Research ArticleRobust Design Optimization of a 4-UPS-S ParallelManipulator for Orientation-Regulating Control System ofSolar Gather Panels

Guohua Cui1 Haidong Zhou1 Haiqiang Zhang1 Dan Zhang2 and Yanwei Zhang1

1College of Equipment Manufacturing Hebei University of Engineering Handan 056038 China2Department of Automotive Mechanical and Manufacturing Engineering University of Ontario Institute of TechnologyOshawa ON Canada L1H 7K4

Correspondence should be addressed to Guohua Cui ghcuihebeueducn

Received 20 October 2014 Revised 6 January 2015 Accepted 29 January 2015

Academic Editor C K Kwong

Copyright copy 2015 Guohua Cui et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a redundantly actuated parallel manipulator 4-UPS-S that is applicable for orientation adjustment in thegathering process of solar power A thorough analysis involving the kinematic issues is performed Inverse kinematic problemsare solved in the close-loop The Jacobian matrix and some performance indexes are analytically derived The multiobjectiveoptimization model is established and the determinacy optimization is completed on the basis of previous research works Six-Sigma robust analysis is performed on the basis of the determinacy optimal solution Results show that 4-UPS-S does not satisfythe quality requirement Therefore it is necessary to implement Six-Sigma robust optimization and select optimial solution ofrobustness to complete the nondeterminacy optimization The research results show that the proposed methodology has a simpleoperation and high optimization efficiency The methodology commodiously obtains robustness parallel manipulator that satisfiesthe quality requirement

1 Introduction

Determinacy analysis and design is a traditional optimizationdesign of system inputs mechanical structures materialproperties manufacturing and installation with no error orwith a constant error value Most of the above parametersin practical engineering application problems are uncertainproblems with some existing errors [1ndash3] In accordancewith certain distributions the actual values fluctuate upand down beside the theoretical values and are not arti-ficially controlled The robust analysis and design are theprocess of studying parameter errors as random variablesResearch on the robust design optimization of the parallelmanipulators is uncommon [4 5] Gao [6] used the robust-ness optimization design of the 3-RPS parallel manipulatorby adopting the Box-Behken experimental design and theANSYS Workbench that generated the initial sample pointsHe further used theKriging interpolation and neural networkmethods to regenerate new sample points Yu [7] analyzedand solved the parallel robot manipulator deviation used in

the sheetmetal assemblyHe further derived a new robustnessdesign index on the basis of the sheet metal assemblydeviation model Meng et al [8] treated the design variablesas random variables and built the robustness optimizationdesign mathematical model with a four-bar manipulatorKato and Muramatsu [9] proposed a method that is theMonte Carlo simulation and particle swarm optimizationmethod to evaluate robustness of adjustable mechanismsAbdellatif et al [10] presented a self-contained approach forthe robust dynamics identification of parallel manipulatorsin terms of uncertain parameters and illustrated the controlaccuracy by numerous experimental investigations Rahmanet al [11] presented the robust design of suspension arm usingstochastic design improvement technique based on MonteCarlo simulationMany scholars employed differentmethodsfor robustness design optimization (eg blind number theory[12] and Monte Carlo simulation method [13 14])

The paper is organized as follows In Section 2 thestructure of the 4-UPS-S parallel manipulator that can be

Hindawi Publishing CorporationJournal of Industrial EngineeringVolume 2015 Article ID 349716 10 pageshttpdxdoiorg1011552015349716

2 Journal of Industrial Engineering

Multi-objective

Robust analysis(6 sigma quality level

analysis)

Robust optimization(6 sigma robust design

optimization)

Determinacyoptimalsolution

Introduce uncertainty elements to compute reliability and quality level of the optimal

solution

Robust design optimization

Determinacy optimization

Establishment and solution of kinematics

Multi-objective optimization model

Achieve more reliable and more robust design

scheme

Start

End

Establishment of multi-objective

optimization model

Nondeterminacyoptimization design

(6 sigma robust design optimization)

optimization

Figure 1 Flow diagram of the Six Sigma robust design and optimization based on Isight

applied in the solar gather panel orientation control sys-tem is described and the reference systems of the paral-lel manipulator are established In Section 3 the inversekinematics of the parallel legs and forward kinematics ofserial passive legs of the manipulator are studied Section 4explains the performance indexes including the maximumpayload capacity maximum and minimum stiffness andkinematic dexterity to evaluate mechanical performancesIn Section 5 a determinacy optimization design is per-formed based on software Isight In Section 6 Six-Sigmaoptimization technique for robust design optimization isanalyzed In Section 7 Six Sigma robust optimization isconducted with consideration uncertain parameters andachieves reliability and quality level Section 8 analyzes andexplains the results of the robust design optimization FinallySection 9 summarizes and arranges contents of this studyTherobustness analysis optimization flow diagram is described inFigure 1

2 Parallel Manipulator Description

The traditional orientation-regulating control manipulatorsof solar gather panels generally have two degrees of free-dom These degrees of freedom regulate the azimuth andmanipulator elevation to track the trajectory of the sunHowever solar equipment cannot achieve the requirement ofmaintaining the azimuth level with changes in the tracking

process (eg solar water heaters)This paper adds a degree offreedom on the basis of the original two degrees of freedomThe additional degree of freedom rotates on its own toregulate the equipment level This study further proposes athree-rotation redundant parallel manipulator (ie 4-UPS-S)for the orientation control system of solar gather panels Theredundant active chain of the 4-UPS-S eliminates the varioussingular configurations of the 3-UPS-S parallel manipulator[15]

The structural schematic of the 4-UPS-S parallel manip-ulator and its application as an adjusting manipulator in thesolar gather panel are shown in Figure 2 The four activeUPS chains contain the hinged points 119860

119894of the bases 119860

and 119861119894moving platform 119861 (119894 = 1 2 3 4) The distance

between the base and moving platform is constant and equalto 500mmThemoving platform is directly connected to thefixed base by a passive spherical joint 119874 Each chain consistsof universal prismatic and spherical joints We assign a fixedCartesian coordinate system 119874-119883

119860119884119860119885119860

and the movingCartesian system 119874-119883

119861119884119861119885119861at the centered point 119874 for

analysis (Figure 2(a)) The 119883119860axis of the fixed coordinate

system is parallel to the1198741198601198601The119885

119860axis is perpendicular to

the base and topThe 119884119860axis is based on the right-hand rule

The119883119861axis of the moving coordinate is parallel to 119874

1198611198611 119885119861

is perpendicular to the moving platformThe 119884119861axis is based

on the right-hand rule The distance between the coordinateorigin and moving platform is ℎ The circumcircle radii of

Journal of Industrial Engineering 3

s

A-A

B1

XA

XAA

O

OA

h

XB

XB

A4

A1

A4

rA

rB

B2

B1

B3B3

B2

B4

B4

A2

A3

YB

YB

YA

YA

hA

hB

OB

OA

ZB

ZA

t middot rB

(a) Schematic of the 4-UPS-S parallel manipulator

Solar gather panels4-UPS-S parallel

Base

manipulator

(b) Application of the 4-UPS-S parallel adjusting manipulator

Figure 2 Schematic of the 4-UPS-S parallel manipulator and its application

points 1198601 1198602 and 119860

3are 119903119860 Point 119860

4is located in the

plane perpendicular to the base and parallel to 1198741198601198603 Point

1198604in the region to the base is calculated as ℎ

119860= 250mm

The projection distance between points 1198601and 119860

4is 119904 The

circumcircle radii of points 1198611 1198612 and 119861

3are 119903119861 Point 119861

4is

located in the 119883119861119874119885119861plane The distance from the point to

the119885119861axis line is (119905 sdot 119903

119861)The distance to themoving platform

is ℎ119861

3 Kinematics Model Solution ofthe Parallel Manipulator

31 Inverse Kinematic Solution According to coordinatetransformation theory and the closed-loop vectormethodwecan obtain the following [16]

l119894=

ABR sdot

BpB119894 minusApA119894 (119894 = 1 2 3 4) (1)

where l119894is the vector for each prismatic actor length ApA119894 is

the position vector of point with respect to the fixed system

and BpB119894 is the position vector of point with respect to themoving system

The 4-UPS-S is a redundantly actuated parallel manipu-lator with three orientation degrees of freedom Accordingto the roll-pitch-yaw rotations the rotation matrix of themoving system with respect to the fixed system is describedas follows

ABR = RPY (120593 120579 120595)

=[[

[

119888120593119888120579 119888120593119904120579119904120595 minus 119904120593119888120595 119888120593119904120579119888120595 + 119904120593119904120595

119904120593119888120579 119904120593119904120579119904120595 + 119888120593119888120595 119904120593119904120579119888120595 minus 119888120593119904120595

minus119904120579 119888120579119904120595 119888120579119888120595

]]

]

(2)

where 120595 120579 and 120593 are the yaw (along the119883 axis) pitch (alongthe 119884 axis) and roll (along the119885 axis) angles respectively 119904120595stands for sin120595 and 119888120595 stands for cos120595

According to the inverse kinematic solution once weobtain the output angles 120595 120579 and 120593 vector l

119894is obtained by

4 Journal of Industrial Engineering

using (1) The square of the length is a function about angles120595 120579 and 120593 Consider

1198972

119894=

1003817100381710038171003817l1198941003817100381710038171003817

2=

10038171003817100381710038171003817

ApB119894 minusApA119894

10038171003817100381710038171003817

2

= Θ119894(120595 120579 120593) (3)

The derivative of both sides of (3) about time leads to thefollowing

A l = B (4)

namely

l = Aminus1B = J (5)

where J = Aminus1B is the Jacobian matrix velocity of the parallelmanipulator with

A =[[

[

21198971

0 0

0 21198972

0

0 0 21198973

]]

]

B =

[[[[[[[[

[

120597Θ1(120595 120579 120593)

120597120595

120597Θ1(120595 120579 120593)

120597120579

120597Θ1(120595 120579 120593)

120597120593

120597Θ2(120595 120579 120593)

120597120595

120597Θ2(120595 120579 120593)

120597120579

120597Θ2(120595 120579 120593)

120597120593

120597Θ3(120595 120579 120593)

120597120595

120597Θ3(120595 120579 120593)

120597120579

120597Θ3(120595 120579 120593)

120597120593

]]]]]]]]

]

= [ 120579 ]119879

(6)

32 Passive Constraint Chain Forward Model The passiveconstraint chain of the 4-UPS-S parallel manipulator isan open-kinematics chain with three-rotation degrees offreedom For analysis we separate the spherical joint intothree revolute joints by adopting the Denavit-Hartenberg(D-H) method We use three rotation angles (ie 120593

1 1205932

and 1205933) to describe its orientation The coordinate system of

the adjacent links is settled by using the D-H method Theoverlap coordinate systemorigin is separated for convenienceof illustration (Figure 3)

The transformation matrix between adjacent links isexpressed in terms of the D-H method

A119894=

[[[[[

[

119888120579119894

minus119904120579119894119888120572119894

119904120579119894119904120572119894

119886119894119888120579119894

119904120579119894

119888120579119894119888120572119894

minus119888120579119894119904120572119894

119886119894119904120579119894

0 119904120572119894

119888120572119894

119889119894

0 0 0 1

]]]]]

]

(7)

The 120579119894 119889119894 119886119894 and 120572

119894parameter values are shown in Table 1

The transformation matrix from the coordinate system119874-119883119860119884119860119885119860to the coordinate system119874-119883

119861119884119861119885119861is described

as follows

T = A0A1A2A3 (8)

O

ZA

ZB

Z1

Z2

Z3

XA

X1

X2X3

XB

YA

Figure 3 D-H notation for the passive constraint chain

Table 1 D-H parameter for the passive constraint chain

120579119894

119889119894

119886119894

120572119894

0 0 0 0 01 90

∘+ 1206011

0 0 90∘

2 90∘+ 1206012

0 0 90∘

3 1206013

0 0 0

Combining (2) and (8) we obtain the following functionrelationship between 120601

1 1206012 and 120601

3and 120595 120579 and 120593

120601119894= Γ119894(120595 120579 120593) (119894 = 1 2 3) (9)

By deviating both sides of (9) we generate the following

Ψ = J119904sdot (10)

where

Ψ = [ 1206011

1206012

1206013]119879

J119904=

[[[[[[[[

[

120597Γ1(120595 120579 120593)

120597120595

120597Γ1(120595 120579 120593)

120597120579

120597Γ1(120595 120579 120593)

120597120593

120597Γ2(120595 120579 120593)

120597120595

120597Γ2(120595 120579 120593)

120597120579

120597Γ2(120595 120579 120593)

120597120593

120597Γ3(120595 120579 120593)

120597120595

120597Γ3(120595 120579 120593)

120597120579

120597Γ3(120595 120579 120593)

120597120593

]]]]]]]]

]

(11)

where J119904is the Jacobian matrix of the middle passive chain

between the input angular velocity and output angular veloc-ity

4 Establishment of the Parallel ManipulatorPerformance Indexes

41 Global Kinematic Dexterity The4-UPS-S parallel manip-ulator is a pure rotationmanipulatorWe adopt the reciprocalof the condition number of the Jacobian matrix to measure

Journal of Industrial Engineering 5

the dexterity and transmission relationship of the manipu-lator between the input and output The reciprocal of thecondition number of the Jacobianmatrix is defined as follows

120581119869=

radic120582max (J119879J)

radic120582min (J119879J) (12)

where 120582max(J119879J) and 120582min(J119879J) are the maximum and mini-mum eigenvalues of J119879J respectively

The global dexterity index is employed to describe themean condition value of the Jacobian matrix in the entireworkspace [17]

119896119895 =

int119882

1120581119869119889119882

int119882

119889119882 (13)

where119882 is the parallel manipulator workspaceThis paper employs the output angle to describe the

orientation workspace of the parallel manipulator Accordingto the application requirements the parallel manipulatorworkspace is expressed as follows

minus30∘le 120595 le 30

∘

minus30∘le 120579 le 30

∘

minus10∘le 120593 le 10

∘

(14)

The global dexterity index is in the range 0 le 119896119895 le

1 A global dexterity index closer to one corresponds tobetter dexterity and control precision We generally calculatethe condition number 120581

119869of the Jacobian matrix in different

workspace positions and orientations to increase indicatoroperabilityThemean of the reciprocal of the Jacobian matrixis calculated as a global kinematic dexterity index 119896119895

42 Global Stiffness Performance Index The friction force ofthe hinges and joints is ignored for brevity According to thevirtual work principle one obtains the following [18]

120591119879 l + 120591119879V Ψ = F119879 (15)

where 120591 = 120594Δl is the actuator force vector applied at eachactuated joint 120591V = 120594VΔ120595 is the passive chain internal torqueand F is the torque applied to the moving platform Themoving platform is assumed to have no gravitational forcesacting on any of the intermediate links The gravitationalforces are generally neglected in industrial applications120594 and120594V are the equivalent stiffness of the active and passive chainsrespectively

Substituting (5) and (10) into (15) yields the following

F = (J119879120594J + J119879119904120594VJ119904)Δ120596 (16)

where

K = J119879120594J + J119879119904120594VJ119904 (17)

where K is the parallel manipulatorrsquos whole stiffness matrix

According to vector extreme calculation theory when themoving platform deformation reaches Δ120596 = Δ120596

119879Δ120596 = 1

the followings are generated

119870max = radic120582max (K119879K) (18)

119870min = radic120582min (K119879K) (19)

where 120582max(K119879K) and 120582min(K119879K) are the maximum andminimum eigenvalues of matrix K119879K respectively

To evaluate the parallel manipulator stiffness perfor-mance in the entire workspace the global stiffness perfor-mance index is similarly defined as follows

119896max =

int119882

119870max 119889119882

int119882

119889119882

119896min =

int119882

119870min 119889119882

int119882

119889119882

(20)

where 119882 is the parallel manipulator workspace We cancompute the stiffness index by using the mean method

Larger global stiffness evaluation indexes 119896max and119896min correspond to better parallel manipulator stiffness

43 Global Maximum Payload Capability Similar to (18) themaximumpayload capability performance index is defined asfollows

119866max = radic120582119865max (G119879G) (21)

where 120582119865max(G119879G) is the maximum eigenvalue of matrix

G119879G The relationship is G = J119879The global maximum payload capability performance

index is written as follows

119892max =

int119882

119866max 119889119882

int119882

119889119882 (22)

where119882 is the parallel manipulator workspaceA greater global maximum payload capability-evaluation

index 119892max corresponds to better parallel manipulatorpayload capability

5 Deterministic Optimization Design Modelof the Parallel Adjusting Manipulator

This study maximizes the global payload capacity and globalmaximum stiffness to improve the parallel adjusting manip-ulator of the loading ability that bears the solar equipmentweight The global kinematic dexterity and minimum stiff-ness are the constraint conditionsThe design variables are ℎ119904 and 119905 (Figure 2(a))

6 Journal of Industrial Engineering

Table 2 Six-Sigma-based robust analysis description

Random design variablesℎ 120583 = 200005 120590 = 1 times 120583

119904 120583 = 129096 120590 = 1 times 120583

119905 120583 = 0799 120590 = 1 times 120583

Random noises The radius 119903119860of the base 120583 = 500mm 120590 = 1 times 120583

The radius 119903119861of the moving platform 120583 = 400mm 120590 = 1 times 120583

Quality constraints The global kinematic dexterity 119896119895 ge 03

The global minimum stiffness 119896min ge 3 times 104

510

530

550

570

590

260000 280000 300000 320000 340000 360000k max

gm

ax

Figure 4 History points of the determinacy optimization

The deterministic optimization design model is repre-sented as

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

119896min ge 3 times 104

(23)

where X = [ℎ 119904 119905]119879 denotes the design variables and

1198911(X) and 119891

2(X) are the global payload capability and global

maximum stiffness index respectivelyWe employed the Simcode components to integrate

