Research Article Fuzzy Logic-Based Techniques for Modeling...

Hindawi Publishing CorporationInternational Journal of Manufacturing EngineeringVolume 2013 Article ID 230463 17 pageshttpdxdoiorg1011552013230463

Research ArticleFuzzy Logic-Based Techniques for Modelingthe Correlation between the Weld Bead Dimensionand the Process Parameters in MIG Welding

Y Surender and Dilip Kumar Pratihar

Soft Computing Lab Mechanical Engineering Department Indian Institute of Technology Kharagpur 721 302 India

Correspondence should be addressed to Dilip Kumar Pratihar dkpramechiitkgpernetin

Received 11 March 2013 Accepted 14 August 2013

Academic Editors G Onwubolu N Rezg and K Salonitis

Copyright copy 2013 Y Surender and D K Pratihar This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Fuzzy logic-based techniques have been developed to model input-output relationships of metal inert gas (MIG) welding processBoth conventional and hierarchical fuzzy logic controllers (FLCs) of Mamdani type have been developed and their performancesare compared The conventional FLC suffers from the curse of dimensionality for handling a large number of variables and ahierarchical FLC was proposed earlier to tackle this problem However in that study both the structure and knowledge base of theFLCwere not optimized simultaneously which has been attempted here Simultaneous optimization of the structure and knowledgebase is a difficult task and to solve it a genetic algorithm (GA)will have to deal with the strings having varied lengths A new schemehas been proposed here to tackle the problem related to crossover of two parents with unequal lengths It is interesting to observethat the conventional FLC yields the best accuracy in predictions whereas the hierarchical FLC can be computationally faster thanothers but at the cost of accuracy Moreover there is no improvement of interpretability by introducing a hierarchical fuzzy systemThus there exists a trade-off between the accuracy obtained in predictions and computational complexity of various FLCs

1 Introduction

Arc welding process plays a vital role in manufacturingDespite thewidespread use of arcwelding for joining themet-als total automation of the process is yet to be achieved and itis so due to the fact that the physics of the problem is not fullyunderstood and quantified Metal inert gas (MIG) welding isone of the most commonly used arc welding processes dueto its low initial cost and high productivity To have the betterknowledge and control of MIG welding process it is neces-sary to determine its input-output relationships MIG weld-ing is a complex process involving multiple variables Thequality of weld bead is decided by its mechanical propertieswhich are dependent on its bothmetallurgical properties andbead geometry which in turn depend on a number of inputprocess parameters such as welding speed voltage wire feedrate gas flow rate nozzle-plate distance torch angle surfacetension of the molten metal and electromagnetic force Sev-eral approaches had been developed by various researchers to

predict bead geometry in welding Those approaches includetheoretical studies statistical regression analysis and softcomputing-based approaches

Rosenthal [1] studied temperature distributions in an inf-inite plate due to amoving point heat source considering theconduction mode of heat transfer He did not relate his studywith the prediction of weld-bead geometry Later on Robertsand Wells [2] theoretically estimated the weld-bead width byconsidering the conduction mode of heat transfer after mak-ing a number of assumptions It might be difficult to model awelding process analytically due to its complexity and inher-ent nonlinearity

Various investigators had tried tomodel the process usingstatistical regression analysis also In this connection thestudies of Yang et al [3] Murugan et al [4 5] and Ganjigattiet al [6 7] are worth mentioning In conventional statisticalregression analysis input-output relationship is determinedresponse-wise (ie one at a time)Thus itmight not be able tocapture the dynamics of the process fully Moreover some of

2 International Journal of Manufacturing Engineering

the responses could be dependent on each other which can-not be determined using this approach

Soft computing-based approaches [8] had also been usedto model the input-output relationships of welding processesby various researchers Li et al [9] developed a control systemfor gas metal arc welding (GMAW) process where an artifi-cial neural network received online arc voltage data and pre-dicted the mode of metal transfer and assessed whether theoperating regime was appropriate for producing good qualitywelds An attempt was also made by Juang et al [10] usingback-propagation neural networks to establish input-outputrelationships of a tungsten inert gas (TIG) welding pro-cess Later on Amarnath and Pratihar [11] could successfullyestablish input-output relationships of the TIG welding pro-cess in both forward and reverse directions using radial basisfunction neural networks (RBFNNs) Experimental data aregenerally associated with imprecision and uncertainty Real-izing the fact that fuzzy logic can deal with the imprecisionand uncertainty Wu et al [12] developed a fuzzy logic systemfor process monitoring and quality evaluation in the GMAWby examining the welding voltage and current distributionsThe system was capable of detecting common disturbancesin the GMAW process by measuring the voltage and currentHong et al [13] designed a neurofuzzy controller which wascoupled with a vision-based system that could stabilize theweld pool online Ganjigatti and Pratihar [14] suggested anapproach for automatic design of fuzzy logic controller (FLC)that could predict bead geometry in MIG welding Singhand Gill [15] developed a model to predict tensile strength offriction welded GI pipes using an adaptive neurofuzzy infer-ence system (ANFIS) Several attempts were made to utilizethe fuzzy andor neurofuzzy systems for modeling of variousmanufacturing processes However those systems sufferedfrom the well-known problem of curse of dimensionality Thenumber of rules increases exponentially with the number ofinput variables and linguistic terms used to represent thosevariables This in turn leads to an increase in computerprocessing time and memory requirement for the storage Toovercome this problem Raju et al [16] proposed a hierarchi-cal structure in which a higher dimensional FLC [17] wasconsidered to be a combination of some lower dimensionalfuzzy logic systems connected in a hierarchical fashion toreduce the number of rules to a linear function of systemvariables Later on Cheong and Lai [18] considered anotherhierarchical structure whichwas not similar to that proposedby Raju et al [16]They designed a symmetrical rule base andthe data base was optimized using an evolutionary algorithmknown as differential evolution Lin et al [19] automated thedesign of Rajursquos hierarchical FLC (HFLC) to control waterlevel in an advanced boiling water reactor using the steepestdescent method To the best of the authorsrsquo knowledge nosignificant study has been reported to model input-outputrelationships of welding process using the HFLC

Building a hierarchical fuzzy system is a difficult taskThisis so due to the fact that we need to define both the architec-ture of the system (ie modules input variables of eachmod-ule and interactions among the modules) and the rulesof each module Chen et al [20] developed an automaticmethod of evolving hierarchical Takagi-Sugeno fuzzy systems

Table 1 Input welding parameters and their ranges

Sl no Input parameters Units Notation Range1 Welding speed cmmin 119878 25ndash462 Arc voltage volts 119881 26ndash313 Wire feed rate mmmin 119865 6ndash84 Gas flow rate litmin 119866 14ndash215 Nozzle to plate distance mm 119863 15ndash236 Torch angle degree 119860 70ndash110

(TS-FS)The hierarchical structure was evolved using a prob-abilistic incremental program evolution (PIPE) with somespecific instructionsThey considered the structure optimiza-tion in a sequential manner and the rule parameters embed-ded in the structure were tuned using an evolutionary pro-gramming (EP)

In the present work both conventional FLCs and HFLCsof Mamdani type have been developed and their perfor-mances are tested for modeling of MIG welding process Anovel approach has been developed for simultaneous opti-mization of the structure and knowledge base of the HFLCsusing a genetic algorithm (GA) [21] The performances ofthe developed approaches have been compared in terms ofaccuracy in predictions and computational time

The paper has been organized as follows Section 2 givesa brief description of the MIG welding process consideredin the present study the tools and techniques utilized in thepresent study have been discussed in Section 3 the developedapproaches are explained in Section 4 results are stated anddiscussed in Section 5 some concluding remarks are madein Section 6 and the scope for future work is suggested inSection 7

2 MIG Welding Process

MIGwelding one of the most popular arc welding processesuses a consumable metal electrode and the molten metalis protected from the atmosphere utilizing the shielding ofan inert gas like argon or helium The weld bead geometrycan be represented by its width (BW) height (BH) andpenetration (BP) and it is dependent on the following inputparameters welding speed (119878) voltage (119881) wire feed rate (119865)gas flow rate (119866) nozzle-plate distance (119863) torch angle (119860)and others Figure 1 displays the experimental set-up used forconducting experiments on the MIG welding process Thenotations of input parameters and their ranges are shown inTable 1 Figure 2 shows a typical weld bead

