CL-Path in B-Spline Form with Global Error Control for 3-Axis Sculptured Surface Machining

Transcription

1 n Inernionl onference on Mechnicl, Proucion n Auomobile Engineering (IMPAE'0) Singpore April 8-9, 0 -Ph in B-Spline Form wih Globl Error onrol for 3-Axis Sculpure Surfce Mchining Mqsoo Ahme. Khn, n Zezhong. hen Absrc Due o he lck of smooh moions n ccure mchining of complex free form shpes using snr liner n circulr moion in convenionl N mchines, new commercil N sysems re equippe wih prmeric curve inerpolion funcion. However, ue o he pproximion of; -ph in AM sysem n ph prmeer in rel-ime, high mchining ccurcy, smooh kinemic n fee re profiles, re ifficul o chieve. This pper presens new meho o represen -ph in B-spline form wih, globlly conrolle ccurcy, n compc represenion of compre o is conemporries. For simulion purpose free form B-spline surfce is selece n ool phs re genere. Keywors Globlly conrolle ccurcy; Sculpure pr surfces; N conrollers; Prmeric curve inerpolion. T I. INTRODUTION O efficienly prouce smooh sculpure pr surfces in N mchining, he curve inerpolor hs recenly been ccepe s n lernive o replce he convenionl liner n circulr inerpolors. Since, hey re poenilly more suible for high-spee n high-ccurcy mchining wih smooh fee re profiles; some of he N conroller mnufcurers (FANU, SIEMENS, HEIDENHAIN n MITSUBISHI) hve evelope curve inerpolion funcion. In prmeric curve inerpolion, s reference, preeermine ph of cuer locion is fe ino he conrollers, n he inerpolor hen clcules, bse on he reference ph, consecuive cuer locions for is insnneous moions uring mchining. The min objecives uring inerpolion re o ensure h he cuer rjecories mch he reference ph wih high fieliy n h he fee re remins he specifie vlue wihou much flucuion. Generlly here re four min sges rele o he mchining of sculpure surfce long priculr prmeric curve; genere poins, bse on he surfce geomery, ool rius n he chorl eviion olernce, o eec n correc gouging/inerference problem for ll poins, Mqsoo Ahme. Khn is in he Deprmen of Inusril n Mnufcuring Engineering, NED Universiy of Engineering n Technology, Universiy Ro, Krchi-7570, Pkisn (phone: (9-0) ; fx: (9-) 99655; e-mil: mqsoohme@neue.eu.pk). Zezhong. hen is in he Deprmen of Mechnicl n Inusril Engineering, oncori Universiy, Monrel, Quebec, n, H3G M8 (emil: zcchen@encs.concori.c). (3) ll vli poins re o be convere o curves wih uni less ph prmeer, (4) he ph prmeer is pproxime for commn generion uring inerpolion in he N conroller. Sge nees one olernce vlue o genere poins wihin he llowble chorl eviion n sge (3) lso nees one olernce o pproxime he ool-ph wihin he permissible fiing error. However, his ouble-olernce pproch only mkes sure h he pproxime curve respecs he iel ool-ph iscree -poins. For genering curve -phs off-line, generlly here re wo pproches. In he firs pproch cubic n quinic splines re use o inerpole he -poins wih ner rc lengh prmeerizion [] [5]. In he secon pproch he poins re inerpole or pproxime by NURBS or B- spline curves [6] [8] wih uni less prmeer. The unique feure of boh pproches is he ccurcy of -ph cn only be gurnee iscree -poins. The inen of curren work is o propose n inegre soluion wih globl error conrol o genere -ph using B-splines in off-line phse. We ssume h ll -poins hve lrey been clcule from ool-ph generion sep. II. ALGORITHM OVERVIEW We ssume h he reer is fmilir wih B-splines [9]. A 3 B-spline surfce S of egree p n egree q in u n v irecions respecively, is efine s Xuv (, ) nu nv p q = Yuv (, ) = P i, jni( un ) j( v) i= 0 j= 0 Zuv (, ) Where P i, j re he conrol poins, n Ni ( u ), N j v re he B-spline bsis funcions. A B-spline curve of egree is efine s n = P knk, k = 0 Where Pk re he conrol poins, n Nk, re he B- spline bsis funcions. Assume ( ) is plnr cubic B- spline curve inerpoling he se of gouging free poins q [ ] T i = ui vi, i = 0,,... N in uv -omin of he p q 39

