hybrid approach of finite element method, kigring ......hybrid approach of finite element method,...

Research ArticleHybrid Approach of Finite Element Method KigringMetamodel and Multiobjective Genetic Algorithm forComputational Optimization of a Flexure Elbow Joint forUpper-Limb Assistive Device

Duc Nam Nguyen1 Thanh-Phong Dao 23 Ngoc Le Chau1 and Van Anh Dang1

1Faculty of Mechanical Engineering Industrial University of Ho Chi Minh City Ho Chi Minh City Vietnam2Division of Computational Mechatronics Institute for Computational Science TonDucThang University Ho ChiMinh City Vietnam3Faculty of Electrical amp Electronics Engineering Ton Duc Thang University Ho Chi Minh City Vietnam

Correspondence should be addressed toThanh-Phong Dao daothanhphongtdtueduvn

Received 30 October 2018 Revised 28 December 2018 Accepted 15 January 2019 Published 27 January 2019

Academic Editor Michele Scarpiniti

Copyright copy 2019 Duc Nam Nguyen et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Modeling for robotic joints is actually complex and may lead to wrong Pareto-optimal solutions Hence this paper develops a newhybrid approach for multiobjective optimization design of a flexure elbow joint The joint is designed for the upper-limb assistivedevice for physically disable people The optimization problem considers three design variables and two objective functions Anefficient hybrid optimization approach of central composite design (CDD) finite elementmethod (FEM) Kigring metamodel andmultiobjective genetic algorithm (MOGA) is developed The CDD is used to establish the number of numerical experiments TheFEM is developed to retrieve the strain energy and the reaction torque of joint And then the Kigring metamodel is used as ablack-box to find the pseudoobjective functions Based on pseudoobjective functions the MOGA is applied to find the optimalsolutions Traditionally an evolutionary optimization algorithm can only find one Pareto front However the proposed approachcan generate 6 Pareto-optimal solutions as near optimal candidates which provides a good decision-maker Based on the userrsquosreal-work problem one of the best optimal solutions is chosen The results found that the optimal strain energy is about 00033mJ and the optimal torque is approximately 58894 Nm Analysis of variance is performed to identify the significant contributionof design variables The sensitivity analysis is then carried out to determine the effect degree of each parameter on the responsesThe predictions are in a good agreement with validations It confirms that the proposed hybrid optimization approach has aneffectiveness to solve for complex optimization problems

1 Introduction

Along with amodern society human people have been facinga fast increase in stroke or accidence Therefore robotics hasreceived a great interest of researchers from academics andindustry If a person is subjected to the stroke themovementrsquosfunction of arm muscles is limited To support the disabledpeople robotic systems are designed and commercialized toassist the upper limb In general physicians use physiother-apy to facilitate rehabilitation process At hospitals doctorsutilize the robots

In the state of art of rehabilitation process assist robotsand rehabilitation devices have been designed and com-mercialized Robotic devices for upper-limb rehabilitationwere proposed for shoulder exercises [1] Mechanical struc-tures and control strategies for exoskeletons for upper-limbexoskeletons were reviewed [2] Bilateral robots for upper-limb stroke rehabilitation were studied [3] A whole armwearable robotic exoskeleton is used for rehabilitation andto assist upper limb [4] A gravity-balanced exoskeleton foractive rehabilitation training of upper limbwas developed [5]Developments of active upper-limb exoskeleton robots are

HindawiComplexityVolume 2019 Article ID 3231914 13 pageshttpsdoiorg10115520193231914

2 Complexity

w

Annular ring 1

Annular ring 2

Annular ring 3

(a)

t3

t2

t1

R3

R2

R1

O

Fixed hole

(b)

Figure 1 Flexure elbow joint (a) 3D model (b) 2D model

reviewed [6] Controller was designed for the angular trajec-tory of human shoulder elbow and wrist joints [7] A passiveupper extremity orthosis for amyoplasia was investigated [8]Bioinspired lower extremity exoskeleton robot was designedfor supporting body gait [9] Even though existing deviceswere designed well they still had shortages such as a heavyweight and large cost The reason is because the devices musttake a motor to generate a moment a gear pairs to transfermotions a coil spring to store and release an elastic energyand submechanical elements Their mechanical elements areassembled based on kinematic joints and this results inundesired clearancesDue to the clearances the rehabilitationsystems may be subject to undesired vibrations that cancause injury back to the disable forearm On the other handbecause of a large size and clearance the existing devices havea complex manufacture and control and expensive cost

Nowadays the patients need a flexible rehabilitationprocess at hospital at therapy center or at home Somepatients with disable muscle in upper-limb desire reasonablecommercial devices for training at homeMotivated from thatrequirement a new flexure elbow joint is designed in thiswork instead of utilizing coil spring The joint can store andrelease the strain energy similar to the function of traditionalsprings The compliant spring is designed based on conceptof compliant mechanism due to a light weight monolithicmanufacture a low cost and high positioning accuracy [10ndash14] The proposed joint can be made from a monolithicstructure by using 3D printer or wire electrical dischargedmachining [15ndash18]

In order to meet the practice requirements the pro-posed joint should have a large strain energy and a widetorque being suitable for various loads A multiobjectiveoptimization design is preferred in this regarding [19 20]In general mathematical equations are formed and then

an evolutionary optimization algorithm is applied to seekthe best solutions However if the established mathematicalequations are wrong the predicted results are unaccurateFor these reasons this study introduces a new data-drivenmultiobjective optimization technique for optimizing theperformances of the proposed joint to decrease the model-ing errors A hybrid integration includes of finite elementmethod Kigring metamodel and multiobjective geneticalgorithm Traditionally most of multiobjective evolutionaryalgorithms (MOEAs) often give a Pareto-optimal front [21]Unlike previous MOEAs the proposed hybrid algorithm cangenerate a lot of Pareto-optimal fronts which provides manyoptimal candidates and then a best solution for real-workpurpose is chosen

The purposes of this paper are to propose a multiobjec-tive optimal design strategy for the flexure elbow joint interms of good static characteristics To improve the staticperformances a hybrid optimization algorithm is developedA validation is performed to evaluate predicted results andefficiency of the proposed approach

2 Description of Mechanical Design

21 Flexure Elbow Joint A flexure elbow joint (FEJ) wasdesigned to allow only one purely rotational motion Thismotion was similar to the motion of actual elbow Forphysically disable people after stoke a movement of muscleis still difficult With the purpose of support the proposedFEJ was used as an elastic joint to store and release a strainenergy during elbow rehabilitation The FEJ was integratedwith the assistive devices the daily life activities of peopleafter an accident or a stroke would be supported easily

Figure 1(a) illustrates a 3D model of the proposed FEJIt included three annular rings separately The ring 1 was

Complexity 3

Figure 2 Meshing model for the flexure elbow joint

an inner elastic element with t1-thickness and this ring wasthinnest among three rings because it could store the strainenergy better The ring 2 was a middle elastic element witht2-thickness The ring 3 was an outer elastic element with t3-thicknessThe thickness of ring 3 was largest in order tomakea good stiffness of the joint followed by t1 and t2

Figure 1(b) gives the basic dimensions of the FEJ Thisjoint was located at three fixed holes In order to makea pure rotation around center O an angle 120572 was appliedCorresponding to a rotational angle a reaction torque Twas appeared Because the FEJ was an elastic element thereaction torquewas therefore proportional to the stiffness androtational angle according to the Hookrsquos law as

119879 = 119870 times 120572 (1)

where T represents the reaction torque K is stiffness of FEJand 120572 is rotational angle

