Performance improvement for optimization of non-linear geometric fitting problem in manufacturing metrology*

Transcription

1 Measurement Scence and Technology 1 Performance mprovement for optmzaton of non-lnear geometrc fttng problem n manufacturng metrology* Govann Moron,Wahyudn P. Syam, and Stefano Petrò Mechancal Engneerng Department, Poltecnco d Mlano, Va La Masa 1, 2156 Mlan, Italy Abstract Product qualty s a man concern n today manufacturng. It s a fundamental requrement for companes to be compettve. To assure such qualty, a dmensonal nspecton to verfy geometrc property of a product has to be carred out. Hgh speed non contact scanners help ths task, by both speedng up acquston speed and ncreasng accuracy through a more complete descrpton of the surface. The algorthms for the management of the measurement data play a crtcal role n ensurng both the measurement accuracy and speed. One of the most fundamental parts of the algorthm s procedure for fttng substtute geometry to the cloud of ponts. Ths artcle addresses ths challenge. Three relevant geometres are selected as case studes: non-lnear least-square fttng of crcle, sphere and cylnder. These geometres are chosen wth consderaton of ther common use n practce; for example the sphere s often adopted as reference artfact for performance verfcaton of coordnate measurng machne (CMM) and cylnder s the most relevant geometry for pn-hole relaton as an assembly feature to construct a complete functonng product. In ths artcle, an mprovement of the ntal pont guess for Levenberg-Marquardt (LM) algorthm by employng Chaos Optmzaton (CO) method s proposed. Ths causes a performance mprovement n the optmzaton of a non-lnear functon fttng the three geometres. The results show that, wth ths combnaton, hgher qualty of fttng results n term of smaller norm of the resduals can be obtaned whle preservng the computatonal cost. Fttng a ncomplete-pont-cloud, whch s a stuaton where the pont cloud do not cover a complete feature e.g. from half of the total part surface, s also nvestgated. Fnally, a case study about fttng a hemsphere s presented. Keywords: Least-square fttng, non-lnear optmzaton, mult-modal functon, steepest-decent, Gauss-Newton, chaos search. * Ths s an extended artcle that were presented as keynote paper at IMEKO 11 th ISMQC 213, Cracow-Kelce, Poland. Correspondng author: Wahyudn P. Syam (wahyudnpermana.syam@polm.t 1. Introducton Qualty of manufacturng product s the man concern n modern world to ncrease compettveness [1]. Qualty nspecton s the procedure to verfy ths geometrc attrbute and can

2 Measurement Scence and Technology 2 be realzed by dmensonal metrology [2]. In dmensonal metrology, least-square (LS) fttng of substtute geometres, after obtanng measurement ponts, s a fundamental step before any geometrc feature may be evaluated [3],[4],[5],[6]. LS fttng algorthm from ponts cloud to basc geometrc feature s deployed after obtanng a set of measured ponts. Hence, the dmensonal measurement, such as the dameter of a crcle or sphere, the angle between two lnes, the dstance between two axes, etc, can be evaluated (fg. 1). Subsequently, a routne for fttng task s a crtcal element n the chan of dmensonal metrology for qualty nspecton. In ths case, fttng the substtute geometry s called functon reconstructon [7]. The lack of pror knowledge about the nomnal geometry to be ftted can sgnfcantly ncrease the dffcultes n the fttng tself. For example, a lack of pror knowledge about the drecton of the axs of a cylnder sgnfcantly ncreases the dffculty of the dentfcaton of the real axs drecton. Ths s the usual condton of reverse engneerng. In addton to the dffcultes of fttng process, a fast fttng process s a strngent requrement snce recent measurement nstruments are able to obtan rapdly thousands or even mllons of ponts n few seconds. In ths case, fttng tme can sgnfcantly ncrease the overall measurement tme, but hgh-speed nspecton s requred to reduce the nspecton cost, and n general, to reduce the product cost [8]. Hence, an accurate and hgh-speed fttng procedure s a strct requrement. In the case of lnear least squares fttng (lke e.g. n the case of plane fttng) robust solutons are avalable. These solutons are based on fndng the drecton cosne of a lne or plane from the Egen vectors of the cloud of ponts. The egenvectors correspond to ts hghest and smallest sngular value, whch can be calculate by means of the Sngular Value Decomposton (SVD), for whch fast and relable algorthms are commonly avalable [6]. But, n the case of non-lnear least squares fttng (lke e.g. n the case of crcle, sphere and cylnder fttng) such smple solutons are not avalable. Ths paper wll propose an mprovement of the current state of art n the feld of non-lnear least squares fttng of geometres. The mprovement s based on a wse choce of the ntal soluton thanks to the applcaton of Chaos Optmzaton (CO). The optmal ntal soluton wll be coupled to a standard (e.g. Levenberg-Marquardt) nonlnear least squares optmzaton algorthm for fndng the fnal fttng geometry. The effectveness of the proposed approach wll be compare to the standard use of Levenberg-Marquardt algorthm n the case of sphere, cylnder, and crcle fttng. 2. Non-lnear Fttng Basc substtute geometres are dvded n two groups: lnear and non-lnear geometry. Ths groupng crteron s based on ther defned parameters. Lne (2D and 3D) and plane fall nto lnear geometry, as ther parameters can be estmated by means of lnear least squares fttng. On the other hand, other basc geometry such as crcle, sphere, cylnder, cone and torus have nonlnear parameters defnng ther shapes. Subsequently, they are categorzed as non-lnear geometres. LS fttng of crcle, sphere and cylnder wll be addressed n ths artcle. Crcle and sphere geometres have many applcatons, for example sphere s a common artfact geometry for calbraton of dmensonal metrology nstruments [9],[1]. In addton, many mechancal products have rotatonal functonalty whch s consttuted by crcular from shafts and holes. Cylnder s a geometry representaton of these shaft-hole systems [11].

