in the YZ plane, where �� and �� are the magnet separations in the Y and Z directions
respectively. Two cases with equivalent relative positions of the magnet centres are shown
here to demonstrate the differences between the 3D and 1D calculations. In Figure 2(a), all
the forces � = [��, ��, ��] are calculated explicitly. This method is relatively complicated,
and the calculations of the forces are required in all three directions. In contrast, the
simplified 1D approach (shown in Figure 2(b)) only requires modelling of the coaxial force (��) using the method described in [33]. In order to obtain forces along the three axes (X, Y,
Z) of the coordinate sytsem, the coaxial force is then projected onto the three axes using the
geometric transformation
�X = ���X��X2 + �Y2 + �Z2�Y = ���Y��X2 + �Y2 + �Z2�Z = ���Z��X2 + �Y2 + �Z2
. (1)
The combination of the coaxial force calculation and the geometric transformation is a
significantly simpler approach compared to the full 3D force calculation. The 1D
approximation method was shown to reduce the calculation time by approximately two orders
of magnitude.
A comparison between the force calculation results using the 3D and the 1D methods is
shown in Figure 3 for the example case shown in Figure 2. Table 1 shows the geometric and
magnetic parameters used in the force calculations.
FFFF ��
��
dY dY
Figure 2: Magnetic forces resulting from horizontal displacement between two magnets, (a) 3D calculation model; (b) 1D approximation model.
�Y ≪ �Z ≈
YZ plane
Y
Z
X
�� ��
��
�� ��
(a) (b)
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Table 1: Physical parameters used in the 3D and 1D force calculations.
Parameter Value
Magnet length 40mm
Magnet width 40mm
Magnet height 20mm
Magnetisation 1.48T
Vertical distance between
magnet centres (�Z) 100mm
Horizontal distance between
magnet centres (�Y) -10mm to 10mm
The 1D approximation to the 3D forces was found to be reasonably accurate with small non-
coaxial displacements (within about 10% of the coaxial displacement). For example, as
shown in Figure 3, the 1D approximation is reasonably accurate when|��| ≤ 0.1��. For the
100mm magnet vertical separation, the horizontal displacement range shown in the figure is −10mm ≤ �� ≤ 10mm. The error between the 1D and 3D methods remains within 0.17N
throughout the -10mm to 10mm displacement range. Despite the small errors in the 1D
approximation method, the magnetic force displacement relationship predicted by the 1D
method fits the trend of the 3D results relatively well. It is not a primary focus in this research
to accurately model the magnetic forces, and the 1D approximation provides a significant
calculation time reduction. Therefore, the small calculation error introduced by the 1D
Figure 3: Comparison between the 3D method and the 1D approximation for calculating ��.
-10 -5 0 5 10-6
-4
-2
0
2
4
6
Horizontal displacement (mm)
Horizonta
l fo
rce (
N)
1D approximation
3D calculation
Relative error
10
approximation method is considered to be an acceptable compromise with computational
speed in this research for investigating the stiffness behaviour of the proposed maglev system.
The aforementioned calculations and comparisons were based on cubic shaped magnets
rather than the cylindrical magnets used in the system model (Figure 1). This is due to the
fact that, to the author’s best knowledge, there was no practical approach available for
calculating the 3D forces between cylindrical magnets at the time this analysis was carried
out. A method of force calculation between cylindrical magnets was only available for
estimating the coaxial forces, which is another reason why the 3D to 1D approximation is
necessary for the modelling and simulation components of this research. It is considered
reasonable to assume that both cubic and cylindrical magnets have similar force-displacement
behaviours in the scenarios analysed here. Therefore, the 1D approximation method is also
considered suitable for cylindrical magnets. Table 2 lists the physical parameters of the
cylindrical magnets used in the design of the maglev isolator.
Table 2: Physical parameters of the cylindrical magnets.
Parameter Value
Magnet diameter 50.8mm
Magnet thickness 25.4mm
Magnetisation 1.48T
Magnet type Nickel plated rare earth magnet
The calculations of magnet forces and torques are based on the following assumptions:
Assumption 1: The magnetisation is uniform across the volume of all magnets.
This assumption ensures that the effective centre of magnetisation is at the geometric centre
of the magnets. Hence, the magnetic forces also originate from the geometric centre of the
magnets. In the subsequent sections, the determination of the nominal operating position of
the floater (the midpoint between two fixed magnets, refer to Figure 1) will be directly related
to the location of the centre of magnetisation.
Assumption 2: All mechanical components have relative magnetic permeability equal to
unity.
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The analysis in the subsequent sections will show the stiffness behaviour of the proposed
maglev vibration isolator in multiple DOFs. In order to realise the favourable force-
displacement relationship of the maglev, it is essential that the floater has no external
interference from the supporting isolator frame, and the flux of the magnetic field is not
distorted. The material selection in the design of the physical isolation system (presented in
Section 5) has ensured, where possible, that all components in the maglev isolator behave
magnetically as if they were in a vacuum.
Assumption 3: Interaction forces between Magnets 1 and 3 and Magnets 2 and 4 are
sufficiently small to be neglected.
For simplicity of the analysis, the force-displacement behaviour between the floater and the
fixed magnets is assumed to be a result of the two adjacent magnet pairs (Magnet 1 and 2,
and Magnets 3 and 4). The total force between Magnets 1 and 3 and between Magnets 2 and
4 was calculated to be 0.0312N with 100mm separation between the top and bottom magnet
pairs. At this separation the total force between the top and bottom magnet pairs is 39.13N.
Therefore, forces between Magnets 1 and 3 and between Magnet 2 and 4 are assumed to have
negligible influence on the floater force-displacement behaviour and were not considered in
the analysis.
3. 6-DOF modelling of the floater forces and torques
In the design of the proposed quasi-zero stiffness maglev system (Figure 1), the floater
assembly consists of two magnets that are rigidly connected via a mechanical linkage, and
vibration isolation is achieved by attaching the payload to the floater assembly. Therefore, the
forces and torques experienced by the floater as a rigid body are of primary interest, as it
governs the vibration isolation performance of the maglev system. The previous sections
have discussed the calculation of the magnetic forces between a pair of magnets. This section
will explain the modelling method used to calculate the forces and torques experienced by the
floater assembly as a rigid body.