Before conducting a computational analysis a 3D FEMmodel of FEJ was built and simulated through finite elementmethod in ANSYS software Coarse mesh was adopted forrigid links while three compliant springs were refined andachieved a fine mesh and better analysis results The facesize method was used for meshing The size of the selectedelements was 03 mm The total number of elements was263009 The number of nodes was 464603 as seen Figure 2To assess the quality of meshing skewness criteria wereadopted [11] Figure 3 shows that the element metric isconcentrated in region less than 06 It means that the qualityof meshing is good

In this analysis a nonlinear finite element model wasused to analyze the strain energy and output torque Thepolyethylene material was chosen for the proposed joint Theproperties of material were given in Table 1 Distributions ofdeformation of the rings and stresswere given in Figure 4Theresults showed that during the operation process a bucklingappears at each spring but it still makes a clearance of threesprings without self-contact deformation among them This

5364400

4000000

3000000

2000000

1000000

000

Num

ber o

f Ele

men

ts

000 013 025 038 050 063 075 088 098

Element Metrics

Tet10

Figure 3 Evaluating the quality of the meshing

guarantees a safety working operation during the assistiveprocess Besides themaximumstress concentration occurredat the middle springs where red color marks as depicted inFigure 4 Rounds and fillets for all springs were created so asto decrease the stress concentration

As depicted in Figure 2 the FEJ was only desired to makea pure rotation around the z-axis Its motion was repliedon the elastic elements Therefore three rings were refinedagain to achieve a good accuracy of analysis The otherswere assigned as coarse meshes because they no need tobe computed As given in Figure 1 there are seven designparameters of the joint including w R1 R2 R3 t1 t2 and t3The factorsw R1 R2 and R3 were assigned as constant valuesbecause they did not influence the responses of the proposedFEJ according to the elastic beam theory Meanwhile thefactors t1 t2 and t3 were considered as the design variablesAccording to the beam theory each elastic element has across-section area of a rectangle the stiffness of that area canbe determined by the following equation [22]

119896119890 = 11986411990811989011990511989034119897119890 = 216104Nm (2)

where E is Youngrsquos modulus of material we is the width teis thickness and le is the length of a cross-section area ofa rectangle According to [22] the shear was assumed to beneglected in (2) and an applied force at free end was exertedon the beam

Compliance of the FEJ was desired for elastic springsIf the compliance was increased the strain energy wasraised well However an increase in stiffness resulted in adecrease in compliance or strain energy As seen in (2) thestiffness was dependent on the width thickness and lengthof elastic spring It means that an increase in we and teleads to raise of stiffness however an increase of the lengthresults in a decrease in the stiffness It could be concludedthat the thicknesses t1 t2 and t3 are the most importantparameters affecting the performances of the joint Theseparameters were selected as the design variables in this studyThe polyethylene was selected as the material for the FEJbecause its light weight The parameters of material anddesign parameters were given as in Table 1

4 Complexity

Table 1 Material polyethylene properties of the joint

Material propertiesDensity Poisonrsquos ratio Youngrsquos modulus Yield strength950 kgm3 042 1100 MPa 25 MPaParameters Unit Dimensiont1 t2 t3 mm VariablesR3 mm 25R2 mm 20R1 mm 15w mm 8

Local stress

Figure 4 Distribution of stress and deformation of the joint

Pulley and string

Arm

Figure 5 Conventional assistive device

22 Primary Application for Upper-Limb Assistive Device Ingeneral a pulley kinematic joint and string are used as agravity-balanced mechanism to support movement of theforearm but costly as illustrated in Figure 5 Instead of usingtraditional system this study suggests a planar compliantsprings As shown in Figure 6 the main components of anew upper-limb assistive device are upper planar springlower planar spring and flexure elbow joint Three proposedsprings make a gravity-balanced system for the forearm inwhich the FEJ is the most important element to generatethe rotational motion for elbow They can be fabricated bywire electrical discharged machining process as a monolithicstructure The forearm can be located by a trap while thehand approaches the handle During rehabilitation process

Trap for forearmHandle

Adjustable ray

Station

Human seat

Flexure Elbow Joint (FEJ)

Fixture for FEJ

Lower spring

Upper spring

Bolt

Motor

Figure 6 3D model of the upper-limb assistive device

the patient is located on the seat or wheelchair A motor isused to generate a moment This rehabilitation system canbe integrated with an intelligent controller to control theassistive process With the support from the system a patientcan pick some objects such pen bottle fruit etcThe gravity-balanced system is aimed at compensating the weight of thehuman arm

Complexity 5

3 Formulation of MultiobjectiveOptimization Problem

Commercialized devices can support well the patients butthe cost is expensive The proposed FEJ was proposed todecrease the motors and actuators Regarding the efficiencyof the system the challenges which have been facing theFEJ includes the following (i) a large strain energy so thatit can store and release the elastic deformation and (ii) alarge torque required to permit a good load capacity Thesetwo requirements were conflicted together a new hybridoptimization algorithm was therefore suggested to trade offthem

31 Design Variables It was noted that the proposed FEJwas very sensitive to the thicknesses Therefore the variousthicknesses were considered as the design variables Thevector of design variables was set as X = [1199051 1199052 1199053]119879

The lower and upper bounds for the design variables wereassigned as follows

07mm le 1199051 le 10mm

06mm le 1199052 le 09mm

05mm le 1199053 le 08mm(3)

where ti (i =1 2 3) represents the thickness of FEJ

32 Objective Functions The multiple quality performancesof FEJ were required as follows (i) the strain energy 1198911(X)was desired as large as possible and (ii) the torque 1198912(X) isrequired as high as possible In summary the optimizationproblem was briefly addressed as

max1198911 (X) (4)

max1198912 (X) (5)

33 Constraints The equivalent stress of the FEJ during theoperation must be under the yield strength of the materialto guarantee the elastic limitation which was described asfollows

119892 (X) le 120590119910119899 (6)

where 119892(X) is the equivalent stress 120590119910 is the yield strength ofproposed material and n is the safety factor A higher safetyis better for the system

The strain energy was expected higher than 0003 mJwhile the torque was to over 500 Nmm

1198911 (X)max ge 0003mJ (7)

1198912 (X) ge 500Nmm (8)

Equations (7)-(8) were determined based on followingmainly important criteria according to the biomechanicalengineering assistive devices are proposed to fulfill clinicalrequirements so as to guarantee a good interaction between

the device and patient [23 24] Such devices must have alightweight and compliance for a comfort wearThe proposedFEJ is compact compliant and lightweight enough for theupper limb During activities of daily living of patients themotionrsquos range of the FEJ is desired as large as possible sothat the patients can handle various objects such as eatingand drinking during their activities [23 24] If the elasticdeformation of the FEJ is enhanced the range of motion isimproved According to the theory of compliant mechanismthe elastic deformation of the FEJ is directly proportional tothe strain energy [25] With the support of the device a largestrain energy is required larger than 0003 mJ so as to reducethe working efforts of patients during their daily living TheFEJ produces a good compliance with a large strain energybut it also requires a high output torqueThe aimof producinga high torque is to enlarge the capability of the device fordifferent patients Elderly with weak muscles require a largestrain energy and a low output torque while young patientswith stronger muscles need a large strain energy and a higheroutput torque [23 24] In this study an output torque over500 Nmm is required to fulfill the clinical requirements ofdifferent patients and decrease working efforts of the users

4 Hybrid Optimization Algorithm

According to a basic optimization problem mathematicalmodels should be formed prior to implement an evolution-ary algorithm The approximate models always have errorsbecause the models are mainly dependent on capability andknowledge of researchers about engineering and mathemat-ics Therefore a solution may be wrong To overcome thislimitation a hybrid approach of RSM FEM Kigring meta-model and MOGA was proposed in this study so as improvethe quality performances of the FEJ Figure 7 illustrates asystematic flowchart for the multiobjective optimization forthe proposed joint