3 Measurement Scence and Technology 3 The bass of LS fttng s the mnmzaton of an objectve functon consttuted by a sum of square of errors. Error s defned as the dfference between estmated and measured value. In dmensonal metrology, error s usually assmlated to the local geometrcal devaton,.e. the dstance between measured ponts and deal substtute geometry (fg 2). LS fttng objectve functon s defned as: n 2 arg mn F p, x d p (1) p 1 Where: F s the dstance functon of ponts x to the ftted geometry. x s a cloud of n ponts sampled on a surface and p s set of parameters on whch a dstance functon d p depends on, so that d p s the dstance of the -th pont from the substtute geometry defned by p. For the crcle, the dstance functon s (Fg. 2a left): 2, x 2 d x r x r x x y y r (2) Where x x, y T s the crcle center, r s the crcle radus, and x, y T and x s L2-norm of a vector x, e.g. v v2 v3 v. Smlarly, for the sphere, the functon can be formulated as: 2 2, x x x y z 2 x s the -th pont, d x r x r y z r (3) Where x x, y, z T s the sphere center, r s the sphere radus, and x, y, z T th pont. The dstance functon of a pont to a cylnder s more complex (see fg. 2a rght): x s the - x d,, r d 3dp2 Axs r x n x n r n (4) where r s the radus of the cylnder and d 3dp2Axs s defned as dstance between 3D pont x to the axs of cylnder (a straght lne). The axs s defned by a pont x belongng to t and a drecton vector n (fg. 2b). One can observe that the objectve functon F whch has to be mnmzed s a non-lnear multmodal functon whch has many local mnma and/or maxma. 3. Levenberg-Marquardt Algorthm.

4 Measurement Scence and Technology 4 Levenberg-Marquardt (LM) algorthm s a well-known approxmaton method for solvng nonlnear least square problems that has applcatons n many felds [12, 13]. The recpe of LM algorthm s the blendng between steepest-decent (gradent search) step method and Gauss- Newton step method. When the current soluton s far from the optmal, the LM method acts lke a steepest-decent method. Then, LM method wll become a Gauss-Newton when the soluton s near optmal. The bass of steepest-decent method s searchng wth regard to the drecton of the gradent. Let F be the functon to optmze, and x k the canddate soluton at step k. Snce n ths case mnmzaton s the problem to solve, the next step n the searchng procedure s: x k 1 x k s F (5) Where F ( F / x, F / y, F / z) s the gradent of the objectve functon as well as the search drecton, and s s the step sze whch determnes how far the next canddate soluton wll be from the current one. Hence, f the s value s set very small, then t wll take longer to reach convergence. Otherwse, f the value of s s very large, there s a hgh probablty that the searchng process wll over-step the optmum value. In Gauss-Newton method, lnearzaton by usng Taylor expanson seres s deployed. The seres n ) T 2 T n are f ( p) f ( p ) ( p p ) f ( p )... ( p p ) f ( p Hgher order term. Commonly, hgher order expansons are not consdered. Not only the algorthm s more effcent to reach the convergence, but also the form s tractable to solve p. By settng f ( p ), the next step of the Gauss-Newton can be calculated as: T T p j 1 p j JdJd Jdd p j (6) J s the Jacoban matrx of Where d p j s the vector of the resdual (dstances) at step j, and d ths vector of dstance functons. Note that Taylor expanson seres s accurate only for a small range of regon, called trust regon. Ths small regon s a regon where the non-lnear estmaton of a functon by usng Taylor expanson s stll reasonably vald. It mples that Gauss-Newton method s vald for searchng through a small area of the neghborhood. Subsequently, the method s effectve when the ntal guess s near the optmum soluton. LM method combnes the advantages of steepest-decent and Gauss-Newton methods. A vector of nput parameters p, whch ncludes the parameters that wll be optmzed, s suppled to the LM algorthm, along wth matrx M whch s A n 3 matrx of all the data ponts, defned as: x y z ; x y z so that an optmzed vector of parameters p s obtaned. The ; n n n, LM method used here s based on the LM used by NIST [6] for ther algorthm testng system. The LM algorthm s: Algorthm 1:Levenberg-Marquardt Algorthm Input: Vector p whch s the ntal guess for the parameter and matrx M whch s the pont cloud to be ftted. Output: Vectorp whch s the ftted parameter