Experiments were conducted previously to collect input-output data according to a full-factorial statistical design[22] Bead-on-plate welding was carried out on steel plate of20mm thickness Welding current was varied in the range of3ndash500A and a reverse polarity was used during the experi-ments It is to be noted that argon was used as the shieldinggas during the welding It is also important to mention thatmainly a spray mode of metal transfer took place during thewelding There were six input process parameters and two

International Journal of Manufacturing Engineering 3

Work table Weld controland wire feeder source

Power Argon cylinder

Weldingtorch

Figure 1 Photograph of the set-up of MIG welding process [22]

levels had been considered for each of them Thus there wasa set of 26 = 64 combinations of input variables The weld-bead samples were cut surface ground using belt grinder andpolished utilizing various grades of sandpaperThe specimenswere again polished using aluminum oxide initially and thenutilizing diamond paste and cloth in a polishing machineThe polished specimens were cleaned with alcohol andmacroetched using 2 nital solution to reveal the geometryof the weld bead and heat-affected zone Photograph of eachmacro-etched samplewas taken and the imagewas convertedinto a pdf The measurement of bead geometry was carriedout and the measured dimensions were also compared withthe results obtained using a microscope to test the accuracyof themeasurement Regression analysis has been carried outto determine nonlinear response equations as given below

BH = 10297885times 1198650318995times 1198630101367

1198780389096 times 119881120688 times 1198660007758 times 1198600264379

BW = 119881155667times 1198650402256times 11986600211693times 1198600112076

10105147times 1198780326531 times 1198630113891

BP = 119881129320times 1198650781304times 1198660112076

100843747times 1198780252858 times 11986300726984 times 11986000726984

(1)

It is interesting to observe from the previous equations thatweld bead-geometric parameters are mainly dependent onthe selection of welding speed and arc voltagecurrent whichexactly match with the experience of the welders

21 Training Data Collection One thousand input-outputdata have been generated using the previous nonlinearregression equations by varying the input process parametersat random within their respective ranges It is to be notedthat sixty-four anchor data points (collected through realexperiments and with the help of which regression analysisis carried out) have also been considered along with the pre-vious mentioned one thousand data during the training ofFLCs Thus the training set consists of 1000 + 64 = 1064 datapoints

Figure 2 A typical weld bead [22] (BH bead height BP bead pen-etration and BW bead width)

I1

I2

I3

I4

In

FLS1FLS2

FLS3

FLSnminus1

O1

O2

O3

Onminus1

Onminus2

= O

Figure 3 A schematic view of the HFLC developed by Raju et al[16]

22 Test Data Collection In order to check the performancesof the developed fuzzy logic techniques twenty-seven testcases (collected through the real experiments) have been con-sidered [22] (refer to Appendix A)

3 Tools and Techniques Used

In this section the tools and techniques used to solve theproblem related to input-output modeling of MIG weldingprocess are going to be discussed

To establish input-output relationships of the said pro-cess Mamdani type of FLC [17] has been adopted It consistsof four modules namely identification of variables fuzzifica-tion inference engine and defuzzification Interested readersmay refer to [8] for a detailed description of the Mamdanitype of FLC It is important to mention that if there are 119899input variables of the process to be controlled and119898 linguisticterms are used to represent each variable there will be atotal of 119898119899 rules of the FLC Center of area method hasbeen utilized in the present work for defuzzification It is alsointeresting to observe that the number of rules increases withthe number of variables and that of linguistic terms used torepresent them Thus computational complexity of the FLCwill be more for the higher values of 119899 and119898

To overcome the above difficulty of the conventionalFLC a hierarchical fuzzy logic controller (HFLC) [16] was


GAstarts

Initial population of

Gen

Optimal FLC

Assign fitness to all strings

Reproduction

Crossover

Mutation

Case

Cal output using FLC

Cal string fitness

No

No

No

Yes

Yes

End

GA-string carries information of

GA-string

Population size

Start trainingcase= 0

Yes

strings gen = 0

gt

gt

max gen

gtmax case

Gen = gen + 1

String = string + 1

Case = case + 1

FLC string = 0

Figure 4 A schematic view of a genetic-fuzzy system

developed in which a higher dimensional FLC is consideredto be a combination of some lower dimensional fuzzy logicsystems (FLSs) connected in a hierarchical fashion Figure 3shows the schematic view of the hierarchical FLC developedby Raju et al [16] In their method two inputs are fed toa subcontroller called FLS and the produced output alongwith the third input is passed to another FLS This processcontinues until all the input variables are utilized Here onlytwo inputs are considered for each FLS If each variableis expressed using 119898 linguistic terms 1198982 rules are to bedesigned for each FLSAs there are (119899minus1) such FLSs1198982 (119899minus1)(in place of119898119899) rules are necessary to develop the FLC com-pletely which clearly indicates that the number of rules of theHFLC increases linearly with the number of input variables 119899

In Rajursquos HFLC [16] the inputs should be passed in sucha way that the most important variable is considered at thefirst level and the second most important variable is put atthe next level and so on However it is not necessary thatthe number of inputs at each subcontroller should always beequal to two The inputs may vary from two to 119899 variables(ie total number of inputs) If all the 119899 variables are passedat a time then it becomes the conventional FLC Thus theHFLC structure is not unique and one may be in dilemma

in choosing a particular type of HFLC structure The chosenstructuremay not be optimal also for a given problemThere-fore the performance of the HFLC depends on (i) structureof the HFLC chosen for modeling (ii) knowledge base of thesubcontrollers (iii) the order in which the inputs are passedto the subcontrollers However designing a suitable hier-archical FLS is a difficult task It is so because both thearchitecture of the system (ie modules input variables ofeach module and interactions among the modules) and therules of each module are to be defined

An FLC does not have an inbuilt optimizer and conse-quently a proper training is to be provided to it with the helpof some training scenarios and an optimization tool say GA[21] The performance of an FLC depends on its knowledgebase consisting of data base (which carries informationrelated tomembership function distributions of the variables)and rule baseTheGAhas been utilized by a number of inves-tigators to improve the performance of the FLC and thedeveloped approaches are popularly known as the genetic-fuzzy systems [23] Figure 4 shows the schematic view of thegenetic-fuzzy system in which a batch mode of training hasbeen adopted with the help of a large number of scenariosusing a GA


4 Developed Approaches

Tomodel input-output relationships of theMIGwelding pro-cess six approaches based on fuzzy logic technique (as dis-cussed previously) have been developed and their perform-ances are compared among themselves

Approach 1 GA-Based Tuning of Rule Base and Data Base ofManually Constructed Conventional FLC [24] Fuzzy logic-(FL-) based controller has been developed to model input-output relationships in MIG welding process There are sixinputs namely welding speed (119878) voltage (119881) wire feedrate (119865) gas flow rate (119866) nozzle-plate distance (119863) andtorch angle (119860) and three outputs such as bead height (BH)bead width (BW) and bead penetration (BP) of the processFigure 5 shows the genetic-fuzzy system developed to modelthe MIG welding process

Manually constructed membership function distribu-tions of the input and output variables for MIG welding pro-cess are shown in Figure 6 For simplicity the shape of themembership function distributions has been assumed to betriangular in nature The range of each variable has beendivided and expressed using three linguistic terms such as LlowMmedium andH highThus there are 36 = 729 rules orinput combinationsThe rule base is designedmanually basedon the designerrsquos knowledge of the process A typical rule ofthe FLC will look as follows

If 119878 is H AND119881 is L AND 119865 is M AND 119866 is M AND119863 isM AND 119860 is M then BH is M BW is L BP is L

A binary-coded GA has been used [21] Ten bits areassigned to represent each of the nine variables carrying base-width information of the triangular membership functiondistributions (represented using 119887 values) Thus 9 times 10 =90 bits are required to represent all the real variables More-over 729 bits are utilized to represent 729 rules (where 1 and 0represent the presence and absence of a rule resp) The GA-string is thus 90 + 729 = 819 bits long Such a GA-string rep-resentation is shown in Figure 7

Here 119887 values indicate the half base-width values of isos-celes and base-width of right angled triangles of the inputand output variables During optimization the 119887 values forwelding speed arc voltage wire feed rate gas flow rate noz-zle-to-plate distance torch angle bead height bead widthand bead penetration have been varied in the ranges of(805 1215) (200 320) (040 090) (180 320) (220380) (1005 2015) (0885 1455) (230 390) and (064118) respectively The fitness of a GA-string 119891 is calculatedas the average of absolute mean deviation in prediction forthe entire set of training scenarios as given below

