Proceedngs of the World Congress on Engneerng 008 Vol I Accessblty Analyss for the Automatc Contact and Non-contact Inspecton on Coordnate Measurng Machnes B. J. Álvarez, P. Fernández, J. C. Rco and G. Valño Abstract The am of ths paper s to determne the vald orentatons of the nspecton devces used n probe operatons and non-contact scannng operatons on a Coordnate Measurng Machne (CMM). The methodology appled s based on the accessblty analyss and the applcaton of ray-tracng technques. Ths analyss wll tae nto account the real shape and geometry of the nspecton devce and the constrants mposed by the CMM on whch t s mounted. Lewse, dfferent algorthms based on computer graphcs have been appled to speed up the searchng of vald orentatons. Index Terms Accessblty, CMM, nspecton, probe, laser strpe, scannng. I. INTRODUCTION Among the actvtes of an automatc process plannng system for nspecton on a Coordnate Measurng Machne (CMM), the determnaton of nspecton devce orentaton wth regard to the part stands out. These nspecton devce orentatons are obtaned from a methodology based on the accessblty analyss [1] and the applcaton of ray-tracng algorthms []. Moreover, dfferent computer graphcs technques le space parttonng and bac-face cullng have been appled n order to speed up the searchng of vald orentatons. The methodology has been appled to nspecton processes based on a touch-trgger probe [] and to non-contact scannng processes based on a laser strpe [4]. In both cases, dfferent constrants have been consdered: real shape and dmensons of the nspecton devces, process parameters and possble orentatons of the motorzed head of the CMM where the nspecton devce has been mounted. Ths motorzed head (PH10MQ) provdes 70 feasble orentaton of the nspecton devce by rotatng at a resoluton of 7.5º about both horzontal and vertcal axes (A, B). Manuscrpt receved March 18, 008. Ths wor s part of the results obtaned n a research project supported by the Spansh Educaton and Scence Mnstry (MEC-04-DPI004-0517) and FEDER. B. J. Álvarez and P. Fernández have collaborated wth the Manufacturng Engneerng Department of the Unversty of Ovedo for developng the research project mentoned above (e-mal author: braulo@unov.es). J. C. Rco s a Professor (e-mal author: jcarlosr@unov.es). G. Valño s a Senor Lecturer (e-mal author: gvr@unov.es). The correspondng address for all authors s: Department of Manufacturng Engneerng, Unversty of Ovedo, Campus de Gjón, 0 Gjón, SPAIN. II. ANALYSIS METHODOLOGY The accessblty analyss for a touch-trgger probe deals wth determnng all the feasble probe orentatons that allow for performng the part nspecton avodng collsons wth the part or any other obstacle n the envronment of the nspecton process. Moreover, the vald orentatons of the non-contact scannng devce wll be obtaned to guarantee the vsblty of the surface to be scanned. The methodology appled n both cases wll be exactly the same. Frst, a local analyss wll be made for each part surface by tang nto account the nspecton devce as an nfnte half-lne whereas the possble nterferences wth the rest of part surfaces or any other obstacle are gnored. Hence, vald orentatons l wll be those whch mae an angle between 0 and π/ wth the normal vector n to the analyzed surface. In a second analyss stage, the orentatons obtaned n the local analyss wll be checed n order to determne f they have or not collson wth the rest of part surfaces or any other obstacle. Ths analyss s called global analyss and t s complex and expensve from a computatonal pont of vew because t nvolves the calculaton of multple nterferences tests. To mae ths calculaton easer, a STL model of the part s used, where each surface s dscretzed by a set of trangles. Thus the global accessblty analyss s reduced to determne f there exst nterferences between orentatons l obtaned n the local analyss and the trangles that compose the STL model of ether the part or the obstacle. Moreover, n order to further reduce the calculaton tme, dfferent computer graphcs technques le space parttonng based on d-tree, ray traversal algorthm, bac-face cullng and ray-trangle ntersecton tests have been also appled [5]. The use of space parttonng