5 Measurement Scence and Technology 5 1: Set. 1 2: DO { decrease T 3: set U JJ T 4: set v J d p T 5: set F d p d p 6: DO { ncrease 7: set H U I dag U 8: solve Hx 9: v T set p p x set F d p d p new ; new new 1: IF converged THEN return p p new 11: UNTIL F new F or stop crteron s true 12: IF Fnew F THEN p p new 13: UNTIL stop crteron s true s LM varable, whch s ncreased and decreased by 1 and.4, respectvely, accordng to NIST suggeston [6]. J s a Jacoban matrx whch elements on ts -th row are d ( p ), whch are the frst order partal dervatves of d respect to each parameter whch has to be estmated for each -th pont. For crcle, the parameters p are x, yof ts center and radus r. For sphere, only one addtonal element z for ts 3D poston of the center s added to the parameters. Fnally, the parameters for cylnder are x, y, z whch s a pont on the axs, havng vector of cosne drecton (normal) n n, n, ), and fnally ts radus r. The number of column of matrx ( 1 2 n3 J corresponds to the number of parameters to be estmated, and the number of rows corresponds to number of ponts the substtute geometry wll be ftted to. The central dea of ths LM method les on the equaton Hx -v. If ths equaton s enlarged nto dag J J I J J x J d p, one can observe that f s zero or small, LM T T T behavor become Gauss-Newton method. In the opposte, f s large, then the off-dagonal T elements of JJwll have less effect such that LM behaves lke steepest-decent method. The term ( T T I dag JJ) s used nstead of DD.Ths s a weghted dstance matrx (dependng on the geometry whch wll be estmated), based on Nash [13] suggeston, such that dag Hx J J I J J x J d p becomes postve defnte. T T T 4. Intal Pont Problem LM teratve method mentoned n the prevous secton depends sgnfcantly on the ntal guess of a set of solutons, p [14]. Ths stuaton s smlar to any other teratve algorthm. The functon to be optmzed s a mult-modal functon wth a complex contour and many local optmums. Subsequently, the rsk exsts that the search s trapped n a local optmum regon. The

6 Measurement Scence and Technology 6 llustraton of mult-model functon s shown n fg. 3 (left) by usng Schwefel functon and N 2 square of the summaton of a crcle dstance functon, whch s F d 1, where d s declared n equaton (2). As mentoned before, the LS nonlnear functon whch to be mnmzed to ft geometres s mult modal. Hence, t has many local mnma, only one beng the global optmum. If the optmzaton process gves a local mnmum soluton, then the soluton s sub-optmal. The searchng procedure can be trapped n local mnma dependng on where the ntal guess s put. Subsequently, as t can be presumed, the result s sgnfcantly affected by the ntal guess [14]. For example, the objectve functon to ft a crcle s shown n fg. 3 (rght). The surface s constructed by varyng the xycenter, poston of the crcle. The dfferent colors show how the surface changes as the canddate radus r changes. Even though n ths case the optmzaton zone s convex, dfferent levels of radus r create dfferent separated optmzaton zones. Ths can trap the searchng process n one of the optmzaton zones. Therefore, t s possble the fnal soluton s not a global optmum, dependng on the ntal soluton. Fg. 4 llustrates how ntal guess as startng soluton affects the fnal results. If the ntal guess s far from optmum, an unexpected fnal result can be obtaned (fg. 4a). On the other hand, a good ntal guess sgnfcantly mproves the fnal soluton reducng the objectve functon value (fg. 4b). 5. Chaos Optmzaton Chaos s defned as a sem-randomness property. Ths property s generated by a nonlnear determnstc equaton. It creates a chaotc dynamc step whch can easly escape from local optma. The concept s dfferent wth usng rejecton-acceptng probablty test n random-based algorthms, such as mprovement heurstc search [15]. Searchng through regularty of chaotc moton, represented by one-dmensonal logstc map, s ts fundamental recpe [16]. Chaos optmzaton (CO) uses these chaos propertes, whch are ergodcty, stochastc property, and regularty [17]. The one-dmensonal logstc map used s: t k 1 t c k (1 t k ) (7) Where 3.56, 4 s a control argument and k s teraton number. Yang [18] recommended c t 1 where t {,.25,.5,.75,1.}. The behavor of equaton (7) becomes chaotc n the sense that ts value s drastcally changed wthn the lmt of c and t k presentng the regularty of chaotc moton. Fg. 5 shows the plot of tme seres of ths functon and pared-plot between two consecutve chaos varables. Ths CO s used to mprove the ntal guess of LM non-lnear fttng teratve method, so that the ntal guess s near the optmal soluton, thus preservng the computaton tme, whch s very mportant when the sample sze s large (mllons of ponts). The combnaton of CO algorthm wth LM algorthm to mprove the ntal guess s as follows: Algorthm 2: Chaos search to mprove the ntal guess n LM method