A free body diagram of the proposed quasi-zero stiffness maglev system is shown in Figure 4,
where !"# and !$% are the nominal (when the floater has zero angular displacement and the
floater COG is at the midpoint between the two centres of the fixed magnets) vertical
separations between the top and bottom magnet pairs, and 2& is the length of the floater
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assembly between the two centres of the floating magnets. In Figure 4, a Cartesian coordinate
system, with its origin at the midpoint between the two fixed magnets (the nominal operating
position), is used to describe the positions of the magnet centres and the poses of the floater
assembly. '( , ') , '* , '+ and ', are the positions of the centres of Magnet 1, Magnet 2,
Magnet 3, Magnet 4, and the floater respectively. -,(., /, 0) represents the angular
displacements of the floater, which are defined from the magnet coordinates as
. = tan4"(5 − 5#6# − 6) / = tan4"(7# − 76# − 6) 0 = tan4"(7# − 75# − 5)
(2)
From the expressions presented previously on the 3D to 1D calculation simplification, it can
be proven that �() has the same direction as the vector connecting the centres of Magnets 1
and 2 ('( − ')), and similarly for �+*. From Eq. (1), it can be shown that the components of
the forces acting between magnet pairs are given by
N
N
N
S
N
Magnet 1 (Fixed)
Magnet 2 (Floating)
Magnet 3 (Floating)
Magnet 4 (Fixed)
α β
γ
Z
Y
X
Figure 4: Free body diagram of the maglev system model.
'((7", 5", 6")
')(7#, 5#, 6#)
'*(7%, 5%, 6%)
'+(7$, 5$, 6$)
COG
',(7, 5, 6)
�()(�"#�, �"#�, �"#�)
�+*(�$%�, �$%�, �$%�)
2l
-,(., /, 0)
!"#
!$%
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�"#� = |�()|∙ (7" − 7#)|'( − ')|�"#� = |9()|∙ (5" − 5#)|'( − ')|�"#� = |9()|∙ (6" − 6#)|'( − ')|�$%� = |9+*|∙ (7% − 7$)|'* − '+|�$%� = |9+*|∙ (5% − 5$)|'* − '+|�$%� = |9+*| ∙ (6% − 6$)|'* − '+| :;
;;;;<;;;;;=
. (3)
To calculate the magnetic forces along the X, Y and Z directions, the coordinates of the four
magnets are required.
Magnets 1 and 4 are two fixed magnets. According to the geometric design of the maglev
system, with the nominal magnet separation ! = !"# = !$%, the distance between the centres
of the fixed magnets is ! + &. This gives
'( = [0 0 ! + &],'+ = [0 0 −! − &] (4)
as the coordinates of the fixed magnets. The coordinates of the floating magnets change with
the pose of the floater (', and -,). However, the coordinates of the floating magnets always
satisfy the following geometric constraints:
7# − 7 = −(7% − 7) 5# − 5 = −(5% − 5) 6# − 6 = −(6% − 6) (7# − 7)# + (5# − 5)# + (6# − 6)# = &#. (5)
Solving Eqs. (2) and (5) for 7#, 5#, 6#, 7%, 5% and 6% yields
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7# = & ∙ tan(/) ∙ > 1tan#(−.) + tan#(/) + 1 + 7
5# = & ∙ tan(−.) ∙ > 11 + tan#(−.) + tan#(−.)tan#(/) + 5
6# = & ∙ > 11 + tan#(−.) + tan#(/) + 6 7% = −& ∙ tan(/) ∙ > 1tan#(−.) + tan#(/) + 1 + 7
5% = −& ∙ tan(−.) ∙ > 11 + tan#(−.) + tan#(−.)tan#(/) + 5
6% = −& ∙ > 11 + tan#(−.) + tan#(/) + 6
(6)
Hence, for any arbitrary floater pose (7, 5, 6, ., /, 0), the coordinates of the floater magnet
centres can be calculated using Eq. (6). Substituting Eqs. (4) and (6) into Eq. (3) gives
�"#� = −|�()|(7 + & ∙ tan(/) ∙ ?)�(& + ! − 6 − & ∙ ?)# + (5 − & ∙ tan(.) ∙ ?)# + (7 + & ∙ tan(/) ∙ ?)# �"#� = −|�()|(5 − & ∙ tan(.) ∙ ?)�(& + ! − 6 − & ∙ ?)# + (5 − & ∙ tan(.) ∙ ?)# + (7 + & ∙ tan(/) ∙ ?)# �"#� = |�()|(& + ! − 6 − & ∙ ?)�(& + ! − 6 − & ∙ ?)# + (5 − & ∙ tan(.) ∙ ?)# + (7 + & ∙ tan(/) ∙ ?)# �$%� = |�+*|(7 − & ∙ tan(/) ∙ ?)�(7 − & ∙ tan(/) ∙ ?)# + (& + ! + 6 − & ∙ ?)# + (5 + & ∙ tan(.) ∙ ?)#
�$%� = |�+*|(5 + & ∙ tan(.) ∙ ?)�(7 − & ∙ tan(/) ∙ ?)# + (& + ! + 6 − & ∙ ?)# + (5 + & ∙ tan(.) ∙ ?)#
�$%� = |�+*|(& + ! + 6 − & ∙ ?)�(7 − & ∙ tan(/) ∙ ?)# + (& + ! + 6 − & ∙ ?)# + (5 + & ∙ tan(.) ∙ ?)#
(7)
where
? = > 1tan(.)# + tan(/)# + 1, (8)
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and the magnitudes of �() and �+* can be determined using the method described in [33].
Therefore, the 3D magnetic forces between the two pairs of magnets can now be calculated
for any given floater pose. Thus, the resultant forces and torques (�� , �� , �� , @A , @B , @C)
experienced by the COG of the floater may be calculated as
�� = �"#� + �$%� �� = �"#� + �$%� �� = �"#� + �$%� @A = �"#� ∙ (6 − 6#) + �$%� ∙ (6 − 6%) @B = �"#� ∙ (6# − 6) + �$%� ∙ (6% − 6) @C = 0. (9)
In Eq. (9), the torque around the floater COG in the γ direction is zero. This is due to the fact
that the resultant force vectors between top and bottom magnet pairs always intersect the Z
axis of the coordinate system shown in Figure 4.
4. 6-DOF levitation stiffness
It was mentioned previously that the proposed maglev isolator can realise quasi-zero stiffness
levitation in the vertical direction, and zero stiffness in the remaining five DOFs. This section
demonstrates the theoretical realisation of the quasi-zero/zero levitation stiffness.
4.1. Quasi-zero stiffness in the vertical direction (Z)
A major advantage of the proposed maglev isolator over traditional isolator designs is the
ability of the maglev system to realise quasi-zero payload support stiffness in the vertical (Z)
direction, while still providing a static payload supporting force. Figure 5(a) shows a case
where the floater motion is in the vertical direction. Using the calculation method outlined in
Section 3, the relationship between the vertical displacement (6 ) and the total vertical
magnetic force �� is shown in Figure 5(b), where ! = 100mm is chosen to be the nominal
separation between the magnets in this case. Table 2 lists the parameters of the magnets used
to obtain the plot data.
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In Figure 5(b), the slope of the tangent line to the force-displacement curve is the stiffness of
the levitation in the vertical direction. It can be seen that at the nominal operating position,
where the COG of the floater is at the midpoint between the two fixed magnets, the stiffness
of the levitation is zero. For a displacement (comparable to the displacement of ordinary
laboratory floor vibration) within a small region around the nominal operating position, the
levitation stiffness remains close to zero. Therefore, in the vertical direction, the magnetic
levitation has quasi-zero stiffness around the nominal operating point.