119891 =1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) (2)

where 119879119894119895and 119874

119894119895represent the target and predicted values

respectively of 119895th output for 119894th training scenario 119873 indi-cates the number of training scenarios 1198991015840 denotes the numberof outputs

The FLC has been trained using a GA offline with thehelp of a batchmode of training involving 1064 scenariosTheGA starts with a population of binary strings (ie solutions)created at random The population of solutions is then mod-ified using the operators namely reproduction crossoverand mutation [8] to hopefully create the better solutions Atournament selection scheme has been utilized in which 119905strings (equal to tournament size) are selected from the pop-ulation at random and the best one in terms of fitness valueis picked Thus a mating pool will be created to participatein the crossover In this operation there is an exchange ofproperties between two parent strings and consequentlytwo children strings will be created In the present work auniform crossover scheme has been adopted The possibilityof occurring crossover is checked at each bit position witha probability of 05 for the head appearing If the headappears at a particular bit position there will be an exchangeof bits between the two parents and consequently twochildren will be created Moreover a bit-wise mutation hasbeen implemented to bring a local change over the currentsolution

Approach 2 Automatic Design of FLC Using a GA [24] InApproach 1 theGA is used to optimize themanually designedKnowledge Base (KB) of the FLC However it might be dif-ficult sometimes to design the KB beforehand due to a lackof knowledge of the process An attempt has been made todesign the FLC automatically using the GA in this approachThe GA-string carries information of the bases of triangularmembership function distributions presence or absence ofthe rules along with the consequent parts It is to be notedthat there are three outputs for each set of input variables andtwo bits are used to represent each output (such as 00 for L 01and 10 forM and 11 forH)Thus six bits are utilized to denoteconsequent parts of a rule A typical GA-string will look as inFigure 8

During optimization the ranges of 119887 values have beenkept the same as those used in Approach 1 and the FLC istrainedwith the help of the same set of 1064 training scenariosused earlierThe fitness of theGA-string has been determinedin the same way as it has been done in Approach 1

Approach 3 GA-Based Tuning of Rule Base and Data Base oftheManually ConstructedHFLC Structure Proposed by Raju etal [16] The schematic diagram of HFLC as proposed by Rajuet al [16] is shown in Figure 3 Asmentioned earlier the inputvariables are considered at different levels according to theircontributions towards the outputThe order of importance ofthe input variables has been kept the same as that inGanjigatti[22] It is important to mention that a significance test isconducted during statistical regression analysis to identify thesignificant and insignificant input variables Moreover theorder of importance of significant input variables can bedecided from either normal probability plot or Pareto-chartof effects on response Interested readers may refer to [25] fora detailed description of the significance test With respect tobead height (BH) it is found to be as follows 119878 119881 119860 119865 119863and 119866 Thus the contributions of welding speed 119878 and gasflow rate 119866 are seen to be the maximum and minimum


GA-based tuning

Knowledge base

Inferenceengine DefuzzificationFuzzification

Outputs

BH

BW

BP

InputsSVFGDA

Fuzzy system

Figure 5 Genetic-fuzzy system to model MIG welding process

Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H

249 350 451 259 280 301 224 3395 455 Arc voltage (volts)Welding speed (cmmin)

Wire feed rate (mmmin) Gas flow rate (litmin) Bead width (mm)

Bead penetration (mm)Torch angle (deg)Nozzle-to-plate distance (mm)

59 65 71 139 160 181 734 1042 135

149 175 201 699 850 1001 137 239 341

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

Inputs Outputs

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

Figure 6 Manually constructed membership function distributions of the input-output variables MIG welding process


10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9

Figure 7

10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9

Consequent part of the outputs

middot middot middot

011010 1100101010 10 0101010110 0101010011

r1 r2 r3 r729

Figure 8

HFLC parameters for BH

10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1

FLS1 FLS2 FLS3 FLS4 FLS5

bI1 bI2 bO1 bI3 bO2 bI4 bO3 bO4 bI6bI5 bO5

HFLC parameters for BW HFLC parameters for BP

bI1 bI1bI2 bI2bO1 bO1bO4 bI6 bI6bO5 bO5

11 0101 0110 101 10 101 011 10001 110 1010 10101 101010

middot middot middot middot middot middot

Rule base for BH Rule base for BW Rule base for BP

1010 010110101 010111010100101 010111000100110

Figure 9

respectively towards the mentioned response Similarly theorders of importance of the input variables are coming outto be 119881 119878 119865 119863 119860 and 119866 and 119860 119881 119878 119865 119866 and 119863 for theoutputs BW and BP respectively As the mentioned order ofimportance is not found to be the same for all the responsesa separate HFLC is constructed for each outputresponse Asthere are six inputs each HFLC will consist of five sub-FLCs

In HFLC each sub-FLC has two inputs represented bythree linguistic terms each and thus a total of 5 times 32 = 45rules are considered for one response For three responsesthe number of rules is coming out to be equal to 3 times 45 =135 Thus in the designed HFLC the total number of rules isreduced from 729 to 135 The rule base is designed manuallybased on the knowledge of the process the designer has


Besides the rules the performances of an HFLC used tomodel one response depend on the membership functiondistributions of six inputs four intermediate outputs and onefinal output that is 6 + 4 + 1 = 11 real variables Ten bits areassigned to represent each of the 33 variables (for threeresponses) The first 330 bits carry the information of 33 realvariables and the next 135 bits represent the rule base forthree approaches The GA-string is 465 bits long which isshown in Figure 9

During optimization the ranges of 119887 values for optimiz-ing the system variables have been kept the same as beforeThehalf base-width values of the triangles used for intermedi-ate outputs have been varied in the range of (040ndash050) Thefitness of the GA-string has been calculated in the same wayas discussed previously The HFLC has been trained with thehelp of the same set of scenarios considered in the previousapproaches

Approach 4 Automatic Design of HFLC Proposed by Rajuet al [16] Using a GA An attempt is made to design the sub-FLCs automatically using a GA In this approach the task ofdesigning good KB of the sub-FLCs is given to the GA andit tries to find an optimal KB through search As discussed inApproach 3 45 rules are used to determine each of the threeresponses Thus there are a total of 3times 45 = 135 rules Twobits are utilized to represent the consequent part of each ruleWith the two bits there could be four possibilities such as 0001 10 and 11 If the sum of two bits is found to be equal to 0(ie 0 + 0 = 0) it indicates L that means the consequent L isrepresented by 00 Similarly both 01 and 10 will denote theconsequent M and 11 represents the consequent HThus theGA-string will consist of 465 (as in Approach 3) + 3 times 45 times 2 =735 bits A particular GA-string can be represented as givenin Figure 10

During optimization the ranges of 119887 values have beenkept the same as those used in Approach 3 and the FLC istrained using the same 1064 training scenarios The fitness ofthe GA-string has been determined in the same way as dis-cussed previously

Approach 5 Structure Optimization and Automatic Design ofHFLC Using a GA Considering Accuracy in Predictions as theFitness In this approach the structure of the FLC has notbeen kept as a fixed oneThe structures investigatedmay varyfrom the hierarchical structure proposed by Raju et al [16]to the structure of conventional FLC For the MIG weldingprocess involving six input variables and for a fixed order ofimportance of the variables sixteen hierarchical structuresmay be possible to design The tree structure representationis used for generating the structure of the HFLC randomly

Here the GA-string carries information of the structureand knowledge base of the HFLC It is important to mentionthat the number of inserted intermediate variables and thatof the rules depend on structural information of the HFLCTherefore the length of GA-stringmay vary in the populationof solutions The first five bits represent the structure of anHFLC and based on that the number of bits required forrepresenting the data base and rule base is determinedFor example let us consider a structure of 2 2 3 2 0

