Torusity Tolerance Verification using Swarm Intelligence

Transcription

1 IEMS Vol. 6, No., pp , December 7. Torusty Tolerance Verfcaton usng Swarm Intellgence Chakguy Prakasvudhsarn Industral Engneerng Program, Srndhorn Internatonal Insttute of Technology Thammasat Unversty, Pathumthan, 111, THAILAND Ext. 18, Emal: Swaporn Kunnapapdeelert Industral Engneerng Program, Srndhorn Internatonal Insttute of Technology Thammasat Unversty, Pathumthan, 111, THAILAND Ext. 18, Emal: Receved Date, October 6; Accepted Date, Aprl 7 Abstract. Measurement technology plays an mportant role n dscrete manufacturng ndustry. Probe-type coordnate measurng machnes (CMMs) are normally used to capture the geometry of part features. The measured ponts are then ft to verfy a specfed geometry by usng the least squares method (LSQ). However, t occasonally overestmates the tolerance zone, whch leads to the rejecton of some good parts. To overcome ths drawback, mnmum zone approaches defned by the ANSI Y14.5M-1994 standard have been extensvely pursued for zone fttng n coordnate form lterature for such basc features as plane, crcle, cylnder and sphere. Meanwhle, complex features such as torus have been left to be dealt-wth by the use of profle tolerance defnton. Ths may be mpractcal when accuracy of the whole profle s desred. Hence, the true devaton model of torus s developed and then formulated as a mnmax problem. Next, a relatvely new and smple populaton based evolutonary approach, partcle swarm optmzaton (PSO), s appled by mtatng the socal behavor of anmals to fnd the mnmum tolerance zone torusty. Smulated data wth specfed torusty zones are used to valdate the devaton model. The torusty results are n close agreement wth the actual torusty zones and also confrm the effectveness of the proposed PSO when compared to those of the LSQ. Keywords: Torusty, Tolerance Verfcaton, Coordnate Metrology, Mnmum Zone Estmaton, Partcle Swarm Optmzaton. 1. INTRODUCTION Coordnate measurng machnes (CMMs) are an extremely powerful metrologcal nstrument. Coupled wth the ad of computer and CMM software, they can automatcally perform complex analyss to verfy manufactured parts conformance to sze and geometrc tolerances such as form, orentaton, and runout. The most wdely used technque for form tolerances analyses n practce s the least squares method (LSQ) due to ts smplcty and robustness (Traband et al., 1989; Shunmugam, 1987; Prakasvudhsarn et al., 3). Also, t can be appled to most geometres qute easly provded that ther respectve dscrepancy models are known. However, the results obtaned by the LSQ do not guarantee the mnmum zones as specfed by the ANSI standard (ASME Y14.5M-1994, 1995). It occasonally overestmates the tolerance zone. Consequently, ths leads to the economcal dsadvantages of rejectng or reworkng some good parts. Therefore, mnmum zone approach has been pursued for zone fttng nstead. Although all basc form tolerances such as straghtness, flatness, crcularty, cylndrcty, and sphercty have been defned and nvestgated based on mnmum zone approach n coordnate form lterature, nterestngly the form tolerances for torus and other complex shapes have been largely gnored due to the lack of ther geometrcal devaton models. They are normally left to be dealt-wth by the use of profle tolerance defnton. Profles such as straght lnes, arcs, and other curved lnes may be appled and the tolerance estmatons of these elements are verfed ndvdually. Such a procedure may be mpractcal n case where accuracy of the : Correspondng Author

2 Torusty Tolerance Verfcaton usng Swarm Intellgence 95 entre profle s a requrement. Ths mples that the common practce, the use of profle, for torus s far from optmal n spte of the suffcent need to nspect them n parts such as outer and nner races n bearngs and torodal contnuous varable transmsson. Hence, to nspect doughnut-shaped feature, a set of mathematcal models for torusty error verfcaton consstng of the true devaton model and ts nonlnear optmzaton counterpart s clearly desred. The true nonlnear devaton model of torus-shaped must frst be developed and then used to establsh the deal feature of torus from actual measurements. The man dea of ths step s smlar to the procedure to obtan the dscrepancy model n the orthogonal leastsquares regresson. Based on mnmum tolerance zone approach, ths deal torus s determned to splt the measured data nto two parts equdstantly, nsde and outsde the assessment torus. In other words, t s utlzed to set up two magnary tor, the outer and the nner tor respect to the deal, to form a tolerance zone torusty. Ths s where the mnmax crteron comes nto play. That s, the chosen deal torus mnmzes the maxmum dstance that an ndvdual pont falls from the deal. Therefore, the normal dscrepancy model and