Chapter ROUNDNESS EVALUATION BY GENETIC ALGORITHMS Michele Lazetta ad Adrea Rossi Departmet of Mechaical, Nuclear ad Productio Egieerig Uiversity of Pisa, Via Diotisalvi 1, 56122 Pisa, Italy ABSTRACT Roudess is oe of the most commo features i machiig, ad various criteria may be used for roudess errors evaluatio. The miimum zoe tolerace (MZT) method produces more accurate solutios tha data fittig methods like least squares iterpolatio. The problem modelig ad the applicatio of Geetic Algorithms (GA) for the roudess evaluatio is reviewed here. Guidelies for the GA parameters selectio are also provided based o computatio experimets. Keywords: Miimum zoe tolerace (MZT), roudess error, geetic algorithm, CMM 1. INTRODUCTION I metrolog the ispectio of maufactured parts ivolves the measuremet of dimesios for coformace to product specificatios (Figure 1). I series or lot productio, feedbacks from statistical aalyses performed o multiple products allow process cotrol for quality improvemet. Product specificatios are associated with toleraces, which represet the acceptable limits for measured parts. Toleraces come from maufacturig requiremets, e.g. assemble parts that fit, or from fuctioal requiremets for use or operatio of the fial product, e.g. rotatio of a wheel, power of a egie. Tel.: +39 050 2218122; fax: +39 050 2218140.
2 Michele Lazetta ad Adrea Rossi Metrology ivolves the acquisitio or samplig of idividual poits by maual istrumets, like aalog or digital calipers, micrometers ad dial gages, or by automated tools like coordiate measurig machies (CMM) or visio systems. Automated tools are equipped with software for post processig of data ad are able to measure complex surfaces. Measuremets ca be liear, such as size, distace, ad depth ad i two or three dimesios, such as surfaces ad volumes. I additio to dimesio tolerace there are form toleraces for two or three dimesioal geometric primitives, like straightess (for a edge, a axis), flatess (for a plae), circularity or roudess (for a circle, a arc) or cylidricity (for a peg, a bar, a hole). The simplest way to assess form toleraces is fidig the belogig geometric primitives by iterpolatio of idividual acquired data poits. Not always liear regressio represets the most accurate estimatio of the form error. Overestimates represet a waste for the rejectio of acceptable parts, while iversely uderestimated form errors may produce defective parts. The estimatio of form errors by o liear methods is a optimizatio problem where metaheuristics, such as geetic algorithms, at coloy systems or eural etworks ca provide more accurate results with respect to liear methods, subject to proper modelig of the mathematical problem. The applicatio of metaheuristics for the estimatio of form error is a active research field ad fial solutios are still far to come, particularly regardig the processig time due to the problem compexity compared to iterpolatio methods, which provide results i fractios of the secod. Product specificatio Tolerace evaluatio Metrology Ispectio Quality cotrol Figure 1. Adjustmet betwee maufacturig toleraces ad product specificatios.