In Figure 5(b), the vertical magnetic force (��, approximately 39N in this case) at the nominal
operating position is the load capacity of the isolator. This vertical force is designed to
balance the weight of the payload and is not the result of elastic deformation of the
supporting structure. The magnitude of the payload supporting force is adjustable by
changing the separations between the top and bottom magnet pairs so that the supporting
force matches the payload weight. Therefore, the maglev system is able to provide quasi-zero
levitation stiffness while still generating a passive magnetic force to support the static weight
of the payload. This feature allows the 6-DOF maglev isolator design to avoid the usual
compromise made between load capacity and isolation performance, which normally exists in
linear vibration isolation systems.
In the preceding analysis of the maglev system, the floater motion was constrained in the
vertical direction. However, it is also important to investigate the sensitivity of the vertical
-30 -20 -10 0 10 20 3020
30
40
50
60
70
80
Vertical displacement of the floater (mm)
Ver
tica
l fo
rce
Fz
(N)
Normaloperatingposition
p=100mm
Slope=0Tangent line
XZ plane
Y
X
Z
p
p
N
S
N
N
S
S
F12Z
F43Z
z
Nominal operating condition
z = 0
Figure 5: Force displacement behavior of the maglev system in the vertical direction (Z). See Table 3 for details of parameters used to define the system, (a): Schematic of the maglev system; (b) Force-
displacement relationship in the vertical direction.
(a)
Nominal operating position
F
(b)
17
magnetic force to the displacements in the remaining five DOFs since external vibrations
propagate in multiple directions. Using the equations derived in Section 3, the vertical
magnetic force, �D, was calculated with combined floater displacements.
The five subplots of Figure 6 show the magnitudes of the vertical force when the Z (vertical)
displacement is combined with the X, Y, α, β and γ displacements respectively. From Figure
6(a) to Figure 6(d), it can be seen in each case considered that the vertical magnetic forces are
dependent on the magnitudes of both of the displacements, and with any given vertical
displacement (6), the vertical force has the maximum value when the displacement in the
non-vertical direction is zero. The cross coupling from the X, Y, α and β directions to the Z
direction result from the magnet axial misalignments introduced by the corresponding
displacements. Motions in these four DOFs effectively increase the distances between the
magnets for both the top and bottom magnet pairs, thus reducing the forces generated
between the magnets. In addition, the axial misalignment between the floater magnets and the
fixed magnets means that only a portion of the total magnetic force is in the vertical direction,
which also explains why the vertical magnetic force is a maximum at zero non-vertical floater
displacements for any given vertical displacement. The cross-coupling effect is undesirable in
this instance where the maglev system is used as a vibration isolation device. The disturbance
to the vertical direction may be introduced from other DOFs through cross-coupling effects.
However, it can be observed from Figure 6 that, under the nominal operating condition (the
floater has zero displacements and poses), the cross-coupling stiffness is quasi-zero between
Z and the four non-vertical DOFs (X, Y, α and β). Therefore, during operation of the maglev
isolator, the floater should be levitated at the nominal operating position to optimise the
vibration isolation performance in multiple DOFs.
Figure 6(e) shows that the vertical force generated by the maglev system is not affected by
the displacement in the γ direction. This can also be observed from the system model in
Figure 4 which shows that the γ displacement does not influence the relative positions
between the two magnet pairs.
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Figure 6: Floater force-displacement behavior of F� with cross-DOF displacements, (a): F� in the ZX plane; (b): F� in the ZY plane; (c): F� in the Zα plane; (d): F� in the Zβ plane; (e): F� in the Zγ plane.
(a) (b)
(c)
(d)
(e)
19
4.2. Zero stiffness in the horizontal directions (X and Y)
As mentioned previously, the proposed maglev system is able to achieve zero levitation
stiffness in the horizontal directions (X and Y). Figure 7 shows a free body diagram of the
floater in the YZ plane with a displacement in the Y direction. The forces experienced by the
floater from both pairs of fixed magnets are resolved as shown in the figure. With floater
displacement only in the Y direction, the magnetic force calculation may be simplified to
E;;;F;;;G�"#� = |F"#| ∙ −5
�5# + !"##�"#� = |F"#| ∙ !
�5# + !"##�$%� = |F$%| ∙ 5
�5# + !$%#�$%� = |F$%| ∙ !
�5# + !$%#
(10)
Under the nominal operating condition, where !"# = !$%, it can be observed that the vector
forces between the two pairs of magnets have the same magnitudes, i.e. |�()| = |�+*|. Hence,
from Eq. (10), it can be shown that
�� = �"#� + �$%� = 0 (11)
F12Y
F12Z
!"#
!$%
2l
5 COG Tα
F43Z
F43
F43Y
N
S
N
N
S
YZ plane
X
Y
Z
F12
Figure 7: Free body diagram of the floater under horizontal displacement
20
for all values of 5. Therefore, the total force experienced by the floater in the Y direction is
zero regardless of the displacement, which leads to the conclusion that the stiffness in the Y
direction remains zero under the nominal operating condition. The derivations for
demonstrating the zero levitation stiffness in the X direction are identical to the preceding
discussion. For a displacement in the X direction, the equations for calculating the relevant
forces in the XZ plane become
E;;;F;;;G�"#� = |F"#| ∙ −7
�7# + !"##�"#� = |F"#| ∙ !
�7# + !"##�$%� = |F$%| ∙ 7
�7# + !$%#�$%� = |F$%| ∙ !
�5# + !$%#
. (12)
Hence, under the nominal operating condition, where |�()| = |�+*|, the total magnetic force
in the X direction can be shown to be equal to zero using the same principles. With evidence
of zero stiffness levitation in both the X and Y directions, it is reasonable to conclude that the
levitation stiffness equals zero in any arbitrary direction within the XY plane.
The realisation of zero levitation stiffness in the horizontal plane (XY plane) requires equal
distances between the two magnet pairs. However, this cannot always be guaranteed during
operation of the isolator due to the multi-DOF nature of the external vibration. Hence, the
force displacement behaviour of the horizontal floater forces was also analysed under
combined displacements. Figure 8 and Figure 9 show the cross-DOF force-displacement
behaviours for displacements in the X and Y directions respectively.
It can be seen from Figure 8(a) and Figure 9(a) that the horizontal force in either the X or Y
direction remains zero for any displacement in the XY plane (the color gradient on the flat
plane is caused by the force calculation uncertainties, which are on the order of 10-14N). This
result agrees with the expression for the zero horizontal stiffness derived previously. Figure
8(b) and Figure 9(b) show that the displacement in the vertical direction changes the zero
stiffness property of the maglev in both the X and Y directions. This is because the vertical
displacement interrupts the condition, |�()| = |�+*|, needed for achieving the zero horizontal
stiffness. As shown by Figure 8(b), the vertical displacement changes the horizontal stiffness
21
linearly from HIJHK = 0N/m at 6 = 0mm to a maximum of
HILHK = 55.033N/m at 6 = 3mm
(displacement constraint from the mechanical isolator design shown in Section 5). At the
magnet nominal separation ( ! = 100 mm) used in this analysis, the vertical payload
supporting force is 39.13N (Figure 5(b)). This corresponds to a payload of approximately 4kg.
