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8/20/2019 Performance Analysis and Optimization of a 5-DOFs Mechanism
1/10
Performance analysis and optimization of a ve-degrees-of-freedomcompliant hybrid parallel micromanipulator
Dan Zhangn, Zhen Gao
Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa, Ontario, Canada
a r t i c l e i n f o
Article history:
Received 7 August 2014
Received in revised form16 January 2015
Accepted 17 January 2015Available online 7 February 2015
Keywords:
Compliant mechanisms
Optimization
Hybrid mechanisms
Performance index
a b s t r a c t
There are generally two main directions for the investigation and development of parallel manipulators,
namely macro/meso stream and micro/nano stream, in which the former one has been thoroughly in-
vestigated in recent decades, while the latter one still remains many performance related open issuesthat signicantly affect their application potentials in critical situations such as high-precision automated
cell manipulation. Improving the overall performance of parallel manipulators is the bridge to connect
the academia and industry for the great development and real-world application. This research is to
develop a novel methodology called performance decomposition and integration for governing the de-
sign optimization process of complicated micromanipulator. A new ve degrees-of-freedom (DOF)
compliant hybrid parallel micromanipulator which is congured with ve identical PSS limbs and one
constraining UPU limb is proposed as a case study. The performance visualization, nite element ana-
lysis, and dimensional optimization are implemented. The proposed methodology is applicable for the
design improvement of different kinds of compliant/parallel mechanisms.
& 2015 Elsevier Ltd. All rights reserved.
1. Introduction
For the past several decades, parallel mechanisms/manip-
ulators can be found for extensive applications including three-
dimensional printers, machine tools, and vehicle simulators,
picking and placing tools, sensors and robots [1–8]. Although a
major portion of these applications are not fully commercialized
and needs further improvement, it has been commonly recognized
with the continuing effort of several decades, parallel manipulator
has become one of the main branches of the family of mechanisms
and robotic systems due to their natural merits many aspects
[9–14].
Regarding performance parallel mechanisms, the global re-
searchers have conducted huge work on from design, analysis to
control [15–18]. However, due to the limitation of capabilities,
conventional manipulators cannot well adapt to the rapid change
of critical applications where reliability, robustness and resilience
are highly demanded. More ef cient methodologies, especially in
micro/nano applications for parallel manipulators, are highly re-
quired to guide the development of compliant manipulators. In
this scenario, a paradigm called as performance decomposition
and integration (PDI) is proposed. Performance decomposition is
necessary to explore the macroscopic characteristics of a
complicated system in a microscopic method. A system may have
many performance indices. Integration is a universal notion thatranges from component level to system level. For a robotic system,
the methodology of performance integration covers the measures
from integrated design and optimization. To explore the overall
performance of a complex system, it will be rstly divided into
several sub-criterions based on PDI. These sub-criterions are in-
vestigated and managed separately. Then, a united index can be
built to examine the comprehensive performance and conse-
quently improve it with performance integration.
As a case study, a 5-DOFs compliant hybrid parallel micro-
manipulator (CHPMM) is proposed. It is congured with ve
identical PSS limbs and one constraining UPU limb. A multi-layer
amplication mechanism based prismatic joint is designed for
each limb and the piezoelectric actuator will be placed at the
center position of the active prismatic joint. In each limb, two
exures based spherical joints are connected either with the
prismatic joint or with the moving stage, respectively. An em-
beddable passive UPU limb is applied to constrain the mobility of
the proposed manipulator into 5DOF. For the content of this paper,
the analysis of kinematic model and Jacobian matrix is conducted.
Three essential performance indices, i.e. dexterity, manipulability
and workspace, are derived and visualized. The nite element
analysis is performed to observe the mechanism behavior. Finally,
dimensional improvement is implemented based on hybrid opti-
mization algorithm.
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/rcim
Robotics and Computer-Integrated Manufacturing
http://dx.doi.org/10.1016/j.rcim.2015.01.002
0736-5845/& 2015 Elsevier Ltd. All rights reserved.
n Corresponding author.
E-mail address: [email protected] (D. Zhang).