Roudess Evaluatio by Geetic Algorithms 3 I the remaider the applicatio of geetic algorithms for roudess evaluatio will be discussed. The approach preseted ca be exteded to other types of form errors. 2. ROUNDNESS ERROR Roudess is the property of beig shaped like or approximately like a circle or cylider. I maufacturig eviromets, variatios o circular features may occur due to: imperfect rotatio, erratic cuttig actio, iadequate lubricatio, tool wear, defective machie parts, chatter, misaligmet of chuck jaws, etc. The out of roudess of circular ad cylidrical parts ca prevet isertio, produce vibratios i rotatig parts, irregular rotatio, oise etc. Form tolerace is evaluated with referece to a ideal geometric feature, i.e. a circle i the case of roudess. The most used criteria to establish the referece circle are: the Least- Squares method (LSQ), the Maximum Iscribed Circle (MIC), the Miimum Circumscribed Circle (MCC) ad the Miimum Zoe Tolerace (MZT). The use of a particular iterpolatio or data fittig method depeds o the required part applicatio, e.g. MIC ad MCC ca be used whe matig a peg ito a hole is ivolved to assess iterferece. The LSQ is oe of the methods used by the coordiate measurig machies for rapidity ad because it is efficiet i computatio with a large umber of measured poits. The roudess error determied by the LSQ is larger tha those determied by other methods, such as the MZT. 23 45678910 11 12 13 18 17 16 15 14 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 58 57 56 55 54 59 60 61 62 63 68 67 66 65 64 69 70 71 72 73 78 77 76 75 74 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 98 97 96 95 94 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 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333 334 335 336 337 338 r(θ) R( ( OC( IC( Ier circle 1234 1235 1236 1237 1238 1229 1230 1231 1232 1233 1224 1225 1226 1227 1228 1219 1220 1221 1222 1223 1214 1215 1216 1217 1218 Medium 1209 1210 1211 1212 1213 1204 1205 1206 1207 1208 1199 1200 1201 1202 1203 1194 1195 1196 1197 1198 circle 1189 1190 1191 1192 1193 1184 1185 1186 1187 1188 1179 1180 1181 1182 1183 1174 1175 1176 1177 1178 1169 1170 1171 1172 1173 1164 1165 1166 1167 1168 1159 1160 1161 1162 1163 1154 1155 1156 1157 1158 1149 1150 1151 1152 1153 1144 1145 1146 1147 1148 1139 1140 1141 1142 1143 1134 1135 1136 1137 1138 1129 1130 1131 1132 1133 1124 1125 1126 1127 1128 1119 1120 1121 1122 1123 1114 1115 1116 1117 1118 1109 1110 1111 1112 1113 Outer circle 1104 1105 1106 1107 1108 1099 1100 1101 1102 1103 1094 1095 1096 1097 1098 1089 1090 1091 1092 1093 1084 1085 1086 1087 1088 1079 1080 1081 1082 1083 1074 1075 1076 1077 1078 1069 1070 1071 1072 1073 1064 1065 1066 1067 1068 1059 1060 1061 1062 1063 1054 1055 1056 1057 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Figure 2. Roudess profile of a real profile with 1800 equally-spaced CMM sample poits ad referece circles: ier, medium, ad outer circles for a give associated derived ceter (.
4 Michele Lazetta ad Adrea Rossi 3. LITERATURE The MZT ca be cosidered the best estimatio of the roudess error because its defiitio meets the stadard defiitio of the roudess error, as reported i ISO 1101 [1]. The MZT determies two cocetric circles that cotai the measured profile ad such that the differece i radii is the least possible value as it is show i Figure 2, where c 1 ad c 2 are two possible ceters of two cocetric circles that iclude the measured poits ad where r 1 ad r 2 are their differece i radii. So, oce foud, the MZ error ca be cosidered the roudess error itself ad the related MZ ceter. However, the MZT is a o liear problem ad several methods to solve this problem have bee proposed i the literature: computatioal geometry techiques ad the solutio of a o liear optimizatio problem. The first approach is, i geeral, very computatioally expesive, especiall whe the umber of data poits is large. Oe of these methods is based o the Vorooi diagram [2]. The secod approach is based o a optimizatio fuctio but the icoveiece is that this fuctio has several local miima. Some examples are: the Chebyshev approximatio [3], the simplex search / liear approximatio [4] [5], the steepest descet algorithm [6], the particle swarm optimizatio (PSO) [7] [8], the simulated aealig (SA) [9], ad geetic algorithms (GAs) [10] [11] [12] [13]. Xiog [14] develops a geeral mathematical theor a model ad a algorithm for differet kids of profiles icludig roudess where the liear programmig method ad exchage algorithm are used. As limaço approximatio is used to represet the circle, the optimality of the solutio is however ot guarateed. Performace of methods have bee reviewed i [15]. A strategy based o geometric represetatio for miimum zoe evaluatio of circles ad cyliders is proposed by Lai ad Che [16]. The strategy employs a o-liear trasformatio to covert a circle ito a lie ad the uses a straightess evaluatio schema to obtai miimum zoe deviatios for the feature cocered. This is a approximatio strategy to miimum zoe circles. M. Wag et al. [17] ad Jywe et al. [18] preset a geeralized o-liear optimizatio procedure based o the developed ecessary ad sufficiet coditios to evaluate roudess error. To meet the stadards the MZ referece circles should pass through at least four poits of the sample poits. This ca occur i two cases: a) whe three poits lie o a circle ad oe poit lies o the other circle (the 1-3 ad the 3-1 criteria); b) whe two poits lie o each of the cocetric circles (the 2-2 criterio). I order to verify these coditios the computatio time icreases expoetially with the dataset size. Gadelmawla [19] use a heuristic approach to drastically reduce the umber of sample poits used by the mi-max 1-3, 3-1 ad 2-2 criteria. Samuel ad Shumugam [20] establish a miimum zoe limaço based o computatioal geometry to evaluate roudess error; with geometric methods, global optima are foud by exhaustively checkig every local miimum cadidate. Moroi ad Petro [15] propose a techique to speed up the exhaustive geeratio of solutios (brute force algorithm) which starts with a sigle poit ad icreases oe sample poit at each step i order to geerate all the possible subsets of poits, util the tolerace zoe of a subset cover the whole dataset (essetial subset).
Roudess Evaluatio by Geetic Algorithms 5 A mesh based method with startig ceter o the LSC, where the covergece depeds o the umber of mesh cross poits, represetig a compromise betwee accuracy ad speed, is proposed by Xiaqig et al. [21]. The strategy to equally-spaced poits sampled o the roudess profile is geerally adopted i the literature. Coversel i previous works the authors developed a crossvalidatio method for small samples to assess the kid of maufacturig sigature o the roudess profile i order to detect critical poits such as peaks ad valleys [22] [23]. They use a strategy where a ext samplig icreasig the poits ear these critical areas of the roudess profile. I [24], some ivestigatios proved that the icrease of the umber of sample poits is effective oly up to a limit umber. Recommeded dataset sizes are give for differet data fittig methods (LSQ, MIC, MCC, MZT) ad for three differet out-of-roudess types (oval, 3-lobig ad 4-lobig). Similar works are [25] ad [26] i which substatially the same results are give. A samplig strategy depeds o the optimal umber of sample poits ad the optimum search-space size for best estimatio accurac particularly with datasets which ivolve thousads of sample poits available by CMM scaig techiques. The samplig strategy tailored for a fast geetic algorithm to solve the MZT problem ca be defied as blid accordig to the classificatio i [27] if it is ot maufacturig specific. By samplig strategy ot oly the umber ad locatio of sample poits o the roudess profile but also their use by the data fittig algorithm is cocered. 4. MZT MODEL I the MZT method, the ukow are the ( coordiates of the associated derived ceter of the miimum zoe referece circles of the roudess profile (MZCI [28]). MZCI is formed by two cocetric circles eclosig the roudess profile, the ier miimum zoe referece circle ad the outer miimum zoe referece circle, havig the least radial separatio. The differece betwee the ier miimum zoe referece circle ad the outer miimum zoe referece circle is the miimum zoe error (MZE). MZE is the target parameter of our optimizatio algorithm as a fuctio of (. Give a extracted circumferetial lie r(θ), with θ (0, 2π], of a sectio perpedicular to the axis of a cylidrical feature, the roudess error R( is defied by: R( = OC( IC(, ( E r ) (1) ( θ where OC( ad IC( are the radii of the referece circles of ceter (, ad E r( θ ) is the area eclosed by r(θ): OC = max r( ) (2) ( θ ( 0,2π ] θ IC = mi r( ) (3) ( θ ( 0,2π ] θ