Hence, with this combination of payload mass and system stiffness, the natural frequency of
the system in the horizontal direction is OPJ = "#Q R ST = 0.5903Hz. This indicates that, even
with the highest induced stiffness, the system can still be expected to provide excellent
vibration isolation in the X direction. Therefore, with small vertical excitations (comparable
to the magnitude of laboratory floor vibration), the induced stiffness in the horizontal
directions has no significant impact on the vibration isolation performance in the X direction.
The same conclusion can be made from Figure 9(b) for the Y direction.
Figure 8(d) and Figure 9(c) show the cross couplings between the β and X directions and the
α and Y directions respectively. The figures show that with a given angular displacement in
the β direction, a constant force in the X direction is created for all values of 7 (displacement
in the X direction). Similarly, a given α displacement will have the same effect in the Y
direction. These cross-coupling effects can also be analytically observed from Figure 10,
which shows that a floater rotation in the α direction creates a horizontal force in the Y
direction, and is calculated as
�� = �"#� + �$%� (13)
for the cross-coupled force in the Y direction, and
�� = �"#� + �$%� (14)
for the cross-coupled force in the X direction. These cross-coupling effects are undesirable in
terms of vibration isolation since they introduce disturbances into the horizontal direction
from the rotational excitations. However, with the small rotational excitations from
laboratory floor vibration, this cross-coupling effect is not expected to introduce significant
disturbance to the performance of the system in the horizontal directions.
22
Figure 8: Floater force-displacement behavior of F� with cross-DOF displacements, (a): F� in the XY plane; (b): F� in the XZ plane; (c): F� in the Xα plane; (d): F� in the Xβ plane; (e): F� in the Xγ plane.
(a) (b)
(c) (d)
(e)
23
Figure 9: Floater force-displacement behavior of F� with cross-DOF displacements, (a): F� in the YX plane; (b): F� in the YZ plane; (c): F� in the Yα plane; (d): F� in the Yβ plane; (e): F� in the Yγ plane.
(e)
(a) (b)
(c) (d)
24
4.3. Zero stiffness in the rotational DOFs (α, β and γ)
The levitation stiffness of the proposed maglev system is also zero in the three rotational
DOFs. An example is shown in Figure 10 where the floater is rotated within the YZ plane by
a rotational displacement (-α). The torque generated by the magnetic forces around the centre
of the floater can be calculated as
@A = �"#� ∙ &X − �$%� ∙ &X, (15)
where &′ is the vertical distance between the centre of the floater and the floater magnet and is
the lever arm for the torque. For simplicity of the analysis, the torque calculation only
considers the mechanical torque created by the horizontal forces on the floater from the fixed
magnets. The magnetic torque [36] between the magnet pairs is neglected due to its small
value. For a horizontal displacement within [-3mm 3mm] (constrained mechanically by the
isolator design presented in Section 5), the largest magnetic torque was calculated to be 2.89
percent of the mechanical torque generated by the horizontal forces (�"#�and�$%�).
It is evident from Figure 10 that when the floater is at the nominal operating condition (equal
nominal separation between the magnet pairs), the horizontal forces, �"#�and �$%�, have the
same magnitude. Therefore,
@A = �"#� ∙ &X − �$%� ∙ &X = 0. (16)
This shows that for any rotational displacement in the α direction, @A = 0. Therefore, the
floater has zero levitation stiffness in the α direction. Due to symmetry, the same principles
can be applied to demonstrate the zero stiffness property of the maglev in the β direction.
25
The preceding analysis assumed zero floater displacement in the remaining DOFs. To
investigate the torque displacement behaviour in the 6-DOF space, the mechanical torque on
the floater was examined for cases with combined displacements. Figure 11 and Figure 12
show the torque-displacement behaviour of the floater for various displacement combinations.
In Figure 11(b) and Figure 12(a), cross coupling is shown between the Y and α directions and
the X and β directions respectively. For a given horizontal displacement in the Y direction, a
constant torque is generated for all angular displacements, ., and with a fixed displacement
in X, a constant torque is created in the β direction for all /. This cross-coupling effect can
also be demonstrated analytically for horizontal displacement, 5, as illustrated in Figure 7.
The mechanical torque created is
@A = �"#� ∙ & − �$%� ∙ &. (17)
Similarly, with an 7 displacement, the torque in the β direction is
@B = −�"#� ∙ & + �$%� ∙ &. (18)
p12
p43 YZ plane
X
Y
Z
N
S
F12
F43 F43Z
F43Y
F12Y
F12Z
COG
FY -α
Figure 10: A schematic of the floater under rotational displacement.
&’&’Tα=0
26
Figure 11: Floater force-displacement behavior of TA with cross-DOF displacements, (a): TA in the αX plane; (b): TA in the αY plane; (c): TA in the αZ plane; (d): Tα in the αβ plane; (e): Tα in the αγ plane.
(e)
(a) (b)
(c)
(d)
27
Figure 12: Floater force-displacement behavior of TB with cross-DOF displacements, (a): TB in the βX
plane; (b): TB in the βY plane; (c): TB in the βZ plane; (d): Tβ in the βα plane; (e): Tβ in the βγ plane.
(a)
(b)
(c)
(d)
(e)
28
In Figure 11(c) and Figure 12(c), a similar cross-coupling effect is shown between the
vertical (Z) and the rotational (α and β) directions. The displacement in the Z direction
increases the rotational stiffness of the maglev linearly from 0Nm/rad to a maximum of
H]̂H_ = 0.4952Nm/rad at 6 = 3mm. For the mechanical design of the isolator, the floater
moment of inertia is a��= 0.0853kgm2 (Table 3). Therefore, in the α direction, the maximum
natural frequency is OPb = "#QR ScJJ = 0.3835Hz, which is an indication of the excellent
rotational vibration isolation that is potentially achievable by the system even at the largest
rotational stiffness induced by cross coupling. Therefore, with the small rotational excitation
that the isolator would experience during normal operation, the introduced rotational stiffness
is not expected to influence the vibration isolation performance of the maglev isolator in the
rotational DOFs.
In the proposed maglev vibration isolation system, the levitation stiffness is zero in the γ
direction. This is due to the fact that the system is axisymmetric and as such there are neither
mechanical constraints nor magnetic force constraints on the floater. The floater is able to
move freely without any interference, and the γ direction is only actively stabilised using the
maglev positioning control (discussed in Section 6). Therefore, theoretically, the maglev
system itself has inherent zero stiffness in the γ direction.