Robotics and Computer-Integrated Manufacturing 34 (2015) 20–29
http://www.sciencedirect.com/science/journal/07365845http://www.elsevier.com/locate/rcimhttp://dx.doi.org/10.1016/j.rcim.2015.01.002mailto:[email protected]://dx.doi.org/10.1016/j.rcim.2015.01.002http://dx.doi.org/10.1016/j.rcim.2015.01.002http://dx.doi.org/10.1016/j.rcim.2015.01.002http://dx.doi.org/10.1016/j.rcim.2015.01.002mailto:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.rcim.2015.01.002&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.rcim.2015.01.002&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.rcim.2015.01.002&domain=pdfhttp://dx.doi.org/10.1016/j.rcim.2015.01.002http://dx.doi.org/10.1016/j.rcim.2015.01.002http://dx.doi.org/10.1016/j.rcim.2015.01.002http://www.elsevier.com/locate/rcimhttp://www.sciencedirect.com/science/journal/07365845
8/20/2019 Performance Analysis and Optimization of a 5-DOFs Mechanism
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2. Conceptual design and kinematic modeling
The computer-aided design (CAD) model of the proposed
CHPMM is shown as Fig. 1. It can be observed that this mechanism
has ve identical limbs which are featured as prismatic–spherical–
spherical structure, within which the prismatic joint is a multi-
layer compliant structure. There is a complicated passive con-
straining limb inside of this mechanism, which is connected with
the base and the moving platform with one universal joint re-
spectively. Between the two universal joints of the embedded
limb, a novel passive prismatic joint is proposed. Five electric-
piezos are mounted at the edge of the base platform to actuate the
external identical limbs. The inside embedded limb constrain the
Fig. 1. The proposed 5-DOF CHPMM: (a) the CAD model and (b) the kinematical
structure.
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
10
2
4
6
8
x 10-4
x (mm)y (mm)
D e x t e r i t y
Fig. 2. The landscapes of dexterity.
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
10
0.2
0.4
0.6
0.8
1
x (mm)y (mm)
M a n i a b i l i t y
Fig. 3. The landscapes of manipulability.
-0.4 -0.3 -0.2 -0.1 0 0.1
0.2 0.3-0.2
0
0.2
0.95
1
1.05
1.1
1.15
1.2
x (mm)y (mm)
z ( m m )
Fig. 4. The landscapes of workspace.
D. Zhang, Z. Gao / Robotics and Computer-Integrated Manufacturing 34 (2015) 20 – 29 21
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mobility of the end-effector into ve degrees-of-freedom. The
whole mechanism is compact and has the capability to fulll ad-
vanced manipulation in micro-range eld such as cell probing and
cell injection. Since for the commonly used automated cell ma-
nipulation, the electric-piezos are congured in three perpendi-
cular directions to control the motion in each axis separately.
However, it is dif cult to perform complicated manipulation in
higher DOFs, i.e. 5DOFs or 6DOFs. Thus, the proposed structure
provides a solution to achieve this.
The kinematical structure of the proposed 5-DOF CHPMM has
two platforms, namely the xed platform B1B2B3B4B5, and the
moving platform P 1P 2P 3P 4P 5, as shown in Fig. 1(b). There are two
coordinate frames, the xed reference frame which is expressed as
O x y z { , , } is attached to the center of the base platform. Another
reference frame is attached to the center of the mobile platform.
A constraining limb with prismatic–universal–universal joints
and each universal joint is attached to the centers of the xed
platform and the moving platform, respectively. The inverse ki-
nematics of the proposed mechanism means to derive the solution
of the actuated joints when the pose of the end-effector is known.
By observing Fig. 1(b), it has,
p x y z i[ , , ] 1, , 5 (1)i i i i T
= = …
r l l i[ cos , sin , 0] 1, , 5 (2)i e ei e ei T θ θ = = …
p x y z [ , , ] (3)T =
b l l i[ cos , sin , 0] 1, , 5 (4)i b bi b bi T θ θ = = …
where p i is the position vector of point P i expressed in the xed
coordinate frame, r i is the position vector of point P i expressed in
the moving coordinate frame, and p is the position vector of point
P expressed in the xed frame, and
⎡
⎣⎢⎤
⎦⎥
[ , , , , ]
, 2 /5 , 4
5 ,
6
5 ,
8
5 (5)
bi b b b b b T
T
1 2 3 4 5 θ θ θ θ θ θ
β π β π
β π
β π
β
=
= + + + +
⎡⎣⎢
⎤⎦⎥
[ , , , , ]
, 2 /5 , 45
, 65
, 85 (6)
pi p p p p p T
T
1 2 3 4 5 θ θ θ θ θ θ
α π α π α π α π α
=
= + + + +
One can then write
p p Qr i, 1, ... , 5, (7)i i= + =
where Q is the rotation matrix, and α , β are the angles on the
base and on the platform respectively. In this case, since there is
no rotation for the moving platform, Q is an identity matrix. Thus
the inverse kinematics can be easily derived through addressing
the closed loop,
′′ ′′ OB O B P P OO i( 1, ... , 5) (8)i i i i ρ = = − − =
Differentiating Eq. (7), one obtains
Fig. 5. The meshing of the proposed CHPMM.