It indicates that there are four subcontrollers (FLSs) with2 2 3 and 2 inputs respectively For one response ten 119887values are required to represent the data base of the HFLCThus a total of 3 times 10 = 30 real 119887 values are to be specified fordetermining three responses Moreover a total of 32 + 32 +33 + 32 = 54 rules are to be considered for determining eachresponseThus there are 3 times 54 = 162 rules in the HFLC usedfor determining three responses As two bits are utilized todenote the consequent of each rule (as done in Approach 4)162 times 2 = 324 bits are required to represent the consequent ofall of them It is to be noted that ten bits are used to denoteeach 119887 valueThus theGA-stringwill consist of 5 + 30times 10 + 3times 54+ 3times 54times 2= 791 bitsThe ranges of systemvariables havebeen kept the same with those considered in the previousapproaches However the variables representing the span ofmembership function distributions of intermediate outputshave been varied in the range of (0001 1001) The stringrepresentation for this approach is shown in Figure 11

The fitness of the GA-string has been defined in the sameway as done in the previous approaches The HFLC hasbeen trained with the help of the same set of 1064 scenariosAs the GA-strings contained in the population of solutionsmay have varied lengths it will be difficult to implementthe conventional crossover operator Thus a special type ofcrossover operator is to be implemented In the present worka crossover operator suitable for variable string lengths hasbeen proposed and its performance has been tested Theproposed crossover operator has been explained below

Simultaneous optimization of the structure and KB of anHFLC lead to a problem in which the GAwill have to handlestrings having varied lengths In such a situation the con-ventional crossover operators cannot be used Let us supposethat the following two parent strings are participating in thecrossover to create two children strings by exchanging theirproperties (see Figure 12)

The first five bits representing the structure of FLC willnot participate in the crossover Let us also assume thatParents 1 and 2 have 119871

1and 119871

2bits respectively and 119871

1lt 1198712

Thus Parent 2 contains (1198712minus1198711)more bits than Parent 1 does

An additional substring of length (1198712minus1198711)will be created for

Parent 1 which is nothing but the complementary substringof (1198712minus1198711) bits counted from the right-hand side of Parent 2

Thus the participating parents in the crossover will look likeas shown in Figure 13

It is to be noted that the bits representing the structurewill not take part in the crossover and for the remainingbits a uniform crossover of 05 probability will be used forthe exchange of properties Due to the crossover Child 1and Child 2 will be created from the Parent 1 and Parent 2respectively After the crossover is done (119871

2minus 1198711) bits will

be deleted from the right-hand side of Child 1 Let us assumethat uniform crossover takes place at 7th 10th 13th 16th and18th positions in the above parent strings Using the aboveprinciple the children strings are found to be as shown inFigure 14

Approach 6 Structure Optimization and Automatic Design ofHFLC Using a GA Considering Both Accuracy and Compact-ness as the GA-FitnessThis approach differs fromApproach 5



10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110

Consequent part of the BWConsequent part of the BH Consequent part of the BP

r1 r2 r3 r45 r1 r2 r3 r45 r1 r2 r3 r45

011010 110010 110100 1010101 0101010110 101010

middot middot middot middot middot middot middot middot middot

Figure 10

Structure

Representation


FLS1 FLS2 FLS3 FLS4

bI1 bI2 bO1 bI3 bO2 bI4 bO3 bI6bI5 bO4

2 2 3 2 0 10 10101 0110 01011 10010 11100 10101 01010 00011 10



11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r54 r1 r2 r3 r54 r1 r2 r3 r54

011010 110010 110100 1010101 0101010110 101010


Figure 11


Table 2 Differences and similarities of six approaches

Approach 1 It deals with conventional Mamdani type FLC of fixed structure A GA is used to tune RB and DB of the manuallyconstructed FLC Accuracy in predictions has been considered as the fitness of a GA-string

Approach 2 Conventional Mamdani type FLC of fixed structure has been dealt A GA is utilized for automatic design of DB and RB ofthe FLC Accuracy in predictions has been considered as the fitness of a GA-string

Approach 3 It deals with Hierarchical FLC of fixed structure A GA has been used to tune RB and DB of the manually constructedHFLC Accuracy in predictions has been considered as the fitness of a GA-string

Approach 4 Hierarchical FLC of fixed structure has been dealt A GA is utilized for automatic design of DB and RB of the HFLC Thefitness of a GA-string is calculated as the accuracy in predictions

Approach 5An FLC of variable structure (may vary from HFLC to conventional FLC) has been considered in this approach A GA hasbeen used for structure optimization and automatic design of FLC Accuracy in predictions has been considered as thefitness of a GA-string

Approach 6 It is the same with Approach 5 but both accuracy in predictions and compactness of the structure have been considered asthe fitness of a GA-string

Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14

in terms of the definition of fitness function only Here bothaccuracy in predictions and number of rules of the HFLChave been considered in the fitness function It is importantto mention that equal weightage has been given to both theaccuracy in predictions and compactness of the HFLC Thefitness of the GA-string 119891 has been calculated as given below

119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)

where 119877 indicates the number of rules present in the HFLCand 119877max is the maximum number of rules to be present inthe conventional FLC The other terms carry usual meaningas explained previously

The differences and similarities of the previously sixapproaches have been highlighted in Table 2

5 Results and Discussion

The performances of the developed approaches have beencompared in predicting input-output relationships of MIGwelding process as discussed below

As the performance of a GA depends on its parametersa careful study has been carried out to determine the set ofoptimal parameters During the parametric study only oneparameter has been varied at a time keeping the others fixedA uniform crossover with probability 119901

119888= 05 has been

utilized in the present study Initially the values of populationsize and maximum number of generations have been keptfixed to 120 and 160 respectively Figure 15(a) shows thevariations of fitness with the probability of mutation 119901

119898 The

GA yields the minimum value of the fitness corresponding to119901119898= 00005Thus 119901

119898has been kept fixed to 000055 for the remaining

part of the parametric study Population size has been variedin the range of (100 320) after keeping the values of 119901

119898and

the maximum number of generations fixed at 000055 and160 respectively (refer to Figure 15(b)) The minimum valueof the fitness is found to occur corresponding to the popula-tion size of 220TheGA is then run for a large number of gen-erations corresponding to 119901

119898= 000055 and a population

size of 220 It is found to converge to a fixedminimumvalue ofthe fitness at 560 generations (refer to Figure 15(c)) From theprevious study the optimum values of mutation probabilitypopulation size and number of generations are found tobe equal to 000055 220 and 560 respectively The opti-mized membership function distributions of the variables


0165

0166

0167

0168

0169

017

0171

0172

0 00002 00004 00006 00008 0001 00012 00014Probability of mutation (Pm)

Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029

0 100 200 300 400 500 600 700 800Number of generations

Fitn

ess v

alue

(c)

Figure 15 Results of GA-parametric study (a) fitness versus probability of mutation (b) fitness versus population size and (c) fitness versusnumber of generations

as obtained using this approach are shown in Figure 16 It isto be noted that only 141 rules are found to be good by the GA(refer to Appendix B) from a total of 729 rules in Approach 1

The performance of the optimized controller has beentested on 27 test cases for making predictions of theresponses namely BH BW andBPThepredicted values havebeen compared to the respective target (experimental) valuesof the responses to determine percent deviation in predictionif any It is to be noted that a slightly better prediction has beenobtained for BH compared to the other two responsesMore-over an underprediction has been observed for the responseBP It might have happened due to the fact that besides someexperimental errors the parameters like surface tension ofthe molten metal electro-magnetic force and others (whichhave not been considered in the present study) have signifi-cant influence on the bead-geometric parameters Moreoverlinear membership function distributions have been con-sidered here for simplicity However nonlinear membershipfunction distributions might have yielded a slightly betterprediction The CPU time has been determined by passingmentioned said 27 test cases for 100 times It has been foundto be equal to 225 seconds for this approach

Similarly the optimized KBs of the FLCs have beenobtained using Approaches 2 through 6 Comparisons havebeen made of their predicted values with their respective

target values of the responses Moreover their computationalcomplexities have been determined in terms of CPU timeThe following subsections deal with the previous compar-isons in detail In Approach 5 where the structure of theHFLC has been optimized considering the accuracy in pre-dictions of the fitness the evolved structure is found to be asthat of a conventional FLC Thus it has been proved that theconventional FLC can provide the best accuracy in predic-tions In Approach 6 the structure of the HFLC has beenoptimized by putting equal weightage to both accuracy andits compactness The evolved structure using Approach 6 forprediction of BH is seen to have a form of (2 2 4 0 0) asshown in Figure 17 The optimized rule base is found to con-tain only 63 rules in Approach 6

