A Discrete Geometry Framework for Geometrical Product Specifications

Transcription

1 A Dscrete Geometry Framework for Geometrcal Product Specfcatons M. Zhang 1, N. Anwer 1, L. Matheu 1, H. B. Zhao, 1 LURPA, Ecole Normale Supereure de Cachan, 61, avenue du Presdent Wlson, Cachan, 9435, France PMA, Katholeke Unverstet Leuven, Celestjnenlaan 300b, Heverlee, 3001, Belgum {mn.zhang, anwer, matheu}@lurpa.ens-cachan.fr Habn.Zhao@mech.kuleuven.be Abstract GeoSpellng as the bass of the Geometrcal Product Specfcatons (GPS) standard [1] enables a comprehensve modelng framework and an unambguous language to descrbe geometrcal varatons durng the overall product lfe cycle. Ths s accomplshed by provdng a set of concepts and operatons based on the fundamental concept of the Skn Model. However, the defnton of GeoSpellng has not been successfully completed. Ths paper presents a novel approach for a formal descrpton of GeoSpellng concepts. In addton to mappng fundamental concepts and operatons to dscrete geometry objects, we nvestgate the use of Monte Carlo Smulaton technques for skn model smulaton when consderng geometrcal specfcatons. The results of the skn model smulatons and vsualzatons are shown and the performances of the descrbed smulaton methods are compared to each other. Keywords: Geometrcal Product Specfcatons, GeoSpellng, Dscrete geometry, Skn model, Mult-Gaussan dstrbuton, Gbbs samplng 1 INTRODUCTION The control of product geometrcal varatons durng the whole development process s an mportant ssue for cost reducton, qualty mprovement and company compettveness n the global manufacturng era []. Durng the desgn phase, geometrc functonal requrements and tolerances are derved from the desgn ntent. The modelng of product shapes and dmensons s now largely supported by geometrc modelng tools. However, permssble geometrcal varatons cannot be ntutvely assessed usng exstng modelng tools, and ths results n the specfcaton uncertanty. In addton, the manufacturng and measurement stages are the man geometrcal varatons generators accordng to the two followng axoms [3-4]: Axom of manufacturng mprecson: all manufacturng processes are nherently mprecse and produce parts that vary. Axom of measurement uncertanty: no measurement can be absolutely accurate and wth every measurement there s some uncertanty about the measured value or measured attrbute. To reduce the total uncertanty, the product geometrcal varatons should be consdered durng the whole product lfe cycle (fgure 1). In order to evaluate product geometrcal varatons and ensure that the fabrcated product can satsfy the functonal requrements, desgners should determne the lmted values that constran product geometrcal varatons. Ths process s now well known as specfcaton. In the context of Dgtal Mock-Ups (DMUs), the desgn process s supported by modelng, smulaton and vsualzaton tools such as CAD systems. The Dgtal Mock-Up, as a dgtal alternatve to constructng physcal parts, should be enrched by geometrcal varaton models to allow testng of desgn errors on assembles and realstc vsualzaton of the product. At the manufacturng level, multple representatons based on smooth surfaces and dscrete representatons (trangular meshes) are consdered. Moreover, an ordered or unordered set of ponts resultng from manufactured part acquston s processed for the purpose of product nspecton. Fgure 1: Geometrcal varatons durng the product lfe cycle. A comprehensve vew of Geometrcal Product Specfcatons should consder multple geometrc representatons, and as well as sutable processng technques and algorthms. The dscrete geometry theory can offer a great support n ths area, snce dscrete geometry s a mathematcal research feld related to geometrcal objects whose nature or property s dscrete. Therefore, t can provde the theory to handle both pont and polyhedral mesh based descrptons. The organzaton of ths paper s as follows. After a comprehensve revew of Geometrcal Product Specfcatons and geometrc tolerancng approaches (secton ) we show ther lmtatons n consderng multple geometrc representatons and non-deal enttes. The prncple of dscrete geometry for GeoSpellng s descrbed n secton 3. Skn model smulaton and vsualzaton are hghlghted n secton 4. Afterwards, the skn model smulatons whch consder geometrc CIRP Desgn Conference 011