7 Measurement Scence and Technology 7 Input: Vector p s the ntal guess for the parameter (1:n-param) Goal: New vector p s the mproved ntal guess by Mn F( p ), p { a, b}, Let p k ( p 1 : p k ), t k ( t 1 : t k ) where : p ( p1,..., pn) and t ( t1,..., tn) k 1, r 3 1: Set k, r, Set max max 2: Produce randomly. t {,1} and {,.25,.5,.75,1.}. 3: k Set t t, t* t, a p MPE, b p MPE where : a ( a,..., an), b ( b,..., bn ) 4: Set p* p ntal guess parameter 5: DO WHILE { r rmax ; DO WHILE { k k max ; 6: r r r r Set p a t ( b a ); k N 2 F 7: calculate IF F k F * THEN k k d ( p ) F* F( p ), p* p, t* t k k 1 8: k k 1; t t (1 t ), {3.56,4} 9: }END k-th teraton; r r 1 1: a b r 1 r r r 11: IF a 12: IF 13: 14: k k 1 p * ( b a ) and 1 r p * ( b a r ) b r 1 a r 1 rmax r b r 1 r THEN a a, {,.5} r r 1 r THEN b b, {,.5} IF r THEN produce t {,1 } by random, k k, t t GOTO(7) ELSE CO s termnated, return p *;} p END r-th teraton; 15: Insert the new p nto Algorthm 1: LM algorthm. To adjust small ergodc range around p *, the parameters are set as. 45 [17], 4 [18], r max 5, and K max 5. The value 4 s set such that a sgnfcant dfference n the long term wll be obtaned from a small change of t. As t can be seen from fg. 5 rght, wth a small change n two consecutve t, a chaotc behavor wll be observed n the tme seres manner (fg. 5 left). The value of K r were chosen to mnmze the overhead computatonal cost max and max n determnng the ntal pont. The statements IF b r 1 b r r 1 r THEN b b, {,.5} a r 1 a r r 1 r THEN a, {,.5} a and IF are to encourage movement farther from the ntal boundng area, set n the begnnng of the search. Wth reference to Fg. 4 (rght), the ntal guess s expected to le on the correct optmzaton zone to fnd the global optmum. 6. Implementaton and Dscusson

8 Measurement Scence and Technology Performance Improvement Ponts wth random error accordng to unform dstrbuton and normal dstrbuton were generated as presented n table 1. For Chaos-LM method, ntal pont guess of the ntal soluton of LM optmzaton teraton was mproved by sendng t to CO method. In LM algorthm, the stoppng rule s set as maxmum teraton = 1 and 1 for the Chaos-LM method. Fg. 6 vsualzes the generated data by plottng the ponts cloud. The algorthm s mplemented n MATLAB and run on an Intel Centrno Core 2 Duo 2.2 GHz. Table 1: Detals of data generaton. Type of Data Number of ponts and Nomnal Parameter Crcle Sphere Cylnder Unform (x,y,z,r)=(15,1 Range (x,y,r)=(15,15,2) mm (x,y,z,r)=(15,15,15,2) mm 5,15,5) and n (µm) (1,1,1) mm Type 1 [-2.2,2,2] 1 pts grd [3x3] grd [25x25] Type 2 [-5,5] 1 pts grd [3x3] grd [25x25] Normal sgma σ Type pts grd [3x3] grd [25x25] Type pts grd [3x3] grd [25x25] The ntal guess of the center of crcle and sphere s the centrod. The centrod locaton for each x,y,z s the average of the ponts x n. The centrod s also the ntal guess of pont on the axs of a cylnder. For the radus, ts ntal estmaton s: 1 (max x mn x ) (max y mn y ) r 2 2 for the crcle and: 1 (max x mn x ) (max y mn y ) (max z mn z ) r 2 3 (8) (9) for the sphere and cylnder. For the specal case of a cylnder, ts ntal guess for cosne drecton of the axs s derved by fttng a 3D lne to the pont clouds. The fttng method s mplemented wth a method accordng to NIST [6].Two levels of sgma for the data devaton were consdered. Type 1 represents only the uncertanty of the nstrument (Maxmum Permssble Error/MPE), whle type 2 smulates the uncertanty due to the part and the nstrument. Type 2 data represents a more realstc stuaton snce an nspected part always contans feature devaton from ts nomnal [19]. Results from 1 runs show that the combnaton of these methods, Chaos and LM, ncreases the accuracy of the fttng process. The ndcaton s that the ftted geometry has less resdual error, n term of the magntude of ther norm of sum of square resduals, whle preservng the computaton cost. Table 2 provdes the complete results of the fttng of full geometry pont clouds both wth only LM method and wth Chaos-LM method. Chaos-LM encourages the ntal

