Proceedings of 2015 International Conference on Quality, Reliability, Risk,
Maintenance, and Safety Engineering (QR2MSE 2015)
July 21-24, 2015, Beijing, China
QR2MSE2015-DRAFT
Put Paper Title Here (e.g.: Sequential Particle Swarm Optimization and
Reliability Assessment of Planar-Type Voice Coil Motor)
Given Name Surname1, Hong-Zhong Huang2,*, …
1. Department of Industrial Engineering,
University of Electronic Science and Technology of China,
Chengdu, Sichuan, 611731, P.R. China
2. Institute of Reliability Engineering,
University of Electronic Science and Technology of China,
Chengdu, Sichuan, 611731, P.R. China
ABSTRACT
Put abstract text here.
Planar-type voice coil motor (VCM) is a key
component in ultra-precision motion of fine stage of
lithography machine. The reliability-based design
optimization method given in this work provides a novel
criterion to ensure performance of Lorentz motors by
evaluating the reliability of force constant. To solve the
reliability based design optimization (RBDO) problem in
discrete space with the speed of decoupled loop in
sequential optimization and reliability assessment (SORA)
for global solution, a Sequential Particle Swarm
Optimization and Reliability Assessment method is
proposed. The reliability boundary shift is put into penalty
function for constrained optimization in fitness evaluation
of particle swarm optimization (PSO). The presented
optimization design model, the geometric parameters of
the studied Planar-type VCM in finite element model are
treated as design variables whereas the thrust force
constant is an output quantity of interests. By using
electromagnetic analysis, the desired requirements of
Lorentz motors are verified.
here. Put body of the paper here. Put body of the paper
here. Put body of the paper here.
1 INTRODUCTION
The high precision position of wafer stage should not
only be measureable, but also be controllable, which is
achieved by motors, bearings and feedback loops [1]. A
two stage strategy, composed of dual-servo stage
including the coarse stage and the fine stage, was
promoted to satisfy the requirement of the wafer stage, the
key subsystem of lithography machine [2]. It needs to
move a long distance (about hundreds millimeters) with
ultra-precision positioning, which is driven by the linear
motor for the coarse stage and the planar-type voice coil
motor for the fine stage [3].
The voice coil motor, as one type of linear Lorentzforce actuator usually used in loudspeaker [4], DVD
drives, is developed to be applied in ultra-precision
motion stages as a mature technology, for that the VCM is
useful in linear precision positioning control systems with
little range by exciting its voice coil with a controlled
current [5]. A voice coil motor, which includes rotary-type
and planar-type, consists of coil, permanent magnetics and
iron yokes. The rotary-type VCM is optimized to increase
the force generation by Halbach Magnetic Circuit, which
could confine the magnetic flux in order to decrease the
thickness of the yoke and to increase the force constant [6,
7]. A type of VCM for digital cameras is designed and
optimized for maximum force constant by using
electromagnetic simulation software of Ansoft Maxwell
[5]. The geometric parameters of VCM have been
optimized by modified MEC model to enhance the
dynamic response [3]. Halbach Magnetic Circuit design,
KEYWORDS: reliability based design optimization;
particle swarm optimization; sequential optimization and
reliability assessment; Planar-type voice coil motor
NOMENCLATURE
Put nomenclature here.
Put body of the paper here. Put body of the paper
here. Put body of the paper here. Put body of the paper
here. Put body of the paper here. Put body of the paper
*
Corresponding Author: hzhuang@uestc.edu.cn
Tel: +86-28-6181252; Fax: +86-28-61830227.
1
simulation technology, and geometric parameters are three
components in the preparation for the optimal design of
VCM.
The reliability based design optimization for high
accuracy stages may takes novel criteria into integrated
circuit manufacture industry, especially for whose failure
mode is different with the traditional machinery. It is
essential to minimize the height of the planar-type VCM
for lightweight design while the force constant needs to be
guaranteed by reliability constraints. Reliability analysis is
divided into analytical methods [8-14] and sampling-based
methods [15-19]. Analytical methods include first order
reliability analysis (FORM) [9-13], second order reliability
analysis (SORM) [14] and so on. The computational cost
of sampling-based method is a limit factor, such as Kriging
and importance sampling[15], Monte Carlo [18], and so
on.