2 tolerances and results comparson are developed n secton 5. The concluson s gven n secton 6. RELATED WORK Many efforts to buld specfcaton models for geometrcal tolerancng have been attempted n recent years. The exstng approaches can be manly classfed nto standard-based and mathematcal models for tolerancng. The standard-based methods rely on techncal drawngs, and are based on the concepts of tolerance features, tolerance zones and datum. Ths geometrcal tolerancng representaton was adopted by ISO , ISO , and ASME Y , and t was the most popular way to descrbe tolerance requrements n the past years. However, ths method cannot keep up wth current tolerance requrements, snce t s based on human nterpretaton and s not convenent to transfer the data what s now a dgtally- based ndustry. Mathematcal models for tolerancng can be classfed nto several groups. The offset zone approach proposed by Requcha [5] obtans the tolerance zone by offsettng the deal feature a certan dstance and ths method s sutable to geometrcal models wth smple shape representatons. Jayaraman and Srnvasan ntroduced the Vrtual Boundary Requrements (VBRs) method [6] to mprove the offset zone method and to defne the vrtual boundary by mathematcal foundatons. The VBRs method consders assembly and materal volume requrements. However, ts shortcomng s that the results are not compatble wth GPS standards and cannot descrbe all knds of tolerances. Hoffman [7] and Turner [8] defned tolerancng models n dfferent dmenson spaces, and Fortn [9] ntroduced the vector tolerancng concept n parameter space, and then Wrtz [10] argued that vector tolerancng should be ncluded n the ISO standards. The shortcomng of the vector tolerancng method s that t s not able to descrbe the tolerance features and the geometrcal varaton requrements. Bourdet and Clement [11] proposed the Small Dsplacement Torsor (SDT) theory, whch can descrbe the tolerancng types by the small rgd dsplacement movement of geometrc features. In contrast, ths method s only approprate for deal features. Clement and Rvere [1] ntroduced the Technologcally and Topologcally Related Surfaces (TTRS) theory. Accordng to TTRS, three-dmensonal surfaces or features are classfed accordng to ther respectve degree of nvarance under the acton of rgd motons. Bascally, seven man features equvalent to knematc lower pars are dentfed: planar feature, cylndrcal feature, revoluton feature, sphercal feature, prsmatc feature, helcoïdal feature and complex feature. Each man feature s then descrbed by a unque mnmum geometrcal reference element (MGRE) that allows postonng n Eucldean space. An MGRE s set as a combnaton of elementary geometrcal objects: pont, lne and plane. TTRS Theory has been adopted by ISO TC13 and successfully mplemented n the CATIA v5 CAD system to manage assembly constrants and tolerance annotatons. All of the methods descrbed above cannot consder nondeal features, and some of them even lead to ambguous nterpretatons. The model of GeoSpellng [13] adopted by ISO allows a unfed descrpton of deal and nondeal features and permts a unque expresson of mathematcal parameterzaton of geometrc features. 3 DISCRETE GEOMETRY FUNDAMENTALS OF GEOSPELLING GeoSpellng proposed by Matheu and Ballu [14] s used to descrbe both deal and non-deal geometrc features [1]. Indeed, t allows the expresson of product specfcatons from functon to verfcaton wth a common language. Ths model s based on geometrcal operatons whch are appled not only to deal features defned by CAD systems, but also to the non-deal features whch can represent a real part. These operatons nclude partton, extracton, fltraton, assocaton, collecton and constructon tems. Dscrete geometry research focuses on basc dscrete geometrcal objects, such as ponts, segments, trangles and other convex dscrete shapes, and t s qute effcent to