9 Measurement Scence and Technology 9 guess of the soluton to move to a better startng pont, thanks to the property of the chaotc moton whch non-repeatedly searches through a set of states n a certan bounded doman [15]. Senstveness of the fnal soluton of LM method to where the ntal guess starts s related to the Taylor approxmaton n the Gauss-Newton method, whch depends hghly on the non-lnearty degree of the neghborhood. The qualty of ths Taylor approxmaton, whch s usually untl frst term approxmaton, decreases for hgher non-lnear functon. Because of ths, a trapped condton durng searchng process can occur. Fg. 7, 8 and 9 propose some vsualzatons of the fttng result for crcle, sphere and cylnder respectvely. From ths, one can observe that the Chaos-LM fttng (fg. 7, 8, and 9 rght) fnally le on the mddle of the pont cloud. Ths s coherent wth the fundamental behavor of least-square fttng whch s an average over the consdered data (n ths case the pont cloud). Plot of the norm of resdual and Central Processng Unt (CPU) tme for crcle, sphere, and cylnder are respectvely presented n fg. 1 and 11. In the specal case of a cylnder fttng result, the computaton tme slghtly ncreases compared to the LM method. Indeed, the mprovement n the sum of squared resduals s sgnfcant. Furthermore, from the graph one can observe that the varaton nterval of the CPU tme for ths cylnder fttng s ntersectng each other, so they are not sgnfcantly dfferent. Random Error Type [µm] U [-2.2,2.2] U [-5,5] N (σ=1.1) N (σ=2.5) Random Error Type [µm] U [-2.2,2.2] U [-5,5] N (σ=1.1) N (σ=2.5) Table 2: Smulaton results of the full-geometrc pont cloud fttng. r (µ±3σ) [mm] Levenberg-Marquardt Algorthm Crcle Sphere Cylnder CPU tme (µ±3σ) [s] r (µ±3σ) [mm] CPU tme (µ±3σ) [s] r (µ±3σ) [mm] CPU tme (µ±3σ) [s] ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.84 r (µ±3σ) [mm] Chaos and Levenberg-Marquardt Algorthm Crcle Sphere Cylnder CPU tme (µ±3σ) [s] r (µ±3σ) [mm] CPU tme (µ±3σ) [s] r (µ±3σ) [mm] CPU tme (µ±3σ) [s] ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.113 A very mportant condton, dffcult to address wth standard approaches to fttng, arses when the cloud of ponts does not cover the whole feature, e.g. only a hemsphere has been sampled. Ths may be due to access lmtaton of the sensor to capture part surface, lke e.g. n the case of laser scannng nstruments, or to the real ncompleteness of the sphere, lke n the case of a crcular groove. Both LM and Chaos-LM methods are appled to half-crcle, half-sphere, and half-cylnder pont clouds. Ths s a sgnfcantly more dffcult stuaton compared to fttng a complete cloud of ponts. One of the reasons s that the estmaton of ntal soluton tends to be

10 Measurement Scence and Technology 1 less accurate snce, n general, the ntal estmaton s based on the symmetrcal propertes of the geometry to be ftted. The data generaton s dentcal to the one consdered for the full-geometry case as presented n table 1. From ths data generaton, half of the cloud of ponts s then dscarded to get the half-geometry. The number of runs n the smulaton and the performance measures of both fttng methods are dentcal to the prevous normal case. Detals of all results of the smulaton runs are presented n table 3. One can observe that the accuracy of fttng halfgeometry pont clouds s sgnfcantly mproved by Chaos-LM method compared to only LM method. Instead, the CPU tme needed for Chaos-LM to have better result s hgher than the one of the LM method. Snce there are ncrements n CPU tme to get a better result of Chaos-LM method, the comparson of the two algorthms n ths case has to be further nvestgated. Graphcal presentaton of the fttng results for half-crcle, half-sphere and half-cylnder are provded n fg. 12, fg.13, and fg. 14 respectvely to ntutvely understand the sgnfcant result of accuracy mprovement by Chaos-LM n the case of half-geometres fttng. Fnally, to graphcally explan the results n table 3, fg. 15 and fg. 16 respectvely plot the norm of resdual r and CPU tme of the half-geometres fttng. Table 3: Smulaton results of the half-geometrc pont cloud fttng. Levenberg-Marquardt Algorthm Random Error Type [µm] r (µ±3σ) [mm] Crcle Sphere Cylnder CPU tme (µ±3σ) [s] r (µ±3σ) [mm] CPU tme (µ±3σ) [s] r (µ±3σ) [mm] CPU tme (µ±3σ) [s] U [-2.2,2.2] ± ± ± ± ± ±.412 U [-5,5] 88.16± ± ± ± ± ±.557 N (σ=1.1) N (σ=2.5) ± ± ± ± ± ± ± ± ± ± ± ±.895 Random Error Type [µm] r (µ±3σ) [mm] Chaos and Levenberg-Marquardt Algorthm Crcle Sphere Cylnder CPU tme (µ±3σ) [s] r (µ±3σ) [mm] CPU tme (µ±3σ) [s] r (µ±3σ) [mm] CPU tme (µ±3σ) [s] U [-2.2,2.2] U [-5,5] N (σ=1.1) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.1623 N (σ=2.5) ± ± ± ± ± ±.627

