CylindricityErrorEvaluationBasedonanImprovedHarmony...

Research ArticleCylindricity Error Evaluation Based on an Improved HarmonySearch Algorithm

Yang Yang 1 Ming Li1 Chen Wang 2 and QingYue Wei1

1Key Laboratory of Intelligent Manufacturing and Robotics School of Mechatronic Engineering and AutomationShanghai University Shanghai 200072 China2Department of Mechanical Engineering Hubei University of Automotive Technology Shiyan 442002 China

Correspondence should be addressed to Yang Yang mryyfiishueducn

Received 30 March 2018 Revised 7 June 2018 Accepted 14 June 2018 Published 19 July 2018

Academic Editor Ricardo Soto

Copyright copy 2018 Yang Yang et al )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

)e cylindricity error is one of the basic form errors in mechanical parts which greatly influences the assembly accuracy andservice life of relevant parts For the minimum zone method (MZM) in international standards there is no specific formula tocalculate the cylindricity error )erefore the evaluation methods of the cylindricity error under the MZM have been widelyconcerned by international scholars To improve the evaluation accuracy and accelerate the iteration speed of the cylindricity animproved harmony search (IHS) algorithm is proposed and applied to compute the cylindricity On the basis of the standardharmony search algorithm the logistic chaotic initialization is introduced into the generation of initial solution to improve thequality of solutions During the iterative process the global and local search capabilities are balanced by adopting the par and bw

operators adaptively After each iteration the Cauchy mutation strategy is adopted to the best solution to further improve thecalculation precision of the IHS algorithm Finally four test functions and three groups of cylindricity error examples were appliedto validity verification of the IHS algorithm the simulation test results show that the IHS algorithm has advantages of thecomputing accuracy and iteration speed compared with other traditional algorithms and it is very effective for the application inthe evaluation of the cylindricity error

1 Introduction

In order to evaluate themanufacturing quality of mechanicalparts the inspection process has always been an importantstep in the whole life cycle of mechanical products As animportant parameter to evaluate the manufacturing accu-racy of the internal and external cylindrical surfaces ofthe rotary part the cylindricity error has become the focusof the research in the field of mechanical measurementCoordinate measuring machine (CMM) is a common tool toinspect the cylindricity error of the mechanical parts whichis widely used because of its high precision )e spatialcoordinates information of the measured point of the part isobtained by the sensor of the CMM)en the measurementdata obtained from the sensor is analyzed through the errorevaluation algorithm to calculate the cylindricity error of the

measured parts At present in order to obtain the cylin-dricity error of measurement data more quickly and accu-rately a lot of research results have been obtained It is ofgreat value in engineering application to evaluate thecylindricity error quickly and accurately which has greatinfluence on the function and life of the product

For the cylindricity error evaluation the least squaremethod (LSM) is widely used in the field of engineeringapplications but the accuracy of LSM is relatively low andcannot be applied in the high-precision evaluation field[1 2] Compared with the LSM the minimum zonemethod (MZM) is one of the techniques that has a higherevaluation accuracy It can meet the minimum zoneprinciple in the international standards and obtain theaccurate results of the measurement data However thereis no specific algorithm in international standards so it has

HindawiScientific ProgrammingVolume 2018 Article ID 2483781 13 pageshttpsdoiorg10115520182483781

received widely attention and research on the MZM byinternational scholars

Based on the initial research results the nonlinearoptimization algorithms [3 4] are the most importantmethods for cylindricity error evaluation under the MZMMoreover the nonlinear optimization algorithms are rel-atively complex and difficult in the construction of themathematical model However with the continuous im-provement of the artificial intelligence technology theintelligent optimization algorithms have developed an ef-fective method to solve such complex nonlinear problemsAnd some of them have been introduced into the cylin-dricity error evaluation because of their simplicity andeffectiveness such as GA [5] and PSO [6 7] In additioncompared with nonlinear optimization algorithms theconstruction of the mathematical model for intelligentoptimization algorithm is also very simple But the ini-tialization parameters of these algorithms have great in-fluence on the calculation results )erefore theimprovement of the accuracy and convergence speed of thecylindricity algorithm has become an important researchtopic at this stage

Harmony search (HS) algorithm is an intelligentoptimization algorithm which simulates the creationprocess of the music [8] It has the advantages of relativelysimple computational principle and strong global searchability so it is very suitable for engineering applicationssuch as the sizing optimization of truss structures andreliability problems [9 10] But for the basic HS algo-rithm there are some problems such as the prematureconvergence in the iteration process and low accuracy inlater iteration

To improve the evaluation accuracy and accelerate theiteration speed of the cylindricity on the basis of thestandard HS algorithm the logistic chaotic initialization isintroduced into the generation of initial solution to im-prove the quality of solutions During the iterative pro-cess the global and local search capabilities are balancedby adopting the par and bw operators adaptively Aftereach iteration the Cauchy mutation strategy is adopted tothe best solution to further improve the calculationprecision of the IHS algorithm Finally four test functionsand three groups of cylindricity error examples wereapplied to validity verification of the IHS algorithm thesimulation test results show that the IHS algorithm hasadvantages of the computing accuracy and iteration speedcompared with other traditional algorithms and it is veryeffective for the application in the evaluation of thecylindricity error

)e organizational structure of this paper is shown asfollows the research background and significance of thepaper are introduced in Section 1 In Section 2 the presentresearch results of this work are described In Section 3the mathematical model of the cylindricity error and theobjective function of the problem are established )ebasic HS algorithm and IHS algorithm are proposed inSections 4 and 5 respectively )e simulation experimentsof the algorithms and cylindricity error analysis arecarried out in Section 6 In the last section of this paper

the conclusion of this work is summarized and the futurework is prospected

2 Related Works

In the research results of the cylindricity error evaluationunder the MZM Carr and Ferreira established the nonlinearmathematical model of the cylindricity and straightness errorsand then used the linear programmingmethod (LPM) to solvethe problem but this method requires a set of appropriateinitial solutions [3] Lai and Chen converted the spatialcylindricity into the plane problems by nonlinear trans-formation and used the parameters adjustment to obtain thecylindricity error [4] Chou and Sun deduced a generalmathematical model of the cylindricity error )e simulatedannealing (SA) algorithm is applied to solve the model thatshows the advantage in the global optimization performance[11] Lai et al used the genetic algorithm (GA) to solve themathematical model of the cylindricity error under the MZMand the experiment proves that the method has good flexi-bility and accuraxcy [5] Weber et al applied the linear ap-proximation technique to the form error evaluation thatincludes the cylindricity error )e method provided an ap-proximate solution for the form error [12] Zhu and Dingcombined the motion geometry and sequence approximationalgorithm to analyze and evaluate the cylindricity error )eexperiments proved the validity of the algorithm [13] Laoet al improved the hyperboloid technique by constructing aninitial axis to establish the cylindrical error evaluation modeland the experiment showed that themethod further improvedthe accuracy of cylindricity evaluation [14] Cui et al andMaoet al applied a particle swarm optimization (PSO) algorithmto cylindricity error evaluation and the calculation accuracywas further improved respectively [6 7] Venkaiah andShunmugam designed a cylindricity error evaluation algo-rithm based on the computational geometry method [15] Beiet al and Guo et al improved the genetic algorithm andapplied it to the cylindricity error assessment respectively[16 17] Li et al used the coordinate transformation toconstruct the cylindricity error mathematical model and themodel has a high evaluation accuracy [18] Luo et al furtherimproved the global optimization ability of an artificial beecolony algorithm by introducing tabu strategy thus increasingthe evaluation speed of the cylindricity error [19] Wen et almodified the particle velocity using a contraction factor andimproved the convergence ability of the PSO )e improvedPSO (IPSO) was applied to the cylindricity and conicity errorswhich have a good flexibility [20] Lei et al designed a methodof cylindricity error evaluation based on the geometry opti-mization searching algorithm )e method can obtain thecylindricity error by setting the hexagon in the measurementpoints [21] Lee et al developed a cylindricity error evaluationbased on support vector machines (SVM) and the algorithmcan be applied in the field of the machine vision system [22]Wen et al utilized a quasiparticle swarm algorithm andcalculated the cylindricity error [23] He et al used kinematicgeometry and a sequential quadratic programming algorithmto solve the cylindricity error of measured parts )e methodhas high stability and accuracy [24] Hermann introduced the