structures le d-trees allows for reducng the number of the trangles to test n the global analyss because t nvolves checng ntersecton exclusvely wth trangles whch can potentally be traversed by each nspecton devce orentaton. The part s parttoned n regons bounded by planes (boundng boxes) and each part trangle s assgned to the regon wthn whch t s located. Then, regons traversed by each nspecton devce orentaton are selected by means of the ray-traversal algorthm that was frst developed and appled by Kaplan []. Before checng ntersecton between each nspecton devce orentaton and all trangles ncluded n the traversed regons prevously determned, the number of ntersecton
Proceedngs of the World Congress on Engneerng 008 Vol I tests can be reduced even more by applyng a bac-face cullng algorthm [5]. Thus, from the ntal set of trangles ncluded n the traversed regons, a subset s extracted that do not nclude those trangles whose vsblty accordng to an analysed nspecton devce orentaton s completely bloced by other trangles. Fnally, the ntersecton test between each undscarded trangle and the nspecton devce orentaton l s carred out. The trangles of the STL format are defned by ts vertces V 0, V 1 and V and normal untary vector n. If the next equaton (Fg. 1) s satsfed: n l = 0 the orentaton l wll be parallel to the supportng plane of the trangle and therefore there wll be no ntersecton. If both the expresson (1) and the next condton are satsfed: n w = 0 then the nspecton devce orentaton l wll be contaned n the plane. In expresson () w s a vector wth orgn at vertex V 0 and end at pont P 0. When ths orentaton ntersects or concdes wth any of the trangle edges, then, ntersecton between devce orentaton and trangle occurs. If none of the prevous relatonshps s fulflled, then there s ntersecton between the nspecton devce orentaton l and the supportng plane of the trangle. The ntersecton pont P can be expressed as (Fg. 1): n w P = P0 l () n l Fnally, t s necessary to chec f ths pont P les nsde the trangle defned by the three vertces V 0, V 1 and V. Ths verfcaton s based on the algorthm developed by Möller and Trumbore [6]. The equaton of the supportng plane of the trangle V 0, V 1 and V can be expressed as: V(,) s t = V + s u+ t v 0 where u and v are two edge vectors of the trangle wth common orgn at V 0. A pont P located on the plane (4) wll be nsde the trangle f there exst values s 0 and t 0 π n V 0 w P 0 v l r u Fg. 1. Intersecton between laser beam l and a trangle facet V 0 V 1 V. P V V 1 (1) () (4) Column Head Machne Touch probe Stylus Tp (a) Touch-trgger probe Column Head Machne Adapter Head Laser (b) Laser strpe system Capsule Sphere Prsm Prsm Sphere Capsule Prsm Fg.. Components of the nspecton devce and ther smplfed models. that satsfes the next equaton: P V = s u+ t v 0 ( s + t 1 ) (5) If the pont les wthn the trangle then there wll be ntersecton and the analyss wll contnue wth another trangle. If there s stll no nterference, the orentaton wll be consdered as vald. III. INTERFERENCE ANALYSIS CONSIDERING REAL DIMENSIONS OF THE INSPECTION DEVICE The orentatons obtaned n prevous sectons are based on an deal representaton of the nspecton devce as an nfnte half-lne. To chec f these orentatons are really vald wll be necessary to tae nto account the real shape and dmensons of the nspecton devce. Therefore, ntersecton between trangles of the STL part model and each of the nspecton devce components must be checed. Fg. shows the components for each of the nspecton devces that have been consdered n ths wor: 1) Touch-trgger probe: column, head, touch probe, stylus and tp (Fg. a). ) Laser strpe system: column, machne head, adapter and laser head (Fg. b). The ntersecton analyss s speeded up by usng a smplfed model of each nspecton devce component (Fg. ). The column and laser head have been modelled by