11 Measurement Scence and Technology 11 Convergence curve analyses are presented n two parts whch are for full- and half- geometres fttng. One should note that n both cases, the value of convergence curve can not reach zero snce error exst on the ponts to be ft due to the smulated perturbaton. The selected type of smulaton s unform dstrbuton n the range of -5µm and 5 µm U [-5,5]. For each group, three convergence curves are shown correspondng to crcle, sphere and cylnder. On the x-axe of the graph, there are two types of teraton. For LM method, t corresponds to LM number of teratons whch ranges from 1 to 1 teratons. For Chaos-LM, ths axs corresponds to the number of chaos teraton n the range from 1 to 1 teratons. In the case of full-geometrc fttng, the convergence rate s much faster n the case of crcle and cylnder. Fg. 17, fg. 18, and fg. 19 depct the convergence curves for the full-geometres case. The convergences rate of LM method for crcle and cylnder fttng are much slower, as the small gradent of the curve compared to the Chaos-LM one denotes. In the case of the sphere, both LM and Chaos-LM method have a smlar convergence rate though Chaos-LM method s faster wth respect to the LM one. Clearer trapped phenomena of LM method n fttng can be observed n the case of half-geometres fttng problem. The convergence curves for the half-geometres are shown n fg. 2, fg. 21, fg. 22 respectvely. From all these three fgures, the LM method does not show mprovement as the number of LM teratons s ncreased. Ths stuaton clearly shows that LM has been trapped n some local optmum regon. It means that, although the number of teraton s ncreased, the result of LM can not gve a better result so one can say t s early converged. On the other hand, a dfferent stuaton can be seen for the Chaos-LM method. Ths method can escape from a local optmum wth an ncrease of the chaos teratons. The reason s that, by usng chaotc movement and ncreasng the number of teratons, more regons are vsted to explore new better feasble soluton that can gve a better result. Chaos-LM can show a sgnfcant mprovement and convergence result wthout a large number of teraton ncrements. From the nvestgaton of the convergence, one can observe that the best number of chaos teratons for Chaos-LM s around 3. The chaos optmzaton to mprove the ntal guess s effectve n LS fttng problem. The chaos search encourages the ntal guess of the soluton to move to a better startng pont that s nearer to the true soluton thanks to the property of the chaotc moton that non-repeatedly searches through a set of states n a certan bounded doman [2]. Wth ths property, the searchng process can cover a wder search space wthn a small number of teraton. Ths s a dfferent property compared to mprovement heurstc search such as genetc algorthm, tabu search, etc [21]. Generally, mprovement heurstc searches algorthm need larger number of teratons to ncrease the vsted feasble soluton around the search space n whch the computatonal cost becomes problematc. Combnatons of Chaos and LM algorthms have a lnear complexty n term of the relaton between number of ponts processed and the ncrease n computaton tme. One can see that they are constructed from two algorthms, whch are the Chaos and the LM algorthm. Each of them contans two nested loops nsde ther algorthm. Snce the two loops of the algorthm are not related to the number of the ponts n, whch are the ponts to be ftted, the relaton of the algorthm steps to ther nputs s only n the calculaton to evaluate the objectve functon through all number of ponts n. Let the total order of the algorthms be f ( n) n1 n2 n n 2n where subscrpt 1 and 2 correspond to algorthm 1 (chaos) and 2 (LM), respectvely. Hence, the

12 Measurement Scence and Technology 12 algorthm effcency s (n) snce n and k n n, f ( n) k. g( ) such that f n) n n ( ). ( 1 2 n 6.2 Case Studes n To test the Chaos-LM method n a real fttng stuaton, a case study was carred out. It s derved from Kawalec and Magdzag [22]. In ther case, a calbrated rng gauge was used snce they focused on comparson of methods to solve the crcle fttng problem. Instead, n the proposed case study, measurement of a calbrated ceramc sphere of a ZEISS CMM for stylus qualfcaton was used as shown n fg. 23. The sphere has a calbrated radus of mm. The CMM machne used was ZEISS PRISMO HTG wth MPEE=2 µm+l/3 µm. The choce of sphere s due to the fact that a 3D geometry fttng can be appled and a sphere s a good example of a common artfact for CMMs. The strategy used to obtan the ponts was by means of scannng strategy. The pont cloud to be ft s a half-geometry. There are two types of pont clouds. The frst type s low densty cloud and the second type s hgh densty (fg. 24). The low densty pont cloud contans 312 ponts. A total of 3435 ponts were collected for hgh densty pont cloud. The ntal parameter of the sphere, for both LM and Chaos-LM method, has been chosen near the optmal. The ntal x and y are from the average of respectvely x- and y- poston of the ponts, and z s selected from maxmum z-poston of the ponts mnus the known nomnal radus of 15 mm. Table 5 summarzes both the fttng results and the devaton from the calbrated radus of the sphere. From the results, the ftted radus of the sphere from Chaos-LM method s much better than the LM one. Note that the results of LM method are already from 5 teratons. Both n the low and hgh densty cases, the procedures are run wth the same number of teraton. In the case of low densty, the accuracy s even better than n the hgh densty one snce for hgh densty, more soluton space s obtaned such that an ncrease of teratons can produce smlar results wth regard to the lower densty result. Vsualzaton of the fttng results s provded n fg. 25 both for low and hgh densty ponts respectvely. Another case study was carred out by measurng and fttng an ndustral-made cylnder work pece made of hardened steel havng nomnal dameter of 6 mm. The measurement procedure s shown n fg. 26a. Total ponts obtaned were 19 ponts by crcular path scannng strategy of three segments. Fg. 26b presents the fttng result. The blue ponts are the obtaned ponts. Meanwhle, the red ponts are the ftted cylnder wth ts axs lne n green. Snce the geometry s symmetry, the selecton for ntal parameters s dentcal to the one set n the smulaton run of secton 6.1 for cylnder case. The results show the mprovement of Chaos-LM fttng. Detals of the results are depcted n table 6. The cylnder s not calbrated. Subsequently, the norm of resdual r s presented to compare the fttng qualty. In ths case, Chaos-LM gves better result. Moreover, from the table, t can be observed that the fttng result by LM s outsde the tolerance lmt, meanwhle the Chaos-LM result s nsde. Table 5: Results of fttng half-calbrated sphere for hgh and low densty pont cloud. x (mm) y (mm) z (mm) radus (mm) devaton from calbrated radus (mm) Calbrated Value