the mnmax crteron are combned to verfy form tolerance of the torodal object. Besdes the need for the set of mathematcal models, an effectve and effcent optmzaton technque should be taken nto consderaton to solve the formulated mo-del for mnmum zone torusty estmaton. A relatvely new algorthm, the partcle swarm optmzaton (PSO), has been ntroduced n the framework of an artfcal socal model. It s a populaton based stochastc optmzaton method that demonstrates appealng propertes such as smplcty, short computer code, fast convergence, consstency results, robustness, and no requrement for gradent nformaton (Kennedy and Eberhart, 1995). Recently, the PSO has been successfully appled to solve a wde range of applcatons (Allahverd and Al-Anz, 6; Lu et al., 6; Lawtrakul and Prakasvudhsarn, 5; Chuanwen and Bompard, 5; Elbeltag et al., 5). However, ts applcaton to form errors evaluaton has not yet been fully realzed. Hence, the PSO s selected to verfy the form conformance of the manufactured torodal object by optmzng the developed mnmax functon. Therefore, to effectvely and effcently nspect doughnut-shaped features, ths research attempts to 1) determne the frst true nonlnear torusty model through an ntegratve nvestgaton of torusty defnton, ts devaton model, and ts optmzaton formulaton and ) nvestgate an applcaton of the PSO for torusty form error evaluaton.. LITERATURE REVIEW Varous technques have been dscussed for varous form tolerances evaluaton based on the mnmum zone concept. They can be roughly classfed nto two categores, computatonal geometry approach and numercal approach. The former approach deals wth algorthms and data structures (Traband et al., 1989; Hong et al., 1991; Le and Lee, 1991; Roy and Zhang, 199, 1994; Roy, 1995). The nformaton of the problems s organzed n such a way that would permt the algorthms to run n the most effectve manner snce they explot the problems structures. However, each algorthm s lmted to a partcular form tolerance and dffcultly expanded to cover other forms, especally when they possess totally dfferent geometrcal characterstcs such as lnearty nature of lne or plane and nonlnearty nature of crcle or cylnder. Some computatonal geometry based methods are convex hull, egenpolygon, and Vorono dagram. The latter approach conssts of usng lnear and nonlnear optmzaton methods such as smplex search, genetc algorthms (GAs), and smulated annealng (SA) wth the correspondng devaton model of each form feature (Shunmugam, 1987; Dhansh and Shunmugam, 1991; Wang, 199; Kanada and Suzuk, 1993; Carr and Ferrera, 1995a, 1995b; La et al., ; Sharma et al., ; Lu et al., 1; Hong et al., 1; Wen and Song, 4). That s, the formulated optmzaton model remans the same for every form feature snce t s based on the same crteron for tolerance zone evaluaton. For example, the mnmax crteron s one of a few varants appled to every basc form wth the respectve dscrepancy model. Hence, ths approach s qute flexble because t can smply be extended to cover varous form tolerances f those forms devaton models are avalable. Its computatonal speed s rather fast even though t s not as computatonally effcent as that of the computatonal geometry based approach. Traband et al. (1989) proposed a methodology based on the convex hull prncple to evaluate form tolerances. Straghtness and flatness were evaluated by adoptng dmensonal (D) and 3 dmensonal (3D) convex hulls, respectvely. Hong et al. (1991) verfed mnmum zone straghtness by usng the concept of geometrcal egen-polygon (EPG). Le and Lee (1991) ntroduced a standard called the mnmum area dfference (MAD) center to measure the tolerance of crcular profle. The MAD center reled on the nformaton of the farthest and nearest neghbor Vorono dagrams and the convex hull concept. Roy and Zhang (199, 1994) proposed a hybrd assessment technque combnng convex hull and Vorono dagrams for determnng the roundness error. Roy (1995) addressed the concepts of the tolerance zones (TZs) and mnmum zones (MZs) for evaluatng form and postonal tolerances. Mnmum zone straghtness and roundness were calculated based on D convex hull whereas flatness and cylndrcty were estmated based on 3D convex hull. In the numercal approach, before the optmzaton model can be formulated, the relatonshp functon of relevant form parameters must geometrcally be deter-