5. Mechanical system of the isolator
In order to validate the theoretical performance of the proposed vibration isolator design, a 6-
DOF maglev vibration isolator was designed and manufactured. This section provides and
overview of the mechanical design of the isolation system.
Figure 13(a) shows a photograph of the constructed isolation system, and Figure 13(b) is a
schematic highlighting the maglev components in the mechanical design. The isolator design
uses the Payload Connectors to attach the payload to the floater for achieving vibration
isolation for the payload. This design allows the isolator to be connected to any mechanical
system that is attachable to the payload connectors. Multiple isolators may also be combined
to suit a wide range of applications. For example, four isolators could be installed at the
corners of a rectangular table top to create a vibration isolation platform.
29
As discussed in Section 4.1 that the load capacity of the isolator can be adjusted through
changing the separations between the permanent magnets. In order to control the magnet
separation, two Magnet Position Control units (Figure 14) driven by DC servo motors, were
installed at the top and bottom of the vibration isolator frame and used to control the positions
of the permanent magnets. A threaded linear drive system was used on each Magnet Position
Control unit to transfer motor rotation to the required linear motion for relocating the magnet
in the vertical direction.
In the maglev vibration isolator, twelve solenoids and six laser sensors are used to control and
monitor the position of the floater respectively in the 6-DOF space. The structure and
working principles of the actuation and sensing systems are discussed in Section 6. The
physical properties of the isolation system are listed in Table 3, and Table 4 shows the
specifications of the isolation system electronics.
Figure 13: Illustration of the components of the 6-DOF maglev vibration isolator, (a): Photograph of the maglev isolation system; (b): Schematic of the maglev system
(a) (b)
v
30
Table 3: Physical properties of the vibration isolation system.
Item Description
Frame material Aluminium 6061-T6
Weight Frame: 19.6kg Floater: 6.4kg
Floater moments of inertia aee = 0.0853kgm2 aff = 0.0851kgm2 agg = 0.0186 kgm2
Dimensions Length: 230mm Width: 230mm Height: 1020mm
Primary magnet Magnet 1 to 4
Ø50.8mm × 25.4mm; Grade: N52; Magnetisation: 1.48T
Actuator magnet (Refer to Figure 15)
Ø19.05mm × 76.2mm; Grade: N52; Magnetisation: 1.48T
Max. allowable vibration Translational: ±3mm; Rotational: ±0.85deg
Table 4: Specification of system electronics.
Item Description
Laser sensor Model: Acuity AR200; Range: 21±6mm; Resolution: 3µm; Output: analogue ±10 V; Max. sampling frequency: 1250Hz
Coil (actuator) Number of turns: 1000; Wire diameter: 0.85mm; Resistance 3.7Ohm; Inductance: 95mH; Sensitivity: 3.04N/A
Actuator amplifier Model: Maxon LSC30/2 4-Q-DC; Operation mode: current control; Max. current output:±2A
Magnet positioning motor Model: Maxon EC 45 flat; Power: 50W; Controller: EPOS2 24/5
Primary Magnet
Timing Pulley
Timing Belt
Threaded Linear Drive
Drive Motor
Figure 14: Detailed view of the Magnet Position Control unit.
31
6. Maglev stabilization system
As previously mentioned, passive magnetic levitation is inherently unstable [25]. Therefore,
an active stabilization system is necessary to enable the operation of the maglev system. In
this design, the maglev stabilization is comprised of a 6-DOF laser position monitoring
system and a 6-DOF actuation system, shown in Figure 15 and Figure 16 respectively.
Floater Assembly
Frame Assembly
Solenoids Actuator Magnets
(a)
Z
Y
X
Figure 16: The Frame Assembly, (a) Assembly with 4 solenoids used to control the floater motion in the XZ plane; (b) Assembly with 12 solenoids for full 6-DOF floater motion control.
(b)
Y
X
Z
Figure 15: 6-DOF laser position monitoring system.
32
The laser position sensing system includes six laser sensors fixed to the frame to provide
displacement data of the levitated floater. The laser sensors are arranged to monitor the
motion of the six points on the floater as shown in Figure 15. The six coordinates (x, y, z, α, β
and γ) of the COG of the floater are then derived based on the displacements measured by the
laser sensors according to
7 = 7" + 7#25 = 5" + 5#26 = 6" + 6#2. = arctan5# − 5"kl/ = arctan 6# − 6"kD0 = arctan7# − 7"km
:;;;;;<;;;;;=
, (19)
where 7n, 5n and 6n represent the displacement readings from the accordingly named sensors,
and Lx = Ly = Lz = 270mm are the distances between each pair of laser monitoring points (e.g.,
Lx is the distance between the laser points of sensors X1 and X2).
The 6-DOF actuation system consists of six pairs of solenoids. Each pair of solenoids is
coupled with one actuator magnet attached to the floater, and is driven in series in order to
double the actuation capacity on each actuator magnet compared to the capacity achievable
from only one solenoid. Figure 16(a) show two pairs of solenoids attached to the frame
assembly in the XZ plane. By driving both pairs of solenoids collectively, forces can be
generated in the Z direction, and by driving the two pairs differentially, torques are generated
in the β direction. The two pairs of actuators in the XZ plane have the ability to control the
floater motion in the two DOFs (Z and β) within the XZ plane. Therefore, with the six pairs
of solenoids installed as shown in Figure 16(b), the floater motion is controllable in all six
DOFs. The relationships between the individual actuation forces from each of the solenoids
and the resultant force/torque experienced by the floater are
33
�X = OXa + OXb + OXc + OXd�Y = OYa + OYb + OYc + OYd�Z = OZa + OZb + OZc + OZd@α = 12 k′X pqOZa + OZbr − qOZc + OZdrs@β = 12 k′Y pqOXa + OXbr − qOXc + OXdrs@γ = 12 k′Z pqOYa + OYbr − qOYc + OYdrs
:;;;<;;;=, (20)
where Ovw (k ∈(X, Y, Z), l ∈(a, b, c, d)) represents the actuation force generated by the
solenoid labelled accordingly, and k′y (m ∈(X, Y, Z)) represents the distance between the
point of actuation and the centre of gravity of the floater on each axis. Due to the fact that
every pair of solenoids is driven in series to generate an equal actuation force, Eq. (20) can
then be simplified to
�X = 2OXa + 2OXc�Y = 2OYa + 2OYc�Z = 2OZa + 2OZc@α = 12 k′X pq2OZar − q2OZcrs@β = 12 k′Y pq2OXar − q2OXcrs@γ = 12 k′Z pq2OYar − q2OYcrs
:;;;<;;;=. (21)
The mechanical design of the actuation system is such that the actuation forces/torques in
each DOF are orthogonal to all the other DOFs, that the actuation system is theoretically
decoupled in the 6-DOF space. This can be seen from Eq. (21) since there always exists a set
of unique solutions to the solenoid forces (fXa, fXc, fYa, fYc, fZa, fZc) for an arbitrary given set of
required actuation forces/torques (��, ��, ��, @A, @B, @C ). Therefore, the actuation system is
decoupled in the 6-DOF space, and SISO (single input single output) control loops may be
used to allow decoupled performance control for stabilising the maglev in each DOF.