The optimization experienced the main steps such asdesign a mechanical structure define design variables andobjective functions build 3D model evaluate initial per-formances and establish the numerical experiments usingresponse surface These steps can be found in detail by [26]

After collecting the numerical data the Kigring meta-model is used to build the relationship between inputsand outputs There are various regression models such asfull 2nd-order polynomials artificial neural network andnonparametric regression In this study Kigring metamodelwas a suitable choice for the estimated database It was con-sidered as a black-box to approximate the complex nonlinearrelationship between design parameters and the qualitiesKigring metamodel was used to find a pseudoobjective func-tions These objective functions were used for the MOGAalgorithm Kigring metamodel is an interpolative Bayesianmetamodeling technique combining a global model withlocal deviations [27]

119910 (119909) = 119891 (119909) + 119911 (119909) (9)

6 Complexity

Define problem

Yes No

Identify design variables and objective functions

Draw 3D-model by FEM

Test initial performances by FEM simulations

Design of experiments usingcentral composite design

CAE AampD

Generate simulated data

Develop regression models using Kigring technique

Setup parameters and constraints for MOGA algorithm

Optimization process

Find Pareto-optimal solution set

Refine range of design variables

Refine objective functions

Validations

Are optimal results satisfied

Response surface amp regression

End

Mechanical design

Yes No

No

Figure 7 The flowchart of multiobjective optimization procedure

where 119910(119909) is a known polynomial function and 119911(119909) is astochastic process with mean zero and nonzero covarianceThe nonzero covariance 119911(119909) is determined

cov (119885 (119909119894) 119885 (119909119895)) = 1205902R ([119877 (119909119894 119909119895)]) (10)

where R is the correlation matrix R(119909119894 119909119895) is the correlationfunction between any two sample points xi and xj TheGaussian correlation function was described as

119877 (119909119894 119909119895) = exp[ 119899sum119896=1

120579119896 (119909119894119896 minus 119909119895119896)2] (11)

where n is the number of design variables and 120579119896 is theunknown correlation parameter to be determined Predictedestimates 119910(119909) at untried values of 119909 are computed

119910 (119909) = 120573 + r119879 (119909)Rminus1 (119910 minus 119891120573) (12)

where y is the column vector of length n that contains thesample data of the responses 119891 is a column vector of length119899 that is filled with ones when 119891(119909) is taken as a constantand r119879(119909) is the correlation vector between a predicted pointx and the sample points 1199091 1199092 119909119873 described as

r119879 (119909) = [119877 (119909 1199091) 119877 (119909 1199092) 119877 (119909 119909119873)]119879 (13)

120573 is estimated by

120573 = (f119879Rminus1f)minus1 f119879Rminus1y (14)

The estimated variance of output model can determined as

1205902 = (y minus f120573)119879Rminus1 (y minus f120573)119873 (15)

The unknown parameters 120579119896 obtained using the maximumlikelihood estimation can be formulated as [27]

max120579119896gt0

Φ (120579119896) = minus[119873 (ln1205902) + ln |R|]2 (16)

where 1205902and |R| are the functions of 120579119896Accuracy of a regression model could be evaluated based

on the four criteria including the coefficient of determi-nation the root mean square error the maximum relativeresidual and the relative root mean square error

The coefficient of determination (R2) with its value in therange [[0 1] was determined as

1198772 = 1 minus sum119898119894=1 (119886119894 minus 119906119894)2sum119898119894=1 (119886119894 minus 119906)2 (17)

Root mean squared error (RMSE) was defined as ameasure of the differences between values predicted by eithera model or an estimator and the observed actual values

119877119872119878119864 = radic 1119898119898sum119894=1

(119886119894 minus 119906119894)2 (18)

Complexity 7

Table 2 Controllable parameters for MOGA algorithm

Parameters ValueNumber of initial samples 100Number of samples per iteration 100Maximum allowable Pareto percentage 70Convergence stability percentage 2Maximum number of iterations 20Maximum number candidates 6Crossover probability 07Mutation probability 001

Relative maximum absolute error (RMAR) was defined asthe difference between the observed values and the predictedvalues and was determined as

119877119872119860119877 = max119894=12119898

119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (19)

Relative average absolute error (RAAE) was determinedas the difference between predicted values by a model or anestimator and the observed true values RAAE measures theaverage of the squares of the errors

119877119860119860119864 = 1119898119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (20)

where m is the number of observations ai and ui denote theactual and predicted respectively

If the regression models were not well established thisstep would be refined by adjusting the range of designvariables Otherwise it would move on the next step

After the estimated objective functions were determinedthe optimization process was implemented by programingmultiobjective genetic algorithm (MOGA) This algorithmwas proposed for multiobjective optimization problem inthis study because it can converge to the global Paretosolutions This algorithm helped to seek a Pareto-optimal setfor multiple objective optimization [26] The MOGA was avariant of the Nondominated Sorted Genetic Algorithm-II(NSGA-II) that replied on controlled elitism concepts It cansolve multiple objectives and constraints At last the MOGAcan find the global optimum solution The controllableparameters of MOGA in this study were given in Table 2These values were chosen through many simulations

If the optimal results were satisfied the extra validationswere conducted to evaluate the robustness and efficiency ofthe proposed hybrid optimization approach The optimiza-tion process was ended herein If they were not well found arefinement was the most important phase to seek the optimalsolutions After optimal candidates were generated and basedon the initial requirements of quality characteristics theresearchers would evaluate the candidates If there was no anycandidate that was satisfied the ranges of quality character-istics or the range of design variables must be controlled orrefined again This step was repeated until the best candidatewas found

5 Results and Discussion

51 Data Collection A 3D FEM model was designed andthen FEA simulations were implemented to collect theperformances of the proposed FEJ The number of numer-ical experiments was determined using (9) The data wereretrieved the FEM a given in Table 3 And then based onKigring regression model each approximated function wasmade for the strain energy and torque The results indicatedthat the stress is still much lower than the yield strength ofpolyethylene This guaranteed a long working fatigue life

52 KigringMetamodels Nowadays there are some commonsurrogate models such as the nonscreening the artificialneural network full second order polynomials and theKigring metamodel In order to establish the regressionmodels the fifteen sets of sample points in Table 3 wereutilizedThe results gave that the coefficient of determinationof the Kigring model is approximately unity This parameterwas better than that of full second-order polynomials andneural network as shown in Table 4The rest metrics such asthe root mean square error the relative maximum absoluteerror and the relative average absolute error of the Kigringmodel were smaller than those of other models Hence theKigring model was adopted for this study

53 Contribution of Design Variable The contribution ofeach parameter to the responses was analyzed by using theanalysis of variance according to 95 confidence intervalsConsidering the strain energy the results showed that twothicknesses t2 and t3 are totally statistical significance withthe p-value less than 005 expect for the thickness t1 with p-value a little higher than 005 It showed that the parametert3 has a largest contribution to the strain energy with 5078(F-value of 2223) followed by parameter t2with contributionof 1316 (F-value of 1367) and parameter t1 has a smallestcontribution of 884 (F-value of 418) as given in Table 5

A similar way to the reaction torque the results showedthat three thicknesses are statistical significance with the p-value less than 005 Besides the result indicated that theparameter t2 has a largest contribution to the reaction torquewith 5653 (F-value of 3403) followed by parameter t1 withcontribution of 2096 (F-value of 1216) and parameter t3has a lowest contribution of 1589 (F-value of 959) as givenin Table 6