3 96 Chakguy Prakasvudhsarn Swaporn Kunnapapdeelert mned. Then, varous technques can be used to search the formulated functon for optmal solutons of basc form features such as straghtness, flatness, crcularty, cylndrcty, and sphercty. Shunmugam (1987) compared the lnear and normal devatons of form tolerances usng the LSQ and mnmum devaton methods. A smplex search algorthm was used n a search procedure for both approaches. Dhansh and Shunmugam (1991) presented an algorthm based on the theory of dscrete and lnear Chebyshev approxmaton to evaluate the form errors. Wang (199) proposed a nonlnear optmzaton method for mnmum zone evaluaton of common form features. The form errors were conceptually determned by mnmzng the maxmum devaton of the sampled ponts, whch led to mnmax problems. They were reformulated nto nonlnearly constraned problems by ntroducng an addtonal varable. Some smple mechansms were suggested to mprove the stablty of the algorthm. Other nonlnear optmzaton technques such as downhll smplex and repettve bracketng methods were also used to evaluate flatness (Kanada and Suzuk, 1993). Carr and Ferrera (1995a) developed a sngle verfcaton methodology that can be appled to the cylndrcty and straghtness of medan lne problems by usng a successve lnear programmng, whch was an extenson of ther straghtness and flatness evaluatons (Carr and Ferrera, 1995b). The form tolerances for complex shapes such as cone and torus were also nvestgated. Prakasvudhsarn and Raman (4) developed the lnear and nonlnear devaton models for concty and used the LSQ and the generalzed reduced gradent algorthm to fnd the concty zones. Subsequently, the lnear devaton models of torus were proposed and ftted by usng the LSQ for torusty evaluaton (Agurre-Cruz and Raman, 5). Some heurstc technques such as GA and SA have also been appled to verfy form tolerances wth good results. GAs were chosen to evaluate the cylndrcty (La et al., ) and also other basc form tolerances (Sharma et al., ). Lu et al. (1) proposed a hybrd approach between GA and geometrc characterzaton method for assessng tolerance specfcatons of straghtness and flatness. Another hybrdzaton of computatonal geometry and SA was used to evaluate straghtness and flatness (Hong et al., 1). Prakasvudhsarn et al. (3) adopted an approach based on the support vector machne algorthm, ν support vector regresson (ν-svr), to estmate the mnmum enclosng zones straghtness and flatness. The presented algorthm attempted to mnmze the ε-nsenstve tube whch was modeled as the tolerance zone of the nspected lnear form features. An algorthm nspred by the mmune system and the evolutonary bology was proposed to evaluate sphercty error (Wen and Song, 4). Another evolutonary method, the PSO, has ganed more nterest due to ts appealng performance n many applcatons n term of convergence rate, consstency results, and dfferentable requrement of the evaluaton functon. The PSO has been ntroduced n the framework of an artfcal socal model to evaluate contnuous nonlnear functons (Eberhart and Kennedy, 1995). It s based on a very smple concept of brd flockng, fsh schoolng, and swarmng theory, exhbtng some popula-ton based stochastc evolutonary computaton. Each partcle fles over the soluton space and adjusts ts trajectory toward ts current velocty, own experence, and swarm s experence. To avod local optma, randomness s also ncorporated nto the computaton of a new velocty. Furthermore, ths phlosophcal framework can be mplemented n a few lnes of computer code. The PSO has been appled to solve a wde range of applcatons. It was used to evolve artfcal neural networks for human tremor analyss (Eberhart and Hu, 1999). It was expanded to handle a mxed-nteger nonlnear optmzaton problem for reactve power and voltage control consderng voltage securty assessment (Yoshda et al., 1999). Furthermore, the PSO was appled to solve a set of wellknown test mnmax problems wth promsng results (Laskar et al., ). To assure the PSO s performance, fve evolutonary-based optmzaton algorthms such as genetc algorthm (GA), memetc algorthms (MA), partcle swarm (PSO), ant-colony optmzaton (ACO), and shuffled frog leapng (SFL) were tested wth both contnuous and dscrete optmzaton problems and then compared n terms of processng tme, success rate, and soluton qualty. The results showed that the PSO algorthm was the second best n terms of processng tme whle performed the best n terms of success rate and qualty of solutons (Elbeltag et al., 5). In summary, the nspecton of torodal object has rarely truly been nvestgated due to ts complexty and hence needs to be tackled to mprove ts effectveness and effcency. Therefore, the purpose of ths study s to develop the currently nonexstent set of true nonlnear mathematcal model for the doughnut-shaped fttng wth mnmax crteron and to apply the promsng algorthm lke the PSO to torusty determnaton. 3. TORUS FORM TOLERANCE DEFINIT- ION AND DEVIATION Tolerances are the total amount from whch a specfed dmenson s permtted to vary. Ths concept s appled not only to sze tolerance but also to geometrc tolerances such as locaton, orentaton, runout, and form. Form tolerances are most frequently appled to sngle feature or porton of a feature. To evaluate form feature, an deal feature s establshed from the actual measurements whle smultaneously constructng a mnmum tolerance zone wthn whch all measurement values must le. The obtaned zone or the devatons of the feature from the deal must be wthn the specfed tolerance. In other words, form tolerances state how far the actual features are permtted to vary from the desgned nomnal form. Common types of form tolerances such as straghtness, flatness, crcularty, sphercty, and