mplement dgtal dscrete processng technques. Therefore, dscrete geometry theores and technques are suted to enhance the data processng capabltes of GeoSpellng. Based on the standard [4], a specfcaton s defned as a condton on a characterstc defned from geometrc features whch are created from a skn model by dfferent operatons. The concepts of "characterstc," "feature," and "operaton" are then mapped to ther underlyng dscrete geometry mathematcal concepts as summarzed n table1. Feature GeoSpellng non-deal feature deal feature Dscrete Geometry pont, segment, trangle, pont set, polylne, mesh geometrc shapes: plane, cylnder, sphere, Characterstc dstance pont to segment, pont to trangle, segment to segment, segment to trangle angle segment to segment, segment to plane, plane to plane Operaton partton segmentaton extracton fltraton assocaton collecton constructon samplng outler removal, flterng approxmaton, nterpolaton unon-boolean operaton ntersecton-boolean operaton Table 1: Concepts mappng between GeoSpellng and dscrete geometry. 3.1 Features In GeoSpellng, features nclude non-deal features and deal features. In dscrete geometry, non-deal features are dscrete shapes, such as ponts, segments, trangles, pont sets, polylnes, and polyhedral meshes. Ideal features are derved from the classfcaton of 3D surfaces based on ther nvarance under the acton of rgd motons. Ideal features can be obtaned by assocaton operatons. 3. Characterstcs In GeoSpellng, characterstcs nclude dstances and angles. Whle n dscrete geometry, dstances are defned between dscrete shapes: pont-pont, pont-segment, pont-trangle, segment-segment, segment-trangle and n a general case to Hausdorff dstances. The angles are related to the three well-known cases: angles between segment-segment, segment-plane, and plane-plane.

3 3.3 Operatons Partton operaton s used to dentfy bounded features [1]. In dscrete geometry ths knd of operaton s called segmentaton. The majorty of pont set segmentaton methods can be classfed nto three categores: edgedetecton method, regon-growng method and hybrd method [15]. The man problem of the edge-based method s that when the ponts are near sharp edges they are qute unrelable. Ths problem means that the edgebased method has a relatvely hgh senstvty to occasonal spurous ponts. The advantage of face-based technques s that they work on a larger number of ponts to reduce the rsk of senstvty to occasonal spurous ponts, and they can dentfy the ponts that belong to each surface, but the man dsadvantage s tme processng. The hybrd method has been developed by combnng the edge-based and regon-based methods together to overcome the lmtatons nvolved n the orgnal methods. Extracton operatons are used to dentfy a fnte number of ponts from a feature wth specfc rules [1]. In dscrete geometry these rules are equvalent to samplng technques. Zhang et al. [16] classfed the extracton strategy nto four categores: grd extracton, stratfed extracton, specal curve extracton and pont extracton. Dependng on the nvarant class of the surface, users can determne the pror extracton strategy. Other extracton strateges were nvestgated n lterature [17], such as Hammersley sequence samplng, the Halton-Zaremba sequence, Algned systematc samplng, and Systematc random samplng. Fltraton operatons are used to dstngush roughness, wavness, form, and so on, by separatng the dfferent wavelength components nto predefned bandwdths [1]. There are already some optons n today s GPS standards, such as polynomal fttng, RC flterng, Gaussan fttng, wavelet flterng, etc. [18]. In dscrete geometry, sgnal processng flterng technques and other technques such as outler removal, based on a certan crteron, are reported n [19]. Assocaton operatons are used to ft deal feature(s) to non-deal feature(s) accordng to specfc rules [1]. In dscrete geometry, assocaton operatons ft deal feature(s) to dscrete geometrc feature(s) accordng to gven crtera, such as the Movng Least Squares (MLS) method. MLS methods take the dstance nfluence nto account when calculatng the assocaton arthmetc [0]. Collecton operatons are used to consder some features together, whch play a functonal role, and constructon s used to buld deal feature(s) from other deal feature(s) [1]. In dscrete geometry unon-boolean operatons and ntersecton-boolean operatons [1] respectvely have the same capablty. 