However, in practice, slight coupling effects may exist between the translational and
rotational DOFs for a number of reasons. The solenoid actuators may not be identical on both
sides of the floater (refer to Figure 16(a)), which can lead to an undesired torque being
generated while the solenoid pairs are driven collectively for vertical actuation. The offset
between the geometric and gravity centres of the floater may also contribute to the cross
coupling, since the actuation is applied with respect to the geometric centre of the floater,
34
while the floater motion is subject to its centre of gravity. These aspects have been considered
in the mechanical design of the isolator to minimize the cross coupling in the actuation
system.
In the maglev isolation system, a dSPACE DS1103 platform is used to execute the controller
designs. Matlab Simulink and dSPACE Control Desk are used to design and configure the
controller algorithms respectively. Figure 17 shows the structure of the maglev stabilization
controller. Six PID controllers are used in parallel for floater position regulation in six DOFs.
The PID type controller was chosen for its ability to directly control the levitation stiffness
and damping. The feedback signals to the PID controllers are the floater displacements
relative to the frame in each DOF (given by Eq. (19)). Therefore, the proportional gains
directly control the levitation stiffness and the derivative gains control the relative damping
of the floater with respect to the frame. In order to isolate ground vibration, low stiffness is
desirable as it reduces the system resonance frequency, and thus, vibration transmissibility.
The gains in the PID controllers were tuned in the frequency domain in order to achieve
minimum system natural frequency and transmissibility.
7. Experimental results and discussions
A number of tests were completed to quantify the performance of each subsystem, as well as
the ground vibration transmissibility of the system. This section will present the experimental
performance of the maglev stabilization controller and the isolator transmissibilities in all six
DOFs.
Figure 17: Structure of the maglev stabilization controller.
Isolator
- +
6-D position
command
[x, y, z, α, β
, γ]
dSPACE DS1103
6 × PID SISO loops
in series with LP
filters (Eq. (22))
Coordinate transformatio
6 × Maxon
LSC30 current
amplifier
6-DOF actuation
6-DOF laser position
monitoring
FX FY FZ
Tα Tβ Tγ
Coordinate Transformation
Eq. (19)
[x1, x2, y1,
y2, z1, z2]
35
7.1. Performance of the maglev stabilization control
The performance of the maglev stabilization control is critical to the overall performance of
the vibration isolator. The active control forces resulting from the levitation stabilization
controller can significantly increase the vibration amplitude on the floater. Therefore, this
self-induced vibration from the levitation stabilization system must be kept within an
acceptable range for the intended application. To avoid excitation of high frequency
structural modes of the floater assembly, a second order low-pass filter is used in series with
each PID controller to attenuate the control signal beyond 200Hz. The transfer function of the
low-pass filters is given as
k{(|) = }#(| + })# , } = 200Hz = 1257rad/s (22)
where k{(|) and } represent the transfer function and the cut off frequency of the filter
respectively. The transfer function of the PID controllers is
�(|) = {n + �n| + ��| (23)
where �(|) represents the controller transfer function, and {n, �n and �� (� ∈ [X, Y, Z, α, β, γ]) represent the proportional, integrative and derivative gains of the ith PID controller.
In order to isolate vibration, low stiffness is desirable for reducing the system resonance
frequency and post-resonance vibration transmissibility. The PID controller gains were tuned
in the frequency domain to achieve minimum system natural frequency and transmissibility
while maintaining the stability of the maglev. Table 5 contains a set of PID gains that has
been found to achieve low natural frequency and low relative damping while also
maintaining stable levitation with acceptable peak response.
36
Table 5: Levitation stabilization controller gains used for transmissibility measurements.
Item Value Item Value
PX 2250 N/m Pα 320 Nm/rad
IX 112.5 N/m•s Iα 240 Nm/rad•s
DX 26.3 N•s/m Dα 3.4 Nm•s/rad
PY 2250 N/m Pβ 320 Nm/rad
IY 112.5 N/m•s Iβ 240 Nm/rad•s
DY 26.3 N•s/m Dβ 3.4 Nm•s/rad
PZ 675 N/m Pγ 64 Nm/rad
IZ 67.5 N/m•s Iγ 24 Nm/rad•s
DZ 6 N•s/m Dγ 1.2 Nm•s/rad
The residual vibration levels on the floater during levitation were measured in six DOFs for
the PID gains listed in Table 5. Figure 18(a) shows the translational vibrations in the X, Y
and Z directions in a quiet laboratory environment, and are comprised of ground-borne
vibration and vibration induced by sensor noise and amplifier noise from the active control
loops. The translational vibrations are compared with the VC-E vibration criterion [37],
which is the most demanding criterion specified for the operation of extremely vibration-
sensitive equipment. The results show that, in the translational directions, the magnitude of
the residual vibration on the floater is below the VC-E threshold, which is an indication of the
satisfactory performance achieved by the maglev stabilization system. The self-induced
vibrations in the rotational DOFs are shown in Figure 18(b). To the best knowledge of the
authors, no vibration standard exists yet to assess the system performance in the rotational
DOFs.
(a) (b)
Figure 18: Levitation induced vibration: (a) in the X, Y and Z directions; (b) in the α, β and γ directions.
100
101
102
-220
-200
-180
-160
-140
-120
Vib
ration V
elo
city
(dB
re.
1 r
ad/s
ec)
Frequency (Hz)
alpha
beta
gamma
100
101
102
-20
0
20
40
60
80
Vib
ration V
elo
city
(dB
re.
1 m
icro
-inch/s
ec)
Frequency (Hz)
X
Y
Z
VC-E
37
7.2. Vibration transmissibility
Apart from the vibration induced by the levitation system, the external vibration
transmissibilities are the most important performance measure of the 6-DOF maglev isolator.
In order to obtain the transmissibility measurements in six DOFs, a testing platform was
designed and constructed. Figure 19 shows a photograph of the testing platform assembly
configured for excitation in the Y direction. The design of the testing platform allows it to be
assembled in five additional ways to allow excitations in the other five DOFs.
The velocity transmissibilities of the isolator were measured in all six DOFs. A set of six
geophones (SENSOR model LF-24) were installed on both the floater and the isolator frame
to quantify the amount of vibration attenuation achieved. Six experiments were conducted to
capture the isolator transfer function in each of the six DOFs. During each measurement, the
isolator frame was excited predominately in one DOF using a high capacity shaker (MB
Dynamics model 110) with a swept sine input signal, and the vibration transmissibilities were
recorded by a Brüel & Kjær Photon+ dynamic signal analyser. Figure 20 shows the measured
transmissibilities of the isolator in six DOFs with the PID controller gains described in Table
5.