MATLAB and calculate the parallel manipulator evaluationindexes The NSGA-II optimization algorithm is adopted forthe present study The population size is 16 rate of crossoveris set to 09 rate of mutation is set to 01 rate of migrationis set to 01 and the generation number is 15 The designvariables constraint conditions and optimization objectivesare selected on the basis of the application requirement [2 19]We run the software platform once the parameter setting iscompleted

The deterministic optimization design is a multiobjectiveoptimization design problem The history points of the twoobjective functions in the solving process are plotted inFigure 4 The two objectives increase or decrease at the sametime The Pareto optimal solution is a convenient solutionfor this multiobjective optimization design problem whenthe global stiffness and global payload capacity obtain themaximum values at the same time

The deterministic optimization design result and initialscheme comparison is presented in Table 3The deterministicoptimization design analysis solution results show that theoptimization objectives (ie global maximum stiffness andglobal payload capacity) have a considerable improvementHowever the parameters (ie ℎ 119904 119905 and 119896119895) are all close tothe constraint boundary at the same time If the uncertaintyfactor influences are considered the deterministic optimiza-tion scheme violates the constraint conditions Thereforeperforming a robust analysis that evaluates the quality andreliability level of the deterministic optimization designscheme is necessary [20 21]

6 Six-Sigma Robust Analysis

In this study some uncertainty factors include the designvariables ℎ 119904 and 119905 The radii 119903

119860and 119903

119861of the base and

moving platform follow the same normal distribution Theoptimal design point value of the deterministic optimizationdesign is considered themean of the randomdesign variablesThe standard deviation is presented in Table 2 120590 = 1 times 120583

represents the 1 coefficient of variation (ie 120590120583 = 001)The robust analysis of the 4-UPS-S parallel manipulator isimplemented by adopting the Six-Sigma component based onIsight software

The Six-Sigma robust analysis results are concretely sum-marized in Table 3 The input variables ℎ and 119905 do not reachthe One Sigma level The reliability is approximately 50which is considered low The global kinematic dexterity andthe global minimum stiffness have a relatively high reliabilityThe minimum stiffness attains a reliability of 9999 butcannot reach the Six-Sigma level The global minimumstiffness Sigma level is 4161 In the long term when the 15120590

quality shift occurs in the system the defective products permillion with a Sigma level of four will sharply increase fromthe short term 63 times 10

minus6 to 6200 times 10minus6 The quality levels

of ℎ and 119905 are low and do not achieve the One Sigma levelTherefore performing a robust optimization for this problemis necessary in the long term

Journal of Industrial Engineering 7

Table 3 Six-Sigma-based robust design and optimization results

Variables Initial schemeDeterminacy optimization scheme Robust optimization scheme

Determinacyoptimal solutions

Six-Sigmaanalysis

Robust optimizationanalysis

Qualitylevel

Input variables

ℎ 300 120583 200005 Sigma level 0676reliability 50099

213330 6356120590 200005 213330

119904 200 120583 129096 Sigma level 7142reliability 100

218936 80120590 129096 218936

119905 06 120583 0799 Sigma level 0755reliability 54980

0683 80120590 000799 000683

Output variables (performance indexes)

Global kinematic dexterity 119896119895 0310 120583 0305 Sigma level 1187reliability 7647

0305 7809120590 0007 0007

Global minimum stiffness 119896min 30045 120583 36880 Sigma level 4161reliability 9999

429917 7563120590 1721 1739

Global maximum payload capability 119892max 538736 120583 591702 574223120590 4276 4014

Global maximum stiffness 119896max 290141 120583 350500 329940120590 5084 4624

7 Six-Sigma Robust Optimization(Uncertainty Optimization)

The essence of the Six-Sigma robust optimization is to addthe mean the variance of the response in optimizationobjectives and the upper quality level limit of the randomvariables in the constraint conditions to satisfy the qualitylevel minimize the mean of the optimal objectives andincrease the robustness and reliability of the mean [22]

This paper sets the lower limit of the Sigma level ofrandom variables and responses to 6120590 The mean of theglobal kinematic dexterity and global minimum stiffness ismaximized whereas the standard deviation of the globalkinematic dexterity and the global minimum stiffness is min-imizedThe robust optimization designmodel is summarizedas follows

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

1198913(X) = 120583 (119892max)

1198914 (X) = 120583 (119896max)

min 1198915 (X) = 120590 (119892max)

1198916 (X) = 120590 (119892max)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

Performance calculation

Robust optimization

Robust analysis

Figure 5 Flow diagram of the Six-Sigma robust design and opti-mization

119896min ge 3 times 104

Sigma Level (ℎ 119904 119905 119896119895 119896min) ge 6

(24)

where 120583(119892max) and 120590(119892max) are the mean and varianceof the maximum payload capability respectively The meanand variance are similar to the others Sigma Level(ℎ 119904119905 119896119895 119896min) denotes the Sigma level value of the parameters(ie ℎ 119904 119905 119896119895 and 119896min)

When setting the optimization objectives the scale factorof the standard deviation should be set to 001 to ensure that

8 Journal of Industrial Engineering

550 555 560 565 570 575 5804

402

404

406

408

41

412

414

Pareto solutionsThe optimal solution

Mean of g max

gm

axVa

rianc

e of

(a) Pareto frontier of the mean and variance of the maximum payloadcapability

305 31 315 32 325 33 3354560

4580

4600

4620

4640

4660

4680

Pareto solutionsThe optimal solution

times105Mean of k max

Varia

nce o

fkm

ax(b) Pareto frontier of the mean and variance of the maximum stiffness

Figure 6 Pareto frontier

the standard deviation and mean have the same order ofmagnitude

The Six-Sigma robust design optimization flow diagramis shown in Figure 5 During the procedure the Six-Sigmacomponent robust analysis is used to analyze the single designpoint robustnessThe optimization design component robustoptimization is employed to perform the optimization designon the basis of robustness analysis If the quality of the singledesign point cannot satisfy the quality level requirementsIsight will continue to improve the optimization variables forthe quality analysis of the next group otherwise the resultswill be exported and the robust optimization is completedThepresent study selects the nondominated sorting evolutionstrategy namely NSGA-II as the optimization algorithm

Some genetic parameters and operators are set as follows[23]

The population size 4The generation number 10Crossover probability 09Crossover distribution index 10Mutation distribution index 20Max failed runs 5Failed run penalty value 1011986430Failed run objective value 1011986430

We obtain the Pareto solutions and frontier after the Six-Sigma robust optimization design The Pareto frontier ofthe mean as well as the variance of the maximum payloadcapacity and maximum stiffness is illustrated in Figure 6The relationship among the optimization objectives from thePareto solutions is observed According to this relationship

and the practical application requirements we select theoptimal solution for the multiobjective design problems andchoose the point of optimal solution (Figure 6) In view ofreliability and quality level theory we can obtain the qualitylevel mean and variance of each response after the robustoptimization through the post-processing functions of Isightsoftware And the results are listed in Table 3

Comparing the quality level with robust optimizationthat is variable quality level of ℎ 119904 119905 is 0676 7142 0755respectively Performance indexes quality level of 119896119895 119896min is1187 4161 respectively After robust optimization the qualitylevel of each response is greater than six

8 Result Analysis of the Six-Sigma Robust ofthe Parallel Adjusting Manipulator

By taking the global kinematic dexterity index as an examplethe probability distribution and Six-Sigma level were com-pared before and after optimization (Figure 7) We intuitivelyobtained the probability distribution mean standard devia-tion and Six-Sigma level of each response (Figure 7)

We rearranged and listed the reliability of each variablebefore and after optimization (Table 4) to concretely illustratethe difference between the reliability and defective qualityper million before and after the robust analysisThe defectiveproducts per million are shown in Table 5

The parameter variables (ie ℎ and 119905) and quality con-straints (ie global kinematic dexterity 119896119895) (Tables 4 and 5)greatly improves after optimization However the reliabilityof the other parameter variables only improved slightlyBy contrast the defective quality per million improvedconsiderably (Table 5) In the long term the adjustingmanipulator designed andmanufacturedwith optimal designvariables has fewer defective products permillion than before

Journal of Industrial Engineering 9

0

20

40

60

80

100

026 028 03 032 034 036

Prob

abili

ty

Quality 1187 sigmaMean 0305Std dev 0007

L bound 03

kj

ndash0722 sigma

(a) Before robust optimization

0

20

40

60

80

100

032 034 036 038 04 042

Prob

abili

ty

Quality 8 sigmaMean 0363

L bound 03ndash9901 sigma

kj

(b) After robust optimization

Figure 7 Probability distribution of the global kinematic dexterity before and after robust optimization

Table 4 Comparison of the reliability before and after optimization

Before robust optimization After robust optimizationℎ 50099 99999119904 100 100119905 54980 100119896119895 76472 100119896min 99997 100

Table 5 Comparison of the defective quality per million before andafter optimization

Before robust optimization After robust optimizationℎ 499002670 2069119890 minus 4

119904 9213119890 minus 7 0119905 450199849 0119896119895 235279150 5773119890 minus 9

119896min 31653 3930119890 minus 8

optimization This result is based on the assumption that thedesign variables manufacturing and installation error of thebase and moving platform can be effectively controlled thusgreatly improving the resistance ability of the uncertaintyfactor disturbance in the whole manufacturing process Datain Tables 4 and 5 show that the Six-Sigma robust optimizationdesign based on Isight is effective and feasible

9 Conclusions

The parallel adjusting manipulator applied in solar gatherpanels is the research objective of this paper Determinacyoptimization analysis was performed with consideration tothe performance indexes By considering the influence of the

uncertainty factors implemented in the robust analysis andoptimization the following conclusions are drawn

(1) The parallel adjusting manipulator was designedby using multiobjective optimization design on thebasis of the inverse kinematic solution The perfor-mance indexes are established the optimal solutionis obtained and the determinacy optimization iscompleted

(2) Robustness analysis is performed on the basis ofdeterminacy optimization design Some parametervariables and performance indexes are close to theconstraint boundary The Sigma level is low and thedefective quality permillion is highThe implementa-tion of the Six-Sigma robust optimization is necessaryto complete the nondeterminacy optimization design

(3) The comparison of the manipulator before and afterthe robust design optimization shows that the newone has a higher Sigma level Furthermore the com-parison shows a lower defective quality per millionthan before optimizationThe optimization objectivesare optimal The mean and standard deviation arein their minimum This result indicates that in thelong term themanipulator after optimization is moresuitable for batch production applications than themanipulator before optimization The robust opti-mization based on Isight is efficient and convenientfor operation with accurate results These results areapplicable to the robust design optimization of large-scale and complex manipulators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

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International Journal of

2 Journal of Industrial Engineering

Multi-objective

Robust analysis(6 sigma quality level

analysis)

Robust optimization(6 sigma robust design

optimization)

Determinacyoptimalsolution

Introduce uncertainty elements to compute reliability and quality level of the optimal

solution

Robust design optimization

Determinacy optimization

Establishment and solution of kinematics

Multi-objective optimization model

Achieve more reliable and more robust design

scheme

Start

End

Establishment of multi-objective

optimization model

Nondeterminacyoptimization design

(6 sigma robust design optimization)

optimization

Figure 1 Flow diagram of the Six Sigma robust design and optimization based on Isight

applied in the solar gather panel orientation control sys-tem is described and the reference systems of the paral-lel manipulator are established In Section 3 the inversekinematics of the parallel legs and forward kinematics ofserial passive legs of the manipulator are studied Section 4explains the performance indexes including the maximumpayload capacity maximum and minimum stiffness andkinematic dexterity to evaluate mechanical performancesIn Section 5 a determinacy optimization design is per-formed based on software Isight In Section 6 Six-Sigmaoptimization technique for robust design optimization isanalyzed In Section 7 Six Sigma robust optimization isconducted with consideration uncertain parameters andachieves reliability and quality level Section 8 analyzes andexplains the results of the robust design optimization FinallySection 9 summarizes and arranges contents of this studyTherobustness analysis optimization flow diagram is described inFigure 1

2 Parallel Manipulator Description

The traditional orientation-regulating control manipulatorsof solar gather panels generally have two degrees of free-dom These degrees of freedom regulate the azimuth andmanipulator elevation to track the trajectory of the sunHowever solar equipment cannot achieve the requirement ofmaintaining the azimuth level with changes in the tracking

process (eg solar water heaters)This paper adds a degree offreedom on the basis of the original two degrees of freedomThe additional degree of freedom rotates on its own toregulate the equipment level This study further proposes athree-rotation redundant parallel manipulator (ie 4-UPS-S)for the orientation control system of solar gather panels Theredundant active chain of the 4-UPS-S eliminates the varioussingular configurations of the 3-UPS-S parallel manipulator[15]

The structural schematic of the 4-UPS-S parallel manip-ulator and its application as an adjusting manipulator in thesolar gather panel are shown in Figure 2 The four activeUPS chains contain the hinged points 119860

119894of the bases 119860

and 119861119894moving platform 119861 (119894 = 1 2 3 4) The distance

between the base and moving platform is constant and equalto 500mmThemoving platform is directly connected to thefixed base by a passive spherical joint 119874 Each chain consistsof universal prismatic and spherical joints We assign a fixedCartesian coordinate system 119874-119883

119860119884119860119885119860

and the movingCartesian system 119874-119883

119861119884119861119885119861at the centered point 119874 for

analysis (Figure 2(a)) The 119883119860axis of the fixed coordinate

system is parallel to the1198741198601198601The119885

119860axis is perpendicular to

the base and topThe 119884119860axis is based on the right-hand rule

The119883119861axis of the moving coordinate is parallel to 119874

1198611198611 119885119861

is perpendicular to the moving platformThe 119884119861axis is based

on the right-hand rule The distance between the coordinateorigin and moving platform is ℎ The circumcircle radii of

Journal of Industrial Engineering 3

s

A-A

B1

XA

XAA

O

OA

h

XB

XB

A4

A1

A4

rA

rB

B2

B1

B3B3

B2

B4

B4

A2

A3

YB

YB

YA

YA

hA

hB

OB

OA

ZB

ZA

t middot rB

(a) Schematic of the 4-UPS-S parallel manipulator

Solar gather panels4-UPS-S parallel

Base

manipulator

(b) Application of the 4-UPS-S parallel adjusting manipulator

Figure 2 Schematic of the 4-UPS-S parallel manipulator and its application

points 1198601 1198602 and 119860

3are 119903119860 Point 119860

4is located in the

plane perpendicular to the base and parallel to 1198741198601198603 Point

1198604in the region to the base is calculated as ℎ

119860= 250mm

The projection distance between points 1198601and 119860

4is 119904 The

circumcircle radii of points 1198611 1198612 and 119861

3are 119903119861 Point 119861

4is

located in the 119883119861119874119885119861plane The distance from the point to

the119885119861axis line is (119905 sdot 119903

119861)The distance to themoving platform

is ℎ119861

3 Kinematics Model Solution ofthe Parallel Manipulator

31 Inverse Kinematic Solution According to coordinatetransformation theory and the closed-loop vectormethodwecan obtain the following [16]

l119894=

ABR sdot

BpB119894 minusApA119894 (119894 = 1 2 3 4) (1)

where l119894is the vector for each prismatic actor length ApA119894 is

the position vector of point with respect to the fixed system

and BpB119894 is the position vector of point with respect to themoving system

The 4-UPS-S is a redundantly actuated parallel manipu-lator with three orientation degrees of freedom Accordingto the roll-pitch-yaw rotations the rotation matrix of themoving system with respect to the fixed system is describedas follows

ABR = RPY (120593 120579 120595)

=[[

[

119888120593119888120579 119888120593119904120579119904120595 minus 119904120593119888120595 119888120593119904120579119888120595 + 119904120593119904120595

119904120593119888120579 119904120593119904120579119904120595 + 119888120593119888120595 119904120593119904120579119888120595 minus 119888120593119904120595

minus119904120579 119888120579119904120595 119888120579119888120595

]]

]

(2)

where 120595 120579 and 120593 are the yaw (along the119883 axis) pitch (alongthe 119884 axis) and roll (along the119885 axis) angles respectively 119904120595stands for sin120595 and 119888120595 stands for cos120595

According to the inverse kinematic solution once weobtain the output angles 120595 120579 and 120593 vector l

119894is obtained by

4 Journal of Industrial Engineering

using (1) The square of the length is a function about angles120595 120579 and 120593 Consider

1198972

119894=

1003817100381710038171003817l1198941003817100381710038171003817

2=

10038171003817100381710038171003817

ApB119894 minusApA119894

10038171003817100381710038171003817

2

= Θ119894(120595 120579 120593) (3)

The derivative of both sides of (3) about time leads to thefollowing

A l = B (4)

namely

l = Aminus1B = J (5)

where J = Aminus1B is the Jacobian matrix velocity of the parallelmanipulator with

A =[[

[

21198971

0 0

0 21198972

0

0 0 21198973

]]

]

B =

[[[[[[[[

[

120597Θ1(120595 120579 120593)

120597120595

120597Θ1(120595 120579 120593)

120597120579

120597Θ1(120595 120579 120593)

120597120593

120597Θ2(120595 120579 120593)

120597120595

120597Θ2(120595 120579 120593)

120597120579

120597Θ2(120595 120579 120593)

120597120593

120597Θ3(120595 120579 120593)

120597120595

120597Θ3(120595 120579 120593)

120597120579

120597Θ3(120595 120579 120593)

120597120593

]]]]]]]]

]

= [ 120579 ]119879

(6)

32 Passive Constraint Chain Forward Model The passiveconstraint chain of the 4-UPS-S parallel manipulator isan open-kinematics chain with three-rotation degrees offreedom For analysis we separate the spherical joint intothree revolute joints by adopting the Denavit-Hartenberg(D-H) method We use three rotation angles (ie 120593

1 1205932

and 1205933) to describe its orientation The coordinate system of

the adjacent links is settled by using the D-H method Theoverlap coordinate systemorigin is separated for convenienceof illustration (Figure 3)