4 SKIN MODEL SIMULATION The skn model s a non-deal surface model. It s a vrtual model magned by desgners when takng nto account dfferent knds of geometrc defects. The man orgnalty of GeoSpellng s to buld geometrc models for tolerancng specfcaton not from nomnal models but from the skn model tself. It can also help desgners to express specfcatons correspondng to manufacturng requrements. Few research studes have focused on the skn model smulaton. Chabert developed a shape dentfcaton method of the skn model usng rgd body moton and Monte Carlo smulaton []. Samper [3] proposed a fnte element analyss method to smulate form defect expressons of skn models. However, there s no unform way to express the skn model nowadays. Therefore, the skn model smulaton method wll be dscussed here. Three dfferent statstcal methods are consdered n ths paper: 1D Gaussan dstrbuton, mult- Gaussan dstrbutons and Gbbs samplng. The detals of each method are explaned below D-Gaussan method In probablty theory, the Gaussan dstrbuton s a contnuous probablty dstrbuton that s often used as a frst approxmaton to descrbe real-valued random varables that tend to cluster around a sngle mean value. The graph of the assocated probablty densty functon s Bell -shaped as showed n fgure. Fgure : The prncple of 1D-Gaussan method. The prncple of the 1D-Gaussan method can be descrbed n fgure. The bell shape reflects the scope of the 1D-Gaussan dstrbuton actng on one pont. It s a random value and t can be calculated by the probablty densty functon of the 1D-Gaussan (Formula 1). In ths formula, the mean value s the nput ponts coordnates, and the varance determnes the wdth of the Gaussan dstrbuton. Ths method uses the Gaussan varable as the devaton value n the drecton of vertex normal (the vertex normal estmaton method s explaned n secton 3.3), then t apples ths calculaton to each pont, the dstrbuton parameters yeld formula, and then the result s llustrated n fgure 3 wth dfferent vews of the skn model of a plane. ( x µ ) 1 σ ( µσ, ) : ( ) σ π X N f x = e (1) u = X µ N(0,1) () σ Where µ s the mean value, σ s the varance, and x s the Gaussan varable. (a) skn model (b) front vew (c) left vew Fgure 3: 1D-Gaussan skn model. 4. Mult-Gaussan method A multvarate Gaussan dstrbuton s used to generate random vectors. The trvarate Gaussan dstrbuton s consdered n ths work. A spatal random vector s defned as X = T X 1 X, X, 3. The probablty densty ( functon of multvarate Gaussan dstrbuton can be expressed as formula 3. X N f x µ, = ( ) : ( ) 1 1 T 1 exp( ( x µ ) ( x µ )) 3 1 ( π ) (3)

4 Where s the covarance matrx, s the determnant value, and µ s the mean vector. The prncple of ths method s descrbed n fgure 4. The ellpsod reflects the scope of 3D-Gaussan dstrbuton actng on one pont. It s a random vector and can be calculated by the probablty densty functon of the 3D- Gaussan (formula 3). In ths formula, the mean value s the pont s coordnates, and the relatonshps among each axs are constraned by the covarance matrx. Fgure 5 dsplays dfferent vews of the skn model of a plane when applyng ths calculaton to each pont. Pont set x z Fgure 4: The prncple of 3D-Gaussan method. y 3D-Gaussan dstrbuton Suppose X = ( X1,..., Xd ) s the random value n R d, and X = ( X1,..., X, X j j 1 j+ 1 Xd s the random,..., ) value of d 1 dmenson. Note that the condtonal densty functon of X j X j s f ( X j X j ). Therefore, the Gbbs sampler select canddate ponts from ths dmenson condtonal dstrbuton. The related process s that, at begnnng, the tme s equal to zero ( t =0) and t has an ntal value X (0). When t s ncreasng ( t =1,,..., T ), then X ( t ) follows a certan functon to generate new pont to replace old one and teratve calculaton s performed untl t converges to the target value. The correspondng pseudo-code s descrbed as follow. 