8 Complexity

Table 3 Design of experiments and numerical data

No t1 t2 t3 Strain energy Torque Stress(mm) (mm) (mm) (mJ) (Nm) (MPa)

1 085 075 065 0001327 5164587 20631262 085 075 05 000323 4439349 19721293 085 075 08 0001369 5781424 21776714 085 06 065 0001047 3870098 20200125 085 09 065 0001503 6502474 22063486 07 075 065 0001159 470434 19723627 1 075 065 0001618 544794 22710868 07 06 05 0000872 3042852 187649 07 06 08 0000869 4025997 187636910 07 09 05 0002742 5165792 182031211 07 09 08 0001216 6398056 198747512 1 06 05 0001021 3366276 205598813 1 06 08 0003013 4796251 214557414 1 09 05 0003585 5843072 205697115 1 09 08 0004526 7623913 2354646

Table 4 The goodness-of-fit of the Kigring response surfaces

Kigring metamodelsParameters Strain energy TorqueCoefficient of determination 1 1Root mean square error 7610minus10 2410minus6

Relative maximum absolute error 0 0Relative average absolute error 0 0

Neural networkParameters Strain energy TorqueCoefficient of determination 088 091Root mean square error 002 9854Relative maximum absolute error 6252 5207Relative average absolute error 2585 2313

Full second order polynomialsParameters Strain energy TorqueCoefficient of determination 093 096Root mean square error 002 5950Relative maximum absolute error 5008 4086Relative average absolute error 2218 1433

54 Optimal Results In order to solve an optimal solutionand tradeoff among between the quality responses theMOGA was then integrated with the FEM the RSM andKigring metamodels During the optimization process theoptimal results were generated automatically And then ifthe optimized solutions are not satisfied an extra adjustmentwas embedded in the proposed methodology It was mainlydependent on the designerrsquos experiences To save computa-tional time and the further adjustment this study proposedtwo ways to gain the candidates The first assignment wasto adjust the range of design parameters The second wasa change in the range of objective functions After imple-menting many computational simulations the results found

that the proposed hybrid methodology could be effectiveonly if the range of two quality responses were limited wellTo suppress the dependence on the expert knowledge twocommon rules were proposed as follows

(i) The lager-the better objective function the rangeof this function should be overcome an allowablethreshold For example the largest strain energy ofFEJ was desired higher than 0003 mJ and the rangeof this response was therefore controlled within theupper range of 0003 mJ as given in Table 7

(ii) A maximum reaction torque was also required beingmore than 500 Nmm and this function was herein

Complexity 9

Table 5 ANOVA of the strain energy

Factor Contribution F-Value P-Valuet1 884 418 0057t2 3129 1367 0003t3 5078 2233 0001

95 Confidence Intervals

Table 6 ANOVA of the torque


95 Confidence intervals

00035

0003

00025

0002

00015

0001

00005

0 01 02 03 04 05 06 07 08 09 1

P12-

Stra

in en

ergy

(mJ)

History of P12 (Maximize)Constraint Lower Bound (P12 gt= 0003 mJ)

Number of points (lowast)

Figure 8 History chart of the strain energy

constrained within the lower range of 500 NmmTheoptimal result could reach the desired value of largerthan 500 Nmm as shown in Table 7

During the optimization process the history chart ofstrain energy and that of reaction torque were retrieved sothat they could be virtualized As seen in Figure 8 the strainenergywas convergent being close to the upper range of 0003mJ As a result the optimal strain energy could be foundlarger than 0003 mJ as the design requirement Meanwhilethe reaction torque was convergent in the upper range of 500Nmm as shown in Figure 9 Hence the optimal torque couldbe found around 570 Nmm

The charts were so noise because the optimal solutionsare constrained in a range so as to achieve the convergenceIt means that the optimal result can be found in the rangeof desired space The curves may be so noise and it istrue that there were so many data points occurring at theoptimum space and the optimal cost functions were foundin the ranges This was a new approach to gain a convergentsolution It allowed choosing a best solution for a real-workproblem

750

700

650

600

550

500

450

400

350

300

Torq

ue (N

m)

History of P7 (Maximize)Constraint Lower Bound (P7 gt= 500 N mm)

0 01 02 03 04 05 06 07 08 09 1


Figure 9 History chart of the torque

The virtual mathematical models were found by usingKigring metamodels To achieve the optimal results forthe FEJ the constraints for two objective functions wereset up There have been six potential candidates generatedfor prefer selection Table 8 shows how to choose a bestcandidate and thiswas dependent onuserrsquos requirementsTheresult revealed that the candidate 6 was chosen as the bestoptimal design because it fully satisfies the mentioned designobjectives (see (3)-(8)) Moreover the equivalent stress wasabout 208 MPa that was still under the yield strength with ahigh safety of 12 This could guarantee a fatigue life and longworking time

55 Analysis of Sensitivity Along with optimization processa sensitivity analysis is also a necessary step to determinean influence of each design parameter on each qualityresponse Commonly there are various techniques whichcan be applied for calculating the sensitivity as the Nelsonmethod Modal method Matrix perturbation method Dif-ferential method and RSM In general the direct differentialmethod takes more time to analyze because it needs toconstruct physical modelsThe sensitivity could be calculatedby following formula

119878119891 = 120597119891119894120597119909119894 (21)

10 Complexity

Table 7 Bounds of the quality responses

Characteristics Constraint type Lower Upper UnitMaximum strain energy Lower lt= Values 0003 NA mJMinimum torque Lower lt= Values 500 NA Nmm

Table 8 Potentially optimal candidates

Candidate t1 t2 t3Strain energy Torque Stress

(mJ) (Nmm) (MPa)Candidate 1 097 090 051 00033 588942 20727Candidate 2 094 089 053 00031 593036 20795Candidate 3 094 090 053 00031 594183 20777Candidate 4 094 090 052 00032 590801 20707Candidate 5 094 089 052 00032 590388 20721Candidate 6 094 089 052 00032 590595 20720

minus50

0

100

Strain energy Torque

t1t2t3

Loca

l sen

sitiv

ity (

)

50

Figure 10 Sensitivity diagram

where 119891119894 119909 are the response and the design variable ithrespectively

The RSM [28] was chosen for this analysis Consideringthe local sensitivity the thickness t3 had a highest influence orsignificant contribution on the strain energy followed by thethickness t2 while the thickness t1 had a lowest contributionIt was noted that a change in t3 would adjust so as achieve thestrain energy as desired Regarding the reaction torque thethickness t2 had a largest influence followed by the thicknesst3 and the thickness t1 had smallest contribution as given inFigure 10

Particularly Figure 11 illustrates the effects of the thick-nesses t1 and t2 on the strain energy and torque as givenin Figures 11(a) and 11(b) respectively The results indicatedthat a decrease in the thickness t1 results in a decrease inthe strain energy But an increase in the thickness t2 leads toraise of the strain energy correspondingly Figure 12(b) showsthe effects of the thicknesses on the reaction torque It isnoted that the torque is linearly proportional correspondingto the thickness This response was changed sharply with anincrease in the thickness

The contribution diagram of thickness t3 was plotted asshown in Figure 12The results revealed that a decrease in theparameter t3 leads to a decrease in the strain energy as seenin Figure 12(a) Meanwhile the torque was sharply increasedwhen t3 was lowered as given in Figure 12(b)

Summary almost the mentioned design parameters hadsignificant contributions on the displacement and safetyfactor This would help designers and researchers to make adecision and meet the requirements of a specific system