The transformation matrix between adjacent links isexpressed in terms of the D-H method

A119894=

[[[[[

[

119888120579119894

minus119904120579119894119888120572119894

119904120579119894119904120572119894

119886119894119888120579119894

119904120579119894

119888120579119894119888120572119894

minus119888120579119894119904120572119894

119886119894119904120579119894

0 119904120572119894

119888120572119894

119889119894

0 0 0 1

]]]]]

]

(7)

The 120579119894 119889119894 119886119894 and 120572

119894parameter values are shown in Table 1

The transformation matrix from the coordinate system119874-119883119860119884119860119885119860to the coordinate system119874-119883

119861119884119861119885119861is described

as follows

T = A0A1A2A3 (8)

O

ZA

ZB

Z1

Z2

Z3

XA

X1

X2X3

XB

YA

Figure 3 D-H notation for the passive constraint chain

Table 1 D-H parameter for the passive constraint chain

120579119894

119889119894

119886119894

120572119894

0 0 0 0 01 90

∘+ 1206011

0 0 90∘

2 90∘+ 1206012

0 0 90∘

3 1206013

0 0 0

Combining (2) and (8) we obtain the following functionrelationship between 120601

1 1206012 and 120601

3and 120595 120579 and 120593

120601119894= Γ119894(120595 120579 120593) (119894 = 1 2 3) (9)

By deviating both sides of (9) we generate the following

Ψ = J119904sdot (10)

where

Ψ = [ 1206011

1206012

1206013]119879

J119904=

[[[[[[[[

[

120597Γ1(120595 120579 120593)

120597120595

120597Γ1(120595 120579 120593)

120597120579

120597Γ1(120595 120579 120593)

120597120593

120597Γ2(120595 120579 120593)

120597120595

120597Γ2(120595 120579 120593)

120597120579

120597Γ2(120595 120579 120593)

120597120593

120597Γ3(120595 120579 120593)

120597120595

120597Γ3(120595 120579 120593)

120597120579

120597Γ3(120595 120579 120593)

120597120593

]]]]]]]]

]

(11)

where J119904is the Jacobian matrix of the middle passive chain

between the input angular velocity and output angular veloc-ity

4 Establishment of the Parallel ManipulatorPerformance Indexes

41 Global Kinematic Dexterity The4-UPS-S parallel manip-ulator is a pure rotationmanipulatorWe adopt the reciprocalof the condition number of the Jacobian matrix to measure

Journal of Industrial Engineering 5

the dexterity and transmission relationship of the manipu-lator between the input and output The reciprocal of thecondition number of the Jacobianmatrix is defined as follows

120581119869=

radic120582max (J119879J)

radic120582min (J119879J) (12)

where 120582max(J119879J) and 120582min(J119879J) are the maximum and mini-mum eigenvalues of J119879J respectively

The global dexterity index is employed to describe themean condition value of the Jacobian matrix in the entireworkspace [17]

119896119895 =

int119882

1120581119869119889119882

int119882

119889119882 (13)

where119882 is the parallel manipulator workspaceThis paper employs the output angle to describe the

orientation workspace of the parallel manipulator Accordingto the application requirements the parallel manipulatorworkspace is expressed as follows

minus30∘le 120595 le 30

∘

minus30∘le 120579 le 30

∘

minus10∘le 120593 le 10

∘

(14)

The global dexterity index is in the range 0 le 119896119895 le

1 A global dexterity index closer to one corresponds tobetter dexterity and control precision We generally calculatethe condition number 120581

119869of the Jacobian matrix in different

workspace positions and orientations to increase indicatoroperabilityThemean of the reciprocal of the Jacobian matrixis calculated as a global kinematic dexterity index 119896119895

42 Global Stiffness Performance Index The friction force ofthe hinges and joints is ignored for brevity According to thevirtual work principle one obtains the following [18]

120591119879 l + 120591119879V Ψ = F119879 (15)

where 120591 = 120594Δl is the actuator force vector applied at eachactuated joint 120591V = 120594VΔ120595 is the passive chain internal torqueand F is the torque applied to the moving platform Themoving platform is assumed to have no gravitational forcesacting on any of the intermediate links The gravitationalforces are generally neglected in industrial applications120594 and120594V are the equivalent stiffness of the active and passive chainsrespectively

Substituting (5) and (10) into (15) yields the following

F = (J119879120594J + J119879119904120594VJ119904)Δ120596 (16)

where

K = J119879120594J + J119879119904120594VJ119904 (17)

where K is the parallel manipulatorrsquos whole stiffness matrix

According to vector extreme calculation theory when themoving platform deformation reaches Δ120596 = Δ120596

119879Δ120596 = 1

the followings are generated

119870max = radic120582max (K119879K) (18)

119870min = radic120582min (K119879K) (19)

where 120582max(K119879K) and 120582min(K119879K) are the maximum andminimum eigenvalues of matrix K119879K respectively

To evaluate the parallel manipulator stiffness perfor-mance in the entire workspace the global stiffness perfor-mance index is similarly defined as follows

119896max =

int119882

119870max 119889119882

int119882

119889119882

119896min =

int119882

119870min 119889119882

int119882

119889119882

(20)

where 119882 is the parallel manipulator workspace We cancompute the stiffness index by using the mean method

Larger global stiffness evaluation indexes 119896max and119896min correspond to better parallel manipulator stiffness

43 Global Maximum Payload Capability Similar to (18) themaximumpayload capability performance index is defined asfollows

119866max = radic120582119865max (G119879G) (21)

where 120582119865max(G119879G) is the maximum eigenvalue of matrix

G119879G The relationship is G = J119879The global maximum payload capability performance

index is written as follows

119892max =

int119882

119866max 119889119882

int119882

119889119882 (22)

where119882 is the parallel manipulator workspaceA greater global maximum payload capability-evaluation

index 119892max corresponds to better parallel manipulatorpayload capability

5 Deterministic Optimization Design Modelof the Parallel Adjusting Manipulator

This study maximizes the global payload capacity and globalmaximum stiffness to improve the parallel adjusting manip-ulator of the loading ability that bears the solar equipmentweight The global kinematic dexterity and minimum stiff-ness are the constraint conditionsThe design variables are ℎ119904 and 119905 (Figure 2(a))

6 Journal of Industrial Engineering

Table 2 Six-Sigma-based robust analysis description

Random design variablesℎ 120583 = 200005 120590 = 1 times 120583

119904 120583 = 129096 120590 = 1 times 120583

119905 120583 = 0799 120590 = 1 times 120583

Random noises The radius 119903119860of the base 120583 = 500mm 120590 = 1 times 120583

The radius 119903119861of the moving platform 120583 = 400mm 120590 = 1 times 120583

Quality constraints The global kinematic dexterity 119896119895 ge 03

The global minimum stiffness 119896min ge 3 times 104

510

530

550

570

590

260000 280000 300000 320000 340000 360000k max

gm

ax

Figure 4 History points of the determinacy optimization

The deterministic optimization design model is repre-sented as

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

119896min ge 3 times 104

(23)

where X = [ℎ 119904 119905]119879 denotes the design variables and

1198911(X) and 119891

2(X) are the global payload capability and global

maximum stiffness index respectivelyWe employed the Simcode components to integrate

MATLAB and calculate the parallel manipulator evaluationindexes The NSGA-II optimization algorithm is adopted forthe present study The population size is 16 rate of crossoveris set to 09 rate of mutation is set to 01 rate of migrationis set to 01 and the generation number is 15 The designvariables constraint conditions and optimization objectivesare selected on the basis of the application requirement [2 19]We run the software platform once the parameter setting iscompleted

The deterministic optimization design is a multiobjectiveoptimization design problem The history points of the twoobjective functions in the solving process are plotted inFigure 4 The two objectives increase or decrease at the sametime The Pareto optimal solution is a convenient solutionfor this multiobjective optimization design problem whenthe global stiffness and global payload capacity obtain themaximum values at the same time

The deterministic optimization design result and initialscheme comparison is presented in Table 3The deterministicoptimization design analysis solution results show that theoptimization objectives (ie global maximum stiffness andglobal payload capacity) have a considerable improvementHowever the parameters (ie ℎ 119904 119905 and 119896119895) are all close tothe constraint boundary at the same time If the uncertaintyfactor influences are considered the deterministic optimiza-tion scheme violates the constraint conditions Thereforeperforming a robust analysis that evaluates the quality andreliability level of the deterministic optimization designscheme is necessary [20 21]

6 Six-Sigma Robust Analysis

In this study some uncertainty factors include the designvariables ℎ 119904 and 119905 The radii 119903

119860and 119903

119861of the base and

moving platform follow the same normal distribution Theoptimal design point value of the deterministic optimizationdesign is considered themean of the randomdesign variablesThe standard deviation is presented in Table 2 120590 = 1 times 120583

represents the 1 coefficient of variation (ie 120590120583 = 001)The robust analysis of the 4-UPS-S parallel manipulator isimplemented by adopting the Six-Sigma component based onIsight software

The Six-Sigma robust analysis results are concretely sum-marized in Table 3 The input variables ℎ and 119905 do not reachthe One Sigma level The reliability is approximately 50which is considered low The global kinematic dexterity andthe global minimum stiffness have a relatively high reliabilityThe minimum stiffness attains a reliability of 9999 butcannot reach the Six-Sigma level The global minimumstiffness Sigma level is 4161 In the long term when the 15120590

quality shift occurs in the system the defective products permillion with a Sigma level of four will sharply increase fromthe short term 63 times 10

minus6 to 6200 times 10minus6 The quality levels

of ℎ and 119905 are low and do not achieve the One Sigma levelTherefore performing a robust optimization for this problemis necessary in the long term

Journal of Industrial Engineering 7

Table 3 Six-Sigma-based robust design and optimization results

Variables Initial schemeDeterminacy optimization scheme Robust optimization scheme

Determinacyoptimal solutions

Six-Sigmaanalysis

Robust optimizationanalysis

Qualitylevel

Input variables

ℎ 300 120583 200005 Sigma level 0676reliability 50099

213330 6356120590 200005 213330

119904 200 120583 129096 Sigma level 7142reliability 100

218936 80120590 129096 218936

119905 06 120583 0799 Sigma level 0755reliability 54980

0683 80120590 000799 000683

Output variables (performance indexes)

Global kinematic dexterity 119896119895 0310 120583 0305 Sigma level 1187reliability 7647

0305 7809120590 0007 0007

Global minimum stiffness 119896min 30045 120583 36880 Sigma level 4161reliability 9999

429917 7563120590 1721 1739

Global maximum payload capability 119892max 538736 120583 591702 574223120590 4276 4014

Global maximum stiffness 119896max 290141 120583 350500 329940120590 5084 4624

7 Six-Sigma Robust Optimization(Uncertainty Optimization)

The essence of the Six-Sigma robust optimization is to addthe mean the variance of the response in optimizationobjectives and the upper quality level limit of the randomvariables in the constraint conditions to satisfy the qualitylevel minimize the mean of the optimal objectives andincrease the robustness and reliability of the mean [22]

This paper sets the lower limit of the Sigma level ofrandom variables and responses to 6120590 The mean of theglobal kinematic dexterity and global minimum stiffness ismaximized whereas the standard deviation of the globalkinematic dexterity and the global minimum stiffness is min-imizedThe robust optimization designmodel is summarizedas follows

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

1198913(X) = 120583 (119892max)

1198914 (X) = 120583 (119896max)

min 1198915 (X) = 120590 (119892max)

1198916 (X) = 120590 (119892max)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

Performance calculation

Robust optimization

Robust analysis

Figure 5 Flow diagram of the Six-Sigma robust design and opti-mization

119896min ge 3 times 104

Sigma Level (ℎ 119904 119905 119896119895 119896min) ge 6

(24)

where 120583(119892max) and 120590(119892max) are the mean and varianceof the maximum payload capability respectively The meanand variance are similar to the others Sigma Level(ℎ 119904119905 119896119895 119896min) denotes the Sigma level value of the parameters(ie ℎ 119904 119905 119896119895 and 119896min)

When setting the optimization objectives the scale factorof the standard deviation should be set to 001 to ensure that

8 Journal of Industrial Engineering

550 555 560 565 570 575 5804

402

404

406

408

41

412

414

Pareto solutionsThe optimal solution

Mean of g max

gm

axVa

rianc

e of

(a) Pareto frontier of the mean and variance of the maximum payloadcapability

305 31 315 32 325 33 3354560

4580

4600

4620

4640

4660

4680

Pareto solutionsThe optimal solution

times105Mean of k max

Varia

nce o

fkm

ax(b) Pareto frontier of the mean and variance of the maximum stiffness

Figure 6 Pareto frontier

the standard deviation and mean have the same order ofmagnitude

The Six-Sigma robust design optimization flow diagramis shown in Figure 5 During the procedure the Six-Sigmacomponent robust analysis is used to analyze the single designpoint robustnessThe optimization design component robustoptimization is employed to perform the optimization designon the basis of robustness analysis If the quality of the singledesign point cannot satisfy the quality level requirementsIsight will continue to improve the optimization variables forthe quality analysis of the next group otherwise the resultswill be exported and the robust optimization is completedThepresent study selects the nondominated sorting evolutionstrategy namely NSGA-II as the optimization algorithm

Some genetic parameters and operators are set as follows[23]

The population size 4The generation number 10Crossover probability 09Crossover distribution index 10Mutation distribution index 20Max failed runs 5Failed run penalty value 1011986430Failed run objective value 1011986430

We obtain the Pareto solutions and frontier after the Six-Sigma robust optimization design The Pareto frontier ofthe mean as well as the variance of the maximum payloadcapacity and maximum stiffness is illustrated in Figure 6The relationship among the optimization objectives from thePareto solutions is observed According to this relationship

and the practical application requirements we select theoptimal solution for the multiobjective design problems andchoose the point of optimal solution (Figure 6) In view ofreliability and quality level theory we can obtain the qualitylevel mean and variance of each response after the robustoptimization through the post-processing functions of Isightsoftware And the results are listed in Table 3

Comparing the quality level with robust optimizationthat is variable quality level of ℎ 119904 119905 is 0676 7142 0755respectively Performance indexes quality level of 119896119895 119896min is1187 4161 respectively After robust optimization the qualitylevel of each response is greater than six

8 Result Analysis of the Six-Sigma Robust ofthe Parallel Adjusting Manipulator

By taking the global kinematic dexterity index as an examplethe probability distribution and Six-Sigma level were com-pared before and after optimization (Figure 7) We intuitivelyobtained the probability distribution mean standard devia-tion and Six-Sigma level of each response (Figure 7)

We rearranged and listed the reliability of each variablebefore and after optimization (Table 4) to concretely illustratethe difference between the reliability and defective qualityper million before and after the robust analysisThe defectiveproducts per million are shown in Table 5

The parameter variables (ie ℎ and 119905) and quality con-straints (ie global kinematic dexterity 119896119895) (Tables 4 and 5)greatly improves after optimization However the reliabilityof the other parameter variables only improved slightlyBy contrast the defective quality per million improvedconsiderably (Table 5) In the long term the adjustingmanipulator designed andmanufacturedwith optimal designvariables has fewer defective products permillion than before

Journal of Industrial Engineering 9

0

20

40

60

80

100

026 028 03 032 034 036

Prob

abili

ty

Quality 1187 sigmaMean 0305Std dev 0007

L bound 03

kj

ndash0722 sigma

(a) Before robust optimization

0

20

40

60

80

100

032 034 036 038 04 042

Prob

abili

ty

Quality 8 sigmaMean 0363

L bound 03ndash9901 sigma

kj

(b) After robust optimization

Figure 7 Probability distribution of the global kinematic dexterity before and after robust optimization

Table 4 Comparison of the reliability before and after optimization

Before robust optimization After robust optimizationℎ 50099 99999119904 100 100119905 54980 100119896119895 76472 100119896min 99997 100

Table 5 Comparison of the defective quality per million before andafter optimization

Before robust optimization After robust optimizationℎ 499002670 2069119890 minus 4

119904 9213119890 minus 7 0119905 450199849 0119896119895 235279150 5773119890 minus 9

119896min 31653 3930119890 minus 8

optimization This result is based on the assumption that thedesign variables manufacturing and installation error of thebase and moving platform can be effectively controlled thusgreatly improving the resistance ability of the uncertaintyfactor disturbance in the whole manufacturing process Datain Tables 4 and 5 show that the Six-Sigma robust optimizationdesign based on Isight is effective and feasible

9 Conclusions

The parallel adjusting manipulator applied in solar gatherpanels is the research objective of this paper Determinacyoptimization analysis was performed with consideration tothe performance indexes By considering the influence of the

uncertainty factors implemented in the robust analysis andoptimization the following conclusions are drawn

(1) The parallel adjusting manipulator was designedby using multiobjective optimization design on thebasis of the inverse kinematic solution The perfor-mance indexes are established the optimal solutionis obtained and the determinacy optimization iscompleted

(2) Robustness analysis is performed on the basis ofdeterminacy optimization design Some parametervariables and performance indexes are close to theconstraint boundary The Sigma level is low and thedefective quality permillion is highThe implementa-tion of the Six-Sigma robust optimization is necessaryto complete the nondeterminacy optimization design

(3) The comparison of the manipulator before and afterthe robust design optimization shows that the newone has a higher Sigma level Furthermore the com-parison shows a lower defective quality per millionthan before optimizationThe optimization objectivesare optimal The mean and standard deviation arein their minimum This result indicates that in thelong term themanipulator after optimization is moresuitable for batch production applications than themanipulator before optimization The robust opti-mization based on Isight is efficient and convenientfor operation with accurate results These results areapplicable to the robust design optimization of large-scale and complex manipulators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

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Submit your manuscripts athttpwwwhindawicom