4 Torusty Tolerance Verfcaton usng Swarm Intellgence 97 cylndrcty are defned by the ANSI standard based on the mnmum zone concept. For nstance, cylndrcty s the entre feature surface durng one revoluton n whch all ponts are on equal dstance from a common axs as shown n Fgure 1. Ths secton only presents the dervaton process of the torusty devaton model. Its mnmum tolerance zone formulaton s dscussed n the subsequent secton. A torus s formed by rotatng a crcle, mnor crcle, about a lne that s n the plane of the crcle, but not ntersectng the crcle as llustrated n Fgure. All center ponts of all revolved crcles form a common core crcle, major crcle, of torus. Torusty s then defned as the entre feature surface durng one revoluton n whch all ponts are on equal dstance from the center ponts on the major crcle. Thus, torusty can be determned by calculatng the normal dstances between measurement ponts and surface of the assessment torus as llustrated n Fgure to Fgure 5. In Fgure 3, gven that L = ( x x, y y, z z ) s the vector between pont Ox (, y, z ) and a measurement pont r, hence, THIS ON THE DRAWING MEANS THIS.5 wde tolerance zone Fgure 1. Specfyng cylndrcty of surface elements (ASME Y14.5M-1994, 1995) Y Z Y Major Radus Major Crcle X Mnor Radus Mnor Crcle RIGHT VIEW Major Crcle c r X TOP VIEW Z X FRONT VIEW Fgure. Torus defnton

5 98 Chakguy Prakasvudhsarn Swaporn Kunnapapdeelert Z ( u, v w ) p =, Y l ( x, y z ) r, O θ ( x, y z ), L X Fgure 3. Assessment of torusty error showng calculaton of the dstance l Z l w ( x, y z ) r, c h a X Fgure 4. Assessment of torusty error showng mnor radus calculaton r ( x, y, z ) d Fgure 5. The orthogonal dstance from a measurement value r( x, y, z ) to the surface of the assessment torus

6 Torusty Tolerance Verfcaton usng Swarm Intellgence 99 l ( ) L p = L and p [( ) ( ) ( ) ] l = x x + y y + z z 1 u + v + w l = u + v + l = 1 [( x x ) u+ ( y y ) v+ ( z z ) w] [ ( x ) ( ) ] x v y y u + [( x ) ( ) ] x w z z u w + [( y y ) w ( z z ) v] (1) () [ ( x x ) v ( y y ) u] + [( x x ) w ( z z ) u] + [( y y ) w ( z z ) v] (3) u + v + w where r s the coordnates of a sampled pont; O( x, y, z ) s the orgn of a local frame (about the center of torus); ( uvw,, ) s a normalzed drecton vector of a torus and ts axes. In Fgure 4, let a represent the normal dstance from a measurement pont to a center pont of a mnor crcle (torus tube); c be the dstance between center ponts of major and mnor crcles; and h or ( z z ) be the lnear dstance along z-axs between a measured pont and xy -plane of torus base. Thus, a = w + h and w = l c. Consequently, a = ( l c) + h and substtute l from Equaton (3), then [( x x ) v ( y y ) u] + [( x x ) w ( z z ) u] + [( y y ) w ( z z ) v] a = c u + v + w + ( z z ) (4) Therefore, an error whch s a normal dstance from a measurement pont r ( x, y, z ) to the surface of the ftted deal torus s d = a r as llustrated n Fgure to Fgure 5. Thus, (5) where r s the mnor radus of the mnor crcle of the deal torus. Wthout loss of generalty, z and w are smpl- fed by equatng to and 1, respectvely. Then, the normal dstance d becomes + + [( x x ) v ( y y ) u] ( x x zu) ( y y z v) c d = u + v + 1 r + z (6) The deal torus can be establshed by determnng values of these relevant parameters, x, y, u, v, c, and r, of the derved devaton model above by usng a crteron for best ft. In ths case, the mnmax crteron was selected snce t conformed to mnmum tolerance zone approach set by the ANSI standard (ASME Y14.5M- 1994, 1995). Ths s a major dfference why the LSQ whch s based on a dfferent crteron, mnmzaton of the sum of the squares of the errors, tends to overestmate the computed tolerance zone. 4. FORM TOLERANCES FORMULATION A form tolerance can be establshed by searchng for the deal form of the nspected feature from collected data ponts whle smultaneously mnmzng the maxmum devaton between measurement values and the ftted form (Wang, 199). As a result, the upper and lower lmts that contan all measured ponts would provde a geometrcal tolerance zone. To follow ths crteron, a mnmax problem can be formulated. For nstance, the bass for mnmum zone straghtness s descrbed as follows: mnmum zone = * mn (max d ) or h = mn (max d ) (7) h h.... r d.. Fgure 6. Mnmum zone straghtness.. Ideal feature d = [( x x) v ( y y) u] [( x x) w ( z z) u] + [( y y) w ( z z) v] + c u + v + w + ( z z ) r where d s a normal dstance from a measurement pont to the deal feature. Fgure 6 llustrates such zone; where dots represent measured data ponts or r, h s a half wdth of zone, and d s the devaton of pont from the deal feature. The mnmax formulaton would result n an deal feature havng equ-dstance to the farthest data ponts on both sdes of ths deal. The mnmum zone obtaned s then compared wth the specfed tolerance lmt for conformance. The above formulaton model can also be appled to cover other form features by usng a proper devaton model for each partcular form.