Figure 19: 6-DOF vibration transmissibility testing platform (current view shows Y direction excitation).
38
In Figure 20, the solid lines represent the measured isolator responses to the external
vibrations in each of the six DOFs, and the dashed lines are the theoretical responses of the
isolation system assuming zero stiffness in all directions. The theoretical zero stiffness
system responses are derived using the control system structure described in Figure 21, which
is a schematic of each PID control loop for stabilizing a single DOF. The control system has
six such loops to stabilize the floater in 6-DOF. The derivation of the theoretical zero-
stiffness responses is based on the following:
• The plant (the isolation system) has negligible inherent damping due to non-
contact magnetic levitation.
• The system responses of the Solenoid Actuator, Actuator Amplifier and Laser
Sensor are linear in the frequency range of measurement in (0.5-20Hz).
Figure 20: Vibration transmissibilities in the six DOFs, (a): X direction; (b): Y direction; (c): Z direction; (d): α direction; (e): β direction; (f): γ direction; (solid curve: measured isolator response; dashed curve: predicted
isolator response with zero inherent stiffness).
100
101
-30
-20
-10
0
10
20
30
Frequency (Hz)
(a)
Tra
nsm
issib
ility
in X
(dB
)
Experimental
Theoretical
100
101
-30
-20
-10
0
10
20
30
Tra
nsm
issib
ility
in Y
(dB
)
Frequency (Hz)
(b)
Experimental
Theoretical
100
101
-30
-20
-10
0
10
20
30
Frequency (Hz)
(c)
Tra
nsm
issib
ility
in Z
(dB
)
Experimental
Theoretical
100
101
-15
-10
-5
0
5
10
15
Frequency (Hz)
(d)
Tra
nsm
issib
ility
in A
lpha (
dB
)
Experimental
Theoretical
100
101
-15
-10
-5
0
5
10
15
Tra
nsm
issib
ility
in B
eta
(dB
)
Frequency (Hz)
(e)
Experimental
Theoretical
100
101
-15
-10
-5
0
5
10
15
Frequency (Hz)
(f)
Tra
nsm
issib
ility
in G
am
ma (
dB
)
Experimental
Theoretical
39
• The laser displacement sensors are attached to the Frame Assembly of the isolator
and hence the motion of the reference points of the laser sensor measurements
have the same amplitude and phase as the external vibration.
Therefore, with zero isolator stiffness in all the DOFs, and in the absence of the feedback
controller, the transfer function of the plant may be simplified to
�n(|) = 1�|# ; ��(|) = 1ann|#, (24)
where � is the mass of the floater,aii is the floater moment of inertia with respect to its COG,
�n(|) and �j(|) (i ∈(X, Y, Z) and j ∈ (α, β, γ)) represent the transfer function of the plant in
the translational and rotational DOFs respectively. With the simplified plant transfer
functions, the frequency response of the controlled levitation to external vibration is
analogous to a linear mass-spring-damper system with the stiffness and damping from the
PID controller. This frequency response is obtainable through the closed-loop transfer
function of the system, written as
@�n(|) = �(|) × k{(|) × �n(|)1 + �(|) × k{(|) × �n(|)@��(|) = �(|) × k{(|) × ��(|)1 + �(|) × k{(|) × ��(|):;<
;=, (25)
where TFi(|) and TFj(|) represent the system transfer functions in the translational and
rotational DOFs respectively. Substituting Eqs. (22), (23) and (24) into Eq. (25) yields
PID LP
Filter
Laser Sensor
Solenoid
Actuator
Actuator
Amplifier Plant + _
+ + External Vibration
Figure 21: Schematic of the individual PID stabilization control loops.
Command Floater Position
40
@�n = }#�n|# + }#{n| + }#�n�|� + 2�}|$ +�}#|% + }#�n|# + }#{n| + }#�n@�� = }#��|# + }#{�| + }#��ann|� + 2ann}|$ + ann}#|% +}#��|# + }#{�| + }#��:;<
;=, (26)
which are the transfer functions used to derive the theoretical system responses in Figure 20
assuming zero inherent stiffness of the maglev system in all six DOFs. In Eq. (26), } is the
cut-off frequency of the low-pass filters; {n , �n , �n and {� , �� , �� are the proportional,
integrative and derivative gains of the PID controllers for stabilising levitation in the
translational and rotational DOFs respectively. The floater mass and moments of inertia are
provided in Table 3, and the PID gains used in the transmissibility tests are listed in Table 5.
The measured isolator responses in all six DOFs shown in Figure 20 reveal similar behaviour
to the predicted isolator response derived with the inherent levitation stiffness assumed to be
zero in all directions. Therefore, it is practically demonstrated that the proposed maglev
isolation system has the ability to realise inherent zero-stiffness levitation in all six DOFs.
Figure 20 also shows that the isolator system response is dominated by the gains of the PID
controllers since the experimental results are comparable to the theoretical response derived
with consideration of the influence of the control system dynamics alone. The PID gains used
during the transmissibility measurements were found to be a suitable combination for both
minimising the vibration transmissibility and maintaining a stable levitation based on the
current mechanical and electronic system used in the isolator prototype. With improved
hardware quality, such as improvements to the uniformity of magnetisation of the magnets,
the mechanical component alignment, the laser sensor resolution, the electronic system noise,
the solenoid actuator performance, and the amplifier performance, the stability of the maglev
system may be achieved with lower controller stiffness. This would potentially result in
improved vibration isolation performance. The potential of the proposed maglev vibration
isolation method is thus currently limited by the quality of the hardware, rather than physical
limits relating to the proposed method.
Figure 20(c) shows a small difference between the resonance frequencies in the experimental
and theoretical results. This is due to the fact that quasi-zero stiffness levitation in the vertical
(Z) direction only exists for small displacements from the nominal operation position. Due to
the relatively low isolator resonance frequency in the Z direction, large amplitude excitations
(approximately ±2.5mm floater displacement relative to the frame assembly) were used in
41
the transmissibility measurements. This level of excitation was necessary to overcome the
initial static friction in the linear bearings used in the testing apparatus, as well as to ensure
high signal to noise ratio in the inertial sensors (geophones). The large floater offset from the
nominal operating position resulted in a small amount of additional stiffness caused by the
vertical magnetic forces, and this effect slightly increased the overall resonance frequency of
maglev isolator in the vertical direction. However, in practical situations of relevance, the
magnitude of typical indoor vibration in the vertical direction is expected to be substantially
smaller than the excitation amplitude used here. Hence, the maglev system is expected to
maintain the quasi-zero stiffness levitation in the vertical direction during normal operation.
According to Figure 20, the experimental and theoretical resonance frequencies for the X, Y
and Z directions are approximately 2.9Hz, 3Hz, 1.8Hz and 3Hz, 3Hz, 1.6Hz respectively.