1. Let = ( - 1) x X t 1 1. Let j s a varable between [ 1, d ]. For j = 1,,..., d, 3. Let usng f ( X x ) to get canddate pont X ( t ), and then update X ( t ). j -j * j ( ) = ( * * t 1 d X t X ( ),..., X ( t and then ncrease t. )) Ths teratve process generates random varables, whch follow the bvarate normal dstrbuton, can smulate the skn model. The result s showed n fgure 7. * j (a) skn model (b) front vew (c) left vew 4.3 Gbbs Method Fgure 5: 3D-Gaussan skn model. The Gbbs samplng algorthm s used to generate a sequence of samples from the jont probablty dstrbuton of two or more random varables. It s an teratve method based on Markov chan Monte Carlo (MCMC) algorthms. It ams to desgn a Markov chan whose statonary dstrbuton s the target dstrbuton. It requres an ntal value of the parameters, and at each teraton, each parameter of nterest s sampled a gven value from the other parameters and data. Once all the parameters of nterest are sampled, the nusance parameters are sampled gven the parameters of nterest and the observed data [4-5]. Ths characterstc of the Gbbs method can make the random dstrbuton of pont set of skn models approxmate a Gaussan dstrbuton. In order to determne the number of teratve runs, the probablty dstrbuton of the normalty assumptons of Gbbs should be consdered. The normal dstrbutons are smulated usng Mntab software for dfferent teratve runs (fgure 6). The results show a good convergence at teratons. (a) skn model (b) front vew (c) left vew Fgure 7: Gbbs Samplng Skn model. 4.4 Skn model vsualzaton Tolerance values are much smaller than features szes, so t s dffcult to vsualze the skn model wth mult-scale geometry. Ths secton proposes to use RGB color scale mappng technque to vsualze the geometrcal devatons on the vertex normal drecton. The skn model can be reflected by a contnuous color strp. Vertex normal estmaton A normal vector s a local geometrc property of a 3D surface, whch s specfc to a gven pont or a planar facet. Many attempts have already been made for relable estmaton of normal vectors from dscrete pont data [15]. Gven a polyhedral mesh surface, the normal vector at a vertex can be estmated as the weghted average of the normal vectors of the adjacent trangle facets around t. Consderng an arbtrary vertex p n a dscrete mesh surface Σ, assumng ts neghbor contans N trangles, then the normal vector at p could be estmated usng formula 4. n p ( ) = ω N = 1 n ω N = 1 n (4) Where, n ( 1,... N) = ndcate the unt normal vector of (a) N=10 (b) N=100 (c) N=10000 Fgure 6: Normalty assumpton of Gbbs method. the th trangle facet. ω ( = 1,... N ) are the weght coeffcents correspondng to the normal vectors of facets f.

5 The method used here for the weght coeffcents computaton consders the nfluence of the area of each adjacent trangle facet and the dstance between the gven vertex and the barycenter of each adjacent facet. Parameter ω can be calculated by formula 5. ω A d = N = 1 A d Where, (5) A ( = 1,... N ) represents the area of the th trangle facet. d ( = 1,... N ) are the dstances between the vertex p and the barycenter of the th trangle facet. Parameter N s the number of all the trangle facets adjacent to the gven vertex. The notatons mentoned n the above formula are descrbed n fgure 8. Fgure 9 shows examples of vertex normal estmaton consderng planar and cylndrcal shapes. 5 CONSTRAINT-BASED SKIN MODEL After creatng the random pont set to smulate the unconstraned skn model, geometrcal and dmensonal tolerances should be consdered to satsfy the specfcaton requrements. The followng secton manly dscusses the form, orentaton and poston tolerance consderatons to enhance the skn model smulaton. Form specfcaton To estmate form specfcaton, the frst step s to determne the tolerance zone drecton usng the Prncpal Component Analyss (PCA) method. Then all of the pont set should belong to the tolerance zone. The prncple of ths method can be descrbed by fgure11, where n s the vector of prncple drecton of the pont set, M 1 and M are two arbtrary ponts, and d s the dstance between these two ponts n the drecton of the vector n. (a) mesh structure (b) vertex normal Fgure 8: Vertex normal estmaton. Fgure 11: Prncple of flatness specfcaton. PCA s a statstcal method for prncpal component analyss by covarance analyss. Consder a dscrete shape P N represented by an arbtrary set of ponts P = [ x, y, z ]. The PCA method computes the prncpal axes of the dscrete shape usng the followng three steps. 