7 1 Chakguy Prakasvudhsarn Swaporn Kunnapapdeelert In other words, the dscrepancy model derved for torodal object should be combned wth ths mnmax crteron and Equaton (6) was then used as an error model, d, n Equaton (7) for torusty verfcaton. Normally, sequental quadratc programmng s a common gradent-based approach for solvng mnmax problems. The qualty of the obtaned soluton s very much dependent upon the ntal soluton and contnuty of the objectve functon (Laskar et al., ). Further, the dervatve nformaton of the objectve functon s requred analytcally or approxmately. Consequently, the gradent-based methods may experence some dffcultes n tryng to reach acceptable zone solutons. An algorthm such as PSO that does not requre the gradent nformaton could allevate these shortcomngs encountered by gradent-based methods. It was also reported to tackle mnmax problems effectvely on a set of wellknown test functons (Laskar et al., ). 5. PARTICLE SWARM OPTIMIZATION (PSO) A relatvely new evolutonary computatonal technque called partcle swarm optmzaton (PSO) was ntroduced by Kennedy and Eberhart (1995). It s nspred by the socal behavor of anmals such as brd flockng n searchng for food. Each partcle fles n hyperspace searchng for the best soluton by adjustng poston and velocty based on ts own flyng experence (pbest) and ts companons experence (gbest). The nerta weght w was later ntroduced to reportedly mprove the PSO optmzer. The PSO has been appled to many applcatons such as optmzaton problems, neural network tranng, travelng salesman problems, and job schedulng. It s very attractve because requrement of gradent nformaton s not needed. Hence, t s unaffected by dscontnutes of the objectve functon. The equatons used consst of flexble and well-balanced mechansms to enhance the global and local exploraton abltes. These allow a thorough search and smultaneously avod the premature convergence. In addton, PSO uses probablstc rules for partcle s movements. Therefore, t s qute robust to local optma. Plus, t can be mplemented easly wth a few lnes of computer code. The steps of the PSO are llustrated below (Eberhart and Sh, 1): 1. Intalze a populaton of partcles wth random postons and veloctes on D dmensons.. Evaluate the desred optmzaton functon n D varables for each partcle. 3. Compare evaluaton wth partcle s prevous best value, pbest[]. If current value s better than pbest[], then pbest[] = current value and pbest locaton, pbestx[][d], s set to the current locaton n d- dmensonal space. 4. Compare evaluaton wth the swarm s prevous best value, (pbest[gbest]). If current value s better than (pbest[gbest]), then gbest = current partcle s array ndex. 5. Change velocty and poston of the partcle accordng to the followng equatons, respectvely: v[][d] = w*v[][d] + c 1 *rand()*(pbestx[][d] presentx[][d]) + c *rand()*(pbestx[gbest][d] presentx[][d]) (8) presentx[][d] = presentx[][d] + v[][d] (9) 6. Loop to step untl a stoppng crteron, a suffceently good evaluaton functon value or a maxmum number of teratons, s met. From Equaton (8) and Equaton (9), v[][d] s a velocty of the th partcle n the d th dmenson; w s nerta weght, pbestx[][d] and pbestx[gbest][d] represent the best prevous poston (the poston gvng the partcle s best ftness value) of the th partcle n the d th dmenson and the best prevous poston (the poston gvng the swarm s best ftness value) of the gbest th partcle n the d th dmenson, respectvely. The current locaton of the th partcle n the d th dmenson s represented as presentx[][d]. As shown n Equaton (8), v[][d] conssts of three terms. The frst term s the momentum of the partcle. It s computed by multplyng the nerta weght wth partcle s prevous velocty. w s a control parameter, whch s used to nfluence the current velocty from prevous velocty. The larger weght mples a global exploraton because the partcle can fly n large area for fndng a good regon. On the contrary, the smaller weght results n refnng the search wthn t. The sutable selecton of the nerta weght should provde the balance between global and local search area. Therefore, the nerta weght should be ntalzed to a large value and then gradually