Fig. 6. The piezo force is applied on one limb with 1 N: (a) total deformation and
(b) elastic strain.
D. Zhang, Z. Gao / Robotics and Computer-Integrated Manufacturing 34 (2015) 20 – 2922
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p p Qr (9)i i= +
Hence, one can write the velocity equation as
At B (10) ρ=
Thus the Jacobian matrix is expressed as
B A J . (11)1= −
3. Performance decomposition
CHPMM, as a complex system, has the characteristics of non-
linearity, multi-parameters and strong coupling. To explore it
overall performance, one of the initial measures is to decompose it
into several critical sub-criterions and visualize and analysis thesesub-criterions separately. As a case study, the indices of dexterity,
manipulability and workspace are landscaped for the further
integration.
The motion isotropy of the CHPMM is highly related with
dexterity whose value can be derived based on the methods of
condition number, determinant, minimum singular value and joint
range availability. The health condition and robustness of Jacobian
can be investigated through condition number. Thus in this work
the condition number of Jacobian matrix is chosen to express the
dexterity. In this case, the range of dexterity is from zero to one. If
zero, it implies an ill-conditioned matrix and singularity. If one, it
implies the CHPMM has isotropic motion characteristics. The
dexterity is de
ned as follows:
Fig. 7. The piezo forces are applied on two limbs with 1 N: (a) total deformation
and (b) elastic strain.
Fig. 8. The piezo forces are applied on three limbs with 1 N: (a) total deformation
and (b) elastic strain.
D. Zhang, Z. Gao / Robotics and Computer-Integrated Manufacturing 34 (2015) 20 – 29 23
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Dexterity Sing J Sing J ( )/ ( ) (12)min max=
where J Sing ( )min and J Sing ( )max are the minimal and maximal
singular values of J , respectively. Fig. 2 shows the distribution of
the dexterity.
The manipulability of the proposed CHPMM is derived as
LMI J J det( ) (13)T = ⋅
The landscape of the manipulability is illustrated in Fig. 3.
Workspace can be roughly divided into joint workspace and
end-effector workspace. If concerning to the input variables, the
motions of all the actuated joints form a joint space whose di-
mension relies on the number of the actuators [19,20]. In most
cases, the end-effector workspace is investigated, since its shape
and volume which are affected by structure parameters,
constraints and actuation strokes are highly related with the per-
formance of the specic application. A simple solution to generate
the workspace is to take advantage of the inverse kinematics and
the constraints of the mechanism. First of all, the end-effector is
located at the home position. Then it will be moved in the space in
a given step length. The inverse kinematic model is conducted is
check of the input variable exceed the motion stroke or achieve
the constraint. If not, continue to move the end-effector to a fur-
ther post and apply the inverse kinematics again. If yes, stop
moving the end-effector in the previous direction. Fig. 4 illustrates
the shape and envelope of the achieved workspace.
Fig. 9. The piezo forces are applied on four limbs with 1 N: (a) total deformation
and (b) elastic strain.
Fig. 10. The piezo forces are applied on ve limbs with 1 N: (a) total deformation
and (b) elastic strain.
Table 1
Modal.
Mode Frequency [Hz]
1 81.38
2 81.695
3 132.43
4 198.515 198.67
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4. Finite element analysis
The overall features of a CHPMM are greatly affected by the
selected material. In this scenario, stainless steel is chosen after a
critical evaluation of vital design criteria as the mechanical body of
the proposed mechanism. The meshing result of the proposed
CHPMM is provided in Fig. 5. The most sensitive parts in each limb
including the cantilever of the prismatic joint and the specic
joints should be rened using the features of face sizing. In FEA,
the pentagon side length is 150 mm, the peripheral radius of the
mobile stage is 37 mm, the height of the stage is 35 mm, and the
length of each arm is 78 mm.
Fig. 11. The total deformation under different natural frequency: (a) Frequency 1; (b) Frequency 2; (c) Frequency 3; (d) Frequency 4; and (e) Frequency 5.