6 FEA Verifications

To evaluate and validate the optimal results of the proposedFEJ a few of FEM tests were carried out The optimal designvariables of Candidate 1 fromTable 8 were used as the optimalvalues It was used to make the 3D model (t1 = 097 mm t2 =09mm t3 =051mm) SomeFEAvalidationswere conductedin ANSYS 18 software Materials boundary conditions andload were similarly in previous stepsThe quality of mesh wasachieved and then the output responses were retrieved

As given in Table 9 the results indicated that the validatedresults were found at the strain energy of 00032 mJ andreaction torque of 59108 Nmm The results found that thepredicted and verification results are in a good agreementIt means that the proposed hybrid optimization method isan efficient approach to solve multiobjective optimizationproblem for the FEJ It can be applied to solve complexoptimization problems It would support well the computa-tional design process of mechanical element for the assistivedevices

7 Conclusions

This paper has presented a new efficient multiobjectiveoptimization approach for a flexure elbow joint The jointwas designed based on connecting in series of the leafsprings The proposed FEJ was used to support one degreeof free rotation of elbow It was integrated into the upper-limb assistive device for disable people The FEJ was themost important element and the static performances was

Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)

Figure 11 Effect diagram of t1 and t2 on (a) the strain energy and (b) torque

Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)


therefore optimized so asmeet the requirements of the deviceTo improve overall static performances including the strainenergy and reaction torque a hybrid optimization approachwas developed This approach was an integration of FEMRSM Kigring metamodel method and MOGA Six optimalcandidates were retrieved and then candidate 1 was chosen asthe optimum solution

ANOVA was employed to identify the significant con-tribution of each variable to the responses The sensitivityanalysis through the RSM was conducted to determineinfluence of each factor It found that the parameter t3 hasa largest contribution to the strain energy with 5078 (F-value of 2223) followed by parameter t2 with contributionof 1316 (F-value of 1367) and parameter t1 has a smallestcontribution of 884 (F-value of 418 It indicated that the

parameter t2 has a largest contribution to the reaction torquewith 5653 (F-value of 3403) followed by parameter t1 withcontribution of 2096 (F-value of 1216) and parameter t3has a lowest contribution of 1589 (F-value of 959) Theresults showed that the optimal results were found at thestrain energy of 00033 mJ and the reaction torque of 58894Nm The predicted results were highly consistent with boththe FEA results It was confirmed that the proposed hybridoptimization approach is effective and reliable to solve forcomplex optimization engineering problems

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

12 Complexity

Table 9 Validation results

Characteristics Prediction FEA verification Error ()Strain energy (mJ) 00033 00032 303Torque (Nmm) 58894 59108 036Stress (MPa) 2072 2106 161

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) undergrant number 10703-201811

References

[1] Y Rosales Luengas R Lopez-Gutierrez S Salazar and RLozano ldquoRobust controls for upper limb exoskeleton real-timeresultsrdquo Proceedings of the Institution of Mechanical EngineersPart I Journal of Systems and Control Engineering vol 232 no7 pp 797ndash806 2018

[2] N Rehmat J Zuo W Meng Q Liu S Q Xie and H LiangldquoUpper limb rehabilitation using robotic exoskeleton systemsa systematic reviewrdquo International Journal of Intelligent Roboticsand Applications vol 2 no 3 pp 283ndash295 2018

[3] B Sheng Y Zhang W Meng C Deng and S Xie ldquoBilateralrobots for upper-limb stroke rehabilitation State of the art andfuture prospectsrdquoMedical Engineering amp Physics vol 38 no 7pp 587ndash606 2016

[4] Y Liu S Guo H Hirata H Ishihara and T Tamiya ldquoDevel-opment of a powered variable-stiffness exoskeleton device forelbow rehabilitationrdquo Biomedical Microdevices vol 20 no 32018

[5] Q Wu X Wang and F Du ldquoDevelopment and analysis of agravity-balanced exoskeleton for active rehabilitation trainingof upper limbrdquo Proceedings of the Institution of MechanicalEngineers Part C Journal of Mechanical Engineering Sciencevol 230 no 20 pp 3777ndash3790 2016

[6] R A R C Gopura D S V Bandara K Kiguchi and G KI Mann ldquoDevelopments in hardware systems of active upper-limb exoskeleton robots A reviewrdquo Robotics and AutonomousSystems vol 75 pp 203ndash220 2016

[7] I Eski and A Kırnap ldquoController design for upper limbmotionusingmeasurements of shoulder elbow andwrist jointsrdquoNeuralComputing and Applications vol 30 no 1 pp 307ndash325 2018

[8] E F Jensen J Raunsbaeligk J N Lund T Rahman J Rasmussenand M N Castro ldquoDevelopment and simulation of a passiveupper extremity orthosis for amyoplasiardquo Journal of Rehabili-tation and Assistive Technologies Engineering vol 5 Article ID205566831876152 2018

[9] C Wang X Wu Y Ma G Wu and Y Luo ldquoA Flexiblelower extremity exoskeleton robot with deep locomotion modeidentificationrdquo Complexity vol 2018 Article ID 5712108 9pages 2018

[10] N Le Chau V A Dang H G Le and T-P Dao ldquoRobustparameter design and analysis of a leaf compliant joint for

micropositioning systemsrdquo Arabian Journal for Science andEngineering vol 42 no 11 pp 4811ndash4823 2017

[11] T-P Dao and S-C Huang ldquoDesign and multi-objective opti-mization for a broad self-amplified 2-DOF monolithic mecha-nismrdquo Sadhana vol 42 no 9 pp 1527ndash1542 2017

[12] X Zhou H Xu J Cheng N Zhao and S Chen ldquoFlexure-basedRoll-to-roll platform A practical solution for realizing large-area microcontact printingrdquo Scientific Reports vol 5 no 1 pp1ndash10 2015

[13] N L Ho T Dao H G Le and N L Chau ldquoOptimal designof a compliant microgripper for assemble system of cell phonevibration motor using a hybrid approach of ANFIS and jayardquoArabian Journal for Science and Engineering vol 44 no 2 pp1205ndash1220 2019

[14] P P Valentini and E Pennestrı ldquoCompliant four-bar linkagesynthesis with second-order flexure hinge approximationrdquoMechanism and Machine Theory vol 128 pp 225ndash233 2018

[15] T-P Dao N L Ho T T Nguyen et al ldquoAnalysis andoptimization of a micro-displacement sensor for compliantmicrogripperrdquo Microsystem Technologies vol 23 no 12 pp5375ndash5395 2017

[16] S Chen M Ling and X Zhang ldquoDesign and experiment ofa millimeter-range and high-frequency compliant mechanismwith two output portsrdquo Mechanism and Machine Theory vol126 pp 201ndash209 2018

[17] A H Sakhaei S Kaijima T L Lee Y Y Tan and M LDunn ldquoDesign and investigation of a multi-material compliantratchet-like mechanismrdquo Mechanism and Machine Theory vol121 pp 184ndash197 2018

[18] L Cao A T Dolovich A Chen and W (Chris) ZhangldquoTopology optimization of efficient and strong hybrid com-pliant mechanisms using a mixed mesh of beams and flexurehinges with strength controlrdquoMechanism and Machine Theoryvol 121 pp 213ndash227 2018

[19] S Tamilselvi S Baskar L AnandapadmanabanV Karthikeyanand S Rajasekar ldquoMulti objective evolutionary algorithm fordesigning energy efficient distribution transformersrdquo Swarmand Evolutionary Computation vol 42 pp 109ndash124 2018

[20] L Vanneschi R Henriques and M Castelli ldquoMulti-objectivegenetic algorithm with variable neighbourhood search forthe electoral redistricting problemrdquo Swarm and EvolutionaryComputation vol 36 pp 37ndash51 2017