decreased toward the end of the search process (Eberhart and Sh, 1998). More detal of nerta weght settng s dscussed n the next secton. The second term s the cognton part because the partcle consults wth ts own best experence, pbestx[][d] presentx[][d]. The thrd term s the socal part snce each partcle consders the shared swarm s best experence, pbestx[gbest][d] presentx[][d]. c 1 and c are the postve constants called acceleraton coeffcents. They ndcate how much the partcle trusts ts own and companons experences. The hgher the constant s, the greater the acceleraton of the partcle wll be. In general, to balance the mpact of the cognton and the socal parts, these two parameters are set to two to gve t a mean of one (Eberhart and Sh, 1998). rand() s a unformly random number generator wthn the (,1) range. These terms, ther parameters, as well as the sharng nformaton mechansm make the PSO less predctable and more flexble to avod local optma, to mprove convergence rate, and to gve consstency results. The system tackles the objectve functon drectly wthout the need for ts gradents. Ths makes t more practcal

8 Torusty Tolerance Verfcaton usng Swarm Intellgence 11 when complex functon that s dffcult to obtan gradents s encountered. Moreover, the PSO s mplctly bult for speed snce only prmtve mathematcal operators are computed. Hence, the PSO s very attractve especally for tolerance zone estmaton that must deal wth complex functon, d, and mnmax crteron. 6. DATA COLLECTION The smulated data were used to verfy the tolerance zone torusty so that the developed devaton model could be valdated and the effectveness of the PSO and the LSQ could be tested wthout concerns of measurement errors such as probe orentaton, probe angle adjustment, and probe compensaton. Table 1 depcts the detals of the fve perfect tor smulated. They assumably represented the preset nomnal values of tor as specfed on engneerng drawng. An arbtrarly specfed error zone was then added to each generated perfect torus. In dong so, two magnary tor were mplemented to cover the perfect torus wth the gap of the specfed error. One of those two tor would le outsde and the other one would le nsde. The measurements were taken by selectng some data ponts from the outer tor and some from the nner one. Together, these taken samples would represent a real manufactured torodal object havng up and down surface around the perfect torus wth theoretcal nomnal values. Table 1. Detals of fve perfect tor tested and the selected actual zones Data set x y u v c r Actual zone A wdely used samplng method n practce, a unform samplng, was appled to collect 64 data ponts from each and every smulated torus. The assumpton that these ponts accurately represented the part surface was held by sectonng each torus normal to ts crcular core for 16 parttons and takng 4 ponts from each secton. Out of these four ponts, two ponts were taken from the surface of the outer torus and the other ponts were chosen from the surface of the nner one. Altogether there would be a total of 64 sampled ponts for each torus. Ths controlled data collecton procedure should ease the computaton of torusty fttng so that the specfed actual zone could be reached. 7. RESULTS AND ANALYSES To establsh deal torus from all measurement values (data ponts collected), four torusty-controlled factors were taken nto consderaton for determnaton of the normal dstances between measurement ponts and surface of the deal torus as llustrated n Fgure to Fgure 5. The frst one was the absolute dstance between major rad of both actual and deal surfaces. The second was the absolute dstance between mnor rad of both surfaces. The thrd was the dstance between the center ponts of both tor. Fnally, the last factor was the dfferent angle between the drecton vectors of both tor depcted as puvw (,, ) n Fgure 3. Together, these factors contrbuted to the dscrepancy model obtaned. Hence, Equaton (6) was resulted and used as an error model n the mnmax problem formulated. The PSO was then appled to solve the obtaned formulaton (Equaton (7) wth d from Equaton (6)) for mnmum tolerance zone torusty. It teratvely attempted to mnmze the desred functon that was the maxmum devaton between data ponts and the searched deal feature. In each teraton, every partcle would search for the values of those sx relevant parameters, x, y, u, v, c, and r, that would contrbute to the devaton of each data pont from the magnary deal torus. The maxmum devaton would then be the evaluaton result n Step of the PSO algorthm and was mnmzed by the PSO. The computaton of the PSO normally depends on populaton sze, nerta weght, maxmum velocty, maxmum and mnmum postons and maxmum number of teraton. The ntal populaton sze was chosen such that t was large enough