2 n Inernionl onference on Mechnicl, Proucion n Auomobile Engineering (IMPAE'0) Singpore April 8-9, 0 surfce S. Le be he imge -curve by mpping ( ) ono he surfce = S ( ( )) in B- spline form. The norml surfce S is efine s he cross prouc of Su = n S v =. To genere he u v norml curve N in B-spline form long ; ( ) is mppe ono norml surfce N = S ( ). Once ( ) n N re known, se of cuer locion poins { Q k }, k = 0,,... M is smple long ( ) n n n B-spline curve L is fi o ll Q k o pproxime he heoreicl -ph. The finl sep which is he min conribuion of he curren work is o boun he error of pproximion globlly for B-spline represenion of he ph by eveloping he error funcion δ in NURBS form. The specifie mchining ccurcy is gurnee long he whole ool-ph inse of only iscree locions n he compc represenion of he ph is chieve wihou scrificing he ccurcy. The following secions illusre he eils of he lgorihm. III. -PATH ON THE SURFAE For he given se of iscree poins long he -ph in prmeric omin from N ool ph plnning, we fi cubic B-spline curve = [ u v ] T h inerpoles hese poins in uv-pln (Fig. ). An explici n conrol poin bse, exc represenion of -ph [ ] T = x y z is obine by mpping ( ) ono he surfce. However, he egree of exc curve is consierbly high n requires subsnil moun of compuions. To overcome hese problems, one opion is o pproxime i by inerpoling finie number of -poins on he surfce. The resul is, he curve oes no in generl lie excly on he surfce, n here re gps beween he surfce n he -ph. These gps cn cuse uner n over cuing of he esign surfce uring mchining. Of course he mximum gp size cn be reuce by fiing hrough more poins, bu requires lrge moun of. In he curren work we hve use he exc represenion of he -ph, by oing he inerpolion in prmeer spce rher hn moel spce, n finlly he -ph is obine by mpping of he D curve compose wih is surfce mpping (Fig. (b)). Inerese reers re referre o [0] for he eils of exc curve on he surfce in B-spline form. IV. -PATH For bll-en mill cuer of rius n ph lying on he surfce, he rue -ph is given by (b) v (,) 00 i x = y = + n. z ( u, v ) i ( u v ) = S, u = v (,) u S( uv,) Fig. Se of -poins for given cuer n surfce quliy, n cubic B-spline curve (-ph) pssing hrough hese poins in uv omin (b) -ph pssing hrough -poins n excly lying on he surfce where, Nx Nx + Ny + Nz nx S Ny u Sv n = = ny =. S u Sv N x Ny Nz nz + + Nz Nx + Ny + Nz Since he close form soluion of equion is no possible, o represen wih B-spline curve, herefore i is pproxime by fiing B-spline o i. Fiing errors re checke o ensure he geomeric shpe of he pproximion oes no evie from he iel ph beyon he given 40