D. Zhang, Z. Gao / Robotics and Computer-Integrated Manufacturing 34 (2015) 20 – 29 25
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5. Strain and deformation
The strain and deformation results reect the performance of
compliance, sensitivity, linearity and verify the motion of the
proposed CHPMM. A selected piezoelectric linear actuator gen-
erates a representative force of 1 N on a single limb of the me-
chanism in Fig. 6. It can be observed that the cantilever nearest to
the spherical joint is subject to the maximal elastic strain with
1.0819105 mm/mm. Besides, the edge of the moving platform
which is closest to the limb with external force produces the
maximal deformation with 2.9967 μm. The force applied at one
limb only has slight effect on other active joints.
When two piezoelectric actuators are acted on two limbs with
1N force respectively, similarly with the preceding situation, the
total deformation and elastic strain is illustrated in Fig. 7. It can befound that cantilever nearest to the ball joint has the maximal
elastic strain is 1.5044105 mm/mm. The edge of the moving
platform which is near to the apply force is experienced the most
deformation with 4.573 μm. It also can be found that the limb on
the opposite side is subject to larger deformation and elastic strain
than the other two limbs.
If there are three piezoelectric linear actuators generates a re-
presentative force of 1N on three limbs in a row of the mechanism
the proposed CHPMM, in Fig. 8, it can be observed that the can-
tilever nearest to the spherical joint of the middle limb of the three
limbs is subject to the maximal elastic strain with 1.7595105
mm/mm. Furthermore, the edge of the moving platform which is
closest to this middle limb produces the maximal deformation
with 5.2497 μm.
In the fourth scenario, when four piezoelectric actuators are
acted on four limbs with 1 N force respectively, it can be found
that maximal elastic strain is 1.7333105 mm/mm and the
maximal deformation is 4.5035 μm. The ball joints are bent to the
side of the limb where no actuator is applied at (Fig. 9).
Fig. 10 shows the results when all the ve external limbs are
actuated with 1N respectively. It seems that the deformation and
the elastic strain are symmetrical on these active limbs. For the
middle passive limb, the deformation and the elastic strain on
each cantilever is also symmetrically distributed. Each cantilever
nearest to the ball joint has the maximal elastic strain with1.4051105 mm/mm. The moving platform is experienced the
maximal deformation with 3.2607 μm.
6. Modal analysis
The natural frequencies with ve possible modes are given in
Table 1. Fig. 11 characterizes the total deformation under different
cases of natural frequencies.
Fig. 12 describes the frequency response of total deformation
and strain when external force is acted on top of the mobile
platform in the opposite direction of z axis with 1 N.
Fig. 13 reects the results when the external force is applied on
the moving platform only in x axis with 1 N.
Fig.12. Frequency response with applied force on the moving platform – case one:
(a) deformation and (b) strain.
Fig.13. Frequency response with applied force on the moving platform – case two:
(a) deformation and (b) strain.
D. Zhang, Z. Gao / Robotics and Computer-Integrated Manufacturing 34 (2015) 20 – 2926
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The last scenario is the results of frequency response for strain
and total deformation if the external force is implemented on the
mobile platform with 1 N in x-, y- and z - axes, respectively
(Fig. 14).
7. Performance integration
The performance of a CHPMM depends on the overall hy-
bridization, which can be called as system hybridization (SH). SH
stands at the higher viewpoint on a system level to investigate themacro/micro conditions thorough the methods such as mechan-
ism hybridization (MH), actuation hybridization (AH) and opti-
mization hybridization (OH). MH takes the merits of serial/parallel
mechanism to develop new structures with the integrated ad-
vantages of parallel mechanism, compliant mechanism, serial
mechanism and even others. For instance, in a non-structure cir-
cumstance, the integrated tensegrity structure and cable driven
parallel mechanism perform a exible autonomous motion. For
adaptive and robotic manufacturing, the integrated of rigid serial
mechanism and rigid parallel mechanism can implement some
complicated machining as a multi-axis CNC robotic machine tool.
For AH, since different actuation system has various features, the
hybrid actuation mode provide a possibility to improve the overall
performance of a hybrid mechanism. For OH, it is focused onbuilding a comprehensive performance index that can consider a
variety of performance indices in an integrated way. The idea of
OH considers the optimization management involved with multi-
objective hybridization, preformation integration and multi-algo-
rithm hybridization. In this scenario, performance integration is
highly related with OH.