It is important to mention that input variables are chosenat different levels of anHFLC according to their contributionstowards an output The most significant variables are passedthrough the first subFLC that is FLS

1 It is also interesting to

observe that Approach 6 has evolved an optimal structure ofthe HFLC for the prediction of BH in which the sequence ofinput variables has turned out to be the same with that men-tioned in Section 4 (ie Approach 3)

51 Comparison of Approaches 1 through 6 in Terms of Deviation in Predictions The performances of the above sixapproaches have been compared in terms of deviation in


Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H



Bead penetration (mm)

Inputs Outputs

Torch angle (deg)Nozzle-to-plate distance (mm)

59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

Figure 16 Optimized membership function distributions of the input and output variables of the FLC Approach 1

S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2

Figure 17 Evolved structure ofHFLCused to predict BH Approach6

predictions of different responses Figures 18(a) 18(b) and18(c) show the plots of percentage deviations in predictionsof BH BW and BP respectively

For BH and BW the values of deviation in predictionsare found to lie on both the positive and negative sides forall the approaches In case of BH the large deviations areobserved for 1st 6th and 26th test cases and the same hasbeen obtained for the 6th and 9th cases for predicting BW byall the approaches It might have come due to simplicity ofthe developed model and some experimental errors It is tobe noted that all the approaches have underpredicted the BPvalues for almost all the test cases It might have happeneddue to the fact that the parameters such as surface tensionof the molten metal and electro-magnetic forces have beenneglected for simplicity of the modeling

52 Comparison of Approaches 1 through 6 in Terms of AverageRMS Deviation in Predictions This subsection compares theperformances of the developed approaches in terms of aver-age root mean squared (RMS) deviation in prediction of


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

Approach 1Approach 2Approach 3


minus5

minus10

minus15

(c)

Figure 18 Plots of percentage deviations (obtained using all six approaches) in predictions of (a) BH (b) BW and (c) BP

Table 3 Comparison of Approaches 1 through 6 in terms of com-putational complexity

ApproachesNo of optimized rules

CPU time(27 cases for 100 runs)

Response-wiseBH BW BP

Approach 1 141 141 141 2250msApproach 2 227 227 227 2740msApproach 3 26 21 30 575msApproach 4 17 21 28 556msApproach 5 251 251 251 2820msApproach 6 63 53 59 402ms

the responses Figure 19 displays RMS deviation values inprediction of individual and combined outputs as obtainedby all six approaches

It is interesting to note that Approach 5 has slightlyoutperformed other approaches in terms of RMS deviation inprediction of the responses As no effort is made for manualdesign of the rule base this approach might be interestingalso Moreover its performance could be further improvedby considering asymmetric and nonlinear membership func-tion distributions Approach 6 has performed better thanApproaches 3 and 4 and has showed good interpolationcapability It is also important to note that both Approaches5 and 6 have outperformed Approach 4 that is the approachproposed by Raju et al [16]

53 Comparison of Approaches 1 through 6 in Terms of Com-putational Complexity Computational complexities of theprevious approaches have been compared in terms of CPUtime values The processing time of an approach dependson the architecture of FLC number of rules present in therule base and others Table 3 displays the number of good


Table 4 Twenty-seven test cases used in this study

Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224

rules identified by all the response-wise approaches and theircomputational complexities measured in terms of CPU timevalues

The previous comparison shows that there is a largereduction in the size of rule base and CPU time for predictingvarious outputs in case of HFLCs (except in Approach 5where the optimized HFLC moves towards the optimizedconventional FLC) On the other hand the conventional FLChas performed slightly better thanHFLC in terms of accuracyin predictions due to more number of optimized rulesThusthere might be a trade-off between good accuracy of conven-tional FLC and low computational complexity of HFLC Ouraim is also to develop a compact FL-based technique whichshould be able to make the predictions as accurately as possi-ble It can be posed as amultiobjective optimization problemas explained below It is important to mention that there willbe no improvement of interpretability by introducing a hier-archical fuzzy system

54 Multiobjective Optimization Using NSGA-II [26] Anattempt has beenmade to determine a Pareto-optimal front ofsolutions using an algorithm named NSGA-II [26] Figure 20displays the obtained Pareto-optimal front of solutions Thealgorithm has identified three distinct and discontinuousregions denoted by Zones A B and C (refer to Figure 20)

It is interesting to note that Zone A shows nondominatedsolutions obtained by the conventional FLC (ie Approaches1 and 2) Zone B consists of the solutions yielded by the pro-posed HFLC (ie Approach 6) and Zone C contains thesolutions of HFLC structure proposed by Raju et al [16] (ieApproach 3)

6 Concluding Remarks

To model input-output relationships of MIG welding pro-cess both conventional and hierarchical FLCs have beendeveloped and their performances are evaluated Out of sixapproaches two are related to conventional FLC and theremaining four belong to HFLCs Conventional FLCs arefound to yield better accuracy in predictions compared to theHFLCs On the other hand HFLC proposed by Raju et al[16] and its automatic implementation (ie Approaches 3 and4) are found to be the most compact ones (hence it has theleast computational complexity) but at the cost of accuracyin predictions However the HFLC proposed by the authors(ie Approach 6) is able to yield some results which can beconsidered as the trade-off between the above two extremeconditions It is important to mention that there is noimprovement of interpretability in the developed hierarchicalfuzzy systems


Table 5 Optimal rule base of conventional FLC obtained usingApproach 1

Sl no 119878 119881 119865 119866 119863 119860 BH BW BP1 H L M M M M M L L2 M H H M M L H H H3 M L M H M L H L H4 H M M L M H L L L5 H M M M L L L L H6 M M M M M M M M M7 H H H H H M L H H8 M M M L L M M M M9 L L M M M L H L H10 H H M H H L L L H11 M H L H M L L M H12 L H M M M L H H H13 L H L L L H L H M14 L H L H L H H H H15 M H L L L H L H L16 L H H H H H H H H17 L M H M M L H H H18 L H H L M L H H H19 H L M L L M M L L20 H H L H M M L M M21 L M H M H L H M H22 H M M L L M L L L23 L H H L L H M H H24 M M L L L M L L L25 H M M H H M L L L26 L L M M H M H L L27 M L M M H H H L L28 M L M H H M H L L29 M M M H H L H L H30 L L H M H L H L H31 H M M L L H L M L32 L H L H H M L H H33 L M M L L M H H H34 H M L M M M L L L35 H H L L M H L L L36 H H M H H M L M M37 H L L H H H L L L38 L H M M M M M H H39 H L L L L L M L L40 H H H H M L L H H41 H H L M M H L M L42 H L H M L L H L H43 L L L H H M H L L44 L L L L H H H L L45 H H M M L L L M H46 H L L H H L M L L47 H M M M H H L L L48 M H M H M L L H H49 M L L M M M M L L

Table 5 Continued

Sl no 119878 119881 119865 119866 119863 119860 BH BW BP50 L L M L L L H L H51 L H M L M H M H M52 L M M M L L H H H53 M H M M L L L H H54 H H H M L L L H H55 M L M M M L H L M56 H L L L L M L L L57 H L M H M L M L M58 H M M M M M L L L59 H H M L L M L M M60 M L H H H H H H L61 L L M M M M H M M62 L L L M H H H L L63 L L L M L L H L H64 M L L M H H M L L65 H L L H L M L L L66 L M H H H L H H H67 H L M H H M M L L68 M L M L L M H L L69 H M L M L M L L L70 L H L M L H L H H71 L H M M H M H H H72 H M M M L M L M M73 H H M M M L L L H74 L M L M M M M M M75 M L L L M M H L L76 L H M H H M M H H77 M L M L L L H L M78 L H L M H H L H L79 L H M L H H H H L80 L L L L H M H L L81 H H H M M M L H H82 H M H M L L M M H83 H L M M L M L L L84 H L L M L H L L L85 M L M L M H H L L86 L H L H L L L H H87 M H L M M M L M M88 H L H M M M H L L89 M L L M M L H L L90 H L M M L L M L M91 M M L M M M L L L92 L H H M L M M H H93 M L M L M M H L L94 H L H L L L H L M95 L M H L L L H H H96 H L L L M H L L L97 H M L M L L L L M98 L H M H M L M H H99 L H M M H H M H M