3 n Inernionl onference on Mechnicl, Proucion n Auomobile Engineering (IMPAE'0) Singpore April 8-9, 0 olernce. In he propose meho for he given ph, surfce finish requiremens n cuing ool efiniion, he poins{ Q k }, k = 0,,... M re smple (Fig. ). Using les squre meho B-spline curve = x T y z is genere (Fig. (b)), o pproxime. x e x x E = ye = L = y y z e z z Smple -poins Q k Qk ( ) = S u, v -ph on he surfce ε R E θ (b) Approxime -ph l -ph ( ) Fig. 3 A -ph, is -ph n error ε The pproximion error ε ( ) is efine s he isnce beween he rue -ph ( ) n he pproxime one ( ). The represenion of he error is ε = R E Here, he squre of pproximion error funcion is efine s ( ) ( ) ( ) r e r e r e ε = x x + y y + z z To simplify equion, se Fig. Smple -poins using n surfce norml N (b) fiing B-spline curve o pproxime -ph. n f = R + E e e e = + x + y + z f = R E cosθ = x x + y y + z z r e r e r e V. ERROR BOUND ON URVE FITTING ERROR To buil he error funcion δ, consier Fig. 3, he ifference R ( ) beween he rue -ph n he -ph is xr R = yr = L zr nx = N = ny nx + ny + nz nz n he ifference E( ) beween he pproxime -ph n he -ph is where θ is he ngle beween R n E cn be simplifie s Tking squre on boh sies, n equion ε f = f (3) ( ε f ) = ( f ) 4 + = ε ε f f 4 f Since ε is normlly very smll herefore; ε ( ) 4 0 ε f + f = 4 f 4

4 n Inernionl onference on Mechnicl, Proucion n Auomobile Engineering (IMPAE'0) Singpore April 8-9, 0 4 ε = f f (4) f ( + )( ) f f f f ε = f f ε = f f f Using equion (3) A. Lemm ( ) f ε = + ε (5) f For given -ph n cuer rius, he rue -ph cnno be represene wih B-spline curve. To represen, B-spline curve is fi o smple -poins on o pproxime i. The pproximion error funcion ε is efine bove. A funcion δ of n upper boun of he error is foun s ε δ = ε f f B. Proof The moule of he ifference R beween he rue ph n he -ph is R, n he moule of he ifference E beween he rue offse n he pproxime curve is E.The following relionship lwys hols, Then we cn ge ( R E ) 0 0 R E R + E Since θ is he ngle beween R n E, generlly, he in-equion 0 cosθ is rue. By muliplying his inequion wih he bove in-equion, we ge which is 0 R E cosθ R + E f 0 f Menwhile, we cn ge n in-equion Therefore, Using equion (5), n f f f f f f + + f f f f ε ε ε = + + f f ε ε δ = = ε f f f f [En] The funcion cn be furher simplifie s f f δ = f f f Since he funcion is lwys equl o or lrger hn he squre of he pproximion error funcion, i is n upper boun funcion of he error funcion. In oher wors, he squre roo of he mximum vlue of he boun funcion is gurnee o be lrger hn ll he error long he -ph. The funcion is rionl funcion; n is numeror n enominor re polynomils wihou squre roo. Thus, i cn be rerepresene s NURBS curve funcion, which is crucil o hve pproximion error globlly boune in he generion of cuer locion ph for sculpure surfce mchining. VI. SIMULATION The propose lgorihm hs been progrmme in Mlb n couple of exmple prs were moele n ese. One of hose simulion resuls is presene n iscusse here. The objecive is o show h our meho cn genere smooh pproxime -ph wih globl error conrol for sculpure surfce mchining n requires less number of conrol poins for he G-coe file. Our meho is no limie o isoprmeric or iso-plnner ool-ph generion mehos (which re less efficien n normlly ope by oher reserchers). For he ske of comprison sme pr ws use o genere B- spline ool-ph using surfce mchining workbench in ATIA. We use NURBS surfce wih egree six in boh irecions n bll en mill cuer of imeer 0.5 inches. The esire chorl eviion long ph is 0.05 inches n he cusp heigh beween ny wo consecuive phs is lso 0.05 inches. Fig. 4 shows he -phs genere by he propose meho. Iniilly, he se of -poins sisfying he olernce vlues re use o genere he B-spline -phs 4