The PSO was extended for reliability based design
optimization with a move to the boundary of reliability
constraints [20]. Furthermore, the features of auto-tuning
and boundary approaching are added in PSO algorithm to
resolve the RBDO problem with the help of subset
simulation [21]. And the position updating may be
according to its own best experience at early iterations,
whereas according to the best experience of whole swarm
in late iterations [22]. Also, the evolutionary algorithm
with the safety-factor method, which is derived from the
Karush–Kuhn–Tucker optimality conditions, eliminates
the need for reliability analysis, however, only the design
variables can be treated as random variables [23]. The
boundary-shifting is used to increase the convergence rate
and particle-position-resetting is used to enhance the
diversity of particles to keep away from local solution
[24]. Here, the boundary shift means to a move towards a
boundary of a constraint function [24], whereas the
boundary shift in SORA is based on reliability
analysis[25], so the boundary shift in SORA is called
reliability boundary shift in order to distinguish.
This paper is organized as follows: In Section 2,
penalty for particle swarm optimization is introduced to
the constrained optimization in discrete solution; In
Section 3, the proposed method for reliability based design
optimization is Sequential Particle Swarm Optimization
and Reliability Assessment Method; In Section 4, the
Planar-type VCM is designed and optimized with
reliability constraints; In Section 5, the results are
examined by perturbation plots and finite element method;
In Section 6, there is a conclusion for the reliability based
design optimization for Planar-type VCM and future work.
searching behavior of birds flocking [26]. In standard PSO
algorithm, a swarm of particles, on behalf of candidate
solutions, are initialized and move through the search
space towards the compromise between the best position
historically and the best among all particles until
converging to the optimum [27]. PSO possesses two
apparent advantages over the genetic algorithms: 1) less
parameters; 2) fast constringency [28]. The new feature of
auto-tuning boundary-approach is applied to PSO in order
to take a move toward the boundary of probabilistic
constraints with non-smooth performance functions of
RBDO [29]. The method is proved to be also useful in
cam design problems such as minimizing the magnitude of
the negative acceleration and the cycle time respectively
[30].
Firstly, the initialization of a population of particles
with random positions and velocities is limited to the
problem space and the maximum and minimum of the
velocities. Secondly, when the fitness of each particle is
evaluated, the assessment of the fitness is compared both
for the best potion of current than the old history
evaluation and the best position for the entire population
concluding positions of all the time. Then, the position
and the velocity will be updated based on three categories
of parameters: two random parameters for random steps;
two constant numbers for the balance between the best
position of the history and the best position of all
particles; one parameter for the influence form last
position [31].
(1) Updating particles
The position of particles is updated by adding a
change velocity:
(1)
xi t  1  xi t   vi t  1
(2) Updating Velocity
Each particle updated the velocity based on the last
velocity and the best position of history and the whole
particle swarm:
vi  t  1  w  vi  t   c1r1  x pbest  xi  t  
(2)
c2 r2  x gbest  xi  t  
where c1 and c2 are acceleration constants, usually equal to
2; r1 and r2 are random numbers uniformly distributed in
the range of (0,1); w is an inertia weight to control the
influence of the previous velocity, which could be an
constant or updated by w(t  1)  w(t )  eta , while eta is a
constant; x pbest represents the best position in history of
the particle, whereas x gbest is the best position in global
candidate solutions.
2
PENALTY
OPTIMIZATION
FOR
PARTICLE
SWARM
2.2 PSO with Penalty Function for Constrained
Optimization
In the process of fitness valuation, the penalty
function method could be used to consider the constraints
for the particle swarm optimization. A typical penalty
function is presented for example, while other penalty
2.1 Particle Swarm Optimization
Particle swarm optimization, which was an
evolutionary computation algorithm
proposed by
Kennedy, is, in fact, an optimizer, based on the food2
function could also be implanted in the assessment in the
fitness to adapt PSO to the constrained optimization. There
are two types of the penalty functions, the interior and
exterior penalty functions. Because the exterior function is
not required to start in feasible solution, it is more applied
in evolutionary algorithm than the interior penalty function
[32]. The fitness is the sum of the cost function or the
objective function and the penalty item   x 
fit ( x)  cos t ( x)    x 
(3)
  x  =   i
(4)
approach (PMA). Whether in RIA or in PMA, reliability
analysis with First Order Reliability Method (FORM) is
transformed into an optimization problem with equality
constraint as limit state function, which is defined in Uspace [33-35].