6 Michele Lazetta ad Adrea Rossi As a CMM scas the roudess profile by samplig a fiite umber,, of equally-spaced poits θ i of the extracted circumferetial lie (θ i = i 2 π, i=1,...,), the OC( ad IC( are evaluated by: OC( = max r( θ ) (4) 2 π θ i, i 1,.., i i = = IC( = mi r( θ ) (5) 2 π θ i, i 1,.., i i = = Figure 2 shows the metioed features for a give (. MZE is evaluated by applyig the MZT data-fittig method to solve the followig optimizatio problem: MZE = mi ( y ) E r ( x, y, θ ) mi max 2π θi = i, i= 1,.., R( = subject to ( E r( θ ) mi r( θi ) i 2π θi = i, i= 1,.., r( θ i ) (6) where Er( θ i ) is a restricted area i the covex evelopmet of the equally-spaced sample poits, i.e. the search space. 5. GENETIC ALGORITHMS FOR ROUNDNESS EVALUATION GAs were proposed for the first time by Hollad [29] ad costitute a class of search methods especially suited for solvig complex optimizatio problems [30]. Geetic algorithms are widely used i research for o-liear problems. They are easily implemeted ad powerful beig a geeral-purpose optimizatio tool. May possible solutios are processed at the same time ad evolve with both elitist ad radom rules, so to quickly coverge to a local optimum which is very close or coicidet to the optimal solutio. Geetic algorithms costitute a class of implicit parallel search methods especially suited for solvig complex optimizatio or o-liear problems. They are easily implemeted ad powerful beig a geeral-purpose optimizatio tool. May possible solutios are processed cocurretly ad evolve with iheritable rules, e.g. the elitist or the roulette wheel selectio, so to quickly coverge to a solutio which is very close or coicidet to the optimal solutio. Geetic algorithms maitai a populatio of ceter cadidates (the idividuals), which are the possible solutios of the MZT problem. The ceter cadidates are represeted by their chromosomes, which are made of pairs of x i ad y i coordiates. Geetic algorithms operate o the x i ad y i coordiates, which represet the iheritable properties of the idividuals by meas of geetic operators. At each geeratio the geetic operators are applied to the selected ceter cadidates from curret populatio i order to create a ew geeratio. The selectio of idividuals depeds o a fitess fuctio, which reflects how well a solutio fulfills the requiremets of the MZT problem, e.g. the objective fuctio.
Roudess Evaluatio by Geetic Algorithms 7 Sharma et al. [2] use a geetic algorithm for MZT of multiple form tolerace classes such as straightess, flatess, roudess, ad cylidricity. There is o eed to optimize the algorithm performace, choosig the parameters ivolved i the computatio, because of the small dataset size (up to 100 sample poits). We et al. [11] implemet a geetic algorithm i real-code, with oly crossover ad reproductio operators applied to the populatio. Thus i this case mutatio operators are ot used. The algorithm proposed is robust ad effective, but it has oly bee applied to small samples. I a geetic algorithm for roudess evaluatio the ceter cadidates are the idividual of the populatio (chromosomes). The search space is a area eclosed by the roudess profile where the ceter cadidates of the iitial populatio are selected for the data fittig algorithm. The area is rectagular because the crossover operator chages the x i ad y i coordiates of the parets to geerate the offspigs [10]. After crossover, the x i ad y i coordiates of parets ad offsprigs are located to the rectagle vertexes. I order to fid the MZ error the search space must iclude the global optima solutio i.e. the MZ ceter. Therefore the cetre of the rectagular area is a estimatio of the MZ ceter evaluated as the mea value of the x i ad y i coordiates of the sampled poits [10], [11], [12], [13]. I [11] the search-space is a square of fixed 0.2 mm side, i [13] it is 5% of the circle diameter ad ceter. I [10], the side is determied by the distace of the farthest poit ad the earest poit from the mea ceter. I [12] it is the rectagle circumscribed to the sample poits. Optimal samplig ad geetic algorithm parameters are listed i Table 1. Table 1. Algorithm parameters accordig to [13] Optimizatio Geometric ad algorithm GA geetic parameters Symbol Value Commet sample size 500 umber of equally spaced sample poits search space E 0.5 iitial populatio radomly selected withi populatio size P s 70 set of chromosomes used i evolvig epoch selectio elitist selectio crossover P c 0.7 oe poit crossover of the pc pop parets gees (i.e. coordiates) at each geeratio mutatio P m 0.07 pm pop idividuals are modified by chagig oe gee (i.e. coordiate) with a radom value stop criterio N 100 the algorithm computes N geeratios after the last best roudess error evaluated rouded off to the fourth decimal digit (0.1 µm)