Proceedngs of the World Congress on Engneerng 008 Vol I straght prsms, the machne head by a sphere and the laser adapter, touch probe and stylus by a capsule. Smlarly to secton, a d-tree algorthm has been mplemented n order to test exclusvely the nterference between each of the nspecton devce components and the trangles ncluded n the part regons that they traverse. To carry out ths tas effectvely, each nspecton devce component s enclosed n a boundng volume and only the part regons that overlap wth that volume are analysed. Then, ntersectons between part trangles ncluded n these regons and the component are checed. Snce several geometrcal shapes have been used to model the components of each nspecton devce, dfferent algorthms for checng ntersectons are appled: 1) Sphere-trangle ntersecton algorthm to analyse nterferences wth the machne head [7]. Ths algorthm smply calculates the mnmum dstance between a pont (sphere centre) and a trangle. If ths dstance s smaller than the radus of the sphere, ntersecton wll occur. ) Prsm-trangle ntersecton algorthm to analyse nterferences wth the column. Ths algorthm derves from the separatng axs theorem whch was ntally developed to determne f two convex polyhedral are dsjont [8]. The theorem states that two convex polyhedra, A and B, are dsjont f they can be separated along ether an axs parallel to a normal of a face of ether A or B, or along an axs formed from the cross product of an edge from A wth an edge from B. The applcaton of ths theorem to a prsm-trangle test nvolves checng ther relatve poston wth regard to dfferent potental separaton axes [9]. ) Capsule-trangle ntersecton algorthms to analyse nterferences wth the touch probe and the stylus-tp [10]. In ths case, ntersecton analyss s based on fndng the mnmum dstance between each edge of the trangle and the capsule lne segment, as well as the mnmum dstance between each extreme pont of the capsule segment and the trangle. If any of these dstances s smaller than the radus of the capsule, then ntersecton wll occur. IV. CLUSTERING From the prevous analyss, the nspecton devce orentatons that are vald for probng each pont or scannng each part trangle have been determned. These orentatons are mathematcally represented by means of a bnary matrx A ( qr, ) where each element corresponds to a combnaton of dscrete values of A and B angles: A ( q, r) 1 f (A = aq, B= br) s a vald orentaton for pont P = (6) 0 f (A = aq, B= br) s a not vald orentaton for pont P To reduce the process operaton tme related to devce orentaton changes, orentatons (a q, b r ) common to the greatest number of ponts to probe (clusters of ponts) or trangles to scan (clusters of trangles) must be found. The classfcaton of ponts or trangles contnues untl no ntersecton can be found between the fnal clusters. -17.5º 18 Probe abstracted by a nfnte half-lne Head Head + Touch probe Head + Touch probe + Stylus and tp Head + Touch probe + Stylus and tp + Column P7 Part STL model P11 P1 P1 P10 P9 s19 s7 P P8 P7 P P5 P4 P6 P5 s9 P1 P11 P1 P16 P9 P6 P14 P P4 P10 P17 P1 P18 P15 P8 P7 P6 P4 P P5 P P1 P P8 Fg.. Accessblty map for the pont P4 on the surface s9.
Proceedngs of the World Congress on Engneerng 008 Vol I The algorthm used for clusterng s smlar to that developed by Vafaeesefat and ElMaraghy [11]. Next, the algorthm s explaned for an nspecton process usng a touch-trgger probe. Each pont P ( = 1,,, n) to be probed s assocated to a bnary matrx of vald orentatons A ( = 1,,, n). Intally ( = 1), the clusters wll be the same as each of the ponts to be probed: C = P. Startng from the clusters C and from the bnary matrces A, a new matrx CI (, j) = A Aj s bult showng the common probe orentatons to the clusters two aganst two. Wth the purpose of creatng clusters whose ponts are assocated wth the greatest number of vald orentatons, the algorthm searches the ndces (s, t) of CI that correspond to the maxmum number of common vald orentatons. After that, clusters Cs and C t assocated to ponts P s and P t respectvely wll be regenerated as follows: C = C C and s s t C t = (7) and the bnary matrx of vald orentaton assocated to the new cluster C wll be A = A A. s s s t Wth these new clusters, matrx A Aj for = t and j = s CI (, j) = for = t and j = t CI s updated to: (8) The clusterng process fnshes when the number of common orentatons correspondng to all the elements above the man dagonal of matrx CI have become zero. A smlar process s used to determne the clusters of trangles T ( = 1,,, n) for a part to be scanned. V. APPLICATION