In RIA, the MPP is u*G  U   0 , calculated by:
minimize U
subject to g  U   0
In PMA, the MPP is u*  t calculated by:
minimize g  U 
subject to U  t
i
The  is dynamic dependent on iteration such as the
i
(5)
j
Where
qi  max(0, gi ) , i j
indicates
the
penalty
coefficient corresponding to i th constraint and
j th
violation level.  i could be equal to 2.
3
SEQUENTIAL
PARTICLE
SWARM
OPTIMIZATION AND RELIABILITY ASSESSMENT
METHOD
Reliability-based design optimization is to offer a
reliable solution considering uncertainty related to
variables, parameters, and models. Design variables may
be discrete due to product standards, whereas the
distributions of the variables may be continuous or
discrete. For example, Diameter of copper coils could be
chosen by several discrete choices, but it is actually
subjected to a continuous distribution.
3.1 Reliability-Based Design Optimization
The general RBDO method can be expressed
Design Variable DV  d, μX 
Minimize f  d, X, P 
S.t. Pr  gi  d, X, P   0   R i
(8)
PMA depends much less on the nonlinear
transformation than RIA, and is adapted to the variety of
distributions, whereas the RIA may fails to bound and
extreme type distributions [34].
The coupling of optimization and the reliability
analysis is classified into three ways: (1)double-loop
method [36]; (2) decoupled method [25]; (3) single-loop
method [37, 38]. Based on benchmark study of numerical
methods for RBDO, with the lower efficiency and more
robustness and accuracy than the single-loop method,
SORA could be suitable for complex structural systems
[39].

iteration or be static value, while  could be
expressed as:
i  i   qi  1
(7)
3.2 Sequential Particle Swarm Optimization and
Reliability Assessment Method
The proposed method, decouples optimization
process and reliability analysis by SORA strategy,
implements performance measure approach for MPPbased reliability analysis, and takes penalty function
method to transform constrained optimization into
unconstrained optimization problems.
In Figure 1, the flowchart of the proposed method is
illustrated with the main processes. In the first cycle of
SORA, deterministic optimization is in the first place,
whereas, the MPP-based reliability analysis is applied for
all particles to find the MPPs. Then, in order to distinguish
from the boundary shift in the augmented particle swarm
optimization, the boundary shift from SORA may be
regarded as reliability boundary shift. Based on the
position of MPPs, the reliability boundary shift is
generated in order to plug the results of reliability analysis
into reliability constraint [25], which is formulated as
(9)
g  μx  s   0
(6)
μ LX  μ X  μ UX , d  R NDV
X  R NRV
i =1,...,NC
where f is the objective function. μ X is the mean value of
the random design vector X; d is the vector of the
deterministic design variables; P is the vector of the
random design parameters. The probabilistic constraint is
Pr  gi  d, X, P   0  , i=1,...,NC ; R i is the reliability
Where, si  μ X ,i  x MPP,i for i th constraint.
And the penalty functions are used to connect the
constraint and the evaluation of the fitness by:
fit  μ x   f  μ x     gi  μ x  si  
(10)
target for the i th constraint; NDV is the number of the
deterministic design variables; NRV is the number of the
random design variables; NC represents the number of the
probabilistic constraints [25].
In RBDO, the assessment of probabilistic constraints
could be solved in two different methods: the reliability
index approach (RIA), or the performance measure
In particular, the fitness for sequential particle swarm
optimization and reliability assessment method is:
fit  μ x   f  μ x     iteration  
   g μ
i
i
3
i
x


 si    max  0, gi  μ x  si    1
i
 (11)
The shift vector contains the reliability information to
switch the boundary in the constrained optimization and
participating in determine the penalty levels for the
distance from the boundary. In this way, it is obvious that
the constrained optimization is turn out to be
unconstrained optimization with a plug in fitness
evaluation, and that the uncertainty is considered by shift
vector, a transfer from reliability analysis to the position
and velocity updating by fitness evaluation.