[21] A Pajares X Blasco J M Herrero and G Reynoso-Meza ldquoAmultiobjective genetic algorithm for the localization of optimaland nearly optimal solutions which are potentially usefulnevMOGArdquoComplexity vol 2018 Article ID 1792420 22 pages2018

[22] A A D BrownMechanical Springs (Engineering Design Guides42) 1981

[23] D J Magermans E K J Chadwick H E J Veeger and F CT van der Helm ldquoRequirements for upper extremity motionsduring activities of daily livingrdquo Clinical Biomechanics vol 20no 6 pp 591ndash599 2005

Complexity 13

[24] I A Murray and G R Johnson ldquoA study of the external forcesandmoments at the shoulder and elbowwhile performing everyday tasksrdquo Clinical Biomechanics vol 19 no 6 pp 586ndash5942004

[25] L L Howell Compliant Mechanisms John Wiley amp Sons 2001[26] N L Chau T Dao and V T Nguyen ldquoAn efficient hybrid

approach of finite element method artificial neural network-based multiobjective genetic algorithm for computational opti-mization of a linear compliant mechanism of nanoindentationtesterrdquoMathematical Problems in Engineering vol 2018 ArticleID 7070868 19 pages 2018

[27] P Jiang C Wang Q Zhou X Shao L Shu and X Li ldquoOp-timization of laser welding process parameters of stainlesssteel 316L using FEM Kriging and NSGA-IIrdquo Advances inEngineering Software vol 99 pp 147ndash160 2016

[28] N L Chau T Dao and V T Nguyen ldquoOptimal design ofa dragonfly-inspired compliant joint for camera positioningsystem of nanoindentation tester based on a hybrid integrationof Jaya-ANFISrdquo Mathematical Problems in Engineering vol2018 Article ID 8546095 16 pages 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

2 Complexity

w

Annular ring 1

Annular ring 2

Annular ring 3

(a)

t3

t2

t1

R3

R2

R1

O

Fixed hole

(b)

Figure 1 Flexure elbow joint (a) 3D model (b) 2D model

reviewed [6] Controller was designed for the angular trajec-tory of human shoulder elbow and wrist joints [7] A passiveupper extremity orthosis for amyoplasia was investigated [8]Bioinspired lower extremity exoskeleton robot was designedfor supporting body gait [9] Even though existing deviceswere designed well they still had shortages such as a heavyweight and large cost The reason is because the devices musttake a motor to generate a moment a gear pairs to transfermotions a coil spring to store and release an elastic energyand submechanical elements Their mechanical elements areassembled based on kinematic joints and this results inundesired clearancesDue to the clearances the rehabilitationsystems may be subject to undesired vibrations that cancause injury back to the disable forearm On the other handbecause of a large size and clearance the existing devices havea complex manufacture and control and expensive cost

Nowadays the patients need a flexible rehabilitationprocess at hospital at therapy center or at home Somepatients with disable muscle in upper-limb desire reasonablecommercial devices for training at homeMotivated from thatrequirement a new flexure elbow joint is designed in thiswork instead of utilizing coil spring The joint can store andrelease the strain energy similar to the function of traditionalsprings The compliant spring is designed based on conceptof compliant mechanism due to a light weight monolithicmanufacture a low cost and high positioning accuracy [10ndash14] The proposed joint can be made from a monolithicstructure by using 3D printer or wire electrical dischargedmachining [15ndash18]

In order to meet the practice requirements the pro-posed joint should have a large strain energy and a widetorque being suitable for various loads A multiobjectiveoptimization design is preferred in this regarding [19 20]In general mathematical equations are formed and then

an evolutionary optimization algorithm is applied to seekthe best solutions However if the established mathematicalequations are wrong the predicted results are unaccurateFor these reasons this study introduces a new data-drivenmultiobjective optimization technique for optimizing theperformances of the proposed joint to decrease the model-ing errors A hybrid integration includes of finite elementmethod Kigring metamodel and multiobjective geneticalgorithm Traditionally most of multiobjective evolutionaryalgorithms (MOEAs) often give a Pareto-optimal front [21]Unlike previous MOEAs the proposed hybrid algorithm cangenerate a lot of Pareto-optimal fronts which provides manyoptimal candidates and then a best solution for real-workpurpose is chosen

The purposes of this paper are to propose a multiobjec-tive optimal design strategy for the flexure elbow joint interms of good static characteristics To improve the staticperformances a hybrid optimization algorithm is developedA validation is performed to evaluate predicted results andefficiency of the proposed approach

2 Description of Mechanical Design

21 Flexure Elbow Joint A flexure elbow joint (FEJ) wasdesigned to allow only one purely rotational motion Thismotion was similar to the motion of actual elbow Forphysically disable people after stoke a movement of muscleis still difficult With the purpose of support the proposedFEJ was used as an elastic joint to store and release a strainenergy during elbow rehabilitation The FEJ was integratedwith the assistive devices the daily life activities of peopleafter an accident or a stroke would be supported easily

Figure 1(a) illustrates a 3D model of the proposed FEJIt included three annular rings separately The ring 1 was

Complexity 3




119879 = 119870 times 120572 (1)




5364400

4000000

3000000

2000000

1000000

000

Num

ber o

f Ele

men

ts

000 013 025 038 050 063 075 088 098

Element Metrics

Tet10




119896119890 = 11986411990811989011990511989034119897119890 = 216104Nm (2)



4 Complexity



Local stress


Pulley and string

Arm




Adjustable ray

Station

Human seat


Fixture for FEJ

Lower spring

Upper spring

Bolt

Motor



Complexity 5





07mm le 1199051 le 10mm

06mm le 1199052 le 09mm

05mm le 1199053 le 08mm(3)



max1198911 (X) (4)

max1198912 (X) (5)


119892 (X) le 120590119910119899 (6)



1198911 (X)max ge 0003mJ (7)

1198912 (X) ge 500Nmm (8)







119910 (119909) = 119891 (119909) + 119911 (119909) (9)

6 Complexity

Define problem

Yes No





CAE AampD








Validations



End

Mechanical design

Yes No

No



cov (119885 (119909119894) 119885 (119909119895)) = 1205902R ([119877 (119909119894 119909119895)]) (10)


119877 (119909119894 119909119895) = exp[ 119899sum119896=1

120579119896 (119909119894119896 minus 119909119895119896)2] (11)


119910 (119909) = 120573 + r119879 (119909)Rminus1 (119910 minus 119891120573) (12)


r119879 (119909) = [119877 (119909 1199091) 119877 (119909 1199092) 119877 (119909 119909119873)]119879 (13)






max120579119896gt0

Φ (120579119896) = minus[119873 (ln1205902) + ln |R|]2 (16)






119877119872119878119864 = radic 1119898119898sum119894=1

(119886119894 minus 119906119894)2 (18)

Complexity 7




119877119872119860119877 = max119894=12119898

119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (19)


119877119860119860119864 = 1119898119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (20)










8 Complexity



1 085 075 065 0001327 5164587 20631262 085 075 05 000323 4439349 19721293 085 075 08 0001369 5781424 21776714 085 06 065 0001047 3870098 20200125 085 09 065 0001503 6502474 22063486 07 075 065 0001159 470434 19723627 1 075 065 0001618 544794 22710868 07 06 05 0000872 3042852 187649 07 06 08 0000869 4025997 187636910 07 09 05 0002742 5165792 182031211 07 09 08 0001216 6398056 198747512 1 06 05 0001021 3366276 205598813 1 06 08 0003013 4796251 214557414 1 09 05 0003585 5843072 205697115 1 09 08 0004526 7623913 2354646