2 Scientific Programming

application of the computation geometry technique to theform errors that include cylindricity [25] Zheng et al adopteda linear programming method to construct the mathematicalmodel of the cylindricity erroremodel was solved by a newsimplex method [26]

On the basis of the above literatures we improved the HSalgorithm through the generation of initial solution thedynamic adjustment of operators and the optimal solutionperturbation to further enhance the optimization ability andthe iterative speed of the basic HS algorithm It is designed toprovide a reliable method for the evaluation of the cylin-dricity error

3 Mathematical Model of CylindricityError Evaluation

According to the concepts in the relevant standard [1 2] thecylindricity error under the MZM can be dened as followsthe area is the radius dierence of two coaxial ideal cylindersthat contain the measured cylinder and when the areareaches the minimum value the radius dierence is thecylindricity error under the MZM As shown in Figure 1 thegraph shows the measurement process of the cylindricityerror For the cylindrical parts due to the manufacturingerrors of the machine tool the actual parts will always havea certain error compared with the ideal parts and the cy-lindrical surface will produce a certain deformationerefore the CMM is used to obtain the coordinates in-formation of the measured parts Figure 2 illustrates theprinciple of the cylindricity error calculation where L is theaxis of the two ideal cylinders and f is the radius dierencewhen f takes the minimum value the value of f is thecylindricity error of the cylinder measured under the MZM

As a space element the mathematical expression of thecylindricity error requires more parameters In order toreduce the amount of computation the relevant literatures

simplied the model by assuming conditions which doesnot really match the reality [5 17] In order to ensure thegenerality of cylindricity error evaluation the generalequation of the cylinder error is established under the MZMto obtain the error It is assumed that the points of themeasured cylinder are pi(xi yi zi)(i 1 2 n) wherepi(xi yi zi) is the coordinates of the measured point in theCartesian coordinate system and n is the number of mea-sured points L is the coaxial cylindrical axis and the point-oriented parametric equation is formula (1) [16] where x0y0 and z0 is the xed point of L and the a b and c are thedirectional parameters respectively For the measured pointpi its distance equation to the space axis L is shown asformula (2) where r is the distance from the point pi to theaxis L and i j and k are the unit vectors of the x y and zdirection

xminus x0a

yi minusy0b

zi minus z0c

(1)

r

i j k

xi minusx0 yi minusy0 zi minus z0a b c

a2 + b2 + c2

radic

(2)

erefore for the intelligent optimization algorithm theobjective function of the problem needs to be establishedBased on the minimum condition of the cylindricity error itcan be converted to a nonlinear minimum problem eobjective function is shown as formula (3) [16] e targetparameters are the value of x0 y0 z0 a b and c when fmetthe minimum value

f x0 y0 z0 a b c( ) min(max(r)minusmin(r)) (3)

Formula (3) describes the mathematical model of thecylindricity error f is the value of the cylindricity errorof the measured part x0 y0 z0 a b and c are the pa-rameters value of the two ideal cylindrical axes when fobtains the minimum value that can satisfy the MZM

Figure 1 e measurement process of the cylindricity error

Measuredcylinder

Idealcylinder

Ideal cylindrical axis L

f

x

y

O

Oprime

z

Figure 2 Schematic of the cylindricity error

Scientic Programming 3

condition )erefore for this nonlinear minimizationproblem it can be solved by the HS algorithm

4 Basic Harmony Search Algorithm

)e HS algorithm is a heuristic intelligent optimizationalgorithm proposed by Geem et al [8] in 2001 It simulatesthe creation process of music to achieve the goal of opti-mization )e audiencersquos evaluation of the music is thefunction fitness value HS algorithm flow includes the pa-rameters initialization the generation of new solutions andthe updating of the harmonymemory library (HMS))eHSalgorithm process is shown as follows

41 Parameter Initialization In the HS algorithm N is thesize of the HM xi is the harmony vector of the HMS (thesolution of problem) xij is the component j of harmony vectori D is the dimension of the problem xl and xu are the lowerand upper values in each dimension the specific expressionis x x|xl le xij lexu i 1 2 N j 1 2 D1113966 1113967 P isthe harmony memory considering the rate par is the pitchadjusting rate and bw is the bandwidth According to theabovementioned variable meaning the harmony vector irsquosachievement is x (xi1 xi2 xiD) i 1 2 N f(x)

is the function fitness value and the optimal solution is thexbest and initialize the harmony memory library by

xij xl+ x

u minus xl

1113872 1113873 times r (4)

where r is a random number whose value is 0 to 1

42 Generation of NewHarmony Vector In this stage a newharmonic vector is generated by three methods which areharmony memory tone adjustment and random genera-tion )e new harmony vector can be expressed asxprime (x1prime x2prime xD

prime ) and each component of xprime is producedby the above three methods Firstly the probability ofHMCR is used to control the new vector component and thepseudocode is shown as Algorithm 1 secondly after con-sidering the probability of memory consideration it needs toconsider probability of the tone adjustment and thepseudocode is shown as Algorithm 2

43 Update the HarmonyMemory Library Update the HMSby comparing the new solution xprime with the worst solution xw

in HM If xprime is better than xw then accept xprime if xprime is worsethan xw then keep xw

44 Termination Condition By setting the number of iter-ations or calculating precision the algorithm is judged to befinished )e pseudocode of the HS algorithm is shown asAlgorithm 3

5 Improved Harmony Search Algorithm

511eMotivation of the IHS HS algorithm is an intelligentoptimization algorithm derived from the creation of themusic For the basic HS the best solution in the HM is an

individual with the best function value Although the HSalgorithm has a good optimization ability it is still easy to fallinto the local optimum According to the principle of thebasic HS algorithm there are three main problems Firstlythe distribution of the initial solution will greatly affect theinitial computational performance of the algorithm Sec-ondly the par and bw of the HS algorithm greatly affect thealgorithm optimization efficiency )irdly the HS algorithmonly updates the worst solution and does not search theoptimal solution )erefore based on the three problemsexisting above in the HS algorithm and combined with theproblems to be solved the initial solution generation dy-namic adjustment of the operator and optimal solutiondisturbance are proposed to increase the diversity of thepopulation and prevent the premature convergence of thealgorithm

If randleP thenxjprime xij xij isin (j 1 2 D)

Elsexjprime xl + rand times (xu minusxl)

End If

ALGORITHM 1 Pseudocode of the first stage

If randlt par thenxjprime xij plusmn rand times bw

End If

ALGORITHM 2 Pseudocode of the second stage

StartSet N HMS D P par bw

Generate HM harmony vector by formula (4)Calculate f(xij)

t 0While tleT doFor j 1 to D doIf randleP then

xjprime xij

If randle par thenxjprime xij plusmn rand times bw

End ifElse

xjprime xl + rand times (xu minusxl)