The evaluation method not only could be used in
particle swarm optimization, but also could be used in
other evolutionary algorithm. The advantage is to fit the
decoupled strategy into the evolutionary algorithm for
reliability design optimization by adding the shift vector in
the penalty function. In this way, other evolutionary
algorithm, such as genetic algorithm and Simulated
Annealing, with their own advantage, could be taken into
SORA strategy.
In the proposed method the advantage of PSO and
SORA are included, since which it owns the ability to
solve engineering problems with the effectiveness from
decoupled strategy in discrete solution space for global
search.
planar voice coil motor. There are three main characters in
this planar voice coil motor: 1) double side magnet circuit;
2) Halbach magnet circuit; 3) moving magnet type 4)
planar voice coil motor. The double side magnet circuits
reduce the force disturbances in z-axis, because the
magnetic flux is in a closed loop. Based on Flemming’s
rule, on account of decreased variation of the direction
between the flux density and current in the moving magnet
type, the fluctuation of force constant is less than that in
the moving coil type. The conventional magnetic circuit
merely consists of vertically-magnetized magnets,
whereas, the Halbach magnetic circuit is made up of a
horizontally-magnetized magnet and two verticallymagnetized magnets. Halbach magnet circuit design could
offer augmentation of magnetic flux, decrease the
thickness of the yoke, and then increase the flux density in
the air gap. The Figure 2 describes the design prototype
for the planar voice coil motor. The arrows on magnet in
Figure 2 indicate magnetization directions of each magnet.
The arrows on coils indicate the current direction.
YOKE
.
MAGNET
.
COIL
.
Initialization
CURRENT
.
Generate
Particles
.
Updating
Reliability
Boundary Shift
vectors
Updating
Velocity
Fitness Value
With Penalty
Function
4.2 Response Surface Method
Response surface approximations of the limit state
functions are used to reduce the calculation expense [40].
The thickness of coil, magnet and yoke are design
variables, and the force constant is the response. X1, X2,
X3 represent the thickness of coil, magnet and yoke
respectively, whereas the k f represents the force constant.
The procedure is simple and the choose of the model is
based on P-value, as expressed in Table 1.
If the P-values of "Prob > F" is less than 0.005, it
indicate that the model terms are significant and could be
kept. However, the P-value of X2^2 is 0.5385, so it is
canceled in the model 2, which improve the F-value in the
same time. The P-value of the model is less than 0.0001,
which means it is significant. As the F-Value of model 2 is
more than that of model 1, the model 2 could be used as
the response surface of the force constant.
The normal probability plot is implied to detect
whether the residuals is subjected to the normal
distribution by a graphical technique, with the assumption
that every variable is independent with each other from the
Figure 3, it is obviously that the points make up an
approximate straight line, which indicates the date set of
the residuals follow a normal distribution.
updating pbest
and gbest
Not
Satisfied
YOKE
Figure 2 Designed Model of Planar-Type Voice Coil
Motor
MPP-based
Reliability
Analysis(MPP)
Updating
Particles
MAGNET
Stopping
Criteria?
satisfied
Output
Figure 1 Flowchart of The Proposed Method
4 DESIGN AND OPTIMIZATION FOR PLANARTYPE VOICE COIL MOTOR
4.1 Halbach Magnetic Circuit for Planar-type VCM
The Planar-type VCM is a moving magnet type with
double side Halbach magnet circuit, also may be called
4
It is useful to test the actual response values versus
the predicted response values by the intuitive graph, in
order to easily provide whether a group of values which
are not predicted perfect with the model. The Figure 4
shows the points are split by a line of 45 degree, which
presents the model is significant.