Table 5 Continued

Sl no 119878 119881 119865 119866 119863 119860 BH BW BP100 H M L H H L L L L101 H M M M M L M L M102 H L H L H H H L L103 L L L L L L H L M104 M M M M M L H L H105 L H H H L L M H H106 M H M L L M L H H107 L H M H H L H H H108 H L M M M L H L L109 H L H H H M H L L110 M L L H H H L L L111 L H M L L L H H H112 M L H M M L H L H113 L M M M M M H H H114 H H M M H H L M L115 M L L L L H L L L116 H L L M M L M L L117 L H L M M M L H H118 M L H L H H H L L119 H M M H H L M L M120 M M M L H H H L L121 L L L M M M H L L122 H L M H M M L L L123 L L M L H H H L L124 H M L M M H L L L125 L M M H H M H M M126 L M H L H L H M H127 L H M M L L H H H128 M M M M H M L H L129 M H M M M M M H H130 H L L L L H L L L131 H L M L M M M L L132 H M L H M H L L M133 H H H M M H L M M134 H H H M H L M M H135 H H H M H H L H M136 M L M H H L H L M137 H H M M L M L H H138 M L M M M M H L L139 L L M L M M H L L140 M H L M M H L H L141 M L H L L L H L H

7 Scope of Future Work

In the present work an attempt has been made to optimizeboth the structure and the knowledge base of the HFLCTheproblems related to forwardmapping ofMIGwelding processhave been tackled using the FL-based techniques Symmetricand linear membership function distributions have beenconsidered to developMamdani-type fuzzy inference systemThe present work can be extended in the following ways

0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6

Bead heightBead width

Bead penetrationCombined output

Figure 19 Comparison of RMS deviations in prediction of individ-ual and combined outputs

044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion

Figure 20 Pareto-optimal front of nondominated solutions formultiobjective optimization problems

(i) reverse mapping problems (in which the processparameters are expressed as the functions of bead-geometric parameters) related to MIG welding pro-cess will be solved in the future by following the sameprocedure

(ii) asymmetrical and nonlinear membership functiondistributions may be considered to improve the accu-racy in predictions

(iii) hierarchical structure related to Takagi- and Sugeno-type fuzzy inference system might be developed totackle the mentioned problem


Appendices

A

See Table 4

B

See Table 5

References

[1] D Rosenthal ldquoMathematical theory of heat distribution duringwelding and cuttingrdquo Welding Journal vol 20 no 5 pp 220ndash234 1941

[2] D K Roberts and A A Wells ldquoFusion welding of aluminumalloysrdquoWelding Journal vol 12 pp 553ndash559 1954

[3] L J Yang R S Chandel and M J Bibby ldquoAn analysis ofcurvilinear regression equations for modeling the submerged-arc welding processrdquo Journal of Materials Processing Tech vol37 no 1ndash4 pp 601ndash611 1993

[4] N Murugan R S Parmar and S K Sud ldquoEffect of submergedarc process variables on dilution and bead geometry in singlewire surfacingrdquo Journal of Materials Processing Tech vol 37 no1ndash4 pp 767ndash780 1993

[5] N Murugan and R S Parmar ldquoEffects of MIG process param-eters on the geometry of the bead in the automatic surfacing ofstainless steelrdquo Journal of Materials Processing Tech vol 41 no4 pp 381ndash398 1994

[6] J P Ganjigatti D K Pratihar and A Roychoudhury ldquoModelingof the MIG welding process using statistical approachesrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 11-12 pp 1166ndash1190 2008

[7] J P Ganjigatti D K Pratihar and A R Choudhury ldquoGlobalversus cluster-wise regression analyses for prediction of beadgeometry inMIGwelding processrdquo Journal ofMaterials Process-ing Technology vol 189 no 1ndash3 pp 352ndash366 2007

[8] D K Pratihar Soft Computing Narosa New Delhi India 2008[9] X Li SW Simpson andM Rados ldquoNeural networks for online

prediction of quality in gas metal arc weldingrdquo Science andTechnology of Welding and Joining vol 5 no 2 pp 71ndash79 2000

[10] S C Juang Y S Tarng and H R Lii ldquoA comparison betweenthe back-propagation and counter-propagation networks in themodeling of the TIG welding processrdquo Journal of MaterialsProcessing Technology vol 75 no 1ndash3 pp 54ndash62 1998

[11] M V Amarnath and D K Pratihar ldquoForward and reversemappings of the tungsten inert gas welding process using radialbasis function neural networksrdquo Proceedings of the Institution ofMechanical Engineers B vol 223 no 12 pp 1575ndash1590 2009

[12] C S Wu T Polte and D Rehfeldt ldquoA fuzzy logic system forprocess monitoring and quality evaluation in GMAWrdquoWeldingJournal vol 80 no 2 supplement pp 33ndashS 2001

[13] L Hong L F M Kee J W J Yu et al ldquoVision-based GTA weldpool sensing and control using neuro-fuzzy logicrdquo Tech RepSIMTECH 2000

[14] J P Ganjigatti and D K Pratihar ldquoForward and reversemodeling in MIG welding process using fuzzy logic-basedapproachesrdquo Journal of Intelligent and Fuzzy Systems vol 19 no2 pp 115ndash130 2008

[15] J Singh and S S Gill ldquoMulti-input single output fuzzy modelto predict tensile strength of radial friction welded GI pipesrdquo

International Journal of Information and Systems Sciences vol4 no 3 pp 462ndash477 2008

[16] G V S Raju J Zhou and R A Kisner ldquoHierarchical fuzzy con-trolrdquo International Journal of Control vol 54 no 5 pp 1201ndash1216 1991

[17] E H Mamdani and S Assilian ldquoAn experiment in linguisticsynthesis with a fuzzy logic controllerrdquo International Journal ofMan-Machine Studies vol 7 no 1 pp 1ndash13 1975

[18] F Cheong and R Lai ldquoDesigning a hierarchical fuzzy logic con-troller using the differential evolution approachrdquo Applied SoftComputing Journal vol 7 no 2 pp 481ndash491 2007

[19] C Lin F-L Jeng C-S Lee and R Raghavan ldquoHierarchicalfuzzy logic water-level control in advanced boiling water reac-torsrdquo Nuclear Technology vol 118 no 3 pp 254ndash262 1997

[20] Y Chen B Yang A Abraham and L Peng ldquoAutomaticdesign of hierarchical Takagi-Sugeno type fuzzy systems usingevolutionary algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 15 no 3 pp 385ndash397 2007

[21] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[22] J P Ganjigatti Application of statistical methods and fuzzy logictechniques to predict bead geometry in welding [PhD thesis] IITKharagpur India 2006

[23] O Cordon F Gomide F Herrera F Hoffmann and L Mag-dalena ldquoTen years of genetic fuzzy systems current frameworkand new trendsrdquo Fuzzy Sets and Systems vol 141 no 1 pp 5ndash312004

[24] D K Pratihar K Deb andA Ghosh ldquoA genetic-fuzzy approachfor mobile robot navigation among moving obstaclesrdquo Interna-tional Journal of Approximate Reasoning vol 20 no 2 pp 145ndash172 1999

[25] D C Montgomery Design and Analysis of Experiments JohnWiley amp Sons New York NY USA 2003

[26] N Srinivas and K Deb ldquoMultiobjective optimization usingnon-dominated sorting in genetic algorithmsrdquo EvolutionaryComputation vol 2 no 3 pp 221ndash248 1994

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the responses could be dependent on each other which can-not be determined using this approach