to cover the search space wthn the teraton lmt based on the tral runs and lterature. The populaton sze of twenty was then selected n ths work. Inerta weght started from.9 and gradually decreased to.4 to balance the global and local exploraton based on a lnear functon of tme (teraton). Ths also contrbuted to mprove convergence rate (Kennedy, 1997). Partcles veloctes on each dmenson were clamped to a maxmum velocty, v max, to control the exploraton ablty of partcles. If v max s too hgh, the PSO facltates global search, and partcles may fly pass good solutons. However, f v max s too small, the PSO facltates local search, and partcles may not explore beyond locally good regons (Kennedy, 1997). Thus, f v[][d] s greater than v max, then v[][d] s equated to v max. Smlarly, f v[][d] s less than -v max, then v[][d] s equated to -v max. In ths study, v max was set at 1% of the dynamc range of the varable n each dmenson. In case of maxmum and mnmum postons of the varables n each dmenson, they were chosen to represent the sutable search space, whch was problem dependence. The selecton of these values could be justfed by consderng the nspected part s relevant specfcaton, x, y, u, v, c, and

9 1 Chakguy Prakasvudhsarn Swaporn Kunnapapdeelert r. The real manufactured torus would vary these varables to some extent. Therefore, the settngs of max- mum and mnmum postons of those varables should not devate from the preset nomnal values (on engneerng drawng) too much n specfyng the boundary of the search space because torusty verfcaton was a geometrcal nspecton of the manufactured part that was always made not very far from the nomnal values. Thus, ths ratonale was realzed and could save some computatonal tme by the PSO. To llustrate the selecton of these settngs, Dataset 5 was used as an example. Its preset nomnal values of x s, y s, u s and v s were all zeroes; c s and r s were set to 33 and 7, respectvely. The maxmum and mnmum postons of x s, y s, u s and v s for all partcles were then specfed as (.1, -.1) whereas those of c s and r s were (33.5, 3.5) and (7., 6.8), respectvely. Ths procedure was also repeated wth all other datasets. They should ensure that the search spaces were never volated and the solutons obtaned were always vald. The last parameter was the maxmum teraton number, whch was set at 3 based on tral runs. Fgure 7 llustrates that the PSO converged very fast. The zone found decreased extremely quckly about 6-7 teratons. Afterward, the curve appears almost flat. Ths mples that the near-optmal zone solutons were reached quckly. The graph of torusty zone obtaned remans flat after teratons. Thus, the maxmum teraton number of 3 was suffcent. Note that the same parameters settngs of the PSO were used for all datasets except maxmum and mnmum postons that were changed to reflect varous dmensons of tor tested. The PSO based torusty fttng algorthm and the LSQ were mplemented n MATLAB 6.5. Every numercal computaton was performed on a PC wth a Pentum IV.4 GHz. The PSO and the LSQ were both tested wth fve sets of smulated data and ther results Table. Torusty tolerance zone obtaned by the PSO and the LSQ Dataset x y u v c r Actual zone PSO LSQ are tabulated n Table. Clearly, the results obtaned by the PSO were equal to the specfed actual zones whereas those obtaned by the LSQ ndcated overestmatons. The optmal tolerance zones were obtaned for every dataset by the PSO based geometrcal fttng algorthm. Ths shows that the proposed devaton model was very effectve. It dd not overestmate or underestmate the tolerance zones wth every tested dataset. The PSO also performed very well. It could fnd the mnmum zone torusty from the developed devaton model wth the mnmax crteron. Under the controlled envronment, the outcomes should valdate and verfy the devaton model presented and also demonstrated that the PSO was very attractve for torusty verfcaton. The LSQ method s generally used to fnd the trend of data under normal dstrbuton condton. Ths assumpton requrement s qute cumbersome to verfy and may not hold n many problems. Moreover, even though the dscrepancy functon of the LSQ was qute senstve to outlers, t stll could not guarantee f the tolerance zone obtaned was, n fact, mnmum. Ths was obvously the case n ths study as well. Consequently, some good parts conformng to the specfcaton would be rejected or reworked. Generally, the gradent-based algorthms are most sutable for a problem wth a smooth objectve functon (contnuous frst and second dervatves). However, many mnmax ft models do not have smooth objectve functons. They may suffer from numercal nstabltes wth respect to convergence because the frst or second dervatves of the objectve functons are not contnuous. The PSO could overcome ths ssue fundamentally. Furthermore, the