gap is, the larger the force constant is. And on behalf of
the request of cooling system in practice, the minimum
value of the Gap is 2mm. In this project, K tar
f is constant
of 9, and F tar is constant of 30, based on the need of
force and the thermal control. To simplify the problem and
from the experience in manufactory, it is reasonable to set
the variables with a standard variance of 0.1, and be
subjected to normal distribution. The discreteness of the
thickness of the coil is provided by the supplier, but X1
still is a normal distributed factor in fact. In section 3, the
model of the force constant decided and calculated. The
PMA method is used for MPP search with t =2 . From
section 4.2, the model 2 for force constant could be
expressed as:
K f =2.16912  0.43450* X1  0.44631*X2 
3.60926* X3  0.028311* X1* X2  0.083270* X1* X3
0.014550* X2* X3  0.039223* X12  0.54656* X32
Table 1 Response Surface of Force Constant
Kf =
Model1
P-value
Model2
P-value
C
* X1
* X2
* X3
* X1 *
X2
* X1 *
X3
* X2 *
X3
* X1^2
* X2^2
* X3^2
F-Value
1.486511
-0.4345
0.632471
3.609263
< 0.0001
< 0.0001
< 0.0001
2.169116
-0.4345
0.446306
3.609263
< 0.0001
< 0.0001
< 0.0001
-0.02831
0.0077
-0.02831
0.0077
-0.08327
< 0.0001
-0.08327
< 0.0001
0.01455
0.039223
-0.01241
-0.54656
576.2221
0.3676
< 0.0001
0.5385
< 0.0001
0.01455
0.039223
None
-0.54656
649.65
0.3670
< 0.0001
None
< 0.0001
(13)
The result of the proposed method for the problem is
in Table 2.
Table 2 The Result of The Proposed Method
Method
μ X1
μX2
μ X3
12.00
11.3632
11.00
10.00
objection
R1
R2
Predicted Response
5.3069
9.00
8.00
PSO_DO
3.40000
7.27345
2.76037
27.46764
0.479134
0.789569
PSO_SORA
3.40000
8.00979
2.74596
28.91150
0.99902
0.97953
7.00
R1 and R2 represent the reliability for the constatint1
and constraint 2 respectively. The probability of each
constraint shows the differences between the RBDO and
the deterministic optimization by Monte Carlo Simulation,
which is a classic evaluation method for reliability
analysis with accurate probability for sufficient numbers
of simulation. The proposed method could guarantee the
reliability constraint with relative small change of the
objective evaluation, which is acceptable for the practical
solution. Different from the traditional reliability design
by the experience of safety factors, the reliability based
design optimization could definitively find the quantitative
solution except for uncertainties of mathematic models.
The PSO not only deal with the discrete solution space,
but could offer global search with an adjustable speed
between the global position and the history position
towards the point of convergence.
Since the solution is based on response surface
method, which may result in uncontrollable errors
occasionally, it is necessary to perturbation analysis for
the force constant and the finite element analysis for the
design regarded as actual response in practice.
6.00
5.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
12.00
Actual Response
Figure 4 Predicted Verse The Actual Response
4.3 Reliability Based Design Optimization of
Planar-Type VCM
Reliability based Design Optimization of Planar-type
VCM is defined as:
Design Variable DV  μ X1 , μ X 2 , μ X 3 
Minimize μ X1 +2*μ X 2 +2*μ X 3 +2*Gap
S.t. Pr g1  0  g2  0   R i
g1  K f  K tar
f ;
(12)
g 2  μ X1 * K f  F tar
1  μ X1  6, and μ X1 is discrete with interval of 0.1
6  μ X 2  9,
1  μ X 3  4,
There are three design variables μ X1 , μ X 2 , μ X3 ,
represent the mean value of thickness of coil, magnet and
yoke respectively. The Gap means the thickness of the gap
in the design, which is important for the force constant.
However, it is a deterministic relevance that the less of the
5
5 PERTURBATION
ELEMENT ANALYSIS
ANALYSIS
AND
FINITE
Force Constant
5.1 Perturbation Analysis for the Force Constant
The perturbation plot for the force constant is applied
to illustrate the effect of the three factors at the design
point in solution space to find the sensitive factor to the
force constant based on response surface. The response of
factors in Figure 5 is plotted by varying just single factor
over all the range while keep the other factors as the
constant at the design point in Design Expert. In this way,
the steeper the slope of the line is, the more sensitive the
factor is to the force constant such as X1, whereas the
flatter the slope of the line is, the less sensitive the factor is
just as the X3. The standard error of the response in Figure
6 is also plotted to show the uncertainty caused by
response error, which is relative low in design point.