Genetic algorithms as a scholastic method that explores all
regions of state space through the iterative usage of operators
including selection, crossover and mutation to chromosomes in
the population to avoid sticking into local niche [21–27]. In this
scenario, a multi-population genetic algorithm is applied for the
performance integration based optimal design of the proposed
CHPMM. Besides, the objective function ObjFun is dened as the
multiplication of each sub-function, i.e. dexterity, manipulabilityand volume of workspace.
ObjFun ObjFun ObjFun ObjFun1 2 3 (14)= × ×
The boundaries of the decision variables are given as follows:
l
l
l
h
40 mm 80 mm
10 mm 40 mm
50 mm 90 mm
15 mm 36 mm
0 0.436 rad
0.349 rad 0.436 rad (15)
b
e
i
α
β
≤ ≤
≤ ≤
≤ ≤
≤ ≤
≤ ≤
≤ ≤
At the beginning of the iteration, the total population pool is
split into four sub-populations with the numbers of chromosomes
Fig. 14. Frequency response with applied force on the moving platform – case
three: (a) deformation and (b) strain.
Table 2
Iteration process.
Iteration # f -Count ObjFun(104) Time of CPU
1 400 3.4824 00:00:00
3 1116 3.7963 00:00:00
6 2190 4.1758 00:00:01
9 3264 4.2747 00:00:01
12 4338 4.304 00:00:06
15 5412 4.3183 00:00:07
18 6488 4.3183 00:00:07
21 7564 4.3198 00:00:09
24 8638 4.3203 00:00:10
27 9712 4.3203 00:00:10
30 10,786 4.3203 00:00:12
33 11,860 4.3203 00:00:12
36 12,934 4.3204 00:00:13
39 14,008 4.3204 00:00:15
40 14,366 4.3204 00:00:15
Table 3
Size of subpopulations.
Iterat io n # Subpopulations #
1 85 15 95 105
3 85 115 95 105
6 111 159 67 63
9 120 174 41 65
12 126 183 25 66
15 126 183 25 6618 129 190 15 66
21 131 194 9 66
24 194 134 6 66
27 194 134 6 66
30 195 134 5 66
33 195 134 5 66
36 117 134 5 144
39 117 134 5 144
40 71 134 5 190
D. Zhang, Z. Gao / Robotics and Computer-Integrated Manufacturing 34 (2015) 20 – 29 27
8/20/2019 Performance Analysis and Optimization of a 5-DOFs Mechanism
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with 85, 115, 95, and 105 respectively. The maximal iterations are
40. Other evolutionary parameters are set as follows:
selection function¼selsus
selection pressure¼1.7
selection gen. gap¼0.9
reinsertion rate¼1
recombination rate¼1
mutation rate¼1mutation range¼0.1, 0.03, 0.01, 0.003
mutation precision¼12
migration rate¼0.1
migration interval¼20
competition rate¼0.4
competition interval¼4
competition subpop minimum¼5
competition div. pressure¼2
The detailed Iteration process and evoluaitonary size of each
subpopulation is given in Tables 2 and 3.
After optimization, the optimal value of objective function is
convergent at 4.3204104 with the optimized dimensional
parameters equal to 54.99 mm, 10 mm, 50 mm, 15 mm,0.0027505 rad, and 0.41112 rad, respectively. The evaluation pro-
cess of the optimal objective function and dimensional parameters
for the proposed CHPMM are obtained in Fig. 15.
8. Conclusions
Performance is a critical topic for the further improvement of
compliant parallel mechanisms. The research attempts to propose
a paradigm called performance decomposition and integration to
manage the overall performance of these mechanisms in a higher
level. A CHPMM is proposed as an example to showcase part of the
principle of PDI in the process of design and optimization. The
proposed method can be well integrated with the methodologyof system hybridization through the specic methods of me-
chanism hybridization, actuation hybridization and optimization
hybridization. For the future work, a physical prototype will
be manufactured to further improve the overall performance in
aspects of manipulation and control under the guidance of the
proposed methods.
Acknowledgments
The authors would like to thank the nancial support from the
Natural Sciences and Engineering Research Council of Canada
(NSERC). The authors gratefully acknowledge the nancial support
from Canada Research Chairs program.
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Fig. 15. The result of multi-objective optimization: (a) objective function; (b) the
dimensional variables; and (c) size of subpopulations.
D. Zhang, Z. Gao / Robotics and Computer-Integrated Manufacturing 34 (2015) 20 – 2928
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