Complexity 9







00035

0003

00025

0002

00015

0001

00005

0 01 02 03 04 05 06 07 08 09 1

P12-

Stra

in en

ergy

(mJ)







750

700

650

600

550

500

450

400

350

300

Torq

ue (N

m)


0 01 02 03 04 05 06 07 08 09 1





119878119891 = 120597119891119894120597119909119894 (21)

10 Complexity






minus50

0

100


t1t2t3

Loca

l sen

sitiv

ity (

)

50







6 FEA Verifications



7 Conclusions


Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3




119879 = 119870 times 120572 (1)




5364400

4000000

3000000

2000000

1000000

000

Num

ber o

f Ele

men

ts

000 013 025 038 050 063 075 088 098

Element Metrics

Tet10




119896119890 = 11986411990811989011990511989034119897119890 = 216104Nm (2)



4 Complexity



Local stress


Pulley and string

Arm




Adjustable ray

Station

Human seat


Fixture for FEJ

Lower spring

Upper spring

Bolt

Motor



Complexity 5





07mm le 1199051 le 10mm

06mm le 1199052 le 09mm

05mm le 1199053 le 08mm(3)



max1198911 (X) (4)

max1198912 (X) (5)


119892 (X) le 120590119910119899 (6)



1198911 (X)max ge 0003mJ (7)

1198912 (X) ge 500Nmm (8)







119910 (119909) = 119891 (119909) + 119911 (119909) (9)

6 Complexity

Define problem

Yes No





CAE AampD








Validations



End

Mechanical design

Yes No

No



cov (119885 (119909119894) 119885 (119909119895)) = 1205902R ([119877 (119909119894 119909119895)]) (10)


119877 (119909119894 119909119895) = exp[ 119899sum119896=1

120579119896 (119909119894119896 minus 119909119895119896)2] (11)


119910 (119909) = 120573 + r119879 (119909)Rminus1 (119910 minus 119891120573) (12)


r119879 (119909) = [119877 (119909 1199091) 119877 (119909 1199092) 119877 (119909 119909119873)]119879 (13)






max120579119896gt0

Φ (120579119896) = minus[119873 (ln1205902) + ln |R|]2 (16)






119877119872119878119864 = radic 1119898119898sum119894=1

(119886119894 minus 119906119894)2 (18)

Complexity 7




119877119872119860119877 = max119894=12119898

119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (19)


119877119860119860119864 = 1119898119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (20)










8 Complexity



1 085 075 065 0001327 5164587 20631262 085 075 05 000323 4439349 19721293 085 075 08 0001369 5781424 21776714 085 06 065 0001047 3870098 20200125 085 09 065 0001503 6502474 22063486 07 075 065 0001159 470434 19723627 1 075 065 0001618 544794 22710868 07 06 05 0000872 3042852 187649 07 06 08 0000869 4025997 187636910 07 09 05 0002742 5165792 182031211 07 09 08 0001216 6398056 198747512 1 06 05 0001021 3366276 205598813 1 06 08 0003013 4796251 214557414 1 09 05 0003585 5843072 205697115 1 09 08 0004526 7623913 2354646










Complexity 9







00035

0003

00025

0002

00015

0001

00005

0 01 02 03 04 05 06 07 08 09 1

P12-

Stra

in en

ergy

(mJ)







750

700

650

600

550

500

450

400

350

300

Torq

ue (N

m)


0 01 02 03 04 05 06 07 08 09 1





119878119891 = 120597119891119894120597119909119894 (21)

10 Complexity






minus50

0

100


t1t2t3

Loca

l sen

sitiv

ity (

)

50







6 FEA Verifications



7 Conclusions


Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity



Local stress


Pulley and string

Arm




Adjustable ray

Station

Human seat


Fixture for FEJ

Lower spring

Upper spring

Bolt

Motor



Complexity 5





07mm le 1199051 le 10mm

06mm le 1199052 le 09mm

05mm le 1199053 le 08mm(3)



max1198911 (X) (4)

max1198912 (X) (5)


119892 (X) le 120590119910119899 (6)



1198911 (X)max ge 0003mJ (7)

1198912 (X) ge 500Nmm (8)







119910 (119909) = 119891 (119909) + 119911 (119909) (9)

6 Complexity

Define problem

Yes No





CAE AampD








Validations



End

Mechanical design

Yes No

No



cov (119885 (119909119894) 119885 (119909119895)) = 1205902R ([119877 (119909119894 119909119895)]) (10)


119877 (119909119894 119909119895) = exp[ 119899sum119896=1

120579119896 (119909119894119896 minus 119909119895119896)2] (11)


119910 (119909) = 120573 + r119879 (119909)Rminus1 (119910 minus 119891120573) (12)


r119879 (119909) = [119877 (119909 1199091) 119877 (119909 1199092) 119877 (119909 119909119873)]119879 (13)






max120579119896gt0

Φ (120579119896) = minus[119873 (ln1205902) + ln |R|]2 (16)






119877119872119878119864 = radic 1119898119898sum119894=1

(119886119894 minus 119906119894)2 (18)

Complexity 7




119877119872119860119877 = max119894=12119898

119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (19)


119877119860119860119864 = 1119898119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (20)










8 Complexity



1 085 075 065 0001327 5164587 20631262 085 075 05 000323 4439349 19721293 085 075 08 0001369 5781424 21776714 085 06 065 0001047 3870098 20200125 085 09 065 0001503 6502474 22063486 07 075 065 0001159 470434 19723627 1 075 065 0001618 544794 22710868 07 06 05 0000872 3042852 187649 07 06 08 0000869 4025997 187636910 07 09 05 0002742 5165792 182031211 07 09 08 0001216 6398056 198747512 1 06 05 0001021 3366276 205598813 1 06 08 0003013 4796251 214557414 1 09 05 0003585 5843072 205697115 1 09 08 0004526 7623913 2354646










Complexity 9







00035

0003

00025

0002

00015

0001

00005

0 01 02 03 04 05 06 07 08 09 1

P12-

Stra

in en

ergy

(mJ)







750

700

650

600

550

500

450

400

350

300

Torq

ue (N

m)


0 01 02 03 04 05 06 07 08 09 1





119878119891 = 120597119891119894120597119909119894 (21)

10 Complexity






minus50

0

100


t1t2t3

Loca

l sen

sitiv

ity (

)

50







6 FEA Verifications



7 Conclusions


Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5





07mm le 1199051 le 10mm

06mm le 1199052 le 09mm

05mm le 1199053 le 08mm(3)



max1198911 (X) (4)

max1198912 (X) (5)


119892 (X) le 120590119910119899 (6)



1198911 (X)max ge 0003mJ (7)

1198912 (X) ge 500Nmm (8)







119910 (119909) = 119891 (119909) + 119911 (119909) (9)

6 Complexity

Define problem

Yes No





CAE AampD








Validations



End

Mechanical design

Yes No

No



cov (119885 (119909119894) 119885 (119909119895)) = 1205902R ([119877 (119909119894 119909119895)]) (10)


119877 (119909119894 119909119895) = exp[ 119899sum119896=1

120579119896 (119909119894119896 minus 119909119895119896)2] (11)


119910 (119909) = 120573 + r119879 (119909)Rminus1 (119910 minus 119891120573) (12)


r119879 (119909) = [119877 (119909 1199091) 119877 (119909 1199092) 119877 (119909 119909119873)]119879 (13)






max120579119896gt0

Φ (120579119896) = minus[119873 (ln1205902) + ln |R|]2 (16)






119877119872119878119864 = radic 1119898119898sum119894=1

(119886119894 minus 119906119894)2 (18)

Complexity 7




119877119872119860119877 = max119894=12119898

119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (19)