RESULTS A. Inspecton Process by means of a Touch-trgger Probe In order to analyze the applcaton results of accessblty and clusterng algorthms to the nspecton process, the part shown n Fg. has been consdered. For ths part, the STL model contans 48 trangles and 8 nspecton ponts located on three dfferent surfaces (s7, s19 and s9). As an example, Fg. shows the accessblty maps for the nspecton pont P4s9 (pont P 4 on surface s9) consderng the dfferent geometrcal abstractons of the probe. As t can be seen, when real dmensons and dfferent probe components are taen nto account, the accessblty map s substantally reduced. For the rest of nspecton ponts, smlar accessblty maps can be obtaned. Furthermore, the applcaton of the clusterng algorthm allows for obtanng the clusters shown n Table I. In ths case seven clusters have been found. B. Scannng process by means of a Laser Strpe System Apart from the nspecton process, the developed methodology allows for determnng the orentatons of a laser strpe system to scan a part. In ths type of scannng systems a laser strpe of nown wdth s projected onto the part surface to be scanned and the reflected beam s detected TABLE I. CLUSTER OF POINTS AND COMMON ORIENTATIONS IN THESE CLUSTERS Cluster Ponts / Surface Orentatons (A, B) 1 P1s19, Ps19, P1s19, P4s7, P4s9, (7.5, 67.5) (45, 67.5) P7s9, P11s9, P1s9 Ps19, P1s9, P5s9, P6s9 (67.5, -60) P4s19, P5s19 (97.5, -90) (97.5, -8.5) (97.5, -75) (97.5, -67.5) (97.5, -60) 4 P6s19, P8s7 (0, -165) (0, 180) (7.5, -17.5) (7.5, -165) (7.5, -157.5) (7.5, -150) (7.5, -14.5) (7.5, -15) (7.5, 157.5) (7.5, 165) (7.5, -17.5) (7.5, 180) (45, -17.5) (45, -165) (45, -157.5) (45, -150) (45, -14.5) (45, -15) (45, -17.5) (45, -10) (45, 180) (5.5, -17.5) (5.5, -157.5) (5.5, -150) (5.5, -14.5) (5.5, -15) (5.5, -17.5) (5.5, -10) (5.5, -11.5) 5 P7s19, P8s19, P9s19, P10s19, (15, 10) (15, 17.5) (.5, 11.5) (.5, 10) (.5, 17.5) P11s19, P6s7, Ps9, Ps9, P8s9, (.5, 15) (.5, 14.5) (.5, 150) (.5, 165) (0, 11.5) P9s9, P10s9, P1s9, P14s9, P15s9, P16s9, P17s9, P18s9 (0, 10) (0, 17.5) (0, 15) (0, 14.5) (0, 150) (0, 165) 6 Ps7, Ps7 (7.5, -15) 7 P7s7 (7.5, -165) (7.5, -157.5) (7.5, -150) (7.5, 165) (7.5, 17.5) (7.5, 180) (15, -17.5) (15, -165) (15, -157.5) (15, -150) (15, -15) (15, -14.5) (15, 150) (15, 157.5) (15, 165) (15, 17.5) (15, 180) (.5, -17.5) (15, -165) (15, -157.5) (15, 17.5) (15, 180) (0, -17.5)
Proceedngs of the World Congress on Engneerng 008 Vol I by a CCD camera. Therefore, not only the ncdent laser beam orentaton has been taen nto account for the accessblty analyss but also the possble occluson due to the nterference of the reflected laser beam wth the part. Fg. 4 shows the laser head orentaton map assocated to the local and global accessblty analyss for trangle 558 of the example part. The darest colour represents the head orentatons (A, B) that concde or are closest to the normal drecton of the trangle analyzed. These are consdered as optmal orentatons. Grey colours represent head orentatons far from the optmal value whch lead to worse scannng qualty. Whte colour represents head orentatons that do not enable to scan the consdered trangles. For a better vsualzaton of the orentaton map, ncrements of 15º have been consdered for angles A and B. For trangle 558 an ncdent laser beam orentaton (A=, B=18) has been selected n order to show the part regons (boundng boxes) that t traverses (Fg. 5). Trangles partally or totally enclosed n these boundng boxes are shown n the fgure before and after applyng the bac-face cullng algorthm. Fg. 6 shows n a grey scale the ten trangle clusters obtaned for the example part. -180 º 0 º +18 Local orentatons (A, B) Trangle 558 Global orentatons (A, B) Fg. 4. Local and global accessblty maps for trangle 558 (a) Trangle 558 and ncdent ray orentaton A0-B180 (b) Part regons traversed by the ncdent ray (c) Trangles wthn the traversed part regons (d) Trangles wthn the traversed part regons after applyng the bac-face cullng algorthm Fg. 5. Dfferent stages to determne the global vsblty map for a trangle (558) and an ncdent laser beam orentaton (A=, B=18). z y x 4 8 1 1 6 5 7 1 9 4 10 y x z Fg. 6. Clusters assocated to the example part.