5 n Inernionl onference on Mechnicl, Proucion n Auomobile Engineering (IMPAE'0) Singpore April 8-9, 0 (perfecly lying on he surfce), hen number of -poins re smple long wih pproxime rc-lenghs o pproxime he iel -ph. The resuling -phs re in B-spline form, which re evlue using previously genere phs n he propose error funcion o boun he mximum curve fiing error long he whole lengh of he ool-ph. Fig. 5 shows he B-spline -phs genere by ATIA for he sme olernce vlues. Z-xis (inch) -ph X-xis (inch) Zoom in Fig. 4 -ph genere by propose meho Pr surfce Y-xis (inch) Tol 3 cubic B-spline segmens re use which requires 90 conrol poins. The propose meho lso nees 3 segmens becuse of sme cusp heigh requiremen bu ol number of conrol poins is only 4 compre o he conrol poins require by ATIA. Therefore, he ool-phs genere by our meho re; smooh, require less for he N file, n he curve fiing error is globlly boune. Z-xis (inch) -ph X-xis (inch) Zoom in Fig. 5 -ph genere by ATIA Pr surfce Y-xis (inch) VII. ONUSION In his reserch, new B-spline ool-ph generion meho is evelope which is pproprie for hree xis sculpure surfce mchining. The min feure of his pproch is he fiing error of pproximing he -phs is globlly boune. Since, he sculpure surfces re complex n ue o he ool-surfce mismch, close-form equions of he iel ool phs for he surfces o no exis. Therefore, iscree - poins nee o be smple bse on chorl eviion olernce o pproxime he iel -ph. The pproxime -ph is si o be ccepble if he mximum pproximion error is wihin he fiing olernce. However, his pproch only mkes sure h he pproxime -curve respecs he iel ool-ph iscree -poins. Therefore, n lgorihm hs been evelope o ensure h he finl -ph oes no evie from he iel ph more hn llowble chorl eviion long he whole ph lengh inse of only iscree locions. The reserch conribuion is he error funcion, which cn be use o esily esime he mximum 3D curve fiing error in orer o globlly conrol he ccurcy. Simulion resul hs emonsre he effeciveness of he propose meho. The pproxime oolphs in B-spline form re; smooh, hve fewer conrol poins, n ll heir fiing errors re wihin he specifie olernce. Therefore, he propose meho hs gre poenil o genere high precision n smooh N ool phs for free form surfce mchining. REFERENES [] Wng, F., n Yng, D..H., 993, Nerly rc-lengh prmeerize quinic-spline inerpolion for precision mchining, ompuer-aie Design, Vol. 5, No. 5, pp [] Yng, D..H., n Wng, F., 99, Quinic spline inerpolor for moion commn generion of compuer-conrolle mchines, The Proceeings of The n Biennil Mechnisms onference, Sepember 3-6, Scosle, A.Z., U.S.A., pp [3] Wng, F., n Wrigh, P.K., 998, Open rchiecure conrollers for mchine ools, Pr : rel ime quinic spline inerpolor, Journl of Mnufcuring Science n Engineering, Trnscions of he ASME, Vol. 0, No., pp [4] Wng, F., Wrigh, P.K., Brsky, B.A., n Yng, D..H., 999, Approximely rc-lengh prmeerize 3 quinic inerpolory splines, Journl of Mechnicl Design, Trnscions Of he ASME, Vol., No. 3, pp [5] Erkorkmz, K., n Alins, Y., 005, Quinic spline inerpolion wih miniml fee flucuion, Journl of Mnufcuring Science n Engineering, Trnscions of he ASME, Vol. 7, No., pp [6] Tosheng, Z., Wong, Y.S., n Mnnn, M.A., 005, A composie B- spline meho for cuer ph generion on free-form surfces In. J. ompuer Applicion in Technology, Vol. 4, No., pp [7] Lrigue,., Thiebu, F., n Mekw, T., 00, N ool ph in erms of B-spline curves, ompuer-aie Design, Vol. 33, No. 4, pp [8] Bey, M., Boujou, S., n Bouzi, N.T., 007, Tool-ph generion for free-form surfces wih B-spline curves, Journl of Mechnicl Engineering, Vol. 53, No., pp [9] Piegl, L. & Tiller, W., 997, "The NURBS book", n eiion, Springer. [0] G. Renner, n V. Weib, 004, Exc n pproxime compuion of B-spline curves on surfces, ompuer-aie Design, Vol. 36, pp