13 Measurement Scence and Technology 13 Hgh Densty LM Method Chaos-LM Method Low Densty LM Method Chaos-LM Method Nomnal Value Table 6: Results of fttng ndustral cylnder. x [mm] y [mm] z [mm] nx ny nz Dameter [mm] ±.1 LM Method Chaos-LM - Method Concludng Remarks r [mm] The problem of fttng non-lner geometres has been addressed. Ths problem s crtcal n dmensonal metrology to assure the qualty of manufacturng products. The geometres consdered are crcle, sphere and cylnder due to ther varous and common use n applcatons such as metrologcal calbraton and mechancal assembly. The ncreasng capablty of modern metrology nstruments n samplng hgh densty clouds of ponts n short tme demands accurate and fast fttng procedures. Both cases of fttng full- and half-geometres are presented. From the fttng of full-geometres, results show that the use of chaos optmzaton to mprove the ntal guess for LM non-lnear least square fttng has sgnfcantly mproved the accuracy of the fttng and kept the computatonal tme small. A slower fttng s observed n the case of halfgeometres. Even though slower, Chaos-LM method can gve the expected results by escapng from local optma. LM method s trapped and early converged n the case of half-geometres fttng, so that no mprovement of the result can be obtaned by ncreasng the number of teratons. Convergent rate effcency of the proposed method s sgnfcantly hgher n the case of ncomplete ponts cloud. It seems that the LM method could have hgher probablty to be trapped n the local optma n ths case. Two real case studes are presented. The case studes are to ft a calbrated sphere from pont clouds representng half of the sphere geometry and to ft an ndustral-made cylnder. The Chaos-LM method gves expected results. Fnally, a note should be hghlghted that a flterng procedure of the pont cloud may be needed before the fttng process s carred out, snce least-square fttng procedure s not completely robust to the outler ponts. The future drecton of ths work s to dentfy the lnk between the non-lnear problem and chaos property such that an adaptve regon boundng and chaotc moton generaton can be determned precsely.

14 Measurement Scence and Technology 14 Acknowledgements Fnancal support to ths work has been provded as part of the project REMS - Rete Lombarda d Eccellenza per la Meccanca Strumentale e Laboratoro Esteso, funded by Lombardy Regon (Italy), CUP: D81J1225. References [1] Loch C H, Van der Heyden L, Van Wassenhove L N, Huchzermeer A, Escalle C, 1999, Industral excellence: management qualty n manufacturng, Sprnger: London. [2] Kunzmann H, Pfefer T, Schmtt R, Schwenke H, Weckenmann A 25 Productve Metrology-Addng Value to Manufacture CIRP Ann. Manuf. Techn. 54(2) [3] Perera P H 212 Cartesan Coordnate Measurng Machnes, n Hocken R. J., Perera, P. H (Eds), Coordnate Measurng Machnes and Systems 2 nd ed, (Florda:CRC Press) [4] Morse E 212 Operatng a Coordnate Measurng Machne, n Hocken R. J., Perera, P. H (Eds), Coordnate Measurng Machnes and Systems 2 nd ed, (Florda:CRC Press) [5] Wlhelm R G, Hocken R, Schwenke H 21 Task Specfc Uncertanty n Coordnate Measurement CIRP Ann. Manuf. Techn. 5(2) [6] Shakarj C M 1998 Least-Squares Fttng Algorthms of the NIST Algorthm Testng System, Journal of Research NIST [7] Hoppe H, DeRose T, Duchamp T, McDonald J, Stuetzle W 1992 Surface reconstructon from unorganzed ponts ACM 26(2) [8] Moron G, Petrò S, Tolo T 211 Early cost estmaton for tolerance verfcaton CIRP Ann. Manuf. Techn. 6(1) [9] ISO 136-4:21 21 Acceptance and re-verfcaton test for coordnate measurng machnes: CMM used n scannng measurng mode. [1] ISO 136-5:21 21 Acceptance and re-verfcaton test for coordnate measurng machnes: CMM usng sngle and multple styluses contactng probng system. [11] Whtney D E 24 Mechancal Assembles: Ther Desgn, Manufacture, and Role n Product Development (USA: Oxford Unversty Press). [12] Marquardt D W 1963 An Algorthm for Least-Squares Estmaton of Non-lnear Parameters, J. Soc. Ind. Appl. Math. 11(2) [13] Nash J C 1979 Compact Numercal Methods for Computers Lnear Algebra and Functon Mnmzaton (Brstol: Adam Hgler Ltd).