End ifEnd forIf f(xprime)ltf(xw) thenxw xprimeEnd ift t+ 1

End whileOutput xbest

ALGORITHM 3 Pseudocode of the HS algorithm


52 Chaos Initialization In the initialization process of theintelligent optimization algorithm different generationmethods will produce different initial solution sequences)e basic HS algorithm generates the initial population ina random way that has some problems such as the unevendistribution of the original individual and thus affects theoptimization performance of the algorithm To make theinitial solution more uniformly distributed and improvethe quality of the initial solution the chaotic initialization isapplied to the standard HS Chaos sequence has the ad-vantages of traversal randomness and regularity and it canbe obtained by a definite equation [27] )e sequence ofinitial solution generated by logistic chaotic [28] mapping isproposed as

xn+1 μxn 1minus xn( 1113857

xprimen xmin + xn xmax minusxmin( 1113857(5)

where μ is 4 and it indicates that the system is a completelychaotic xmin and xmax are the upper and lower limits of thevariable in solution space respectively xn is a initializationvariable generated by chaotic in (01] and xn

prime is the variablethat is mapped to the range of value between xmin and xmaxby xn

53 New Parameters of par and bw In the basic harmonysearch algorithm par and bw are the two key control pa-rameters )e global and local search ability of the algorithmis mainly controlled by par and the convergence speed of thealgorithm is adjusted by bw However par and bw areconstant values and they are impossible to adjust the globaland local search ability effectively in the iterative process ofthe algorithm which leads to the algorithm falling into thelocal optimum

In order to improve the optimization ability and iterativespeed of the basic HS the par and bw are adjusted bya dynamic factor For par a smaller value is taken at thebeginning of the iteration to increase the global explorationability and the local search ability is enhanced by a largervalue at the end of the algorithm For bw a large value istaken at the beginning of the iteration to increase the di-versity of the population and a smaller value is taken in thelate iteration inspired by a simplified artificial fish warmalgorithm (SAFSA) [29] )e new par and bw are shown as

par 1

1 + exp(5minus 10(tT))

bw bw0 lowast exp minus30 timest

T1113874 1113875

101113888 1113889

(6)

where bw0 is the initial value of 05 and t is the currentnumber of iterations

54 CauchyMutation Strategy In the iterative process of theharmonic search algorithm the optimal solution usuallyswings between the historical optimal solutions so it needsto further increase the search space of the optimal solutionand avoid the algorithm falling into the local optimum In

order to improve the accuracy of the algorithm the Cauchymutation strategy is introduced into the algorithm in eachiterative process the global optimal solution is mutated andthe mutation prevents the optimal solution oscillation andaccelerates the convergence speed of the algorithm β isdefined as the random number of Cauchy (0 1) distributionas shown in the following formulae

xijprime xbestj + βlowastxbestj (7)

β tan(π lowast (rand(0 1)minus 05)) (8)

55 Algorithm Steps of IHS Based on the correspondingimprovement methods the proposed IHS algorithm processis shown below

(1) Parameters initializationN is the size of theHMD isthe dimension of the problem xl and xu are the

x0 rand(1 D)

For i 1 to Nx(i) 4lowastx(iminus 1)(1minusx(iminus 1))

Endxprime xmin + (xmax minusxmin)lowastx

End

ALGORITHM 4 Pseudocode of population initialization


x(0) rand(1 D)

For i 1 to Nx(i) 4lowast x(iminus 1)(1minusx(iminus 1))


Endt 0While tleT

For i 1 to DIf randleP

xjprime xij xij isin (j 1 2 D)

If randle 1(1 + exp(5minus 10(tT)))

xjprime xij plusmn rand times bw0 lowast exp(minus30 times (tT)10)

EndElse


EndEndIf f(xprime)ltf(xw)

xw xprimeEnd

Update xbest as formulae (7) and (8)End whileOutput xbest

ALGORITHM 5 Pseudocode of IHS

Scientific Programming 5

upper and lower limits of the variables P is theharmony memory considering the rate par is thepitch adjusting rate and bw is the bandwidth )engo to step (2)

(2) Chaos initialization As described in Section 52random sequence individuals which number is N aregenerated and the fitness function value of eachindividual is calculated the generated N harmonyvectors are deposited into the harmony memory asthe initial solution vectors )en go to step (3)

(3) Generate a new harmony vector As described inSection 42 a new harmony is generated by harmonymemory tone adjustment and random generationrespectively where par and bw are replaced bySection 52 )en go to step (4)

(4) Update the HMS as described in Section 43 )engo to step (5)

(5) Cauchy mutation strategy According to Section 53the best solution xbest is updated shown as formulae(7) and (8) )en go to step (6)

(6) Iteration stopping criterion If the calculation satisfiesthe stopping criterion the calculation is completedand the optimal solution is output if not return tostep (3) )e pseudocode of IHS is shown asAlgorithm 5

6 Experiment and Discussion

In order to verify the validity of the IHS algorithm for thecalculation of the cylindricity error this paper has selectedfour test functions and three groups of examples to test theperformance of the IHS algorithm )e computer system inthis experiment is Win 7 64 bit the processor is i5-4200ucpu160GHz the installation memory is 400GBeach group experiment is calculated 30 times experimentiterations of each group is 500 the population quantity is 50the dimension of problem D is 6 bw0 is 05 P is 09 and theupper and lower bounds of variables are [minus30 30]

61 Performance Test To test the performance of the IHSfour standard test functions are adopted As shown in Ta-ble 1 sphere function is a simple one-peak function thatmainly tests the iteration speed of the algorithm Rosenbrockfunction is a relatively complex single-peak function to testthe optimal performance Rastrigin function and Griewankfunction are both multimodal test functions But theGriewank function is relatively complex than the Rastriginfunction and they are commonly used to test the global

search capability of the algorithm )e images of the abovefour test functions are shown as Figures 3ndash6

To test the computing performance of the IHS the HSfirefly algorithm (FA) dragonfly algorithm (DA) artificialbee colony (ABC) and modified harmony search (MHS)algorithm are applied to the test as contrast algorithms)e parameters of the IHS are shown in the followingthe number of initial population is 50 the dimension of theproblem is 6 the number of global iteration is 500 and thealgorithm is tested 30 times

)e experimental results are shown in Table 2 For thetest results of the test functions compared with the HS FADA ABC and MHS algorithms the IHS algorithm has thelowest mean value that represents highest computing pre-cision and the lowest standard deviation that represents thehigh computational stability in sphere and Rosenbrockfunctions For the test results of Rastrigin and Griewank IHScan converge to the global optimal solution while otheralgorithms fall into the local optimal )erefore IHS algo-rithm has good optimization ability compared with otheralgorithms

Figures 7 9 11 and 13 describe the iterative convergencecurves of FA DA ABC HS MHS and IHS Meanwhile theoptimal solution boxplot of the algorithms under each testfunction is also drawn It can see that the IHS algorithmconverges faster and converges to the optimal solution onRastrigin and Griewank test functions And as it also can beseen from Figures 8 10 12 and 14 that the IHS algorithmhas a concentrated distribution of optimization results in 30times which shows has good stability Based on the results ofthe test function of the six algorithms it can been seen thatthe IHS algorithm is better than the other five algorithmsfrom the comparison of the experimental curves inFigures 7ndash14

Table 3 shows the statistical analysis results of theWilcoxon-rank test of the algorithms )e P value is theWilcoxon test value of IHS algorithm with other five al-gorithms at confidence level of 005 According to the resultsin Table 3 the calculated P values are all less than 005 andthere are significant differences between IHS and the otherfive algorithms under these four test functions )ereforethe conclusion is of statistical significance

62 Cylindricity Error Experiment 1 To verify the effec-tiveness of the IHS algorithm in the cylindricity errorevaluation the data of the measured cylindrical parts wereobtained by the CMM and are shown in Table 4 HS and IHSalgorithms are used to calculate the cylindricity error

Table 1 Four test functions

Number Name Function Range Minf1 Sphere f5(x) 1113936

ni1x

2i [minus30 30] 0

f2 Rosenbrock f1(x) 1113936nminus1i1 [100(xi+1 minusx2

i )2 + (xi minus 1)2] [minus30 30] 0

f3 Rastrigin f2(x) 111393630i1[x2

i minus 10 cos 2πxi + 10] [minus512 512] 0

f4 Griewank f3(x) (14000)1113936ni1x

2i minus1113937

ni1(xi

i

radic) + 1 [minus10 10] 0


Table 2 Results of test functions (D 6)