8 Michele Lazetta ad Adrea Rossi 6. GA OPERATION Selectio: durig this operatio, a solutio has a probability of beig selected accordig to its fitess. Some of the commo selectio mechaisms are: the roulette wheel procedure, the Touramet selectio, ad the elitist selectio. With this latter, the idividuals are ordered o the basis of their fitess fuctio; the best idividuals produce offsprig. The ext geeratio will be composed of the best chromosomes chose betwee the set of offsprig ad the previous populatio. Crossover: this operator allows to create ew idividuals as offsprig of two parets by iheritig gees from parets with high fitess. The possibility for this operator to be applied depeds o the crossover probability. There are differet crossover types: the oe poit ad multiple poit crossover, ad other sophisticated oes. I the proposed GA was used the arithmetic crossover mechaism, which geerates offsprig as a compoet-wise liear combiatio of the parets. Mutatio: a ew idividual is created by makig modificatios to oe selected idividual. I geetic algorithms, mutatio is a source of variabilit ad is applied i additio to selectio ad crossover. This method prevets the search to be trapped oly i local solutios. The relative parameter is the mutatio probabilit that is the probability that oe idividual is mutated. Stop criterio: the algorithm has a iterative behavior ad eeds a stop coditio to ed the computatio. Possible criteria iclude: overcomig a predefied threshold for the fitess fuctio or iteratio umber or their combiatios. I the proposed GA, the stop criterio is cotrolled by the umber N of the iteratios if o improvemet i the solutio occurs. 7. TESTING ALGORITHMS To aalyze the behavior of a algorithm with the MZT method, dataset with kow MZ error are available. These datasets are geerated with NPL Chebyshev best fit circle certified software [31]. The use of certified software has the followig beefits it produces radomly distributed error makes the results more geeral, because the results achieved with the geetic algorithm are ot maufacturig sigature specific; the circle ceter is kow, so it allows estimatig the circular profile ceter is computed as a average value of the measurig poits coordiates [11] the average MZ ceter foud by the algorithm Datasets produced by certified software have a user-selected ceter ad radius. For performace assessmet, the algorithm is usually executed o a dataset several tes of times for each test, ad the average MZ error ad the average computatio time are computed ad compared with the omial MZ error. A example of executio i show i Figure 3, which displays the average ad stadard error.
Roudess Evaluatio by Geetic Algorithms 9 0.2 0.18 0.16 0.02 0.06 0.12 0.18 0.14 Average MZE (mm) 0.12 0.1 0.08 0.06 0.04 0.02 0 5 10 25 50 100 datset size (x10 2 ) Figure 3. Average MZE with error bars for large datasets ad differet roudess errors. I a typical experimetal approach, a first raw estimatio of the circular profile ceter is ecessary as a startig poit for a more accurate roudess evaluatio. A first estimatio of the circular profile ceter is computed as a average value of the measurig poits coordiates i expressio (1). SUMMARY The applicatio of a geetic algorithm for the roudess evaluatio of circular profiles usig the MZT method has bee described. The GA approach described ad the parameters provided may solve most roudess measuremet eeds, both for small ad large datasets (up to several thousads datapoits). The listed literature may serve as a guidelie for optimal GA parameters estimatio o ew problems. ACKNOWLEDGEMENT Part of this work has bee derived from Alessadro Meo ad Luca Profumo s project work for the class of automatio of productio processes of the master degree i automatio egieerig at the school of egieerig at Pisa Uiversity.
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