Vibration attenuation achieved experimentally at 10Hz is -17dB, -15dB and -28dB in the X,
Y and Z directions. These performance measures are comparable to the top of the range
active vibration isolation products offered in the market, such as products from Newport [3]
and Herzan [4]. In addition, the maglev isolator proposed in this paper achieves passive
payload weight support using forces from permanent magnets, which allows the system to be
stabilised with minimal power consumption. Experiments have shown that the maglev only
consumes less than 1W of electricity for levitating a mass of 6.693kg (floater weight with the
added geophones for measurements), much of which is due to parasitic losses in the linear
power stage of the actuator amplifiers.
As mentioned previously, the performance of the developed isolation system is dominated by
the characteristics of the maglev stabilization controller. The vibration isolation performance
may be improved with system hardware of higher quality. As the proposed maglev system is
actively stabilised, the performance of the isolator is able to be actively tuned to satisfy the
requirements for different applications. For example, low controller gains can be used to
address ground vibration isolation, while high controller gains may be used to realise low
isolator compliance to eliminate on board equipment vibrations.
In Figure 20(a), the peak response of the isolator in the X direction shows a larger amplitude
than the theoretical response. This may be caused by a cross-coupling effect existing on the
vibration testing platform. For excitation frequencies above 3Hz, the limited rigidity of the
excitation platform is not able to constrain the excitation energy to be purely within a single
42
DOF. This phenomenon can also be observed in Figure 22 as the coherences of most of the
cross-coupling measurements show a significant increase beyond 3Hz. The cross coupled
excitation energy is transmitted into the floater through the complex mechanical linkages of
the excitation platform, and results in increased floater vibration at the resonance frequency
in the X direction.
Figure 20(d) and Figure 20(e) show the frequency responses of the isolator in the α and β
directions. The unpredicted behaviour of the isolator responses at low frequencies is a result
of coupling with the gravitational field. During the measurements of the rotational
transmissibility of the isolator, large excitation angles were used to obtain accurate
measurements at low frequencies. This caused a large misalignment between the magnetic
payload supporting force and the gravitational field, which resulted in the unexpected floater
movements at low frequencies.
7.3. 6-DOF isolator cross coupling
The cross-coupled transmissibility of the proposed maglev vibration isolator was measured
between all six orthogonal DOFs through six experiments. In each experiment, the isolator
was excited in predominately one DOF, and the vibration velocity of the floater was recorded
in all six DOFs. Figure 22 shows the measured transmissibility between each pair of DOFs.
The transmissibility plots are mapped from the column index to the row index; for example,
the amplitude of cross transmissibility from the Y direction to the α direction is shown by the
plot in column two and row seven. The coherence of each measurement is shown in the
coherence (Coh) plot located directly below each transmissibility amplitude (Amp) plot. Plots
in Figure 22 are arranged as:
• First quadrant (plots in row 1 to 6 and column 4 to 6):
Transmissibility measurements from the translational DOFs to the rotational DOFs
• Second quadrant (plots in row 1 to 6 and column 1 to 3):
Transmissibility measurements from the translational DOFs to the translational DOFs
• Third quadrant (plots in row 7 to 12 and column 1 to 3):
Transmissibility measurements from the rotational DOFs to the translational DOFs
• Fourth quadrant (plots in row 7 to 12 and column 4 to 6):
Transmissibility measurements from the translational DOFs to the rotational DOFs
43
100
101
-80-60-40-20
020
Am
p (
dB
)
100
101
0
0.5
1
Coh
100
101
-80-60-40-20
020
Am
p (
dB
)
100
101
0
0.5
1
Coh
100
101
-80-60-40-20
020
Am
p (
dB
)
100
101
0
0.5
1
Co
h
100
101
-80-60-40-20
020
Am
p (
dB
)
100
101
0
0.5
1
Coh
100
101
-80-60-40-20
020
Am
p (
dB
)
100
101
0
0.5
1
Coh
100
101
-80-60-40-20
020
Am
p (
dB
)
100
101
0
0.5
1
Frequency (Hz)
Coh
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
Frequency (Hz)
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
Frequency (Hz)
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
Frequency (Hz)
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
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101
0
0.5
1
100
101
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020
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101
0
0.5
1
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101
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020
100
101
0
0.5
1
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101
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020
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101
0
0.5
1
Frequency (Hz)
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
100
101
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020
100
101
0
0.5
1
100
101
-80-60-40-20
020
100
101
0
0.5
1
Frequency (Hz)
Figure 22: Measurements of the cross coupling between DOFs. Horizontally across represents the input axis, vertically represents the response axis (FFT properties: DC coupled, 128 averages, 1600 lines,
Hanning window, 24,000Hz sampling frequency).
X (m) Y (m) Z (m) α (rad) β (rad) γ (rad)
X (
m)
Y (
m)
Z (
m)
α (
rad
) β (
rad
) γ
(rad
)
44
It can be seen that the diagonal plots (X to X, Y to Y, Z to Z, α to α, β to β and γ to γ) in the
second and fourth quadrants have significantly higher amplitudes (by approximately 40 dB
on average) than the cross-DOF transmissibilities. This indicates that each translational DOF
is highly decoupled from the other translational DOFs, and each rotational DOF is also
decoupled from the other rotational DOFs. As discussed in Section 4, the maglev system has
cross coupling between the X and β directions, as well as between the Y and α directions.
These cross couplings can be observed in the measurements shown by the plots in the first
and third quadrants. Plots X to β, β to X, Y to α and α to Y all show relatively high (close to 1)
values of coherence, which indicate the cross-coupling effects between the X and β, and the
Y and α directions. The cross couplings shown from the α and β directions into the Z
direction are caused by the misalignments between the directions of the payload supporting
force and gravity. As discussed previously, these misalignments are a result of the large
amplitudes of excitation used to capture isolator transmissibilities at low frequencies.
In Figure 22, the coherence curves of all the cross transmissibility measurements show values
close to 1 beyond 3Hz. This is due to the finite stiffness of the testing platform which allowed
excitation energy in the testing DOF to couple into other DOFs of the testing platform.
8. Conclusion
This paper presents the theory and implementation of a high performance 6-DOF vibration
isolator based on the quasi-zero stiffness maglev system. The theories and experimental
validations presented in this paper reveal the potential of the proposed maglev system as a 6-
DOF vibration isolator. It was experimentally demonstrated that the maglev system is able to
realise inherent quasi-zero stiffness levitation in the vertical direction while providing a static
payload supporting force, and the inherent stiffness of levitation in the remaining five DOFs
are also zero. Although other quasi-zero stiffness isolator designs have been reported in the
literature, the vast majority of them are only able to provide quasi-zero stiffness in one DOF.
The vibration transmissibility measurements have also demonstrated the low resonance
frequencies achieved in multiple DOFs, which further indicates the good performance of the
system as a vibration isolator. It is noteworthy that the performance of the proposed system is
a combined result of both mechanical and electrical systems. Therefore, the performance of
the proposed system can potentially be further improved through embedding hardware with
higher quality and less noise.
45
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