15 Measurement Scence and Technology 15 [14] Rardn R L 26 Optmzaton n Operaton Research (New York: Addson-Wesley). [15] Syam W P and Al-Harkan I M 21 Comparson of Three Meta Heurstcs to Optmze Hybrd Flow Shop Schedulng Problem wth Parallel Machnes, World Academy of Scence, Engneerng and Technology [16] Luo Y Z, Tang G J, Zhou N L 28 Hybrd approach for solvng systems of nonlnear equatons usng chaos optmzaton and quas-newton method Appled Soft Computng 8(2) [17] Tavazoe M S and Haer M 27 An optmzaton algorthm based on chaotc behavor and fractal nature Comput. Appl. Math. 26(2) [18] Yang D, L G, Cheng G 29 On the effcency of chaos optmzaton algorthms for global optmzaton Chaos, Soltons & Fractals 34(4) [19] Kruth J P, Van Gestel N, Bleys P, Welkenhuyzen F. 29 Uncertanty determnaton for CMMs by Monte Carlo smulaton ntegratng feature form devatons CIRP Ann. Manuf. Techn. 58(1) [2] Jang B L W 1998 Optmzng complex functons by chaos search. Cybern. Syst. 29(4) [21] Trafals T B and Kasap S 22 A novel metaheurstc approach for contnuous global optmzaton J. Global. Optm [22] Kawalec A and Magdzak M 212 Usablty assessment of selected method of optmzaton for some measurement task n coordnate measurement technque Measurement

16 Measurement Scence and Technology 16 Fgure 1: Crtcal step of geometrcal fttng process n metrology. Fgure 2: (a) Defnton of pont dstance for crcle (sphere) and cylnder, (b) Defnton of pont dstance for 3D-lne.

17 Measurement Scence and Technology 17 Fgure 3: Illustraton of Schwefel and Square of crcle dstance multmodal functon. Fgure 4: Dfferent ntal solutons affect the fnal soluton. (a) Intal guess s far from optmal, (b) ntal guess s near optmal.

18 Measurement Scence and Technology 18 Fgure 5: Logstc map. (a) tme-seres plot of logstc map, (b) pared-plot between two consecutve chaos varables. Fgure 6: Illustraton of data generated for crcle, sphere, and cylnder.

19 Measurement Scence and Technology 19 Fgure 7: Substtute full crcle fttng results of (a) LM method and (b) Chaos-LM. Fgure 8: Substtute full sphere fttng results of (a) LM method and (b) Chaos-LM.

20 Measurement Scence and Technology 2 Fgure 9: Substtute full cylnder fttng results of (a) LM method and (b) Chaos-LM.

21 Measurement Scence and Technology 21 Fgure 1: Norm of resdual of LM method and Chaos-LM method for full pont cloud geometry fttng. Fgure 11: CPU tme of LM method and Chaos-LM method for full pont cloud geometry fttng.

22 Measurement Scence and Technology 22 Fgure 12: Substtute Half crcle pont cloud fttng results of (a) LM method and (b) Chaos-LM. Fgure 13: Substtute Half sphere pont cloud fttng results of (a) LM method and (b) Chaos-LM.

23 Measurement Scence and Technology 23 Fgure 14: Substtute Half cylnder pont cloud fttng results of (a) LM method and (b) Chaos- LM.

24 Measurement Scence and Technology 24 Fgure 15: Norm of resdual of LM method and Chaos-LM method for Half pont cloud geometry fttng.

25 Measurement Scence and Technology 25 Fgure 16: CPU tme of LM method and Chaos-LM method for Half pont cloud geometry fttng.

26 Measurement Scence and Technology 26 Fgure 17: Convergent rate for fttng full crcle pont cloud. Fgure 18: Convergent rate for fttng full sphere pont cloud.

27 Measurement Scence and Technology 27 Fgure 19: Convergent rate for fttng full cylnder pont cloud. Fgure 2: Convergent rate for fttng half crcle pont cloud.

28 Measurement Scence and Technology 28 Fgure 21: Convergent rate for fttng half sphere pont cloud. Fgure 22: Convergent rate for fttng half cylnder pont cloud.

29 Measurement Scence and Technology 29 Fgure 23: Measurement of calbrated ceramc sphere wth Brdge-type CMM. Fgure 24: Obtaned pont cloud. (a) Low densty, (b) Hgh densty.

30 Measurement Scence and Technology 3 Fgure 25: Sphere fttng of (a) low densty, (b) hgh densty. Fgure 26: (a) Measurement of ndustral cylnder, (b) The fttng results.