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DistributedSensor Networks

International Journal of

Journal of Industrial Engineering 3

s

A-A

B1

XA

XAA

O

OA

h

XB

XB

A4

A1

A4

rA

rB

B2

B1

B3B3

B2

B4

B4

A2

A3

YB

YB

YA

YA

hA

hB

OB

OA

ZB

ZA

t middot rB

(a) Schematic of the 4-UPS-S parallel manipulator

Solar gather panels4-UPS-S parallel

Base

manipulator

(b) Application of the 4-UPS-S parallel adjusting manipulator

Figure 2 Schematic of the 4-UPS-S parallel manipulator and its application

points 1198601 1198602 and 119860

3are 119903119860 Point 119860

4is located in the

plane perpendicular to the base and parallel to 1198741198601198603 Point

1198604in the region to the base is calculated as ℎ

119860= 250mm

The projection distance between points 1198601and 119860

4is 119904 The

circumcircle radii of points 1198611 1198612 and 119861

3are 119903119861 Point 119861

4is

located in the 119883119861119874119885119861plane The distance from the point to

the119885119861axis line is (119905 sdot 119903

119861)The distance to themoving platform

is ℎ119861

3 Kinematics Model Solution ofthe Parallel Manipulator

31 Inverse Kinematic Solution According to coordinatetransformation theory and the closed-loop vectormethodwecan obtain the following [16]

l119894=

ABR sdot

BpB119894 minusApA119894 (119894 = 1 2 3 4) (1)

where l119894is the vector for each prismatic actor length ApA119894 is

the position vector of point with respect to the fixed system

and BpB119894 is the position vector of point with respect to themoving system

The 4-UPS-S is a redundantly actuated parallel manipu-lator with three orientation degrees of freedom Accordingto the roll-pitch-yaw rotations the rotation matrix of themoving system with respect to the fixed system is describedas follows

ABR = RPY (120593 120579 120595)

=[[

[

119888120593119888120579 119888120593119904120579119904120595 minus 119904120593119888120595 119888120593119904120579119888120595 + 119904120593119904120595

119904120593119888120579 119904120593119904120579119904120595 + 119888120593119888120595 119904120593119904120579119888120595 minus 119888120593119904120595

minus119904120579 119888120579119904120595 119888120579119888120595

]]

]

(2)

where 120595 120579 and 120593 are the yaw (along the119883 axis) pitch (alongthe 119884 axis) and roll (along the119885 axis) angles respectively 119904120595stands for sin120595 and 119888120595 stands for cos120595

According to the inverse kinematic solution once weobtain the output angles 120595 120579 and 120593 vector l

119894is obtained by

4 Journal of Industrial Engineering

using (1) The square of the length is a function about angles120595 120579 and 120593 Consider

1198972

119894=

1003817100381710038171003817l1198941003817100381710038171003817

2=

10038171003817100381710038171003817

ApB119894 minusApA119894

10038171003817100381710038171003817

2

= Θ119894(120595 120579 120593) (3)

The derivative of both sides of (3) about time leads to thefollowing

A l = B (4)

namely

l = Aminus1B = J (5)

where J = Aminus1B is the Jacobian matrix velocity of the parallelmanipulator with

A =[[

[

21198971

0 0

0 21198972

0

0 0 21198973

]]

]

B =

[[[[[[[[

[

120597Θ1(120595 120579 120593)

120597120595

120597Θ1(120595 120579 120593)

120597120579

120597Θ1(120595 120579 120593)

120597120593

120597Θ2(120595 120579 120593)

120597120595

120597Θ2(120595 120579 120593)

120597120579

120597Θ2(120595 120579 120593)

120597120593

120597Θ3(120595 120579 120593)

120597120595

120597Θ3(120595 120579 120593)

120597120579

120597Θ3(120595 120579 120593)

120597120593

]]]]]]]]

]

= [ 120579 ]119879

(6)

32 Passive Constraint Chain Forward Model The passiveconstraint chain of the 4-UPS-S parallel manipulator isan open-kinematics chain with three-rotation degrees offreedom For analysis we separate the spherical joint intothree revolute joints by adopting the Denavit-Hartenberg(D-H) method We use three rotation angles (ie 120593

1 1205932

and 1205933) to describe its orientation The coordinate system of

the adjacent links is settled by using the D-H method Theoverlap coordinate systemorigin is separated for convenienceof illustration (Figure 3)

The transformation matrix between adjacent links isexpressed in terms of the D-H method

A119894=

[[[[[

[

119888120579119894

minus119904120579119894119888120572119894

119904120579119894119904120572119894

119886119894119888120579119894

119904120579119894

119888120579119894119888120572119894

minus119888120579119894119904120572119894

119886119894119904120579119894

0 119904120572119894

119888120572119894

119889119894

0 0 0 1

]]]]]

]

(7)

The 120579119894 119889119894 119886119894 and 120572

119894parameter values are shown in Table 1

The transformation matrix from the coordinate system119874-119883119860119884119860119885119860to the coordinate system119874-119883

119861119884119861119885119861is described

as follows

T = A0A1A2A3 (8)

O

ZA

ZB

Z1

Z2

Z3

XA

X1

X2X3

XB

YA

Figure 3 D-H notation for the passive constraint chain

Table 1 D-H parameter for the passive constraint chain

120579119894

119889119894

119886119894

120572119894

0 0 0 0 01 90

∘+ 1206011

0 0 90∘

2 90∘+ 1206012

0 0 90∘

3 1206013

0 0 0

Combining (2) and (8) we obtain the following functionrelationship between 120601

1 1206012 and 120601

3and 120595 120579 and 120593

120601119894= Γ119894(120595 120579 120593) (119894 = 1 2 3) (9)

By deviating both sides of (9) we generate the following

Ψ = J119904sdot (10)

where

Ψ = [ 1206011

1206012

1206013]119879

J119904=

[[[[[[[[

[

120597Γ1(120595 120579 120593)

120597120595

120597Γ1(120595 120579 120593)

120597120579

120597Γ1(120595 120579 120593)

120597120593

120597Γ2(120595 120579 120593)

120597120595

120597Γ2(120595 120579 120593)

120597120579

120597Γ2(120595 120579 120593)

120597120593

120597Γ3(120595 120579 120593)

120597120595

120597Γ3(120595 120579 120593)

120597120579

120597Γ3(120595 120579 120593)

120597120593

]]]]]]]]

]

(11)

where J119904is the Jacobian matrix of the middle passive chain

between the input angular velocity and output angular veloc-ity

4 Establishment of the Parallel ManipulatorPerformance Indexes

41 Global Kinematic Dexterity The4-UPS-S parallel manip-ulator is a pure rotationmanipulatorWe adopt the reciprocalof the condition number of the Jacobian matrix to measure

Journal of Industrial Engineering 5

the dexterity and transmission relationship of the manipu-lator between the input and output The reciprocal of thecondition number of the Jacobianmatrix is defined as follows

120581119869=

radic120582max (J119879J)

radic120582min (J119879J) (12)

where 120582max(J119879J) and 120582min(J119879J) are the maximum and mini-mum eigenvalues of J119879J respectively

The global dexterity index is employed to describe themean condition value of the Jacobian matrix in the entireworkspace [17]

119896119895 =

int119882

1120581119869119889119882

int119882

119889119882 (13)

where119882 is the parallel manipulator workspaceThis paper employs the output angle to describe the

orientation workspace of the parallel manipulator Accordingto the application requirements the parallel manipulatorworkspace is expressed as follows

minus30∘le 120595 le 30

∘

minus30∘le 120579 le 30

∘

minus10∘le 120593 le 10

∘

(14)

The global dexterity index is in the range 0 le 119896119895 le

1 A global dexterity index closer to one corresponds tobetter dexterity and control precision We generally calculatethe condition number 120581

119869of the Jacobian matrix in different

workspace positions and orientations to increase indicatoroperabilityThemean of the reciprocal of the Jacobian matrixis calculated as a global kinematic dexterity index 119896119895

42 Global Stiffness Performance Index The friction force ofthe hinges and joints is ignored for brevity According to thevirtual work principle one obtains the following [18]

120591119879 l + 120591119879V Ψ = F119879 (15)

where 120591 = 120594Δl is the actuator force vector applied at eachactuated joint 120591V = 120594VΔ120595 is the passive chain internal torqueand F is the torque applied to the moving platform Themoving platform is assumed to have no gravitational forcesacting on any of the intermediate links The gravitationalforces are generally neglected in industrial applications120594 and120594V are the equivalent stiffness of the active and passive chainsrespectively

Substituting (5) and (10) into (15) yields the following

F = (J119879120594J + J119879119904120594VJ119904)Δ120596 (16)

where

K = J119879120594J + J119879119904120594VJ119904 (17)

where K is the parallel manipulatorrsquos whole stiffness matrix

According to vector extreme calculation theory when themoving platform deformation reaches Δ120596 = Δ120596

119879Δ120596 = 1

the followings are generated

119870max = radic120582max (K119879K) (18)

119870min = radic120582min (K119879K) (19)

where 120582max(K119879K) and 120582min(K119879K) are the maximum andminimum eigenvalues of matrix K119879K respectively

To evaluate the parallel manipulator stiffness perfor-mance in the entire workspace the global stiffness perfor-mance index is similarly defined as follows

119896max =

int119882

119870max 119889119882

int119882

119889119882

119896min =

int119882

119870min 119889119882

int119882

119889119882

(20)

where 119882 is the parallel manipulator workspace We cancompute the stiffness index by using the mean method

Larger global stiffness evaluation indexes 119896max and119896min correspond to better parallel manipulator stiffness

43 Global Maximum Payload Capability Similar to (18) themaximumpayload capability performance index is defined asfollows

119866max = radic120582119865max (G119879G) (21)

where 120582119865max(G119879G) is the maximum eigenvalue of matrix

G119879G The relationship is G = J119879The global maximum payload capability performance

index is written as follows

119892max =

int119882

119866max 119889119882

int119882

119889119882 (22)

where119882 is the parallel manipulator workspaceA greater global maximum payload capability-evaluation

index 119892max corresponds to better parallel manipulatorpayload capability

5 Deterministic Optimization Design Modelof the Parallel Adjusting Manipulator

This study maximizes the global payload capacity and globalmaximum stiffness to improve the parallel adjusting manip-ulator of the loading ability that bears the solar equipmentweight The global kinematic dexterity and minimum stiff-ness are the constraint conditionsThe design variables are ℎ119904 and 119905 (Figure 2(a))

6 Journal of Industrial Engineering

Table 2 Six-Sigma-based robust analysis description

Random design variablesℎ 120583 = 200005 120590 = 1 times 120583

119904 120583 = 129096 120590 = 1 times 120583

119905 120583 = 0799 120590 = 1 times 120583

Random noises The radius 119903119860of the base 120583 = 500mm 120590 = 1 times 120583

The radius 119903119861of the moving platform 120583 = 400mm 120590 = 1 times 120583

Quality constraints The global kinematic dexterity 119896119895 ge 03

The global minimum stiffness 119896min ge 3 times 104

510

530

550

570

590

260000 280000 300000 320000 340000 360000k max

gm

ax

Figure 4 History points of the determinacy optimization

The deterministic optimization design model is repre-sented as

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

119896min ge 3 times 104

(23)

where X = [ℎ 119904 119905]119879 denotes the design variables and

1198911(X) and 119891

2(X) are the global payload capability and global

maximum stiffness index respectivelyWe employed the Simcode components to integrate

MATLAB and calculate the parallel manipulator evaluationindexes The NSGA-II optimization algorithm is adopted forthe present study The population size is 16 rate of crossoveris set to 09 rate of mutation is set to 01 rate of migrationis set to 01 and the generation number is 15 The designvariables constraint conditions and optimization objectivesare selected on the basis of the application requirement [2 19]We run the software platform once the parameter setting iscompleted

The deterministic optimization design is a multiobjectiveoptimization design problem The history points of the twoobjective functions in the solving process are plotted inFigure 4 The two objectives increase or decrease at the sametime The Pareto optimal solution is a convenient solutionfor this multiobjective optimization design problem whenthe global stiffness and global payload capacity obtain themaximum values at the same time

The deterministic optimization design result and initialscheme comparison is presented in Table 3The deterministicoptimization design analysis solution results show that theoptimization objectives (ie global maximum stiffness andglobal payload capacity) have a considerable improvementHowever the parameters (ie ℎ 119904 119905 and 119896119895) are all close tothe constraint boundary at the same time If the uncertaintyfactor influences are considered the deterministic optimiza-tion scheme violates the constraint conditions Thereforeperforming a robust analysis that evaluates the quality andreliability level of the deterministic optimization designscheme is necessary [20 21]

6 Six-Sigma Robust Analysis

In this study some uncertainty factors include the designvariables ℎ 119904 and 119905 The radii 119903

119860and 119903

119861of the base and

moving platform follow the same normal distribution Theoptimal design point value of the deterministic optimizationdesign is considered themean of the randomdesign variablesThe standard deviation is presented in Table 2 120590 = 1 times 120583

represents the 1 coefficient of variation (ie 120590120583 = 001)The robust analysis of the 4-UPS-S parallel manipulator isimplemented by adopting the Six-Sigma component based onIsight software

The Six-Sigma robust analysis results are concretely sum-marized in Table 3 The input variables ℎ and 119905 do not reachthe One Sigma level The reliability is approximately 50which is considered low The global kinematic dexterity andthe global minimum stiffness have a relatively high reliabilityThe minimum stiffness attains a reliability of 9999 butcannot reach the Six-Sigma level The global minimumstiffness Sigma level is 4161 In the long term when the 15120590

quality shift occurs in the system the defective products permillion with a Sigma level of four will sharply increase fromthe short term 63 times 10

minus6 to 6200 times 10minus6 The quality levels

of ℎ and 119905 are low and do not achieve the One Sigma levelTherefore performing a robust optimization for this problemis necessary in the long term

Journal of Industrial Engineering 7

Table 3 Six-Sigma-based robust design and optimization results

Variables Initial schemeDeterminacy optimization scheme Robust optimization scheme

Determinacyoptimal solutions

Six-Sigmaanalysis

Robust optimizationanalysis

Qualitylevel

Input variables

ℎ 300 120583 200005 Sigma level 0676reliability 50099

213330 6356120590 200005 213330

119904 200 120583 129096 Sigma level 7142reliability 100

218936 80120590 129096 218936

119905 06 120583 0799 Sigma level 0755reliability 54980

0683 80120590 000799 000683

Output variables (performance indexes)

Global kinematic dexterity 119896119895 0310 120583 0305 Sigma level 1187reliability 7647

0305 7809120590 0007 0007

Global minimum stiffness 119896min 30045 120583 36880 Sigma level 4161reliability 9999

429917 7563120590 1721 1739

Global maximum payload capability 119892max 538736 120583 591702 574223120590 4276 4014

Global maximum stiffness 119896max 290141 120583 350500 329940120590 5084 4624

7 Six-Sigma Robust Optimization(Uncertainty Optimization)

The essence of the Six-Sigma robust optimization is to addthe mean the variance of the response in optimizationobjectives and the upper quality level limit of the randomvariables in the constraint conditions to satisfy the qualitylevel minimize the mean of the optimal objectives andincrease the robustness and reliability of the mean [22]

This paper sets the lower limit of the Sigma level ofrandom variables and responses to 6120590 The mean of theglobal kinematic dexterity and global minimum stiffness ismaximized whereas the standard deviation of the globalkinematic dexterity and the global minimum stiffness is min-imizedThe robust optimization designmodel is summarizedas follows

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

1198913(X) = 120583 (119892max)

1198914 (X) = 120583 (119896max)

min 1198915 (X) = 120590 (119892max)

1198916 (X) = 120590 (119892max)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

Performance calculation

Robust optimization

Robust analysis

Figure 5 Flow diagram of the Six-Sigma robust design and opti-mization

119896min ge 3 times 104

Sigma Level (ℎ 119904 119905 119896119895 119896min) ge 6

(24)

where 120583(119892max) and 120590(119892max) are the mean and varianceof the maximum payload capability respectively The meanand variance are similar to the others Sigma Level(ℎ 119904119905 119896119895 119896min) denotes the Sigma level value of the parameters(ie ℎ 119904 119905 119896119895 and 119896min)

When setting the optimization objectives the scale factorof the standard deviation should be set to 001 to ensure that

8 Journal of Industrial Engineering

550 555 560 565 570 575 5804

402

404

406

408

41

412

414

Pareto solutionsThe optimal solution

Mean of g max

gm

axVa

rianc

e of

(a) Pareto frontier of the mean and variance of the maximum payloadcapability

305 31 315 32 325 33 3354560

4580

4600

4620

4640

4660

4680

Pareto solutionsThe optimal solution

times105Mean of k max

Varia

nce o

fkm

ax(b) Pareto frontier of the mean and variance of the maximum stiffness

Figure 6 Pareto frontier

the standard deviation and mean have the same order ofmagnitude

The Six-Sigma robust design optimization flow diagramis shown in Figure 5 During the procedure the Six-Sigmacomponent robust analysis is used to analyze the single designpoint robustnessThe optimization design component robustoptimization is employed to perform the optimization designon the basis of robustness analysis If the quality of the singledesign point cannot satisfy the quality level requirementsIsight will continue to improve the optimization variables forthe quality analysis of the next group otherwise the resultswill be exported and the robust optimization is completedThepresent study selects the nondominated sorting evolutionstrategy namely NSGA-II as the optimization algorithm

Some genetic parameters and operators are set as follows[23]

The population size 4The generation number 10Crossover probability 09Crossover distribution index 10Mutation distribution index 20Max failed runs 5Failed run penalty value 1011986430Failed run objective value 1011986430