Algorithm Function Minimum value Maximum value Mean value Standard deviationFA

f1

240Eminus 06 746Eminus 06 479Eminus 06 341Eminus 11DA 104Eminus 14 239Eminus 07 428Eminus 08 567Eminus 14ABC 339Eminus 12 393Eminus 11 187Eminus 11 225Eminus 21HS 185Eminus 06 302Eminus 05 117Eminus 05 732Eminus 10MHS 117Eminus 14 494Eminus 10 557Eminus 11 215Eminus 19IHS 140Eminus 17 427Eminus 17 255Eminus 17 183Eminus 26FA

f2

111Eminus 02 281Eminus 02 196Eminus 02 200Eminus 04DA 836E+ 00 1 22E+ 01 937E+ 00 118E+ 01ABC 127E+ 00 314E+ 00 229E+ 00 252E+ 00HS 956Eminus 01 380E+ 00 148E+ 00 113E+ 00MHS 345E+ 00 551E+ 00 465E+ 00 389E+ 00IHS 317Eminus 03 171Eminus 02 804Eminus 03 175Eminus 05FA

f3

779Eminus 04 193Eminus 03 122Eminus 03 126Eminus 06DA 995E+ 00 162E+ 01 183E+ 01 386E+ 01ABC 350E+ 00 727E+ 00 600E+ 00 969E+ 00HS 938Eminus 01 331E+ 00 195E+ 00 555E+ 00MHS 231Eminus 04 198Eminus 02 673Eminus 03 454Eminus 04IHS 0 0 0 0FA

f4

246Eminus 02 210Eminus 01 104Eminus 01 327Eminus 02DA 242Eminus 01 804Eminus 01 423Eminus 01 249Eminus 01ABC 141Eminus 01 200Eminus 01 170Eminus 01 444Eminus 03HS 133Eminus 02 188Eminus 01 444Eminus 01 911Eminus 05MHS 117Eminus 08 237Eminus 06 859Eminus 07 617Eminus 12IHS 0 0 0 0

010

20

40

5 10

f(x1x 2

) 60

5x2

80

0x1

100

0ndash5 ndash5ndash10 ndash10

Figure 5 Rastrigin function

010

05

1

5 10

f(x1x 2

) 15

5x2

2

0x1

25


550 0

Figure 6 Griewank function

040

500

20 40

1000

f(x1x 2

)

20

1500

x20

x 1

2000


Figure 3 Sphere function

040

2

4

20 40

(times107)

f(x1x 2

) 6

20x2

8

0x 1

10


Figure 4 Rosenbrock function


e calculation results of HS and IHS algorithmsare listed in Table 5 It can be seen that the calculationresult of HS is 001941mm e calculation result of IHSis 001938mm and the cylinder axis coordinates are13004 501066 149803 minus511827E minus 05 minus0000733209and minus0816712 its result is smaller than HS in the listedalgorithms so the accuracy of the cylindricity error ishighest As can be seen from Figure 15 the iteration curveof the HS and IHS algorithms are respectively describedAs seen in Figure 8 when the number of iterations is set to64 times the IHS algorithm is convergence and for HSthe number of iterations is 390 times e IHS has a fasteriterative speed than HS

63 Cylindricity Error Experiment 2 To test the performanceof the IHS algorithm in the cylindricity evaluation the dataobtained from correlation literature are used to experiment

[21]e evaluation results of the HS and IHS algorithms arecompared with the other literatures Table 6 is reproducedfrom Lei et al [21] (under the Creative Commons Attri-bution Licensepublic domain)

e calculation results of the IHS and other relatedliterature algorithms are listed in Table 7 It can be seen thatthe calculation results of other algorithms are less than theMIC e results of the GOS algorithm in the paper are00319mm e calculation results of HS and IHS are003204mm and 003188mm respectively and the cylinderaxis coordinates of the IHS are 00120278 001206 60522minus0000008223 0000137151 and minus0853656 their result ofIHS is the smallest in the listed algorithms so the accuracyof the cylindricity error is the highest e iteration curvesof HS and IHS are shown in Figure 16 e convergencetimes of IHS are at 57 iterations and HS times is at 365iterations e convergence speed of the IHS algorithm isfaster than that of the HS algorithm

FADAABC

HSMHSIHS

10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of s

pher

e

50 100 150 200 250 300 350 400 450 5000Iterations

Figure 7 Curve of sphere function

FA DA ABC HS MHS IHSAlgorithms

10ndash18

10ndash16

10ndash14

10ndash12

10ndash10

10ndash8

10ndash6

10ndash4

Valu

es

Figure 8 Boxplot of sphere function

0 50 100 150 200 250 300 350 400 450 500

FADAABC

HSMHSIHS

Iterations

10ndash4

10ndash2

100

102

104

106

108

1010

Aver

age v

alue

of R

osen

broc

k

Figure 9 Curve of Rosenbrock function

10ndash3

10ndash2

10ndash1

100

101

102

Valu

es

DA ABC HS MHS IHSFAAlgorithms

Figure 10 Boxplot of Rosenbrock function

8 Scientic Programming

Iterations

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of R

astr

igin

50 100 150 200 250 300 350 400 450 5000

FADAABC

HSMHSIHS

Figure 11 Curve of Rastrigin function


0

2

4

6

8

10

12

14

16

Valu

es

Figure 12 Boxplot of Rastrigin function

10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of G

riew

ank

50 100 150 200 250 300 350 400 450 5000Iterations

FADAABC

HSMHSIHS

Figure 13 Curve of Griewank function

0

01

02

03

04

05

06

07

08

Valu

es


Figure 14 Boxplot of Griewank function

Table 3 P value of the Wilcoxon-rank test

Function FA DA ABC HS MHSf1 182Eminus 04 182Eminus 04 182Eminus 04 182Eminus 04 182Eminus 04f2 585Eminus 04 585Eminus 04 585Eminus 04 585Eminus 04 585Eminus 04f3 639Eminus 05 639Eminus 05 639Eminus 05 639Eminus 05 639Eminus 05f4 638Eminus 05 638Eminus 05 638Eminus 05 638Eminus 05 638Eminus 05

Table 4 Measurement data (mm)

n x y z

1 126722 17869 499912 219325 142729 5013 258783 51659 54 222705 minus39311 55 126857 minus78611 500026 35711 minus37504 500027 0135 48405 58 40705 142751 499999 126777 178729 9999510 219385 142734 10000611 2587 51675 1012 222712 minus39328 1013 12689 minus7866 9996614 35762 minus37559 9999915 01314 48369 10000216 40694 142654 1017 126799 178697 1518 219398 142659 15000919 258742 5173 150520 222676 minus39255 1521 126891 minus78421 14999622 3588 minus37526 15000523 01364 48404 14999424 40696 142729 1525 126832 178678 20000126 219382 14268 19999527 258706 51725 2028 222654 minus39283 20000129 126813 minus78449 20000830 35785 minus37515 2031 01296 48316 19999332 4073 142676 200008


Table 5 Cylindricity error of data (mm)

Method x0 y0 z0 a b c f

HS 130038 500857 12642 minus340985Eminus 05 minus0000517393 minus0576862 001941IHS 13004 501066 149803 minus511827Eminus 05 minus0000733209 minus0816712 001938

HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations

Figure 15 Cylindricity error iteration curve of HS and IHSalgorithms

Table 6 Measurement data (mm) [21]

n x y z

1 minus7169 minus3421 3052 minus14324 minus31887 3053 minus25509 minus23877 3054 minus33683 minus9283 3055 minus34167 7378 3056 minus25051 24377 3057 minus71 34238 3058 8071 34034 3059 19797 2886 30510 28945 19647 30511 34891 2595 30512 34168 minus7484 30513 28525 minus20224 30514 18113 minus29908 30515 7971 minus34044 30516 minus6778 minus34290 30517 minus11322 minus33073 43318 minus20262 minus28475 43319 minus24016 minus25389 43320 minus31014 minus16098 43321 minus34584 minus4991 43322 minus30614 16878 43323 minus1946 29045 43324 minus5802 34484 43325 10248 33444 43326 23395 26026 43327 32027 14043 433

Table 6 Continued

n x y z

28 34985 0374 43329 33364 minus10449 43330 27418 minus21716 43331 17678 minus30164 43332 minus0706 minus34942 43333 minus14539 minus31801 60834 minus22717 minus26563 60835 minus26655 minus22607 60836 minus3159 minus14943 60837 minus34892 1967 60838 minus29347 18995 60839 minus1904 29322 60840 minus6241 34407 60841 6174 34448 60842 14594 31802 60843 28945 19636 60844 34887 2506 60845 31458 minus15279 60846 20538 minus28299 60847 6489 minus34352 60848 minus737 minus34173 60849 minus15817 minus31168 8150 minus28573 minus20125 8151 minus33084 minus11238 8152 minus34611 4838 8153 minus25488 23924 8154 minus1638 30884 8155 minus0788 34969 8156 10206 33457 8157 21025 27971 8158 30269 17524 8159 3393 8513 8160 34787 minus3557 8161 31626 minus14895 8162 18362 minus29755 8163 34 minus34794 8164 minus5318 minus34548 8165 minus8484 minus33909 97566 minus18607 minus29583 97567 minus26163 minus23175 97568 minus33973 minus8156 97569 minus3449 571 97570 minus29015 19502 97571 minus17564 30231 97572 minus3842 34765 97573 15389 31424 97574 29428 18935 97575 34054 8042 97576 34931 minus2026 97577 33204 minus11008 97578 29818 minus18273 97579 21451 minus27629 97580 15791 minus31216 975


64 Cylindricity Error Experiment 3 To further test theperformance of the IHS algorithm in cylindricity errorevaluation the data from other correlation literature areapplied to the experiment [3] e computing results of theHS and IHS algorithms are compared with the other relevantliteratures Table 8 is reproduced from Carr and Ferreira [3](under the Creative Commons Attribution Licensepublicdomain)

e calculation results of IHS and other related literaturealgorithms are listed in Table 9 It can be seen that thecalculation results of other algorithms are less than the LSMe calculation results of HS and IHS are 018748mm and0183957mm respectively and the cylinder axis coordinatesof the IHS are 0005953 0024811 7583571 00060120028294 and minus9706070 their result of IHS is the smallest inthe listed algorithms so the accuracy of the cylindricity erroris the highest e iteration curves of HS and IHS are shownin Figure 17 e convergence times of IHS are at 48 iter-ations and HS times is at 285 iteration times e con-vergence speed of the IHS algorithm is faster than that of theHS algorithm

7 Conclusion and Future Work

To improve the evaluation accuracy and accelerate the it-eration speed of the cylindricity the harmony search

algorithm is applied to the evaluation of the cylindricityerror An improved harmony search algorithm is proposedto solve the problems of early premature convergence of the



LSC [21] 00125 00123 305 minus00000040172 0000013612 1 00366MZC [21] 00118 00170 305 minus00000025775 minus000016886 1 00324MIC [21] 00121 00076 305 minus0000012014 0000074796 1 00367MCC [21] 00110 00165 305 0000004866 minus000014041 1 00335GOS [21] 00123 00170 305 minus0000010192 minus000016365 1 00319HS 00121121 00113201 653792 minus0000005753 0000109724 minus069781 003204IHS 00120278 001206 60522 minus0000008223 0000137151 minus0853656 003188

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations

Figure 16 Cylindricity error iteration curves of the HS and IHS algorithms


n x y z

1 60051121 0002953 39461342 minus57932024 15399312 159830173 5743213 17488707 203659424 55022756 minus23936632 115050625 291801 minus52423113 10371636 minus58861558 minus11113569 201344827 minus44597179 40113733 20052678 minus23247383 minus55406652 176692999 34041568 minus49309081 1580786310 minus34084135 minus49427745 1247998111 50684216 minus32022045 2286594112 57318676 17619539 2208245713 minus4040813 minus44485701 2269231514 minus3983837 44994386 741116715 minus10261352 minus59146784 2260067516 53919844 26493193 1894904217 minus8540012 59442972 1309234218 minus59369089 8361285 713323319 minus38029817 46404843 499521620 47946099 minus3592538 27276243


basic harmony search algorithm and easy to get into localoptimization e logistic chaotic initialization is introducedinto the generation of initial solution to improve the quality ofsolutions During the iterative process the global and localsearch capabilities are balanced by adopting the par and bwoperators adaptively After each iteration the Cauchy muta-tion strategy is adopted to the best solution to further improvethe calculation precision of the IHS algorithm e resultsshow that the improved harmony search algorithm is verysuitable for the accurate evaluation of the cylindricity error

With the continuous development of big data and ar-ticial intelligence technology we will carry on a depthanalysis to the cylindricity error data of the parts to nd theinternal relation between the manufacturing informationand cylindricity error so as to further optimize the pro-cessing technology and improve the manufacturing accuracyof the parts

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concenticts of interestregarding the publication of this paper

Acknowledgments

is paper was supported by the National High TechnologyResearch and Development Program of China (no2015AA043003)

References

[1] ASME Y 145M Dimensioning and Tolerancing e Amer-ican Society of Mechanical Engineers New York NY USA1994

[2] ISOTS 12180-1 Geometrical Product Specication GSPCylindricity Part 1 Vocabulary and Parameters of CylindricityForm International Organization for StandardizationGeneva Switzerland 2011

[3] K Carr and P Ferreira ldquoVerication of form tolerances partII cylindricity and straightness of a median linerdquo PrecisionEngineering vol 17 no 2 pp 144ndash156 1995

[4] J-Y Lai and I-H Chen ldquoMinimum zone evaluation of circlesand cylindersrdquo International Journal of Machine Tools andManufacture vol 36 no 4 pp 435ndash451 1996

[5] H-Y Lai W-Y Jywe C-K Chen and C-H Liu ldquoPrecisionmodeling of form errors for cylindricity evaluation usinggenetic algorithmsrdquo Precision Engineering vol 24 no 4pp 310ndash319 2000

[6] C-C Cui F-G Huang R-C Zhang and B Li ldquoResearch oncylindricity evaluation based on the particle swarm optimi-zation algorithmrdquo Optics and Precision Engineering vol 14no 2 pp 254ndash260 2006



LSM [3] 008246 011153 000101 minus000490 minus000546 099997 021197LPM [3] 001065 004692 000014 minus000062 minus000292 100000 018396SA [11] 0091257 0096517 minus 0004542 0004802 0999978 0184590IGA [30] 0011853 0047689 0 minus0000674 0002960 1 0184274PSO-DE [31] 0010650 0046918 0 minus0000619 minus0002915 1 018397196HS 0032054 00338191 10996 minus00536994 minus00848952 155462 018748IHS 0005953 0024811 7583571 0006012 0028294 minus9706070 0183957

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations

Figure 17 Cylindricity error iteration curves of HS and IHS algorithms


[7] J Mao H-W Zheng Y-L Cao and S Xu ldquoMethod forcylindricity errors evaluation using particle swarm optimi-zation algorithmrdquo Transactions of the Chinese Society forAgricultural Machinery vol 38 no 2 pp 146ndash149 2007