The influential factors X2 and X3 in the perturbation
plot are compared with each other in the contour plot
Figure 7 and Figure 8 respectively, with a constant factor
X3, which is the same in the deterministic optimization
and RBDO. The plots of the perturbation analysis of the
force constant show the reliability design, which keep the
distance from the deterministic constraints.
Figure 7 Response Surface of The Model with One
Constant Factor
Force Constant
Standard Error of Force COnstant
5.2 Finite Element Analysis
Finite element analysis is applied to examine whether
the results are reasonable in practice. In reality, the design
is calculated by Ansoft Maxwell software. The mean
values of the random design variables are put into the final
design in finite element analysis. The plots of magnetic
flux density in coil and in yoke are plotted in the Figure 9
and Figure 10 respectively.
Deviation from Reference Point
Figure 8 Standard Error of Response with One Constant
Factor
Standard Error of Force Constant
Figure 5 The Perturbation of The Model
Figure 9 Magnetic Flux Density In Coil
Deviation from Reference Point
Figure 6 Perturbation of The Standard Error of Response
6
proposed method is well applied in the process to produce
the optimal design subjected to reliability constraints.
6 CONCLUSIONS
This paper design and optimize a planar-type voice
coil motor considering uncertainty. The reliability
boundary shift, added into the penalty function in fitness
evaluation, is the key process to take the reliability
constraints into the penalty for particle swarm
optimization by SORA strategy. The proposed method
improves the reliability constrained optimization in
discrete space with the speed from SORA strategy and the
global solution form particle swarm optimization. The
optimized Planar-type voice coil motor is useful in the fine
stage and other applications where a large acceleration
speed and high accuracy are needed. In the future work,
the reliability based topology for the motor will be applied
for lightweight while the force constant is guaranteed.
Figure 10 Magnetic Flux Density In Yoke
The Halbach Magnetic Circuit design restricts more
energy in the middle of the Coil instead of out in the space,
which is evident in Figure 9 and Figure 10. There are
tremendous variances of magnetic flux density in small
distance, whether in the yoke or in the coil, expressed as
the rapid changes of colors, which present the
concentration of magnetic flux density distribution. Figure
9 illustrates the distribution of magnetic flux density in
coil, in which there is a potential optimization in future to
put more current in central place to generate more power
and reduce the weight. Figure 10 demonstrates the
function of the yoke to keep more energy in the motor, in
which the yoke could be reshaped by topology technique
in the future design to put more thickness in the central
place.
In Table 3, the simulation result of finite element
analysis is presented. The force constant satisfies the initial
deterministic design with a margin for reliability
constraints. The results of RBDO is not much too
conservative to afford, instead, the cost is worthy to nearly
improve fifty percentages and thirty percentages in
reliability constraints in Table 2, respectively. It not only
proves the availability of the proposed method, but also
the useful tool of the response surface method, which is
simple but robust in many practical engineering problems.
Even though other directions of the force are not
considered in the initial work, which need to be improved
in the future work to reduce the forces in the other
directions to enhance the performance of motion with high
precision, they are relative low and keep little disturbance
to the moving direction, enough to be endured or even
eliminated by the control system. Also, there may be some
errors from the simulation, which may be different with the
actual measurements.
ACKNOWLEDGMENTS
Put acknowledgments here.
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[7] H. Z. Huang, J. Qu, and M. J. Zuo, “Geneticalgorithm-based optimal apportionment of reliability
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[8] R. Braun, Collaborative Optimization: An
Architecture for Large-scale Distributed Design, Ph.D.
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[9] UKHSE, United Kingdom Health and Safety
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Table 3 Results of Finite Element Analysis
Force_axis
F_x
F_y
F_z
F_mag
Force value
0.082503
9.1172
0.11423
9.1183
The Perturbation Analysis and Finite Element
Analysis testify that the optimized design is reasonable and
simulation result is acceptable for the reliability based
design optimization solution of VCM. In this way, the
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