We obtain the Pareto solutions and frontier after the Six-Sigma robust optimization design The Pareto frontier ofthe mean as well as the variance of the maximum payloadcapacity and maximum stiffness is illustrated in Figure 6The relationship among the optimization objectives from thePareto solutions is observed According to this relationship

and the practical application requirements we select theoptimal solution for the multiobjective design problems andchoose the point of optimal solution (Figure 6) In view ofreliability and quality level theory we can obtain the qualitylevel mean and variance of each response after the robustoptimization through the post-processing functions of Isightsoftware And the results are listed in Table 3

Comparing the quality level with robust optimizationthat is variable quality level of ℎ 119904 119905 is 0676 7142 0755respectively Performance indexes quality level of 119896119895 119896min is1187 4161 respectively After robust optimization the qualitylevel of each response is greater than six

8 Result Analysis of the Six-Sigma Robust ofthe Parallel Adjusting Manipulator

By taking the global kinematic dexterity index as an examplethe probability distribution and Six-Sigma level were com-pared before and after optimization (Figure 7) We intuitivelyobtained the probability distribution mean standard devia-tion and Six-Sigma level of each response (Figure 7)

We rearranged and listed the reliability of each variablebefore and after optimization (Table 4) to concretely illustratethe difference between the reliability and defective qualityper million before and after the robust analysisThe defectiveproducts per million are shown in Table 5

The parameter variables (ie ℎ and 119905) and quality con-straints (ie global kinematic dexterity 119896119895) (Tables 4 and 5)greatly improves after optimization However the reliabilityof the other parameter variables only improved slightlyBy contrast the defective quality per million improvedconsiderably (Table 5) In the long term the adjustingmanipulator designed andmanufacturedwith optimal designvariables has fewer defective products permillion than before

Journal of Industrial Engineering 9

0

20

40

60

80

100

026 028 03 032 034 036

Prob

abili

ty

Quality 1187 sigmaMean 0305Std dev 0007

L bound 03

kj

ndash0722 sigma

(a) Before robust optimization

0

20

40

60

80

100

032 034 036 038 04 042

Prob

abili

ty

Quality 8 sigmaMean 0363

L bound 03ndash9901 sigma

kj

(b) After robust optimization

Figure 7 Probability distribution of the global kinematic dexterity before and after robust optimization

Table 4 Comparison of the reliability before and after optimization

Before robust optimization After robust optimizationℎ 50099 99999119904 100 100119905 54980 100119896119895 76472 100119896min 99997 100

Table 5 Comparison of the defective quality per million before andafter optimization

Before robust optimization After robust optimizationℎ 499002670 2069119890 minus 4

119904 9213119890 minus 7 0119905 450199849 0119896119895 235279150 5773119890 minus 9

119896min 31653 3930119890 minus 8

optimization This result is based on the assumption that thedesign variables manufacturing and installation error of thebase and moving platform can be effectively controlled thusgreatly improving the resistance ability of the uncertaintyfactor disturbance in the whole manufacturing process Datain Tables 4 and 5 show that the Six-Sigma robust optimizationdesign based on Isight is effective and feasible

9 Conclusions

The parallel adjusting manipulator applied in solar gatherpanels is the research objective of this paper Determinacyoptimization analysis was performed with consideration tothe performance indexes By considering the influence of the

uncertainty factors implemented in the robust analysis andoptimization the following conclusions are drawn

(1) The parallel adjusting manipulator was designedby using multiobjective optimization design on thebasis of the inverse kinematic solution The perfor-mance indexes are established the optimal solutionis obtained and the determinacy optimization iscompleted

(2) Robustness analysis is performed on the basis ofdeterminacy optimization design Some parametervariables and performance indexes are close to theconstraint boundary The Sigma level is low and thedefective quality permillion is highThe implementa-tion of the Six-Sigma robust optimization is necessaryto complete the nondeterminacy optimization design

(3) The comparison of the manipulator before and afterthe robust design optimization shows that the newone has a higher Sigma level Furthermore the com-parison shows a lower defective quality per millionthan before optimizationThe optimization objectivesare optimal The mean and standard deviation arein their minimum This result indicates that in thelong term themanipulator after optimization is moresuitable for batch production applications than themanipulator before optimization The robust opti-mization based on Isight is efficient and convenientfor operation with accurate results These results areapplicable to the robust design optimization of large-scale and complex manipulators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

4 Journal of Industrial Engineering

using (1) The square of the length is a function about angles120595 120579 and 120593 Consider

1198972

119894=

1003817100381710038171003817l1198941003817100381710038171003817

2=

10038171003817100381710038171003817

ApB119894 minusApA119894

10038171003817100381710038171003817

2

= Θ119894(120595 120579 120593) (3)

The derivative of both sides of (3) about time leads to thefollowing

A l = B (4)

namely

l = Aminus1B = J (5)

where J = Aminus1B is the Jacobian matrix velocity of the parallelmanipulator with

A =[[

[

21198971

0 0

0 21198972

0

0 0 21198973

]]

]

B =

[[[[[[[[

[

120597Θ1(120595 120579 120593)

120597120595

120597Θ1(120595 120579 120593)

120597120579

120597Θ1(120595 120579 120593)

120597120593

120597Θ2(120595 120579 120593)

120597120595

120597Θ2(120595 120579 120593)

120597120579

120597Θ2(120595 120579 120593)

120597120593

120597Θ3(120595 120579 120593)

120597120595

120597Θ3(120595 120579 120593)

120597120579

120597Θ3(120595 120579 120593)

120597120593

]]]]]]]]

]

= [ 120579 ]119879

(6)

32 Passive Constraint Chain Forward Model The passiveconstraint chain of the 4-UPS-S parallel manipulator isan open-kinematics chain with three-rotation degrees offreedom For analysis we separate the spherical joint intothree revolute joints by adopting the Denavit-Hartenberg(D-H) method We use three rotation angles (ie 120593

1 1205932

and 1205933) to describe its orientation The coordinate system of

the adjacent links is settled by using the D-H method Theoverlap coordinate systemorigin is separated for convenienceof illustration (Figure 3)

The transformation matrix between adjacent links isexpressed in terms of the D-H method

A119894=

[[[[[

[

119888120579119894

minus119904120579119894119888120572119894

119904120579119894119904120572119894

119886119894119888120579119894

119904120579119894

119888120579119894119888120572119894

minus119888120579119894119904120572119894

119886119894119904120579119894

0 119904120572119894

119888120572119894

119889119894

0 0 0 1

]]]]]

]

(7)

The 120579119894 119889119894 119886119894 and 120572

119894parameter values are shown in Table 1

The transformation matrix from the coordinate system119874-119883119860119884119860119885119860to the coordinate system119874-119883

119861119884119861119885119861is described

as follows

T = A0A1A2A3 (8)

O

ZA

ZB

Z1

Z2

Z3

XA

X1

X2X3

XB

YA

Figure 3 D-H notation for the passive constraint chain

Table 1 D-H parameter for the passive constraint chain

120579119894

119889119894

119886119894

120572119894

0 0 0 0 01 90

∘+ 1206011

0 0 90∘

2 90∘+ 1206012

0 0 90∘

3 1206013

0 0 0

Combining (2) and (8) we obtain the following functionrelationship between 120601

1 1206012 and 120601

3and 120595 120579 and 120593

120601119894= Γ119894(120595 120579 120593) (119894 = 1 2 3) (9)

By deviating both sides of (9) we generate the following

Ψ = J119904sdot (10)

where

Ψ = [ 1206011

1206012

1206013]119879

J119904=

[[[[[[[[

[

120597Γ1(120595 120579 120593)

120597120595

120597Γ1(120595 120579 120593)

120597120579

120597Γ1(120595 120579 120593)

120597120593

120597Γ2(120595 120579 120593)

120597120595

120597Γ2(120595 120579 120593)

120597120579

120597Γ2(120595 120579 120593)

120597120593

120597Γ3(120595 120579 120593)

120597120595

120597Γ3(120595 120579 120593)

120597120579

120597Γ3(120595 120579 120593)

120597120593

]]]]]]]]

]

(11)

where J119904is the Jacobian matrix of the middle passive chain

between the input angular velocity and output angular veloc-ity

4 Establishment of the Parallel ManipulatorPerformance Indexes

41 Global Kinematic Dexterity The4-UPS-S parallel manip-ulator is a pure rotationmanipulatorWe adopt the reciprocalof the condition number of the Jacobian matrix to measure

Journal of Industrial Engineering 5

the dexterity and transmission relationship of the manipu-lator between the input and output The reciprocal of thecondition number of the Jacobianmatrix is defined as follows

120581119869=

radic120582max (J119879J)

radic120582min (J119879J) (12)

where 120582max(J119879J) and 120582min(J119879J) are the maximum and mini-mum eigenvalues of J119879J respectively

The global dexterity index is employed to describe themean condition value of the Jacobian matrix in the entireworkspace [17]

119896119895 =

int119882

1120581119869119889119882

int119882

119889119882 (13)

where119882 is the parallel manipulator workspaceThis paper employs the output angle to describe the

orientation workspace of the parallel manipulator Accordingto the application requirements the parallel manipulatorworkspace is expressed as follows

minus30∘le 120595 le 30

∘

minus30∘le 120579 le 30

∘

minus10∘le 120593 le 10

∘

(14)

The global dexterity index is in the range 0 le 119896119895 le

1 A global dexterity index closer to one corresponds tobetter dexterity and control precision We generally calculatethe condition number 120581

119869of the Jacobian matrix in different

workspace positions and orientations to increase indicatoroperabilityThemean of the reciprocal of the Jacobian matrixis calculated as a global kinematic dexterity index 119896119895

42 Global Stiffness Performance Index The friction force ofthe hinges and joints is ignored for brevity According to thevirtual work principle one obtains the following [18]

120591119879 l + 120591119879V Ψ = F119879 (15)

where 120591 = 120594Δl is the actuator force vector applied at eachactuated joint 120591V = 120594VΔ120595 is the passive chain internal torqueand F is the torque applied to the moving platform Themoving platform is assumed to have no gravitational forcesacting on any of the intermediate links The gravitationalforces are generally neglected in industrial applications120594 and120594V are the equivalent stiffness of the active and passive chainsrespectively

Substituting (5) and (10) into (15) yields the following

F = (J119879120594J + J119879119904120594VJ119904)Δ120596 (16)

where

K = J119879120594J + J119879119904120594VJ119904 (17)

where K is the parallel manipulatorrsquos whole stiffness matrix

According to vector extreme calculation theory when themoving platform deformation reaches Δ120596 = Δ120596

119879Δ120596 = 1

the followings are generated

119870max = radic120582max (K119879K) (18)

119870min = radic120582min (K119879K) (19)

where 120582max(K119879K) and 120582min(K119879K) are the maximum andminimum eigenvalues of matrix K119879K respectively

To evaluate the parallel manipulator stiffness perfor-mance in the entire workspace the global stiffness perfor-mance index is similarly defined as follows

119896max =

int119882

119870max 119889119882

int119882

119889119882

119896min =

int119882

119870min 119889119882

int119882

119889119882

(20)

where 119882 is the parallel manipulator workspace We cancompute the stiffness index by using the mean method

Larger global stiffness evaluation indexes 119896max and119896min correspond to better parallel manipulator stiffness

43 Global Maximum Payload Capability Similar to (18) themaximumpayload capability performance index is defined asfollows

119866max = radic120582119865max (G119879G) (21)

where 120582119865max(G119879G) is the maximum eigenvalue of matrix

G119879G The relationship is G = J119879The global maximum payload capability performance

index is written as follows

119892max =

int119882

119866max 119889119882

int119882

119889119882 (22)

where119882 is the parallel manipulator workspaceA greater global maximum payload capability-evaluation

index 119892max corresponds to better parallel manipulatorpayload capability

5 Deterministic Optimization Design Modelof the Parallel Adjusting Manipulator

This study maximizes the global payload capacity and globalmaximum stiffness to improve the parallel adjusting manip-ulator of the loading ability that bears the solar equipmentweight The global kinematic dexterity and minimum stiff-ness are the constraint conditionsThe design variables are ℎ119904 and 119905 (Figure 2(a))

6 Journal of Industrial Engineering

Table 2 Six-Sigma-based robust analysis description

Random design variablesℎ 120583 = 200005 120590 = 1 times 120583

119904 120583 = 129096 120590 = 1 times 120583

119905 120583 = 0799 120590 = 1 times 120583

Random noises The radius 119903119860of the base 120583 = 500mm 120590 = 1 times 120583

The radius 119903119861of the moving platform 120583 = 400mm 120590 = 1 times 120583

Quality constraints The global kinematic dexterity 119896119895 ge 03

The global minimum stiffness 119896min ge 3 times 104

510

530

550

570

590

260000 280000 300000 320000 340000 360000k max

gm

ax

Figure 4 History points of the determinacy optimization

The deterministic optimization design model is repre-sented as

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

119896min ge 3 times 104

(23)

where X = [ℎ 119904 119905]119879 denotes the design variables and

1198911(X) and 119891

2(X) are the global payload capability and global

maximum stiffness index respectivelyWe employed the Simcode components to integrate

MATLAB and calculate the parallel manipulator evaluationindexes The NSGA-II optimization algorithm is adopted forthe present study The population size is 16 rate of crossoveris set to 09 rate of mutation is set to 01 rate of migrationis set to 01 and the generation number is 15 The designvariables constraint conditions and optimization objectivesare selected on the basis of the application requirement [2 19]We run the software platform once the parameter setting iscompleted

The deterministic optimization design is a multiobjectiveoptimization design problem The history points of the twoobjective functions in the solving process are plotted inFigure 4 The two objectives increase or decrease at the sametime The Pareto optimal solution is a convenient solutionfor this multiobjective optimization design problem whenthe global stiffness and global payload capacity obtain themaximum values at the same time

The deterministic optimization design result and initialscheme comparison is presented in Table 3The deterministicoptimization design analysis solution results show that theoptimization objectives (ie global maximum stiffness andglobal payload capacity) have a considerable improvementHowever the parameters (ie ℎ 119904 119905 and 119896119895) are all close tothe constraint boundary at the same time If the uncertaintyfactor influences are considered the deterministic optimiza-tion scheme violates the constraint conditions Thereforeperforming a robust analysis that evaluates the quality andreliability level of the deterministic optimization designscheme is necessary [20 21]

6 Six-Sigma Robust Analysis

In this study some uncertainty factors include the designvariables ℎ 119904 and 119905 The radii 119903

119860and 119903

119861of the base and

moving platform follow the same normal distribution Theoptimal design point value of the deterministic optimizationdesign is considered themean of the randomdesign variablesThe standard deviation is presented in Table 2 120590 = 1 times 120583

represents the 1 coefficient of variation (ie 120590120583 = 001)The robust analysis of the 4-UPS-S parallel manipulator isimplemented by adopting the Six-Sigma component based onIsight software

The Six-Sigma robust analysis results are concretely sum-marized in Table 3 The input variables ℎ and 119905 do not reachthe One Sigma level The reliability is approximately 50which is considered low The global kinematic dexterity andthe global minimum stiffness have a relatively high reliabilityThe minimum stiffness attains a reliability of 9999 butcannot reach the Six-Sigma level The global minimumstiffness Sigma level is 4161 In the long term when the 15120590

quality shift occurs in the system the defective products permillion with a Sigma level of four will sharply increase fromthe short term 63 times 10

minus6 to 6200 times 10minus6 The quality levels

of ℎ and 119905 are low and do not achieve the One Sigma levelTherefore performing a robust optimization for this problemis necessary in the long term

Journal of Industrial Engineering 7

Table 3 Six-Sigma-based robust design and optimization results

Variables Initial schemeDeterminacy optimization scheme Robust optimization scheme

Determinacyoptimal solutions

Six-Sigmaanalysis

Robust optimizationanalysis

Qualitylevel

Input variables

ℎ 300 120583 200005 Sigma level 0676reliability 50099

213330 6356120590 200005 213330

119904 200 120583 129096 Sigma level 7142reliability 100

218936 80120590 129096 218936

119905 06 120583 0799 Sigma level 0755reliability 54980

0683 80120590 000799 000683

Output variables (performance indexes)

Global kinematic dexterity 119896119895 0310 120583 0305 Sigma level 1187reliability 7647

0305 7809120590 0007 0007

Global minimum stiffness 119896min 30045 120583 36880 Sigma level 4161reliability 9999

429917 7563120590 1721 1739

Global maximum payload capability 119892max 538736 120583 591702 574223120590 4276 4014

Global maximum stiffness 119896max 290141 120583 350500 329940120590 5084 4624

7 Six-Sigma Robust Optimization(Uncertainty Optimization)

The essence of the Six-Sigma robust optimization is to addthe mean the variance of the response in optimizationobjectives and the upper quality level limit of the randomvariables in the constraint conditions to satisfy the qualitylevel minimize the mean of the optimal objectives andincrease the robustness and reliability of the mean [22]

This paper sets the lower limit of the Sigma level ofrandom variables and responses to 6120590 The mean of theglobal kinematic dexterity and global minimum stiffness ismaximized whereas the standard deviation of the globalkinematic dexterity and the global minimum stiffness is min-imizedThe robust optimization designmodel is summarizedas follows

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

1198913(X) = 120583 (119892max)

1198914 (X) = 120583 (119896max)

min 1198915 (X) = 120590 (119892max)

1198916 (X) = 120590 (119892max)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

Performance calculation

Robust optimization

Robust analysis

Figure 5 Flow diagram of the Six-Sigma robust design and opti-mization

119896min ge 3 times 104

Sigma Level (ℎ 119904 119905 119896119895 119896min) ge 6

(24)

where 120583(119892max) and 120590(119892max) are the mean and varianceof the maximum payload capability respectively The meanand variance are similar to the others Sigma Level(ℎ 119904119905 119896119895 119896min) denotes the Sigma level value of the parameters(ie ℎ 119904 119905 119896119895 and 119896min)

When setting the optimization objectives the scale factorof the standard deviation should be set to 001 to ensure that