[8] Z W Geem J H Kim and G V Loganathan ldquoA newheuristic optimization algorithm harmony searchrdquo Simula-tion vol 76 no 2 pp 60ndash68 2001

[9] S O Degertekin ldquoImproved harmony search algorithms forsizing optimization of truss structuresrdquo Computers andStructures vol 92-93 no 3 pp 229ndash241 2012

[10] D-X Zou L-Q Gao J-H Wu S Li and Y Li ldquoA novelglobal harmony search algorithm for reliability problemsrdquoComputers and Industrial Engineering vol 58 no 2pp 307ndash316 2010

[11] S-Y Chou and C-W Sun ldquoAssessing cylindricity for obliquecylindrical featuresrdquo International Journal of Machine Toolsand Manufacture vol 40 no 3 pp 327ndash341 2000

[12] T Weber S Motavalli B Fallahi and S-H Cheraghid ldquoAunified approach to form error evaluationrdquo Precision Engi-neering vol 26 no 3 pp 269ndash278 2002

[13] L-M Zhu and H Ding ldquoApplication of kinematic geometryto computational metrology distance function based hier-archical algorithms for cylindricity evaluationrdquo InternationalJournal of Machine Tools and Manufacture vol 43 no 2pp 203ndash215 2003

[14] Y-Z Lao H-W Leong F P Preparata and G SinghldquoAccurate cylindricity evaluation with axis-estimation pre-processingrdquo Precision Engineering vol 27 no 4 pp 429ndash4372003

[15] N Venkaiah and M S Shunmugam ldquoEvaluation of form datausing computational geometric techniquesmdashpart II cylin-dricity errorrdquo International Journal of Machine Tools andManufacture vol 47 no 7-8 pp 1237ndash1245 2007

[16] G-X Bei P-H Lou X-Y Wang H-Y Zhu and H DuldquoCylindricity error evaluation based on genetic algorithmsrdquoJournal of Shandong University vol 38 no 2 pp 33ndash36 2008

[17] H Guo D-J Lin J-Z Pan and S-W Jiang ldquoCylindricityerror evaluation based on multi-population genetic algo-rithmrdquo Journal of Engineering Graphics vol 29 no 4pp 48ndash53 2008

[18] J-S Li X-Q Lei Y-J Xue and M-D Duan ldquoEvaluationalgorithm of cylindricity error based on coordinate trans-formationrdquo China Mechanical Engineering vol 20 no 16pp 1983ndash1987 2009

[19] J Luo J-J Lu W-M Chen et al ldquoCylindricity error eval-uation using artificial bee colony algorithm with tabu strat-egyrdquo Journal of Chongqing University vol 32 no 12pp 1482ndash1485 2009

[20] X-L Wen J-C Huang D-H Sheng and F-L WangldquoConicity and cylindricity error evaluation using particleswarm optimizationrdquo Precision Engineering vol 34 no 2pp 338ndash344 2010

[21] X-Q Lei H-W Song Y-J Xue J-S Li J Zhou andM-D Duan ldquoMethod for cylindricity error evaluation usinggeometry optimization searching algorithmrdquo Measurementvol 44 no 9 pp 1556ndash1563 2011

[22] K Lee S Cho and S Asfour ldquoWeb-based algorithm forcylindricity evaluation using support vector machine learn-ingrdquo Computers and Industrial Engineering vol 60 no 2pp 228ndash235 2011

[23] X-L Wen Y-B Zhao D-X Wang and J Pan ldquoAdaptiveMonte Carlo and GUM methods for the evaluation ofmeasurement uncertainty of cylindricity errorrdquo PrecisionEngineering vol 37 no 4 pp 856ndash864 2013

[24] G-Y He P-P Liu and K Wang ldquoEvaluation of the cylin-dricity error based on the sequential quadratic programmingalgorithmrdquo Mechanical Science and Technology for AerospaceEngineering vol 33 no 12 pp 1845ndash1849 2014

[25] G Hermann ldquoApplication of computational geometry incoordinate measurementrdquo in Proceedings of the 10th JubileeIEEE International Symposium on Applied ComputationalIntelligence and Informatics pp 511ndash516 IEEE TimisoaraRomania May 2015

[26] P Zheng J-Q Wu and L-N Zhang ldquoResearch of the on-lineevaluating the cylindricity error technology based on thenew generation of GPSrdquo Procedia Engineering vol 174pp 402ndash409 2017

[27] B Li and W-S Jiang ldquoChaos optimization method and itsapplicationrdquo Control theory and application vol 14 no 4pp 613ndash615 1997

[28] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40no 5 pp 2557ndash2568 2009

[29] L-G Wang Y Hong and D-M Yu ldquoSimplified artificial fishswarm algorithmrdquo Journal of Chinese Computer Systemsvol 30 no 8 pp 1663ndash1667 2009

[30] X-L Wen and A-G Song ldquoAn improved genetic algorithmfor planar and spatial straightness error evaluationrdquoInternational Journal of Machine Tools and Manufacturevol 43 no 11 pp 1157ndash1162 2003

[31] X-C Zhang X-Q Jiang and P J Scott ldquoA reliable method ofminimum zone evaluation of cylindricity and conicity fromcoordinate measurement datardquo Precision Engineering vol 35no 3 pp 484ndash489 2011


Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems


Volume 2018


ReconfigurableComputing



Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018


Civil EngineeringAdvances in


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia


Biomedical Imaging



Engineering Mathematics


RoboticsJournal of



Computational Intelligence and Neuroscience


Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018


Human-ComputerInteraction

Advances in


Scientic Programming

Submit your manuscripts atwwwhindawicom

received widely attention and research on the MZM byinternational scholars

Based on the initial research results the nonlinearoptimization algorithms [3 4] are the most importantmethods for cylindricity error evaluation under the MZMMoreover the nonlinear optimization algorithms are rel-atively complex and difficult in the construction of themathematical model However with the continuous im-provement of the artificial intelligence technology theintelligent optimization algorithms have developed an ef-fective method to solve such complex nonlinear problemsAnd some of them have been introduced into the cylin-dricity error evaluation because of their simplicity andeffectiveness such as GA [5] and PSO [6 7] In additioncompared with nonlinear optimization algorithms theconstruction of the mathematical model for intelligentoptimization algorithm is also very simple But the ini-tialization parameters of these algorithms have great in-fluence on the calculation results )erefore theimprovement of the accuracy and convergence speed of thecylindricity algorithm has become an important researchtopic at this stage

Harmony search (HS) algorithm is an intelligentoptimization algorithm which simulates the creationprocess of the music [8] It has the advantages of relativelysimple computational principle and strong global searchability so it is very suitable for engineering applicationssuch as the sizing optimization of truss structures andreliability problems [9 10] But for the basic HS algo-rithm there are some problems such as the prematureconvergence in the iteration process and low accuracy inlater iteration

To improve the evaluation accuracy and accelerate theiteration speed of the cylindricity on the basis of thestandard HS algorithm the logistic chaotic initialization isintroduced into the generation of initial solution to im-prove the quality of solutions During the iterative pro-cess the global and local search capabilities are balancedby adopting the par and bw operators adaptively Aftereach iteration the Cauchy mutation strategy is adopted tothe best solution to further improve the calculationprecision of the IHS algorithm Finally four test functionsand three groups of cylindricity error examples wereapplied to validity verification of the IHS algorithm thesimulation test results show that the IHS algorithm hasadvantages of the computing accuracy and iteration speedcompared with other traditional algorithms and it is veryeffective for the application in the evaluation of thecylindricity error