Proceedngs of the World Congress on Engneerng 008 Vol I VI. CONCLUSIONS Most of the accessblty analyss presented n other wors only deal wth a lmted number of the nspecton devce orentatons, smple parts wth only planar surfaces or specfc geometrcal shapes, smplfed devce representatons or a short number of ponts n the case of nspecton by touch-trgger probe. However, n ths paper, a new methodology for accessblty analyss s presented whch allows for overcomng the prevous lmtatons: 1) The methodology has been extended to the nspecton process by means of a touch-trgger probe and the scannng process by means of a laser strpe system. ) All the possble orentatons (70) of the nspecton devce are taen nto consderaton. ) The use of the STL model permts the applcaton of the developed system to any type of part, regardless of ts shape and complexty. 4) The real shape and dmensons of the nspecton devce are consdered for the analyss. 5) The mplemented algorthms based on Computer Graphcs reduce computaton tme and consequently can deal wth a hgh number of nspecton ponts and complex surfaces. 6) Moreover, a clusterng algorthm s appled that effcently groups the nspecton ponts and trangles of the STL of the part to be scanned n order to reduce the number of probe orentaton changes. The developed system has been appled to dfferent parts wth satsfactory results demonstratng the applcaton n practce. Future research wll concentrate on developng new algorthms that further reduce computaton tme, and on generatng the nspecton paths from the orentatons obtaned n the clusterng process. REFERENCES [1] A. J. Spyrd and A. A. G. Requcha, Accessblty analyss for the automatc nspecton of mechancal parts by coordnate measurng machnes, Proc. of the IEEE Int. Conf. on Robotcs and Automaton, 1990, pp. 184-189. [] M. Kaplan, Space-tracng: A constant tme ray-tracer, Proceedngs of the SIGGRAPH 85, 1985, pp. 149-158. [] A. Lmaem and H. A. ElMaraghy, CATIP: A Computer-Aded tactle nspecton plannng system, Int. J. of Prod. Res., vol. 7(), 1999 pp. 447-465. [4] K. H. Lee and H. -P. Par, Automated nspecton plannng of free-form shape parts by laser scannng, Robot. Comput. Integr. Manuf., vol. 16(4), 000, pp. 01-10. [5] J. D. Foley, A. van Dam, S. K. Fener and J. F. Hughes, Computer Graphcs, Prncples and Practce, Cornell: Addson-Wesley, 1995, ch. 15, pp. 66-664. [6] T. A. Möller and B. Trumbore, Fast, mnmum storage ray/trangle ntersecton, Journal of Graphcs Tools, vol. (1), 1997, pp. 1-8. [7] P. J. Schneder and D. H. Eberly, Geometrcal Tools for computer graphcs, San Francsco: Morgan Kaufmann Publshers Inc., 00, ch. 10, pp. 76-8. [8] S. Gottschal, M. C. Ln and D. Manocha, OBBTree: a herarchcal structure for rapd nterference detecton, Proc. of the SIGGRAPH 96, 1996, pp. 171-180. [9] T. A. Möller, Fast D trangle-box overlap testng. Journal of Graphcs Tools, vol. 6(1), 001, pp.9. [10] D. H. Eberly, D Game engne desgn. A practcal approach to real-tme computer graphcs, San Dego: Morgan Kaufmann Publshers Inc., 000, ch., pp. 5-57. [11] A. Vafaeesefat and H. A. ElMaraghy, Automated accessblty analyss and measurement clusterng for CMMs, Int. J. Prod. Res., vol. 8(10), 000, pp. 15-1.