8 Journal of Industrial Engineering

550 555 560 565 570 575 5804

402

404

406

408

41

412

414

Pareto solutionsThe optimal solution

Mean of g max

gm

axVa

rianc

e of

(a) Pareto frontier of the mean and variance of the maximum payloadcapability

305 31 315 32 325 33 3354560

4580

4600

4620

4640

4660

4680

Pareto solutionsThe optimal solution

times105Mean of k max

Varia

nce o

fkm

ax(b) Pareto frontier of the mean and variance of the maximum stiffness

Figure 6 Pareto frontier

the standard deviation and mean have the same order ofmagnitude

The Six-Sigma robust design optimization flow diagramis shown in Figure 5 During the procedure the Six-Sigmacomponent robust analysis is used to analyze the single designpoint robustnessThe optimization design component robustoptimization is employed to perform the optimization designon the basis of robustness analysis If the quality of the singledesign point cannot satisfy the quality level requirementsIsight will continue to improve the optimization variables forthe quality analysis of the next group otherwise the resultswill be exported and the robust optimization is completedThepresent study selects the nondominated sorting evolutionstrategy namely NSGA-II as the optimization algorithm

Some genetic parameters and operators are set as follows[23]

The population size 4The generation number 10Crossover probability 09Crossover distribution index 10Mutation distribution index 20Max failed runs 5Failed run penalty value 1011986430Failed run objective value 1011986430

We obtain the Pareto solutions and frontier after the Six-Sigma robust optimization design The Pareto frontier ofthe mean as well as the variance of the maximum payloadcapacity and maximum stiffness is illustrated in Figure 6The relationship among the optimization objectives from thePareto solutions is observed According to this relationship

and the practical application requirements we select theoptimal solution for the multiobjective design problems andchoose the point of optimal solution (Figure 6) In view ofreliability and quality level theory we can obtain the qualitylevel mean and variance of each response after the robustoptimization through the post-processing functions of Isightsoftware And the results are listed in Table 3

Comparing the quality level with robust optimizationthat is variable quality level of ℎ 119904 119905 is 0676 7142 0755respectively Performance indexes quality level of 119896119895 119896min is1187 4161 respectively After robust optimization the qualitylevel of each response is greater than six

8 Result Analysis of the Six-Sigma Robust ofthe Parallel Adjusting Manipulator

By taking the global kinematic dexterity index as an examplethe probability distribution and Six-Sigma level were com-pared before and after optimization (Figure 7) We intuitivelyobtained the probability distribution mean standard devia-tion and Six-Sigma level of each response (Figure 7)

We rearranged and listed the reliability of each variablebefore and after optimization (Table 4) to concretely illustratethe difference between the reliability and defective qualityper million before and after the robust analysisThe defectiveproducts per million are shown in Table 5

The parameter variables (ie ℎ and 119905) and quality con-straints (ie global kinematic dexterity 119896119895) (Tables 4 and 5)greatly improves after optimization However the reliabilityof the other parameter variables only improved slightlyBy contrast the defective quality per million improvedconsiderably (Table 5) In the long term the adjustingmanipulator designed andmanufacturedwith optimal designvariables has fewer defective products permillion than before

Journal of Industrial Engineering 9

0

20

40

60

80

100

026 028 03 032 034 036

Prob

abili

ty

Quality 1187 sigmaMean 0305Std dev 0007

L bound 03

kj

ndash0722 sigma

(a) Before robust optimization

0

20

40

60

80

100

032 034 036 038 04 042

Prob

abili

ty

Quality 8 sigmaMean 0363

L bound 03ndash9901 sigma

kj

(b) After robust optimization

Figure 7 Probability distribution of the global kinematic dexterity before and after robust optimization

Table 4 Comparison of the reliability before and after optimization

Before robust optimization After robust optimizationℎ 50099 99999119904 100 100119905 54980 100119896119895 76472 100119896min 99997 100

Table 5 Comparison of the defective quality per million before andafter optimization

Before robust optimization After robust optimizationℎ 499002670 2069119890 minus 4

119904 9213119890 minus 7 0119905 450199849 0119896119895 235279150 5773119890 minus 9

119896min 31653 3930119890 minus 8

optimization This result is based on the assumption that thedesign variables manufacturing and installation error of thebase and moving platform can be effectively controlled thusgreatly improving the resistance ability of the uncertaintyfactor disturbance in the whole manufacturing process Datain Tables 4 and 5 show that the Six-Sigma robust optimizationdesign based on Isight is effective and feasible

9 Conclusions

The parallel adjusting manipulator applied in solar gatherpanels is the research objective of this paper Determinacyoptimization analysis was performed with consideration tothe performance indexes By considering the influence of the

uncertainty factors implemented in the robust analysis andoptimization the following conclusions are drawn

(1) The parallel adjusting manipulator was designedby using multiobjective optimization design on thebasis of the inverse kinematic solution The perfor-mance indexes are established the optimal solutionis obtained and the determinacy optimization iscompleted

(2) Robustness analysis is performed on the basis ofdeterminacy optimization design Some parametervariables and performance indexes are close to theconstraint boundary The Sigma level is low and thedefective quality permillion is highThe implementa-tion of the Six-Sigma robust optimization is necessaryto complete the nondeterminacy optimization design

(3) The comparison of the manipulator before and afterthe robust design optimization shows that the newone has a higher Sigma level Furthermore the com-parison shows a lower defective quality per millionthan before optimizationThe optimization objectivesare optimal The mean and standard deviation arein their minimum This result indicates that in thelong term themanipulator after optimization is moresuitable for batch production applications than themanipulator before optimization The robust opti-mization based on Isight is efficient and convenientfor operation with accurate results These results areapplicable to the robust design optimization of large-scale and complex manipulators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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RotatingMachinery

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Navigation and Observation

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DistributedSensor Networks

International Journal of

Journal of Industrial Engineering 5

the dexterity and transmission relationship of the manipu-lator between the input and output The reciprocal of thecondition number of the Jacobianmatrix is defined as follows

120581119869=

radic120582max (J119879J)

radic120582min (J119879J) (12)

where 120582max(J119879J) and 120582min(J119879J) are the maximum and mini-mum eigenvalues of J119879J respectively

The global dexterity index is employed to describe themean condition value of the Jacobian matrix in the entireworkspace [17]

119896119895 =

int119882

1120581119869119889119882

int119882

119889119882 (13)

where119882 is the parallel manipulator workspaceThis paper employs the output angle to describe the

orientation workspace of the parallel manipulator Accordingto the application requirements the parallel manipulatorworkspace is expressed as follows

minus30∘le 120595 le 30

∘

minus30∘le 120579 le 30

∘

minus10∘le 120593 le 10

∘

(14)

The global dexterity index is in the range 0 le 119896119895 le

1 A global dexterity index closer to one corresponds tobetter dexterity and control precision We generally calculatethe condition number 120581

119869of the Jacobian matrix in different

workspace positions and orientations to increase indicatoroperabilityThemean of the reciprocal of the Jacobian matrixis calculated as a global kinematic dexterity index 119896119895

42 Global Stiffness Performance Index The friction force ofthe hinges and joints is ignored for brevity According to thevirtual work principle one obtains the following [18]

120591119879 l + 120591119879V Ψ = F119879 (15)

where 120591 = 120594Δl is the actuator force vector applied at eachactuated joint 120591V = 120594VΔ120595 is the passive chain internal torqueand F is the torque applied to the moving platform Themoving platform is assumed to have no gravitational forcesacting on any of the intermediate links The gravitationalforces are generally neglected in industrial applications120594 and120594V are the equivalent stiffness of the active and passive chainsrespectively

Substituting (5) and (10) into (15) yields the following

F = (J119879120594J + J119879119904120594VJ119904)Δ120596 (16)

where

K = J119879120594J + J119879119904120594VJ119904 (17)

where K is the parallel manipulatorrsquos whole stiffness matrix

According to vector extreme calculation theory when themoving platform deformation reaches Δ120596 = Δ120596

119879Δ120596 = 1

the followings are generated

119870max = radic120582max (K119879K) (18)

119870min = radic120582min (K119879K) (19)

where 120582max(K119879K) and 120582min(K119879K) are the maximum andminimum eigenvalues of matrix K119879K respectively

To evaluate the parallel manipulator stiffness perfor-mance in the entire workspace the global stiffness perfor-mance index is similarly defined as follows

119896max =

int119882

119870max 119889119882

int119882

119889119882

119896min =

int119882

119870min 119889119882

int119882

119889119882

(20)

where 119882 is the parallel manipulator workspace We cancompute the stiffness index by using the mean method

Larger global stiffness evaluation indexes 119896max and119896min correspond to better parallel manipulator stiffness

43 Global Maximum Payload Capability Similar to (18) themaximumpayload capability performance index is defined asfollows

119866max = radic120582119865max (G119879G) (21)

where 120582119865max(G119879G) is the maximum eigenvalue of matrix

G119879G The relationship is G = J119879The global maximum payload capability performance

index is written as follows

119892max =

int119882

119866max 119889119882

int119882

119889119882 (22)

where119882 is the parallel manipulator workspaceA greater global maximum payload capability-evaluation

index 119892max corresponds to better parallel manipulatorpayload capability

5 Deterministic Optimization Design Modelof the Parallel Adjusting Manipulator

This study maximizes the global payload capacity and globalmaximum stiffness to improve the parallel adjusting manip-ulator of the loading ability that bears the solar equipmentweight The global kinematic dexterity and minimum stiff-ness are the constraint conditionsThe design variables are ℎ119904 and 119905 (Figure 2(a))

6 Journal of Industrial Engineering

Table 2 Six-Sigma-based robust analysis description

Random design variablesℎ 120583 = 200005 120590 = 1 times 120583

119904 120583 = 129096 120590 = 1 times 120583

119905 120583 = 0799 120590 = 1 times 120583

Random noises The radius 119903119860of the base 120583 = 500mm 120590 = 1 times 120583

The radius 119903119861of the moving platform 120583 = 400mm 120590 = 1 times 120583

Quality constraints The global kinematic dexterity 119896119895 ge 03

The global minimum stiffness 119896min ge 3 times 104

510

530

550

570

590

260000 280000 300000 320000 340000 360000k max

gm

ax

Figure 4 History points of the determinacy optimization

The deterministic optimization design model is repre-sented as

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

119896min ge 3 times 104

(23)

where X = [ℎ 119904 119905]119879 denotes the design variables and

1198911(X) and 119891

2(X) are the global payload capability and global

maximum stiffness index respectivelyWe employed the Simcode components to integrate

MATLAB and calculate the parallel manipulator evaluationindexes The NSGA-II optimization algorithm is adopted forthe present study The population size is 16 rate of crossoveris set to 09 rate of mutation is set to 01 rate of migrationis set to 01 and the generation number is 15 The designvariables constraint conditions and optimization objectivesare selected on the basis of the application requirement [2 19]We run the software platform once the parameter setting iscompleted

The deterministic optimization design is a multiobjectiveoptimization design problem The history points of the twoobjective functions in the solving process are plotted inFigure 4 The two objectives increase or decrease at the sametime The Pareto optimal solution is a convenient solutionfor this multiobjective optimization design problem whenthe global stiffness and global payload capacity obtain themaximum values at the same time

The deterministic optimization design result and initialscheme comparison is presented in Table 3The deterministicoptimization design analysis solution results show that theoptimization objectives (ie global maximum stiffness andglobal payload capacity) have a considerable improvementHowever the parameters (ie ℎ 119904 119905 and 119896119895) are all close tothe constraint boundary at the same time If the uncertaintyfactor influences are considered the deterministic optimiza-tion scheme violates the constraint conditions Thereforeperforming a robust analysis that evaluates the quality andreliability level of the deterministic optimization designscheme is necessary [20 21]

6 Six-Sigma Robust Analysis

In this study some uncertainty factors include the designvariables ℎ 119904 and 119905 The radii 119903

119860and 119903

119861of the base and

moving platform follow the same normal distribution Theoptimal design point value of the deterministic optimizationdesign is considered themean of the randomdesign variablesThe standard deviation is presented in Table 2 120590 = 1 times 120583

represents the 1 coefficient of variation (ie 120590120583 = 001)The robust analysis of the 4-UPS-S parallel manipulator isimplemented by adopting the Six-Sigma component based onIsight software

The Six-Sigma robust analysis results are concretely sum-marized in Table 3 The input variables ℎ and 119905 do not reachthe One Sigma level The reliability is approximately 50which is considered low The global kinematic dexterity andthe global minimum stiffness have a relatively high reliabilityThe minimum stiffness attains a reliability of 9999 butcannot reach the Six-Sigma level The global minimumstiffness Sigma level is 4161 In the long term when the 15120590

quality shift occurs in the system the defective products permillion with a Sigma level of four will sharply increase fromthe short term 63 times 10

minus6 to 6200 times 10minus6 The quality levels

of ℎ and 119905 are low and do not achieve the One Sigma levelTherefore performing a robust optimization for this problemis necessary in the long term

Journal of Industrial Engineering 7

Table 3 Six-Sigma-based robust design and optimization results

Variables Initial schemeDeterminacy optimization scheme Robust optimization scheme

Determinacyoptimal solutions

Six-Sigmaanalysis

Robust optimizationanalysis

Qualitylevel

Input variables

ℎ 300 120583 200005 Sigma level 0676reliability 50099

213330 6356120590 200005 213330

119904 200 120583 129096 Sigma level 7142reliability 100

218936 80120590 129096 218936

119905 06 120583 0799 Sigma level 0755reliability 54980

0683 80120590 000799 000683

Output variables (performance indexes)

Global kinematic dexterity 119896119895 0310 120583 0305 Sigma level 1187reliability 7647

0305 7809120590 0007 0007

Global minimum stiffness 119896min 30045 120583 36880 Sigma level 4161reliability 9999

429917 7563120590 1721 1739

Global maximum payload capability 119892max 538736 120583 591702 574223120590 4276 4014

Global maximum stiffness 119896max 290141 120583 350500 329940120590 5084 4624

7 Six-Sigma Robust Optimization(Uncertainty Optimization)

The essence of the Six-Sigma robust optimization is to addthe mean the variance of the response in optimizationobjectives and the upper quality level limit of the randomvariables in the constraint conditions to satisfy the qualitylevel minimize the mean of the optimal objectives andincrease the robustness and reliability of the mean [22]

This paper sets the lower limit of the Sigma level ofrandom variables and responses to 6120590 The mean of theglobal kinematic dexterity and global minimum stiffness ismaximized whereas the standard deviation of the globalkinematic dexterity and the global minimum stiffness is min-imizedThe robust optimization designmodel is summarizedas follows

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

1198913(X) = 120583 (119892max)

1198914 (X) = 120583 (119896max)

min 1198915 (X) = 120590 (119892max)

1198916 (X) = 120590 (119892max)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

Performance calculation

Robust optimization

Robust analysis

Figure 5 Flow diagram of the Six-Sigma robust design and opti-mization

119896min ge 3 times 104

Sigma Level (ℎ 119904 119905 119896119895 119896min) ge 6

(24)

where 120583(119892max) and 120590(119892max) are the mean and varianceof the maximum payload capability respectively The meanand variance are similar to the others Sigma Level(ℎ 119904119905 119896119895 119896min) denotes the Sigma level value of the parameters(ie ℎ 119904 119905 119896119895 and 119896min)

When setting the optimization objectives the scale factorof the standard deviation should be set to 001 to ensure that

8 Journal of Industrial Engineering

550 555 560 565 570 575 5804

402

404

406

408

41

412

414

Pareto solutionsThe optimal solution

Mean of g max

gm

axVa

rianc

e of

(a) Pareto frontier of the mean and variance of the maximum payloadcapability

305 31 315 32 325 33 3354560

4580

4600

4620

4640

4660

4680

Pareto solutionsThe optimal solution

times105Mean of k max

Varia

nce o

fkm

ax(b) Pareto frontier of the mean and variance of the maximum stiffness

Figure 6 Pareto frontier

the standard deviation and mean have the same order ofmagnitude

The Six-Sigma robust design optimization flow diagramis shown in Figure 5 During the procedure the Six-Sigmacomponent robust analysis is used to analyze the single designpoint robustnessThe optimization design component robustoptimization is employed to perform the optimization designon the basis of robustness analysis If the quality of the singledesign point cannot satisfy the quality level requirementsIsight will continue to improve the optimization variables forthe quality analysis of the next group otherwise the resultswill be exported and the robust optimization is completedThepresent study selects the nondominated sorting evolutionstrategy namely NSGA-II as the optimization algorithm

Some genetic parameters and operators are set as follows[23]

The population size 4The generation number 10Crossover probability 09Crossover distribution index 10Mutation distribution index 20Max failed runs 5Failed run penalty value 1011986430Failed run objective value 1011986430

We obtain the Pareto solutions and frontier after the Six-Sigma robust optimization design The Pareto frontier ofthe mean as well as the variance of the maximum payloadcapacity and maximum stiffness is illustrated in Figure 6The relationship among the optimization objectives from thePareto solutions is observed According to this relationship

and the practical application requirements we select theoptimal solution for the multiobjective design problems andchoose the point of optimal solution (Figure 6) In view ofreliability and quality level theory we can obtain the qualitylevel mean and variance of each response after the robustoptimization through the post-processing functions of Isightsoftware And the results are listed in Table 3

Comparing the quality level with robust optimizationthat is variable quality level of ℎ 119904 119905 is 0676 7142 0755respectively Performance indexes quality level of 119896119895 119896min is1187 4161 respectively After robust optimization the qualitylevel of each response is greater than six

8 Result Analysis of the Six-Sigma Robust ofthe Parallel Adjusting Manipulator

By taking the global kinematic dexterity index as an examplethe probability distribution and Six-Sigma level were com-pared before and after optimization (Figure 7) We intuitivelyobtained the probability distribution mean standard devia-tion and Six-Sigma level of each response (Figure 7)