)e organizational structure of this paper is shown asfollows the research background and significance of thepaper are introduced in Section 1 In Section 2 the presentresearch results of this work are described In Section 3the mathematical model of the cylindricity error and theobjective function of the problem are established )ebasic HS algorithm and IHS algorithm are proposed inSections 4 and 5 respectively )e simulation experimentsof the algorithms and cylindricity error analysis arecarried out in Section 6 In the last section of this paper

the conclusion of this work is summarized and the futurework is prospected

2 Related Works

In the research results of the cylindricity error evaluationunder the MZM Carr and Ferreira established the nonlinearmathematical model of the cylindricity and straightness errorsand then used the linear programmingmethod (LPM) to solvethe problem but this method requires a set of appropriateinitial solutions [3] Lai and Chen converted the spatialcylindricity into the plane problems by nonlinear trans-formation and used the parameters adjustment to obtain thecylindricity error [4] Chou and Sun deduced a generalmathematical model of the cylindricity error )e simulatedannealing (SA) algorithm is applied to solve the model thatshows the advantage in the global optimization performance[11] Lai et al used the genetic algorithm (GA) to solve themathematical model of the cylindricity error under the MZMand the experiment proves that the method has good flexi-bility and accuraxcy [5] Weber et al applied the linear ap-proximation technique to the form error evaluation thatincludes the cylindricity error )e method provided an ap-proximate solution for the form error [12] Zhu and Dingcombined the motion geometry and sequence approximationalgorithm to analyze and evaluate the cylindricity error )eexperiments proved the validity of the algorithm [13] Laoet al improved the hyperboloid technique by constructing aninitial axis to establish the cylindrical error evaluation modeland the experiment showed that themethod further improvedthe accuracy of cylindricity evaluation [14] Cui et al andMaoet al applied a particle swarm optimization (PSO) algorithmto cylindricity error evaluation and the calculation accuracywas further improved respectively [6 7] Venkaiah andShunmugam designed a cylindricity error evaluation algo-rithm based on the computational geometry method [15] Beiet al and Guo et al improved the genetic algorithm andapplied it to the cylindricity error assessment respectively[16 17] Li et al used the coordinate transformation toconstruct the cylindricity error mathematical model and themodel has a high evaluation accuracy [18] Luo et al furtherimproved the global optimization ability of an artificial beecolony algorithm by introducing tabu strategy thus increasingthe evaluation speed of the cylindricity error [19] Wen et almodified the particle velocity using a contraction factor andimproved the convergence ability of the PSO )e improvedPSO (IPSO) was applied to the cylindricity and conicity errorswhich have a good flexibility [20] Lei et al designed a methodof cylindricity error evaluation based on the geometry opti-mization searching algorithm )e method can obtain thecylindricity error by setting the hexagon in the measurementpoints [21] Lee et al developed a cylindricity error evaluationbased on support vector machines (SVM) and the algorithmcan be applied in the field of the machine vision system [22]Wen et al utilized a quasiparticle swarm algorithm andcalculated the cylindricity error [23] He et al used kinematicgeometry and a sequential quadratic programming algorithmto solve the cylindricity error of measured parts )e methodhas high stability and accuracy [24] Hermann introduced the








xminus x0a

yi minusy0b

zi minus z0c

(1)

r

i j k


a2 + b2 + c2

radic

(2)





Measuredcylinder

Idealcylinder


f

x

y

O

Oprime

z








xij xl+ x

u minus xl

1113872 1113873 times r (4)












End If



End If





xjprime xij


End ifElse













par 1



T1113874 1113875

101113888 1113889

(6)








x0 rand(1 D)



End



x(0) rand(1 D)



Endt 0While tleT





EndElse



xw xprimeEnd





















ni1x

2i [minus30 30] 0


i )2 + (xi minus 1)2] [minus30 30] 0



f4 Griewank f3(x) (14000)1113936ni1x

2i minus1113937

ni1(xi

i





f1


f2


f3


f4


010

20

40

5 10

f(x1x 2

) 60

5x2

80

0x1

100



010

05

1

5 10

f(x1x 2

) 15

5x2

2

0x1

25


550 0


040

500

20 40

1000

f(x1x 2

)

20

1500

x20

x 1

2000



040

2

4

20 40

(times107)

f(x1x 2

) 6

20x2

8

0x 1

10








FADAABC

HSMHSIHS

10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of s

pher

e

50 100 150 200 250 300 350 400 450 5000Iterations



10ndash18

10ndash16

10ndash14

10ndash12

10ndash10

10ndash8

10ndash6

10ndash4

Valu

es


0 50 100 150 200 250 300 350 400 450 500

FADAABC

HSMHSIHS

Iterations

10ndash4

10ndash2

100

102

104

106

108

1010

Aver

age v

alue

of R

osen

broc

k


10ndash3

10ndash2

10ndash1

100

101

102

Valu

es




Iterations

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of R

astr

igin

50 100 150 200 250 300 350 400 450 5000

FADAABC

HSMHSIHS



0

2

4

6

8

10

12

14

16

Valu

es


10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of G

riew

ank

50 100 150 200 250 300 350 400 450 5000Iterations

FADAABC

HSMHSIHS


0

01

02

03

04

05

06

07

08

Valu

es






n x y z






HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z


Table 6 Continued

n x y z











10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in










xminus x0a

yi minusy0b

zi minus z0c

(1)

r

i j k


a2 + b2 + c2

radic

(2)





Measuredcylinder

Idealcylinder


f

x

y

O

Oprime

z








xij xl+ x

u minus xl

1113872 1113873 times r (4)












End If



End If





xjprime xij


End ifElse













par 1



T1113874 1113875

101113888 1113889

(6)








x0 rand(1 D)



End



x(0) rand(1 D)



Endt 0While tleT





EndElse



xw xprimeEnd





















ni1x

2i [minus30 30] 0


i )2 + (xi minus 1)2] [minus30 30] 0



f4 Griewank f3(x) (14000)1113936ni1x

2i minus1113937

ni1(xi

i





f1


f2


f3


f4


010

20

40

5 10

f(x1x 2

) 60

5x2

80

0x1

100



010

05

1

5 10

f(x1x 2

) 15

5x2

2

0x1

25


550 0


040

500

20 40

1000

f(x1x 2

)

20

1500

x20

x 1

2000



040

2

4

20 40

(times107)

f(x1x 2

) 6

20x2

8

0x 1

10








FADAABC

HSMHSIHS

10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of s

pher

e

50 100 150 200 250 300 350 400 450 5000Iterations



10ndash18

10ndash16

10ndash14

10ndash12

10ndash10

10ndash8

10ndash6

10ndash4

Valu

es


0 50 100 150 200 250 300 350 400 450 500

FADAABC

HSMHSIHS

Iterations

10ndash4

10ndash2

100

102

104

106

108

1010

Aver

age v

alue

of R

osen

broc

k


10ndash3

10ndash2

10ndash1

100

101

102

Valu

es




Iterations

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of R

astr

igin

50 100 150 200 250 300 350 400 450 5000

FADAABC

HSMHSIHS



0

2

4

6

8

10

12

14

16

Valu

es


10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of G

riew

ank

50 100 150 200 250 300 350 400 450 5000Iterations

FADAABC

HSMHSIHS


0

01

02

03

04

05

06

07

08

Valu

es






n x y z






HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z


Table 6 Continued

n x y z











10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in









xij xl+ x

u minus xl

1113872 1113873 times r (4)












End If



End If





xjprime xij


End ifElse













par 1



T1113874 1113875

101113888 1113889

(6)








x0 rand(1 D)



End



x(0) rand(1 D)



Endt 0While tleT





EndElse



xw xprimeEnd





















ni1x

2i [minus30 30] 0


i )2 + (xi minus 1)2] [minus30 30] 0



f4 Griewank f3(x) (14000)1113936ni1x

2i minus1113937

ni1(xi

i





f1


f2


f3


f4


010

20

40

5 10

f(x1x 2

) 60

5x2

80

0x1

100



010

05

1

5 10

f(x1x 2

) 15

5x2

2

0x1

25


550 0


040

500

20 40

1000

f(x1x 2

)