Soft computing-based approaches [8] had also been usedto model the input-output relationships of welding processesby various researchers Li et al [9] developed a control systemfor gas metal arc welding (GMAW) process where an artifi-cial neural network received online arc voltage data and pre-dicted the mode of metal transfer and assessed whether theoperating regime was appropriate for producing good qualitywelds An attempt was also made by Juang et al [10] usingback-propagation neural networks to establish input-outputrelationships of a tungsten inert gas (TIG) welding pro-cess Later on Amarnath and Pratihar [11] could successfullyestablish input-output relationships of the TIG welding pro-cess in both forward and reverse directions using radial basisfunction neural networks (RBFNNs) Experimental data aregenerally associated with imprecision and uncertainty Real-izing the fact that fuzzy logic can deal with the imprecisionand uncertainty Wu et al [12] developed a fuzzy logic systemfor process monitoring and quality evaluation in the GMAWby examining the welding voltage and current distributionsThe system was capable of detecting common disturbancesin the GMAW process by measuring the voltage and currentHong et al [13] designed a neurofuzzy controller which wascoupled with a vision-based system that could stabilize theweld pool online Ganjigatti and Pratihar [14] suggested anapproach for automatic design of fuzzy logic controller (FLC)that could predict bead geometry in MIG welding Singhand Gill [15] developed a model to predict tensile strength offriction welded GI pipes using an adaptive neurofuzzy infer-ence system (ANFIS) Several attempts were made to utilizethe fuzzy andor neurofuzzy systems for modeling of variousmanufacturing processes However those systems sufferedfrom the well-known problem of curse of dimensionality Thenumber of rules increases exponentially with the number ofinput variables and linguistic terms used to represent thosevariables This in turn leads to an increase in computerprocessing time and memory requirement for the storage Toovercome this problem Raju et al [16] proposed a hierarchi-cal structure in which a higher dimensional FLC [17] wasconsidered to be a combination of some lower dimensionalfuzzy logic systems connected in a hierarchical fashion toreduce the number of rules to a linear function of systemvariables Later on Cheong and Lai [18] considered anotherhierarchical structure whichwas not similar to that proposedby Raju et al [16]They designed a symmetrical rule base andthe data base was optimized using an evolutionary algorithmknown as differential evolution Lin et al [19] automated thedesign of Rajursquos hierarchical FLC (HFLC) to control waterlevel in an advanced boiling water reactor using the steepestdescent method To the best of the authorsrsquo knowledge nosignificant study has been reported to model input-outputrelationships of welding process using the HFLC

Building a hierarchical fuzzy system is a difficult taskThisis so due to the fact that we need to define both the architec-ture of the system (ie modules input variables of eachmod-ule and interactions among the modules) and the rulesof each module Chen et al [20] developed an automaticmethod of evolving hierarchical Takagi-Sugeno fuzzy systems

Table 1 Input welding parameters and their ranges

Sl no Input parameters Units Notation Range1 Welding speed cmmin 119878 25ndash462 Arc voltage volts 119881 26ndash313 Wire feed rate mmmin 119865 6ndash84 Gas flow rate litmin 119866 14ndash215 Nozzle to plate distance mm 119863 15ndash236 Torch angle degree 119860 70ndash110

(TS-FS)The hierarchical structure was evolved using a prob-abilistic incremental program evolution (PIPE) with somespecific instructionsThey considered the structure optimiza-tion in a sequential manner and the rule parameters embed-ded in the structure were tuned using an evolutionary pro-gramming (EP)

In the present work both conventional FLCs and HFLCsof Mamdani type have been developed and their perfor-mances are tested for modeling of MIG welding process Anovel approach has been developed for simultaneous opti-mization of the structure and knowledge base of the HFLCsusing a genetic algorithm (GA) [21] The performances ofthe developed approaches have been compared in terms ofaccuracy in predictions and computational time

The paper has been organized as follows Section 2 givesa brief description of the MIG welding process consideredin the present study the tools and techniques utilized in thepresent study have been discussed in Section 3 the developedapproaches are explained in Section 4 results are stated anddiscussed in Section 5 some concluding remarks are madein Section 6 and the scope for future work is suggested inSection 7

2 MIG Welding Process

MIGwelding one of the most popular arc welding processesuses a consumable metal electrode and the molten metalis protected from the atmosphere utilizing the shielding ofan inert gas like argon or helium The weld bead geometrycan be represented by its width (BW) height (BH) andpenetration (BP) and it is dependent on the following inputparameters welding speed (119878) voltage (119881) wire feed rate (119865)gas flow rate (119866) nozzle-plate distance (119863) torch angle (119860)and others Figure 1 displays the experimental set-up used forconducting experiments on the MIG welding process Thenotations of input parameters and their ranges are shown inTable 1 Figure 2 shows a typical weld bead

Experiments were conducted previously to collect input-output data according to a full-factorial statistical design[22] Bead-on-plate welding was carried out on steel plate of20mm thickness Welding current was varied in the range of3ndash500A and a reverse polarity was used during the experi-ments It is to be noted that argon was used as the shieldinggas during the welding It is also important to mention thatmainly a spray mode of metal transfer took place during thewelding There were six input process parameters and two




Weldingtorch



BH = 10297885times 1198650318995times 1198630101367



10105147times 1198780326531 times 1198630113891

BP = 119881129320times 1198650781304times 1198660112076


(1)




I1

I2

I3

I4

In

FLS1FLS2

FLS3

FLSnminus1

O1

O2

O3

Onminus1

Onminus2

= O








GAstarts


Gen

Optimal FLC


Reproduction

Crossover

Mutation

Case


Cal string fitness

No

No

No

Yes

Yes

End


GA-string

Population size


Yes

strings gen = 0

gt

gt

max gen

gtmax case

Gen = gen + 1

String = string + 1

Case = case + 1

FLC string = 0














119891 =1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) (2)

where 119879119894119895and 119874








GA-based tuning

Knowledge base


Outputs

BH

BW

BP

InputsSVFGDA

Fuzzy system


Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




59 65 71 139 160 181 734 1042 135

149 175 201 699 850 1001 137 239 341

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

Inputs Outputs

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)



10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9

Figure 7

10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9



011010 1100101010 10 0101010110 0101010011

r1 r2 r3 r729

Figure 8


10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110

Figure 9














1and 119871


1lt 1198712











10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r45 r1 r2 r3 r45 r1 r2 r3 r45

011010 110010 110100 1010101 0101010110 101010


Figure 10

Structure

Representation


FLS1 FLS2 FLS3 FLS4


2 2 3 2 0 10 10101 0110 01011 10010 11100 10101 01010 00011 10



11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r54 r1 r2 r3 r54 r1 r2 r3 r54

011010 110010 110100 1010101 0101010110 101010


Figure 11









Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14


119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)






119888= 05 has been


119898 The




119898and





0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Weldingtorch



BH = 10297885times 1198650318995times 1198630101367



10105147times 1198780326531 times 1198630113891

BP = 119881129320times 1198650781304times 1198660112076


(1)




I1

I2

I3

I4

In

FLS1FLS2

FLS3

FLSnminus1

O1

O2

O3

Onminus1

Onminus2

= O








GAstarts


Gen

Optimal FLC


Reproduction

Crossover

Mutation

Case


Cal string fitness

No

No

No

Yes

Yes

End


GA-string

Population size


Yes

strings gen = 0

gt

gt

max gen

gtmax case

Gen = gen + 1

String = string + 1

Case = case + 1

FLC string = 0














119891 =1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) (2)

where 119879119894119895and 119874








GA-based tuning

Knowledge base


Outputs

BH

BW

BP

InputsSVFGDA

Fuzzy system


Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




59 65 71 139 160 181 734 1042 135

149 175 201 699 850 1001 137 239 341

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

Inputs Outputs

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)



10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9

Figure 7

10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9



011010 1100101010 10 0101010110 0101010011

r1 r2 r3 r729

Figure 8


10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110

Figure 9














1and 119871


1lt 1198712











10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r45 r1 r2 r3 r45 r1 r2 r3 r45

011010 110010 110100 1010101 0101010110 101010


Figure 10

Structure

Representation


FLS1 FLS2 FLS3 FLS4


2 2 3 2 0 10 10101 0110 01011 10010 11100 10101 01010 00011 10



11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r54 r1 r2 r3 r54 r1 r2 r3 r54

011010 110010 110100 1010101 0101010110 101010


Figure 11









Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14


119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)






119888= 05 has been


119898 The




119898and





0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










GAstarts


Gen

Optimal FLC


Reproduction

Crossover

Mutation

Case


Cal string fitness

No

No

No

Yes

Yes

End


GA-string

Population size


Yes

strings gen = 0

gt

gt

max gen

gtmax case

Gen = gen + 1

String = string + 1

Case = case + 1

FLC string = 0














119891 =1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) (2)

where 119879119894119895and 119874








GA-based tuning

Knowledge base


Outputs

BH

BW

BP

InputsSVFGDA

Fuzzy system


Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




59 65 71 139 160 181 734 1042 135

149 175 201 699 850 1001 137 239 341

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

Inputs Outputs

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)



10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9

Figure 7

10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9



011010 1100101010 10 0101010110 0101010011

r1 r2 r3 r729

Figure 8


10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110

Figure 9














1and 119871


1lt 1198712











10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r45 r1 r2 r3 r45 r1 r2 r3 r45