1. The orgn of the prncpal coordnates system s determned as the centrod of P N whch s calculated by formula 6. T Fgure 9: Example of vertex normal estmaton on dscrete shapes. RGB mappng technque The geometrcal devatons between the smulated skn model and the ntal pont set are computed by projectng the devaton vectors on the vertces normal. A contnuous RGB color scale s then used to vsualze the skn model (fgure 10). 1 N o = p ( p P ) pca N (6) N = 1. The covarance matrx s defned by formula 7. cov N T = ( )( ) ( ) (7) pca pca N = 1 M p o p o p P 3. Egenvalues and egenvectors are estmated. The frst prncpal axs s the egenvector correspondng to the largest egenvalue. The two other prncpal axes are obtaned from the remanng egenvectors. The tolerance zone drecton s determned usng the PCA method, and the pont set should satsfy the tolerance zone constrant (formula 8). m n m n (8) Max( ) Mn( ) t flatness (a) pont set (b) color scale Fgure 10: Skn model wth color scale. Where m s the vector of th pont, and n s the vector of tolerance drecton, t flatness s the flatness tolerance value. Orentaton and poston specfcaton Besdes form constrants, the orentaton and poston constrants must also be consdered. For these two constrants, the tolerance drecton s the same as the

6 normal drecton of the datum plane and all the ponts should be wthn the tolerance zone. The parallelsm specfcaton (fgure 1) satsfes the constrants n formula 9. = m n m n (9) d Max ( ) Mn ( ) t parallelsm Fgure 1: Prncple of parallelsm specfcaton. Where n s the normal drecton of the datum plane, M s an arbtrary pont, and d s the dstance between two ponts n the drecton of vector n. For poston specfcaton (fgure 13), the related constrant s descrbed by formula 10. t t poston poston = (, ) [, + ] (10) A d Dst m P a a Where n s the normal drecton of the datum plane, M s an arbtrary pont, and d s the dstance between the pont to datum plane P A n the drecton of vector n, and a s the nomnal dstance value of poston tolerance. Comparsons The skn model s created usng the three dfferent smulaton methods proposed n ths paper. A pont cloud of a plane composed by 73 ponts s the reference test, and the specfcaton constrants are flatness, parallelsm and poston tolerances whch are equal to 0.01 mm, 0.0 mm and 0.05 mm respectvely. The 50 skn models are generated and the man statstcal characterstcs are computed for comparson. In the Gbbs samplng method, the teratve tme s equal to The comparson tems nclude the average devaton value, the lmt value, and the processng tme. The dstrbuton of the devaton values s computed usng Mntab statstcal software. The result s shown n fgure 14. Correspondng standard devatons are summarzed n table. From ths table, t can be deduced that the Gbbs method offers the closest smulaton result to the target value, but t s much more tme consumng. Items 1D-Gaussan 3D-Gaussan Gbbs Average Value Standard devaton Maxmum value Mnmum value Tme <0.001 s <0.001 s 0.97s Table : Results of comparson of the three methods. Fgure 13: Prncple of poston specfcaton. (a) 1D-Gaussan method (b) 3D-Gaussan method (c) Gbbs method Fgure 14: Statstcal dstrbuton results. 6 SUMMARY AND CONCLUDING REMARKS In ths paper, we saw that dscrete geometry for GeoSpellng provdes a new mathematcal framework for Geometrcal Product Specfcatons. Startng from the fundamental concepts of the skn model and non-deal features, skn model smulaton and vsualzaton usng 1D Gaussan, 3D-Gaussan and Gbbs method s developed and compared. Wth new foundatons for Geometrcal Product Specfcatons, ths paper concludes that dscrete

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