20

1500

x20

x 1

2000



040

2

4

20 40

(times107)

f(x1x 2

) 6

20x2

8

0x 1

10








FADAABC

HSMHSIHS

10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of s

pher

e

50 100 150 200 250 300 350 400 450 5000Iterations



10ndash18

10ndash16

10ndash14

10ndash12

10ndash10

10ndash8

10ndash6

10ndash4

Valu

es


0 50 100 150 200 250 300 350 400 450 500

FADAABC

HSMHSIHS

Iterations

10ndash4

10ndash2

100

102

104

106

108

1010

Aver

age v

alue

of R

osen

broc

k


10ndash3

10ndash2

10ndash1

100

101

102

Valu

es




Iterations

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of R

astr

igin

50 100 150 200 250 300 350 400 450 5000

FADAABC

HSMHSIHS



0

2

4

6

8

10

12

14

16

Valu

es


10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of G

riew

ank

50 100 150 200 250 300 350 400 450 5000Iterations

FADAABC

HSMHSIHS


0

01

02

03

04

05

06

07

08

Valu

es






n x y z






HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z


Table 6 Continued

n x y z











10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in











par 1



T1113874 1113875

101113888 1113889

(6)








x0 rand(1 D)



End



x(0) rand(1 D)



Endt 0While tleT





EndElse



xw xprimeEnd





















ni1x

2i [minus30 30] 0


i )2 + (xi minus 1)2] [minus30 30] 0



f4 Griewank f3(x) (14000)1113936ni1x

2i minus1113937

ni1(xi

i





f1


f2


f3


f4


010

20

40

5 10

f(x1x 2

) 60

5x2

80

0x1

100



010

05

1

5 10

f(x1x 2

) 15

5x2

2

0x1

25


550 0


040

500

20 40

1000

f(x1x 2

)

20

1500

x20

x 1

2000



040

2

4

20 40

(times107)

f(x1x 2

) 6

20x2

8

0x 1

10








FADAABC

HSMHSIHS

10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of s

pher

e

50 100 150 200 250 300 350 400 450 5000Iterations



10ndash18

10ndash16

10ndash14

10ndash12

10ndash10

10ndash8

10ndash6

10ndash4

Valu

es


0 50 100 150 200 250 300 350 400 450 500

FADAABC

HSMHSIHS

Iterations

10ndash4

10ndash2

100

102

104

106

108

1010

Aver

age v

alue

of R

osen

broc

k


10ndash3

10ndash2

10ndash1

100

101

102

Valu

es




Iterations

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of R

astr

igin

50 100 150 200 250 300 350 400 450 5000

FADAABC

HSMHSIHS



0

2

4

6

8

10

12

14

16

Valu

es


10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of G

riew

ank

50 100 150 200 250 300 350 400 450 5000Iterations

FADAABC

HSMHSIHS


0

01

02

03

04

05

06

07

08

Valu

es






n x y z






HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z


Table 6 Continued

n x y z











10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





















ni1x

2i [minus30 30] 0


i )2 + (xi minus 1)2] [minus30 30] 0



f4 Griewank f3(x) (14000)1113936ni1x

2i minus1113937

ni1(xi

i





f1


f2


f3


f4


010

20

40

5 10

f(x1x 2

) 60

5x2

80

0x1

100



010

05

1

5 10

f(x1x 2

) 15

5x2

2

0x1

25


550 0


040

500

20 40

1000

f(x1x 2

)

20

1500

x20

x 1

2000



040

2

4

20 40

(times107)

f(x1x 2

) 6

20x2

8

0x 1

10








FADAABC

HSMHSIHS

10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of s

pher

e

50 100 150 200 250 300 350 400 450 5000Iterations



10ndash18

10ndash16

10ndash14

10ndash12

10ndash10

10ndash8

10ndash6

10ndash4

Valu

es


0 50 100 150 200 250 300 350 400 450 500

FADAABC

HSMHSIHS

Iterations

10ndash4

10ndash2

100

102

104

106

108

1010

Aver

age v

alue

of R

osen

broc

k


10ndash3

10ndash2

10ndash1

100

101

102

Valu

es




Iterations

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of R

astr

igin

50 100 150 200 250 300 350 400 450 5000

FADAABC

HSMHSIHS



0

2

4

6

8

10

12

14

16

Valu

es


10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of G

riew

ank

50 100 150 200 250 300 350 400 450 5000Iterations

FADAABC

HSMHSIHS


0

01

02

03

04

05

06

07

08

Valu

es






n x y z






HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z


Table 6 Continued

n x y z











10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






f1


f2


f3


f4


010

20

40

5 10

f(x1x 2

) 60

5x2

80

0x1

100



010

05

1

5 10

f(x1x 2

) 15

5x2

2

0x1

25


550 0


040

500

20 40

1000

f(x1x 2

)

20

1500

x20

x 1

2000



040

2

4

20 40

(times107)

f(x1x 2

) 6

20x2

8

0x 1

10








FADAABC

HSMHSIHS

10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of s

pher

e

50 100 150 200 250 300 350 400 450 5000Iterations



10ndash18

10ndash16

10ndash14

10ndash12

10ndash10

10ndash8

10ndash6

10ndash4

Valu

es


0 50 100 150 200 250 300 350 400 450 500

FADAABC

HSMHSIHS

Iterations

10ndash4

10ndash2

100

102

104

106

108

1010

Aver

age v

alue

of R

osen

broc

k


10ndash3

10ndash2

10ndash1

100

101

102

Valu

es




Iterations

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of R

astr

igin

50 100 150 200 250 300 350 400 450 5000

FADAABC

HSMHSIHS



0

2

4

6

8

10

12

14

16

Valu

es


10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of G

riew

ank

50 100 150 200 250 300 350 400 450 5000Iterations

FADAABC

HSMHSIHS


0

01

02

03

04

05

06

07

08

Valu

es






n x y z






HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z


Table 6 Continued

n x y z











10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in








FADAABC

HSMHSIHS

10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of s

pher

e

50 100 150 200 250 300 350 400 450 5000Iterations



10ndash18

10ndash16

10ndash14

10ndash12

10ndash10

10ndash8

10ndash6

10ndash4

Valu

es


0 50 100 150 200 250 300 350 400 450 500

FADAABC

HSMHSIHS

Iterations

10ndash4

10ndash2

100

102

104

106

108

1010

Aver

age v

alue

of R

osen

broc

k


10ndash3

10ndash2

10ndash1

100

101

102

Valu

es




Iterations

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of R

astr

igin

50 100 150 200 250 300 350 400 450 5000

FADAABC

HSMHSIHS



0

2

4

6

8

10

12

14

16

Valu

es


10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of G

riew

ank

50 100 150 200 250 300 350 400 450 5000Iterations

FADAABC

HSMHSIHS


0

01

02

03

04

05

06

07

08

Valu

es






n x y z






HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z


Table 6 Continued

n x y z











10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




Iterations

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of R

astr

igin

50 100 150 200 250 300 350 400 450 5000

FADAABC

HSMHSIHS



0

2

4

6

8

10

12

14

16

Valu

es


10ndash20

10ndash15

10ndash10

10ndash5

100

105

Aver

age v

alue

of G

riew

ank

50 100 150 200 250 300 350 400 450 5000Iterations

FADAABC

HSMHSIHS


0

01

02

03

04

05

06

07

08

Valu

es






n x y z






HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z


Table 6 Continued

n x y z











10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in







HSIHS

10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z


Table 6 Continued

n x y z











10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in












10ndash2

10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations



n x y z





Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






Data Availability




Acknowledgments


References










10ndash1

100

101

102

Aver

age v

alue

of c

ylin

dric

ity

HSIHS

50 100 150 200 250 300 350 400 450 5000Iterations


































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in



































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




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