011010 110010 110100 1010101 0101010110 101010


Figure 10

Structure

Representation


FLS1 FLS2 FLS3 FLS4


2 2 3 2 0 10 10101 0110 01011 10010 11100 10101 01010 00011 10



11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r54 r1 r2 r3 r54 r1 r2 r3 r54

011010 110010 110100 1010101 0101010110 101010


Figure 11









Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14


119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)






119888= 05 has been


119898 The




119898and





0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















119891 =1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) (2)

where 119879119894119895and 119874








GA-based tuning

Knowledge base


Outputs

BH

BW

BP

InputsSVFGDA

Fuzzy system


Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




59 65 71 139 160 181 734 1042 135

149 175 201 699 850 1001 137 239 341

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

Inputs Outputs

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)



10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9

Figure 7

10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9



011010 1100101010 10 0101010110 0101010011

r1 r2 r3 r729

Figure 8


10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110

Figure 9














1and 119871


1lt 1198712











10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r45 r1 r2 r3 r45 r1 r2 r3 r45

011010 110010 110100 1010101 0101010110 101010


Figure 10

Structure

Representation


FLS1 FLS2 FLS3 FLS4


2 2 3 2 0 10 10101 0110 01011 10010 11100 10101 01010 00011 10



11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r54 r1 r2 r3 r54 r1 r2 r3 r54

011010 110010 110100 1010101 0101010110 101010


Figure 11









Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14


119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)






119888= 05 has been


119898 The




119898and





0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










GA-based tuning

Knowledge base


Outputs

BH

BW

BP

InputsSVFGDA

Fuzzy system


Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




59 65 71 139 160 181 734 1042 135

149 175 201 699 850 1001 137 239 341

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

Inputs Outputs

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)



10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9

Figure 7

10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9



011010 1100101010 10 0101010110 0101010011

r1 r2 r3 r729

Figure 8


10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110

Figure 9














1and 119871


1lt 1198712











10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r45 r1 r2 r3 r45 r1 r2 r3 r45

011010 110010 110100 1010101 0101010110 101010


Figure 10

Structure

Representation


FLS1 FLS2 FLS3 FLS4


2 2 3 2 0 10 10101 0110 01011 10010 11100 10101 01010 00011 10



11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r54 r1 r2 r3 r54 r1 r2 r3 r54

011010 110010 110100 1010101 0101010110 101010


Figure 11









Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14


119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)






119888= 05 has been


119898 The




119898and





0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9

Figure 7

10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100

Rule base (729bits)

Inputs Outputs

b1 b2 b3 b4 b5 b6 b7 b8 b9



011010 1100101010 10 0101010110 0101010011

r1 r2 r3 r729

Figure 8


10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110

Figure 9














1and 119871


1lt 1198712











10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r45 r1 r2 r3 r45 r1 r2 r3 r45

011010 110010 110100 1010101 0101010110 101010


Figure 10

Structure

Representation


FLS1 FLS2 FLS3 FLS4


2 2 3 2 0 10 10101 0110 01011 10010 11100 10101 01010 00011 10



11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r54 r1 r2 r3 r54 r1 r2 r3 r54

011010 110010 110100 1010101 0101010110 101010


Figure 11









Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14


119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)






119888= 05 has been


119898 The




119898and





0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















1and 119871


1lt 1198712











10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r45 r1 r2 r3 r45 r1 r2 r3 r45

011010 110010 110100 1010101 0101010110 101010


Figure 10

Structure

Representation


FLS1 FLS2 FLS3 FLS4


2 2 3 2 0 10 10101 0110 01011 10010 11100 10101 01010 00011 10



11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r54 r1 r2 r3 r54 r1 r2 r3 r54

011010 110010 110100 1010101 0101010110 101010


Figure 11









Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14


119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)






119888= 05 has been


119898 The




119898and





0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











10 10101 0110 01011 10010 11100 10101 01010 00011 01011 1100 1





11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r45 r1 r2 r3 r45 r1 r2 r3 r45

011010 110010 110100 1010101 0101010110 101010


Figure 10

Structure

Representation


FLS1 FLS2 FLS3 FLS4


2 2 3 2 0 10 10101 0110 01011 10010 11100 10101 01010 00011 10



11 0101 0110 101 10 101 011 10001 110 1010 10101 101010



1010 010110101 010111010100101 010111000100110


r1 r2 r3 r54 r1 r2 r3 r54 r1 r2 r3 r54

011010 110010 110100 1010101 0101010110 101010


Figure 11









Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14


119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)






119888= 05 has been


119898 The




119898and





0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















Parent 1

Parent 2

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

Figure 12

2 2 2 2 2

2 2 2 3 0

1 0 1 0 1

0 1 1 0 0

1

1 1 1 0 1 1 0 1 1

0 0 1 0 0 1 0 0Parent 1I

Parent 2I

Figure 13

Child 1

Child 2

2 2 2 2 2

2 2 2 3 0

1 1 1 0 0

0 0 1 0 1

1

1 1 0 0 1 0 0 0 1

Figure 14


119891 = 05 times1

119873

119873

sum

119894=1

(1

1198991015840

1198991015840

sum

119895=1

(10038161003816100381610038161003816119879119894119895minus 119874119894119895

10038161003816100381610038161003816)) + 05 times

119877

119877max

(3)






119888= 05 has been


119898 The




119898and





0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0165

0166

0167

0168

0169

017

0171

0172


Fitn

ess v

alue

(a)

016016101620163016401650166016701680169

100 120 140 160 180 200 220 240 260 280 300 320 340Population size

Fitn

ess v

alue

(b)

015

017

019

021

023

025

027

029


Fitn

ess v

alue

(c)











Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Bead height (mm)

00

10 ML HML HML H

ML HML HML H

ML HML HML H




Inputs Outputs


59 6772 7645 139 1705 202 734 10033 12726

149 1868 2246 699 884 1069 137 204 271

00

10

00

10

00

10

00

10

00

10

00

10

00

10

00

10

L LowM MediumH High

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)

(120583)


S

V

A

F

D

G

BH

FLS1FLS2

FLS3

O1

O2






0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of test case

Dev

iatio

n in

pre

dict

ion

()

0

5

10

15

minus5

minus10

minus15

minus20

minus25

minus30

minus35

(a)

Number of test case

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()

minus5

minus10

minus15

minus20

minus25

(b)

Number of test case

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Dev

iatio

n in

pre

dict

ion

()



minus5

minus10

minus15

(c)












Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Run no 119878 119881 119865 119866 119863 119860 BH BW BP1 40 27 64 17 18 92 261 946 1722 30 28 68 18 17 87 345 944 2333 40 29 66 19 19 84 272 1002 2394 35 27 66 19 17 87 333 874 2235 40 29 68 16 17 87 258 1029 2556 40 27 68 17 16 96 283 1088 2557 35 29 64 17 16 96 294 978 2398 30 27 66 16 16 96 336 911 2449 30 29 64 17 16 96 291 887 20910 35 27 68 19 17 87 331 920 22411 30 27 64 17 17 87 354 824 23012 40 28 68 17 17 87 322 870 20313 40 28 64 18 16 96 266 913 23814 40 27 64 17 17 87 304 844 21215 30 29 68 16 17 87 322 1008 25116 30 28 66 17 17 87 340 952 22617 35 27 64 16 19 84 300 894 22318 35 29 66 17 17 87 289 1032 26719 35 28 64 18 16 96 288 988 19620 35 29 68 19 17 87 295 1076 26821 30 28 68 16 16 96 311 997 22622 35 28 64 19 17 87 297 993 22123 35 28 66 16 18 92 289 993 21024 40 29 64 17 18 92 256 1087 22525 30 29 66 19 16 96 301 1047 22126 30 27 68 17 17 87 376 900 21827 40 28 66 17 16 96 278 946 224










Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Table 5 Continued



Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table 5 Continued




0

01

02

03

04

05

06

07

08

09

Appr

oach

1

Appr

oach

2

Appr

oach

3

Appr

oach

4

Appr

oach

5

Appr

oach

6




044

042

04

038

036

034

032

03

028

026

0240 005 01 015 02 025 03 035

Zone C Zone B

Zone A

RRmax

Aver

age a

bsol

ute d

evia

tion






Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Appendices

A

See Table 4

B

See Table 5

References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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