We rearranged and listed the reliability of each variablebefore and after optimization (Table 4) to concretely illustratethe difference between the reliability and defective qualityper million before and after the robust analysisThe defectiveproducts per million are shown in Table 5

The parameter variables (ie ℎ and 119905) and quality con-straints (ie global kinematic dexterity 119896119895) (Tables 4 and 5)greatly improves after optimization However the reliabilityof the other parameter variables only improved slightlyBy contrast the defective quality per million improvedconsiderably (Table 5) In the long term the adjustingmanipulator designed andmanufacturedwith optimal designvariables has fewer defective products permillion than before

Journal of Industrial Engineering 9

0

20

40

60

80

100

026 028 03 032 034 036

Prob

abili

ty

Quality 1187 sigmaMean 0305Std dev 0007

L bound 03

kj

ndash0722 sigma

(a) Before robust optimization

0

20

40

60

80

100

032 034 036 038 04 042

Prob

abili

ty

Quality 8 sigmaMean 0363

L bound 03ndash9901 sigma

kj

(b) After robust optimization

Figure 7 Probability distribution of the global kinematic dexterity before and after robust optimization

Table 4 Comparison of the reliability before and after optimization

Before robust optimization After robust optimizationℎ 50099 99999119904 100 100119905 54980 100119896119895 76472 100119896min 99997 100

Table 5 Comparison of the defective quality per million before andafter optimization

Before robust optimization After robust optimizationℎ 499002670 2069119890 minus 4

119904 9213119890 minus 7 0119905 450199849 0119896119895 235279150 5773119890 minus 9

119896min 31653 3930119890 minus 8

optimization This result is based on the assumption that thedesign variables manufacturing and installation error of thebase and moving platform can be effectively controlled thusgreatly improving the resistance ability of the uncertaintyfactor disturbance in the whole manufacturing process Datain Tables 4 and 5 show that the Six-Sigma robust optimizationdesign based on Isight is effective and feasible

9 Conclusions

The parallel adjusting manipulator applied in solar gatherpanels is the research objective of this paper Determinacyoptimization analysis was performed with consideration tothe performance indexes By considering the influence of the

uncertainty factors implemented in the robust analysis andoptimization the following conclusions are drawn

(1) The parallel adjusting manipulator was designedby using multiobjective optimization design on thebasis of the inverse kinematic solution The perfor-mance indexes are established the optimal solutionis obtained and the determinacy optimization iscompleted

(2) Robustness analysis is performed on the basis ofdeterminacy optimization design Some parametervariables and performance indexes are close to theconstraint boundary The Sigma level is low and thedefective quality permillion is highThe implementa-tion of the Six-Sigma robust optimization is necessaryto complete the nondeterminacy optimization design

(3) The comparison of the manipulator before and afterthe robust design optimization shows that the newone has a higher Sigma level Furthermore the com-parison shows a lower defective quality per millionthan before optimizationThe optimization objectivesare optimal The mean and standard deviation arein their minimum This result indicates that in thelong term themanipulator after optimization is moresuitable for batch production applications than themanipulator before optimization The robust opti-mization based on Isight is efficient and convenientfor operation with accurate results These results areapplicable to the robust design optimization of large-scale and complex manipulators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Industrial Engineering

Table 2 Six-Sigma-based robust analysis description

Random design variablesℎ 120583 = 200005 120590 = 1 times 120583

119904 120583 = 129096 120590 = 1 times 120583

119905 120583 = 0799 120590 = 1 times 120583

Random noises The radius 119903119860of the base 120583 = 500mm 120590 = 1 times 120583

The radius 119903119861of the moving platform 120583 = 400mm 120590 = 1 times 120583

Quality constraints The global kinematic dexterity 119896119895 ge 03

The global minimum stiffness 119896min ge 3 times 104

510

530

550

570

590

260000 280000 300000 320000 340000 360000k max

gm

ax

Figure 4 History points of the determinacy optimization

The deterministic optimization design model is repre-sented as

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

119896min ge 3 times 104

(23)

where X = [ℎ 119904 119905]119879 denotes the design variables and

1198911(X) and 119891

2(X) are the global payload capability and global

maximum stiffness index respectivelyWe employed the Simcode components to integrate

MATLAB and calculate the parallel manipulator evaluationindexes The NSGA-II optimization algorithm is adopted forthe present study The population size is 16 rate of crossoveris set to 09 rate of mutation is set to 01 rate of migrationis set to 01 and the generation number is 15 The designvariables constraint conditions and optimization objectivesare selected on the basis of the application requirement [2 19]We run the software platform once the parameter setting iscompleted

The deterministic optimization design is a multiobjectiveoptimization design problem The history points of the twoobjective functions in the solving process are plotted inFigure 4 The two objectives increase or decrease at the sametime The Pareto optimal solution is a convenient solutionfor this multiobjective optimization design problem whenthe global stiffness and global payload capacity obtain themaximum values at the same time

The deterministic optimization design result and initialscheme comparison is presented in Table 3The deterministicoptimization design analysis solution results show that theoptimization objectives (ie global maximum stiffness andglobal payload capacity) have a considerable improvementHowever the parameters (ie ℎ 119904 119905 and 119896119895) are all close tothe constraint boundary at the same time If the uncertaintyfactor influences are considered the deterministic optimiza-tion scheme violates the constraint conditions Thereforeperforming a robust analysis that evaluates the quality andreliability level of the deterministic optimization designscheme is necessary [20 21]

6 Six-Sigma Robust Analysis

In this study some uncertainty factors include the designvariables ℎ 119904 and 119905 The radii 119903

119860and 119903

119861of the base and

moving platform follow the same normal distribution Theoptimal design point value of the deterministic optimizationdesign is considered themean of the randomdesign variablesThe standard deviation is presented in Table 2 120590 = 1 times 120583

represents the 1 coefficient of variation (ie 120590120583 = 001)The robust analysis of the 4-UPS-S parallel manipulator isimplemented by adopting the Six-Sigma component based onIsight software

The Six-Sigma robust analysis results are concretely sum-marized in Table 3 The input variables ℎ and 119905 do not reachthe One Sigma level The reliability is approximately 50which is considered low The global kinematic dexterity andthe global minimum stiffness have a relatively high reliabilityThe minimum stiffness attains a reliability of 9999 butcannot reach the Six-Sigma level The global minimumstiffness Sigma level is 4161 In the long term when the 15120590

quality shift occurs in the system the defective products permillion with a Sigma level of four will sharply increase fromthe short term 63 times 10

minus6 to 6200 times 10minus6 The quality levels

of ℎ and 119905 are low and do not achieve the One Sigma levelTherefore performing a robust optimization for this problemis necessary in the long term

Journal of Industrial Engineering 7

Table 3 Six-Sigma-based robust design and optimization results

Variables Initial schemeDeterminacy optimization scheme Robust optimization scheme

Determinacyoptimal solutions

Six-Sigmaanalysis

Robust optimizationanalysis

Qualitylevel

Input variables

ℎ 300 120583 200005 Sigma level 0676reliability 50099

213330 6356120590 200005 213330

119904 200 120583 129096 Sigma level 7142reliability 100

218936 80120590 129096 218936

119905 06 120583 0799 Sigma level 0755reliability 54980

0683 80120590 000799 000683

Output variables (performance indexes)

Global kinematic dexterity 119896119895 0310 120583 0305 Sigma level 1187reliability 7647

0305 7809120590 0007 0007

Global minimum stiffness 119896min 30045 120583 36880 Sigma level 4161reliability 9999

429917 7563120590 1721 1739

Global maximum payload capability 119892max 538736 120583 591702 574223120590 4276 4014

Global maximum stiffness 119896max 290141 120583 350500 329940120590 5084 4624

7 Six-Sigma Robust Optimization(Uncertainty Optimization)

The essence of the Six-Sigma robust optimization is to addthe mean the variance of the response in optimizationobjectives and the upper quality level limit of the randomvariables in the constraint conditions to satisfy the qualitylevel minimize the mean of the optimal objectives andincrease the robustness and reliability of the mean [22]

This paper sets the lower limit of the Sigma level ofrandom variables and responses to 6120590 The mean of theglobal kinematic dexterity and global minimum stiffness ismaximized whereas the standard deviation of the globalkinematic dexterity and the global minimum stiffness is min-imizedThe robust optimization designmodel is summarizedas follows

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

1198913(X) = 120583 (119892max)

1198914 (X) = 120583 (119896max)

min 1198915 (X) = 120590 (119892max)

1198916 (X) = 120590 (119892max)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

Performance calculation

Robust optimization

Robust analysis

Figure 5 Flow diagram of the Six-Sigma robust design and opti-mization

119896min ge 3 times 104

Sigma Level (ℎ 119904 119905 119896119895 119896min) ge 6

(24)

where 120583(119892max) and 120590(119892max) are the mean and varianceof the maximum payload capability respectively The meanand variance are similar to the others Sigma Level(ℎ 119904119905 119896119895 119896min) denotes the Sigma level value of the parameters(ie ℎ 119904 119905 119896119895 and 119896min)

When setting the optimization objectives the scale factorof the standard deviation should be set to 001 to ensure that

8 Journal of Industrial Engineering

550 555 560 565 570 575 5804

402

404

406

408

41

412

414

Pareto solutionsThe optimal solution

Mean of g max

gm

axVa

rianc

e of

(a) Pareto frontier of the mean and variance of the maximum payloadcapability

305 31 315 32 325 33 3354560

4580

4600

4620

4640

4660

4680

Pareto solutionsThe optimal solution

times105Mean of k max

Varia

nce o

fkm

ax(b) Pareto frontier of the mean and variance of the maximum stiffness

Figure 6 Pareto frontier

the standard deviation and mean have the same order ofmagnitude

The Six-Sigma robust design optimization flow diagramis shown in Figure 5 During the procedure the Six-Sigmacomponent robust analysis is used to analyze the single designpoint robustnessThe optimization design component robustoptimization is employed to perform the optimization designon the basis of robustness analysis If the quality of the singledesign point cannot satisfy the quality level requirementsIsight will continue to improve the optimization variables forthe quality analysis of the next group otherwise the resultswill be exported and the robust optimization is completedThepresent study selects the nondominated sorting evolutionstrategy namely NSGA-II as the optimization algorithm

Some genetic parameters and operators are set as follows[23]

The population size 4The generation number 10Crossover probability 09Crossover distribution index 10Mutation distribution index 20Max failed runs 5Failed run penalty value 1011986430Failed run objective value 1011986430

We obtain the Pareto solutions and frontier after the Six-Sigma robust optimization design The Pareto frontier ofthe mean as well as the variance of the maximum payloadcapacity and maximum stiffness is illustrated in Figure 6The relationship among the optimization objectives from thePareto solutions is observed According to this relationship

and the practical application requirements we select theoptimal solution for the multiobjective design problems andchoose the point of optimal solution (Figure 6) In view ofreliability and quality level theory we can obtain the qualitylevel mean and variance of each response after the robustoptimization through the post-processing functions of Isightsoftware And the results are listed in Table 3

Comparing the quality level with robust optimizationthat is variable quality level of ℎ 119904 119905 is 0676 7142 0755respectively Performance indexes quality level of 119896119895 119896min is1187 4161 respectively After robust optimization the qualitylevel of each response is greater than six

8 Result Analysis of the Six-Sigma Robust ofthe Parallel Adjusting Manipulator

By taking the global kinematic dexterity index as an examplethe probability distribution and Six-Sigma level were com-pared before and after optimization (Figure 7) We intuitivelyobtained the probability distribution mean standard devia-tion and Six-Sigma level of each response (Figure 7)

We rearranged and listed the reliability of each variablebefore and after optimization (Table 4) to concretely illustratethe difference between the reliability and defective qualityper million before and after the robust analysisThe defectiveproducts per million are shown in Table 5

The parameter variables (ie ℎ and 119905) and quality con-straints (ie global kinematic dexterity 119896119895) (Tables 4 and 5)greatly improves after optimization However the reliabilityof the other parameter variables only improved slightlyBy contrast the defective quality per million improvedconsiderably (Table 5) In the long term the adjustingmanipulator designed andmanufacturedwith optimal designvariables has fewer defective products permillion than before

Journal of Industrial Engineering 9

0

20

40

60

80

100

026 028 03 032 034 036

Prob

abili

ty

Quality 1187 sigmaMean 0305Std dev 0007

L bound 03

kj

ndash0722 sigma

(a) Before robust optimization

0

20

40

60

80

100

032 034 036 038 04 042

Prob

abili

ty

Quality 8 sigmaMean 0363

L bound 03ndash9901 sigma

kj

(b) After robust optimization

Figure 7 Probability distribution of the global kinematic dexterity before and after robust optimization

Table 4 Comparison of the reliability before and after optimization

Before robust optimization After robust optimizationℎ 50099 99999119904 100 100119905 54980 100119896119895 76472 100119896min 99997 100

Table 5 Comparison of the defective quality per million before andafter optimization

Before robust optimization After robust optimizationℎ 499002670 2069119890 minus 4

119904 9213119890 minus 7 0119905 450199849 0119896119895 235279150 5773119890 minus 9

119896min 31653 3930119890 minus 8

optimization This result is based on the assumption that thedesign variables manufacturing and installation error of thebase and moving platform can be effectively controlled thusgreatly improving the resistance ability of the uncertaintyfactor disturbance in the whole manufacturing process Datain Tables 4 and 5 show that the Six-Sigma robust optimizationdesign based on Isight is effective and feasible

9 Conclusions

The parallel adjusting manipulator applied in solar gatherpanels is the research objective of this paper Determinacyoptimization analysis was performed with consideration tothe performance indexes By considering the influence of the

uncertainty factors implemented in the robust analysis andoptimization the following conclusions are drawn

(1) The parallel adjusting manipulator was designedby using multiobjective optimization design on thebasis of the inverse kinematic solution The perfor-mance indexes are established the optimal solutionis obtained and the determinacy optimization iscompleted

(2) Robustness analysis is performed on the basis ofdeterminacy optimization design Some parametervariables and performance indexes are close to theconstraint boundary The Sigma level is low and thedefective quality permillion is highThe implementa-tion of the Six-Sigma robust optimization is necessaryto complete the nondeterminacy optimization design

(3) The comparison of the manipulator before and afterthe robust design optimization shows that the newone has a higher Sigma level Furthermore the com-parison shows a lower defective quality per millionthan before optimizationThe optimization objectivesare optimal The mean and standard deviation arein their minimum This result indicates that in thelong term themanipulator after optimization is moresuitable for batch production applications than themanipulator before optimization The robust opti-mization based on Isight is efficient and convenientfor operation with accurate results These results areapplicable to the robust design optimization of large-scale and complex manipulators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Industrial Engineering 7

Table 3 Six-Sigma-based robust design and optimization results

Variables Initial schemeDeterminacy optimization scheme Robust optimization scheme

Determinacyoptimal solutions

Six-Sigmaanalysis

Robust optimizationanalysis

Qualitylevel

Input variables

ℎ 300 120583 200005 Sigma level 0676reliability 50099

213330 6356120590 200005 213330

119904 200 120583 129096 Sigma level 7142reliability 100

218936 80120590 129096 218936

119905 06 120583 0799 Sigma level 0755reliability 54980

0683 80120590 000799 000683

Output variables (performance indexes)

Global kinematic dexterity 119896119895 0310 120583 0305 Sigma level 1187reliability 7647

0305 7809120590 0007 0007

Global minimum stiffness 119896min 30045 120583 36880 Sigma level 4161reliability 9999

429917 7563120590 1721 1739

Global maximum payload capability 119892max 538736 120583 591702 574223120590 4276 4014

Global maximum stiffness 119896max 290141 120583 350500 329940120590 5084 4624

7 Six-Sigma Robust Optimization(Uncertainty Optimization)

The essence of the Six-Sigma robust optimization is to addthe mean the variance of the response in optimizationobjectives and the upper quality level limit of the randomvariables in the constraint conditions to satisfy the qualitylevel minimize the mean of the optimal objectives andincrease the robustness and reliability of the mean [22]

This paper sets the lower limit of the Sigma level ofrandom variables and responses to 6120590 The mean of theglobal kinematic dexterity and global minimum stiffness ismaximized whereas the standard deviation of the globalkinematic dexterity and the global minimum stiffness is min-imizedThe robust optimization designmodel is summarizedas follows

max 1198911 (X) = 119892max (ℎ 119904 119905)

1198912 (X) = 119896max (ℎ 119904 119905)

1198913(X) = 120583 (119892max)

1198914 (X) = 120583 (119896max)

min 1198915 (X) = 120590 (119892max)

1198916 (X) = 120590 (119892max)

st 200mm le ℎ le 450mm

120mm le 119904 le 300mm

04 le 119905 le 08

119896119895 ge 03

Performance calculation

Robust optimization

Robust analysis

Figure 5 Flow diagram of the Six-Sigma robust design and opti-mization

119896min ge 3 times 104

Sigma Level (ℎ 119904 119905 119896119895 119896min) ge 6

(24)

where 120583(119892max) and 120590(119892max) are the mean and varianceof the maximum payload capability respectively The meanand variance are similar to the others Sigma Level(ℎ 119904119905 119896119895 119896min) denotes the Sigma level value of the parameters(ie ℎ 119904 119905 119896119895 and 119896min)

When setting the optimization objectives the scale factorof the standard deviation should be set to 001 to ensure that