global mechansm n PSO and global strategy used (makng several runs from several ntal solutons to avod local optma traps) should help mprove the solutons obtaned. In addton, specfc requrements on the mode of data collecton for any of these forms were not needed. That s, the locaton of the measurements could be anywhere on the surface. Ths mples that the data ponts were not necessarly collected at sectons perpendcular to the crcular core of a torus. They could be spread around coverng the entre surface of nspected torus so that they would assumedly represent ths part. The form tolerances for complex shapes lke ths are typcally left to be dealt-wth by the use of profle tolerance defnton. Ths really s the soluton of two D problems rather than the 3D soluton. Ths procedure results n sgnfcant nconsstences and may be mpractcal n cases where accuracy of the whole profle s a requrement. The PSO theoretcally may requre qute a computatonal tme when the sze of dataset s qute large (n thousands or more). However, n dscrete measurement where dataset s usually n tens or hundreds, the tme taken does not have much effect. Furthermore, the PSO requres only prmtve mathematcal operators and uses memory array to handle varables and solutons. These make t very fast to determne the near-optmal torusty

10 Torusty Tolerance Verfcaton usng Swarm Intellgence 13 Table 3. Average computatonal tme n seconds of both methods for ten runs Dataset PSO s tme LSQ s tme tolerance zone. The average processng tme of both methods for ten runs s depcted n Table 3. The tme taken by the PSO was about.5 second to reach the preset maxmum teraton of 3 and the LSQ s computatonal tme was around.8 second. Clearly, the LSQ was more effcent snce theoretcally t was an analytcal method that requred only substtuton of relevant values for the respectve varables nto ts closed form soluton. Even though the PSO s executon tme was longer, t was stll consdered very fast, about a half of a second for computng 64 collected data ponts. In fact, ths tme could be further saved by decreasng the preset maxmum teraton number snce the PSO could converge very rapdly as shown n Fgure 7. Fgure 7. Convergence of PSO for Dataset 5 wth the closer looks of decreasng torusty from -1 and from -3 teratons Therefore, the PSO based zone estmaton could effectvely and effcently solve two-sded mnmax torusty fttng leadng to mnmum tolerance zone specfed by the ANSI standard (ASME Y14.5M-1994, 1995). Wth accuracy of torusty zones, ease of algorthm and programmng, and fast processng tme, the models and algorthm proposed were very attractve. 8. CONCLUSION Form tolerances nspecton plays an mportant role n ndustry. The complex forms such as torus are normally dealt-wth by the use of profles of ndvdual features whch may not be very accurate when combned. Hence, a new method for fndng the mnmum enclosng zone of torus was proposed n ths work. The presented method addressed the devaton model of torus and ts zone estmaton usng the PSO under the assumpton that the data ponts collected accurately represented the manufactured part surface. The true nonlnear devaton model of torusty was derved for the frst tme. The torusty zones obtaned clearly ndcate that the true nonlnear devaton model served ts purpose very well and t should be used, nstead of profles or approxmated lnear model, for torusty verfcaton wthout any specfc requrements durng data collecton process. The PSO was next appled for determnaton of mnmum zone torusty due to ts smplcty n concept and programmng, short computer code, and no requrement of gradent nformaton. Hence, t was unaffected by dscontnuty of objectve functons presented n the mnmax problem. It also requred only prmtve mathematcal operators. Coupled wth the PSO s global mechansm and global strategy used, the obtaned results showed that the PSO algorthm provded very good solutons, especally when compared wth those of the LSQ. Therefore, the PSO demonstrated much potental n fndng mnmum enclosng zone and consequently was very attractve for adopton n practce. The tme taken to collect data and nformaton obtaned from these data should have sgnfcant mpact for torus nspecton n terms of accuracy and cost. Ideally, ths verfcaton requres nformaton of entre feature. Normally, large sample sze s preferred but the nspecton tme would also ncrease. Therefore, effcent data collecton consstng of samplng strateges; sample sze and samplng locaton, should be nvestgated to mnmze the nspecton tme and hence cost whle mantanng the hgh level of accuracy of torusty nspecton. In addton, systematc parameters selecton of the PSO for form tolerance analyss wll certanly enhance ts ease of use and should be nvestgated further.

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