119877119860119860119864 = 1119898119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (20)










8 Complexity



1 085 075 065 0001327 5164587 20631262 085 075 05 000323 4439349 19721293 085 075 08 0001369 5781424 21776714 085 06 065 0001047 3870098 20200125 085 09 065 0001503 6502474 22063486 07 075 065 0001159 470434 19723627 1 075 065 0001618 544794 22710868 07 06 05 0000872 3042852 187649 07 06 08 0000869 4025997 187636910 07 09 05 0002742 5165792 182031211 07 09 08 0001216 6398056 198747512 1 06 05 0001021 3366276 205598813 1 06 08 0003013 4796251 214557414 1 09 05 0003585 5843072 205697115 1 09 08 0004526 7623913 2354646










Complexity 9







00035

0003

00025

0002

00015

0001

00005

0 01 02 03 04 05 06 07 08 09 1

P12-

Stra

in en

ergy

(mJ)







750

700

650

600

550

500

450

400

350

300

Torq

ue (N

m)


0 01 02 03 04 05 06 07 08 09 1





119878119891 = 120597119891119894120597119909119894 (21)

10 Complexity






minus50

0

100


t1t2t3

Loca

l sen

sitiv

ity (

)

50







6 FEA Verifications



7 Conclusions


Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

Define problem

Yes No





CAE AampD








Validations



End

Mechanical design

Yes No

No



cov (119885 (119909119894) 119885 (119909119895)) = 1205902R ([119877 (119909119894 119909119895)]) (10)


119877 (119909119894 119909119895) = exp[ 119899sum119896=1

120579119896 (119909119894119896 minus 119909119895119896)2] (11)


119910 (119909) = 120573 + r119879 (119909)Rminus1 (119910 minus 119891120573) (12)


r119879 (119909) = [119877 (119909 1199091) 119877 (119909 1199092) 119877 (119909 119909119873)]119879 (13)






max120579119896gt0

Φ (120579119896) = minus[119873 (ln1205902) + ln |R|]2 (16)






119877119872119878119864 = radic 1119898119898sum119894=1

(119886119894 minus 119906119894)2 (18)

Complexity 7




119877119872119860119877 = max119894=12119898

119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (19)


119877119860119860119864 = 1119898119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (20)










8 Complexity



1 085 075 065 0001327 5164587 20631262 085 075 05 000323 4439349 19721293 085 075 08 0001369 5781424 21776714 085 06 065 0001047 3870098 20200125 085 09 065 0001503 6502474 22063486 07 075 065 0001159 470434 19723627 1 075 065 0001618 544794 22710868 07 06 05 0000872 3042852 187649 07 06 08 0000869 4025997 187636910 07 09 05 0002742 5165792 182031211 07 09 08 0001216 6398056 198747512 1 06 05 0001021 3366276 205598813 1 06 08 0003013 4796251 214557414 1 09 05 0003585 5843072 205697115 1 09 08 0004526 7623913 2354646










Complexity 9







00035

0003

00025

0002

00015

0001

00005

0 01 02 03 04 05 06 07 08 09 1

P12-

Stra

in en

ergy

(mJ)







750

700

650

600

550

500

450

400

350

300

Torq

ue (N

m)


0 01 02 03 04 05 06 07 08 09 1





119878119891 = 120597119891119894120597119909119894 (21)

10 Complexity






minus50

0

100


t1t2t3

Loca

l sen

sitiv

ity (

)

50







6 FEA Verifications



7 Conclusions


Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7




119877119872119860119877 = max119894=12119898

119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (19)


119877119860119860119864 = 1119898119898sum119894=1

10038161003816100381610038161003816100381610038161003816 119886119894 minus 11990611989411988611989410038161003816100381610038161003816100381610038161003816 times 100 (20)










8 Complexity



1 085 075 065 0001327 5164587 20631262 085 075 05 000323 4439349 19721293 085 075 08 0001369 5781424 21776714 085 06 065 0001047 3870098 20200125 085 09 065 0001503 6502474 22063486 07 075 065 0001159 470434 19723627 1 075 065 0001618 544794 22710868 07 06 05 0000872 3042852 187649 07 06 08 0000869 4025997 187636910 07 09 05 0002742 5165792 182031211 07 09 08 0001216 6398056 198747512 1 06 05 0001021 3366276 205598813 1 06 08 0003013 4796251 214557414 1 09 05 0003585 5843072 205697115 1 09 08 0004526 7623913 2354646










Complexity 9







00035

0003

00025

0002

00015

0001

00005

0 01 02 03 04 05 06 07 08 09 1

P12-

Stra

in en

ergy

(mJ)







750

700

650

600

550

500

450

400

350

300

Torq

ue (N

m)


0 01 02 03 04 05 06 07 08 09 1





119878119891 = 120597119891119894120597119909119894 (21)

10 Complexity






minus50

0

100


t1t2t3

Loca

l sen

sitiv

ity (

)

50







6 FEA Verifications



7 Conclusions


Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity



1 085 075 065 0001327 5164587 20631262 085 075 05 000323 4439349 19721293 085 075 08 0001369 5781424 21776714 085 06 065 0001047 3870098 20200125 085 09 065 0001503 6502474 22063486 07 075 065 0001159 470434 19723627 1 075 065 0001618 544794 22710868 07 06 05 0000872 3042852 187649 07 06 08 0000869 4025997 187636910 07 09 05 0002742 5165792 182031211 07 09 08 0001216 6398056 198747512 1 06 05 0001021 3366276 205598813 1 06 08 0003013 4796251 214557414 1 09 05 0003585 5843072 205697115 1 09 08 0004526 7623913 2354646










Complexity 9







00035

0003

00025

0002

00015

0001

00005

0 01 02 03 04 05 06 07 08 09 1

P12-

Stra

in en

ergy

(mJ)







750

700

650

600

550

500

450

400

350

300

Torq

ue (N

m)


0 01 02 03 04 05 06 07 08 09 1





119878119891 = 120597119891119894120597119909119894 (21)

10 Complexity






minus50

0

100


t1t2t3

Loca

l sen

sitiv

ity (

)

50







6 FEA Verifications



7 Conclusions


Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9







00035

0003

00025

0002

00015

0001

00005

0 01 02 03 04 05 06 07 08 09 1

P12-

Stra

in en

ergy

(mJ)







750

700

650

600

550

500

450

400

350

300

Torq

ue (N

m)


0 01 02 03 04 05 06 07 08 09 1





119878119891 = 120597119891119894120597119909119894 (21)

10 Complexity






minus50

0

100


t1t2t3

Loca

l sen

sitiv

ity (

)

50







6 FEA Verifications



7 Conclusions


Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity






minus50

0

100


t1t2t3

Loca

l sen

sitiv

ity (

)

50







6 FEA Verifications



7 Conclusions


Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11

Stra

in en

ergy

(mJ)

0003

0002

0001

007

08

09

1085

08075

07

065

00028

00024

0002

00016

00012

00008

00004

0

t1 (mm) t 2(mm)

(a)

Torq

ue (N

m)

07

08

09

1085

08075

07

065

600

500

400

640

600

560

520

480

440

400

360

t1 (mm) t 2(mm)

(b)


Stra

in en

ergy

(mJ) 0003

00025

0002

00015

0001

000320003

000280002600024000220002

000180001600014000120001

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(a)

Torq

ue (N

m)

600

550

500

450

610590570550530510490470450430410

07

08

09

1 08

07

06

05

t1 (mm)t 3(mm)

(b)





Data Availability


12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity





Acknowledgments


References

























Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018








Complexity 13













Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








hybrid approach of finite element method, kigring ......hybrid approach of finite element method,...

Documents