8 Journal of Industrial Engineering

550 555 560 565 570 575 5804

402

404

406

408

41

412

414

Pareto solutionsThe optimal solution

Mean of g max

gm

axVa

rianc

e of

(a) Pareto frontier of the mean and variance of the maximum payloadcapability

305 31 315 32 325 33 3354560

4580

4600

4620

4640

4660

4680

Pareto solutionsThe optimal solution

times105Mean of k max

Varia

nce o

fkm

ax(b) Pareto frontier of the mean and variance of the maximum stiffness

Figure 6 Pareto frontier

the standard deviation and mean have the same order ofmagnitude

The Six-Sigma robust design optimization flow diagramis shown in Figure 5 During the procedure the Six-Sigmacomponent robust analysis is used to analyze the single designpoint robustnessThe optimization design component robustoptimization is employed to perform the optimization designon the basis of robustness analysis If the quality of the singledesign point cannot satisfy the quality level requirementsIsight will continue to improve the optimization variables forthe quality analysis of the next group otherwise the resultswill be exported and the robust optimization is completedThepresent study selects the nondominated sorting evolutionstrategy namely NSGA-II as the optimization algorithm

Some genetic parameters and operators are set as follows[23]

The population size 4The generation number 10Crossover probability 09Crossover distribution index 10Mutation distribution index 20Max failed runs 5Failed run penalty value 1011986430Failed run objective value 1011986430

We obtain the Pareto solutions and frontier after the Six-Sigma robust optimization design The Pareto frontier ofthe mean as well as the variance of the maximum payloadcapacity and maximum stiffness is illustrated in Figure 6The relationship among the optimization objectives from thePareto solutions is observed According to this relationship

and the practical application requirements we select theoptimal solution for the multiobjective design problems andchoose the point of optimal solution (Figure 6) In view ofreliability and quality level theory we can obtain the qualitylevel mean and variance of each response after the robustoptimization through the post-processing functions of Isightsoftware And the results are listed in Table 3

Comparing the quality level with robust optimizationthat is variable quality level of ℎ 119904 119905 is 0676 7142 0755respectively Performance indexes quality level of 119896119895 119896min is1187 4161 respectively After robust optimization the qualitylevel of each response is greater than six

8 Result Analysis of the Six-Sigma Robust ofthe Parallel Adjusting Manipulator

By taking the global kinematic dexterity index as an examplethe probability distribution and Six-Sigma level were com-pared before and after optimization (Figure 7) We intuitivelyobtained the probability distribution mean standard devia-tion and Six-Sigma level of each response (Figure 7)

We rearranged and listed the reliability of each variablebefore and after optimization (Table 4) to concretely illustratethe difference between the reliability and defective qualityper million before and after the robust analysisThe defectiveproducts per million are shown in Table 5

The parameter variables (ie ℎ and 119905) and quality con-straints (ie global kinematic dexterity 119896119895) (Tables 4 and 5)greatly improves after optimization However the reliabilityof the other parameter variables only improved slightlyBy contrast the defective quality per million improvedconsiderably (Table 5) In the long term the adjustingmanipulator designed andmanufacturedwith optimal designvariables has fewer defective products permillion than before

Journal of Industrial Engineering 9

0

20

40

60

80

100

026 028 03 032 034 036

Prob

abili

ty

Quality 1187 sigmaMean 0305Std dev 0007

L bound 03

kj

ndash0722 sigma

(a) Before robust optimization

0

20

40

60

80

100

032 034 036 038 04 042

Prob

abili

ty

Quality 8 sigmaMean 0363

L bound 03ndash9901 sigma

kj

(b) After robust optimization

Figure 7 Probability distribution of the global kinematic dexterity before and after robust optimization

Table 4 Comparison of the reliability before and after optimization

Before robust optimization After robust optimizationℎ 50099 99999119904 100 100119905 54980 100119896119895 76472 100119896min 99997 100

Table 5 Comparison of the defective quality per million before andafter optimization

Before robust optimization After robust optimizationℎ 499002670 2069119890 minus 4

119904 9213119890 minus 7 0119905 450199849 0119896119895 235279150 5773119890 minus 9

119896min 31653 3930119890 minus 8

optimization This result is based on the assumption that thedesign variables manufacturing and installation error of thebase and moving platform can be effectively controlled thusgreatly improving the resistance ability of the uncertaintyfactor disturbance in the whole manufacturing process Datain Tables 4 and 5 show that the Six-Sigma robust optimizationdesign based on Isight is effective and feasible

9 Conclusions

The parallel adjusting manipulator applied in solar gatherpanels is the research objective of this paper Determinacyoptimization analysis was performed with consideration tothe performance indexes By considering the influence of the

uncertainty factors implemented in the robust analysis andoptimization the following conclusions are drawn

(1) The parallel adjusting manipulator was designedby using multiobjective optimization design on thebasis of the inverse kinematic solution The perfor-mance indexes are established the optimal solutionis obtained and the determinacy optimization iscompleted

(2) Robustness analysis is performed on the basis ofdeterminacy optimization design Some parametervariables and performance indexes are close to theconstraint boundary The Sigma level is low and thedefective quality permillion is highThe implementa-tion of the Six-Sigma robust optimization is necessaryto complete the nondeterminacy optimization design

(3) The comparison of the manipulator before and afterthe robust design optimization shows that the newone has a higher Sigma level Furthermore the com-parison shows a lower defective quality per millionthan before optimizationThe optimization objectivesare optimal The mean and standard deviation arein their minimum This result indicates that in thelong term themanipulator after optimization is moresuitable for batch production applications than themanipulator before optimization The robust opti-mization based on Isight is efficient and convenientfor operation with accurate results These results areapplicable to the robust design optimization of large-scale and complex manipulators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Industrial Engineering

550 555 560 565 570 575 5804

402

404

406

408

41

412

414

Pareto solutionsThe optimal solution

Mean of g max

gm

axVa

rianc

e of

(a) Pareto frontier of the mean and variance of the maximum payloadcapability

305 31 315 32 325 33 3354560

4580

4600

4620

4640

4660

4680

Pareto solutionsThe optimal solution

times105Mean of k max

Varia

nce o

fkm

ax(b) Pareto frontier of the mean and variance of the maximum stiffness

Figure 6 Pareto frontier

the standard deviation and mean have the same order ofmagnitude

The Six-Sigma robust design optimization flow diagramis shown in Figure 5 During the procedure the Six-Sigmacomponent robust analysis is used to analyze the single designpoint robustnessThe optimization design component robustoptimization is employed to perform the optimization designon the basis of robustness analysis If the quality of the singledesign point cannot satisfy the quality level requirementsIsight will continue to improve the optimization variables forthe quality analysis of the next group otherwise the resultswill be exported and the robust optimization is completedThepresent study selects the nondominated sorting evolutionstrategy namely NSGA-II as the optimization algorithm

Some genetic parameters and operators are set as follows[23]

The population size 4The generation number 10Crossover probability 09Crossover distribution index 10Mutation distribution index 20Max failed runs 5Failed run penalty value 1011986430Failed run objective value 1011986430

We obtain the Pareto solutions and frontier after the Six-Sigma robust optimization design The Pareto frontier ofthe mean as well as the variance of the maximum payloadcapacity and maximum stiffness is illustrated in Figure 6The relationship among the optimization objectives from thePareto solutions is observed According to this relationship

and the practical application requirements we select theoptimal solution for the multiobjective design problems andchoose the point of optimal solution (Figure 6) In view ofreliability and quality level theory we can obtain the qualitylevel mean and variance of each response after the robustoptimization through the post-processing functions of Isightsoftware And the results are listed in Table 3

Comparing the quality level with robust optimizationthat is variable quality level of ℎ 119904 119905 is 0676 7142 0755respectively Performance indexes quality level of 119896119895 119896min is1187 4161 respectively After robust optimization the qualitylevel of each response is greater than six

8 Result Analysis of the Six-Sigma Robust ofthe Parallel Adjusting Manipulator

By taking the global kinematic dexterity index as an examplethe probability distribution and Six-Sigma level were com-pared before and after optimization (Figure 7) We intuitivelyobtained the probability distribution mean standard devia-tion and Six-Sigma level of each response (Figure 7)

We rearranged and listed the reliability of each variablebefore and after optimization (Table 4) to concretely illustratethe difference between the reliability and defective qualityper million before and after the robust analysisThe defectiveproducts per million are shown in Table 5

The parameter variables (ie ℎ and 119905) and quality con-straints (ie global kinematic dexterity 119896119895) (Tables 4 and 5)greatly improves after optimization However the reliabilityof the other parameter variables only improved slightlyBy contrast the defective quality per million improvedconsiderably (Table 5) In the long term the adjustingmanipulator designed andmanufacturedwith optimal designvariables has fewer defective products permillion than before

Journal of Industrial Engineering 9

0

20

40

60

80

100

026 028 03 032 034 036

Prob

abili

ty

Quality 1187 sigmaMean 0305Std dev 0007

L bound 03

kj

ndash0722 sigma

(a) Before robust optimization

0

20

40

60

80

100

032 034 036 038 04 042

Prob

abili

ty

Quality 8 sigmaMean 0363

L bound 03ndash9901 sigma

kj

(b) After robust optimization

Figure 7 Probability distribution of the global kinematic dexterity before and after robust optimization

Table 4 Comparison of the reliability before and after optimization

Before robust optimization After robust optimizationℎ 50099 99999119904 100 100119905 54980 100119896119895 76472 100119896min 99997 100

Table 5 Comparison of the defective quality per million before andafter optimization

Before robust optimization After robust optimizationℎ 499002670 2069119890 minus 4

119904 9213119890 minus 7 0119905 450199849 0119896119895 235279150 5773119890 minus 9

119896min 31653 3930119890 minus 8

optimization This result is based on the assumption that thedesign variables manufacturing and installation error of thebase and moving platform can be effectively controlled thusgreatly improving the resistance ability of the uncertaintyfactor disturbance in the whole manufacturing process Datain Tables 4 and 5 show that the Six-Sigma robust optimizationdesign based on Isight is effective and feasible

9 Conclusions

The parallel adjusting manipulator applied in solar gatherpanels is the research objective of this paper Determinacyoptimization analysis was performed with consideration tothe performance indexes By considering the influence of the

uncertainty factors implemented in the robust analysis andoptimization the following conclusions are drawn

(1) The parallel adjusting manipulator was designedby using multiobjective optimization design on thebasis of the inverse kinematic solution The perfor-mance indexes are established the optimal solutionis obtained and the determinacy optimization iscompleted

(2) Robustness analysis is performed on the basis ofdeterminacy optimization design Some parametervariables and performance indexes are close to theconstraint boundary The Sigma level is low and thedefective quality permillion is highThe implementa-tion of the Six-Sigma robust optimization is necessaryto complete the nondeterminacy optimization design

(3) The comparison of the manipulator before and afterthe robust design optimization shows that the newone has a higher Sigma level Furthermore the com-parison shows a lower defective quality per millionthan before optimizationThe optimization objectivesare optimal The mean and standard deviation arein their minimum This result indicates that in thelong term themanipulator after optimization is moresuitable for batch production applications than themanipulator before optimization The robust opti-mization based on Isight is efficient and convenientfor operation with accurate results These results areapplicable to the robust design optimization of large-scale and complex manipulators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Industrial Engineering 9

0

20

40

60

80

100

026 028 03 032 034 036

Prob

abili

ty

Quality 1187 sigmaMean 0305Std dev 0007

L bound 03

kj

ndash0722 sigma

(a) Before robust optimization

0

20

40

60

80

100

032 034 036 038 04 042

Prob

abili

ty

Quality 8 sigmaMean 0363

L bound 03ndash9901 sigma

kj

(b) After robust optimization

Figure 7 Probability distribution of the global kinematic dexterity before and after robust optimization

Table 4 Comparison of the reliability before and after optimization

Before robust optimization After robust optimizationℎ 50099 99999119904 100 100119905 54980 100119896119895 76472 100119896min 99997 100

Table 5 Comparison of the defective quality per million before andafter optimization

Before robust optimization After robust optimizationℎ 499002670 2069119890 minus 4

119904 9213119890 minus 7 0119905 450199849 0119896119895 235279150 5773119890 minus 9

119896min 31653 3930119890 minus 8

optimization This result is based on the assumption that thedesign variables manufacturing and installation error of thebase and moving platform can be effectively controlled thusgreatly improving the resistance ability of the uncertaintyfactor disturbance in the whole manufacturing process Datain Tables 4 and 5 show that the Six-Sigma robust optimizationdesign based on Isight is effective and feasible

9 Conclusions

The parallel adjusting manipulator applied in solar gatherpanels is the research objective of this paper Determinacyoptimization analysis was performed with consideration tothe performance indexes By considering the influence of the

uncertainty factors implemented in the robust analysis andoptimization the following conclusions are drawn

(1) The parallel adjusting manipulator was designedby using multiobjective optimization design on thebasis of the inverse kinematic solution The perfor-mance indexes are established the optimal solutionis obtained and the determinacy optimization iscompleted

(2) Robustness analysis is performed on the basis ofdeterminacy optimization design Some parametervariables and performance indexes are close to theconstraint boundary The Sigma level is low and thedefective quality permillion is highThe implementa-tion of the Six-Sigma robust optimization is necessaryto complete the nondeterminacy optimization design

(3) The comparison of the manipulator before and afterthe robust design optimization shows that the newone has a higher Sigma level Furthermore the com-parison shows a lower defective quality per millionthan before optimizationThe optimization objectivesare optimal The mean and standard deviation arein their minimum This result indicates that in thelong term themanipulator after optimization is moresuitable for batch production applications than themanipulator before optimization The robust opti-mization based on Isight is efficient and convenientfor operation with accurate results These results areapplicable to the robust design optimization of large-scale and complex manipulators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Industrial Engineering

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51175143)

References

[1] O Linda and M Manic ldquoUncertainty-robust design of intervaltype-2 fuzzy logic controller for delta parallel robotrdquo IEEETransactions on Industrial Informatics vol 7 no 4 pp 661ndash6702011

[2] R Kelaiaia O Company and A Zaatri ldquoMultiobjective opti-mization of a linear Delta parallel robotrdquo Mechanism andMachine Theory vol 50 pp 159ndash178 2012

[3] D D Ivezic and T B Petrovic ldquoRobust IMC controllerswith optimal setpoints tracking and disturbance rejection forindustrial boilerrdquo Journal of Mechanical Engineering vol 56 no9 pp 565ndash574 2010

[4] S Evgeny B Odellia and M Michael ldquoRobust optimization ofsystemdesignrdquoProcediaComputer Science vol 28 pp 489ndash4962014

[5] K Takeo and M Masatoshi ldquoRobust design method for adjust-able mechanismsrdquo Journal of Mechanics Engineering and Auto-mation vol 4 pp 16ndash24 2014

[6] P Gao The robust optimization design of minimally invasivespinal surgery guidance equipment based on the 3-RPS parallelmechanism [MS thesis] Hebei University of EngineeringHandan China 2013

[7] H J YuResearch on Parallel Robot Based on Flexible Fixtures forAutomotive Sheet Metal Assembly Harbin Institute of Technol-ogy Harbin China 2010

[8] X J Meng C Zhang and Z X Shi ldquoOptimization design forrobustness of linkagesrdquo Journal of Machine Design vol 24 no6 pp 31ndash33 2004

[9] T Kato and M Muramatsu ldquoRobust design method foradjustable mechanismsrdquo Journal of Mechanics Engineering andAutomation vol 4 pp 16ndash24 2011

[10] H Abdellatif B Heimann and J Kotlarski On the RobustDynamics Identification of Parallel Manipualtors Methodologyand Experiments I-Tech Education and Publishing ViennaAustria 2008

[11] MM Rahman HMMohyaldeenMMNoor K Kadirgamaand R A Bakar ldquoRobust design of suspension arm based onstochastic design improvementrdquo in Proceedings of the 2nd Inter-national Conference on Mechanical and Electronics Engineering(ICMEE rsquo10) pp V1460ndashV1464 August 2010

[12] H P Li B Q Shi and W M Zhang ldquoRobust optimizationdesign of a machine based on the blind number theoryrdquo Journalof University of Science and Technology Beijing vol 28 no 12pp 1178ndash1181 2006

[13] S Thrun D Fox W Burgard and F Dellaert ldquoRobust MonteCarlo localization for mobile robotsrdquo Artificial Intelligence vol128 no 1-2 pp 99ndash141 2001

[14] V Eric Robust Monte Carlo Methods for Light Transport Simu-lation Stanford University 1997

[15] G H Cui D Zhang H D Zhou and Y W Zhang ldquoOperatingdexterity optimization and analysis of a 3-DOF parallel manip-ulator for a tunnel segment assembly systemrdquo InternationalJournal of Mechanics and Materials in Design 2014

[16] G Cheng P Xu D-H Yang H Li and H-G Liu ldquoAnalysingkinematics of a novel 3CPS parallel manipulator based on

rodrigues parametersrdquo Journal of Mechanical Engineering vol59 no 5 pp 291ndash300 2013

[17] C Gosselin and J Angeles ldquoGlobal performance index forthe kinematic optimization of robotic manipulatorsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 113no 3 pp 220ndash226 1991

[18] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[19] Z Ren D Zhang and C S Koh ldquoMulti-objective optimizationapproach to reliability-based robust global optimization of elec-tromagnetic devicerdquoThe International Journal for ComputationandMathematics in Electrical andElectronic Engineering vol 33no 1-2 pp 191ndash200 2014

[20] Y Luo H Wen and H Li ldquoMultidisciplinary robust opti-mization design of multi-functional open-air hydraulic drillrdquoAppliedMechanics andMaterials vol 44-47 pp 1135ndash1140 2011

[21] A P Bowling J E Renaud J T Newkirk N M Patel andH Agarwal ldquoReliability-based design optimization of roboticsystem dynamic performancerdquo Journal of Mechanical DesignTransactions of the ASME vol 129 no 4 pp 449ndash454 2007

[22] A Csebfalvi ldquoA new theoretical approach for robust truss opti-mization with uncertain load directionsrdquo Mechanics BasedDesign of Structures and Machines vol 42 no 4 pp 442ndash4532014

[23] D S Lee L F Gonzalez J Periaux and K Srinivas ldquoRobustdesign optimisation using multi-objective evolutionary algo-rithmsrdquo Computers and Fluids vol 37 no 5 pp 565ndash583 2008

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International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of