Development of Machinable Ellipses by NURBS Curves

Transcription

1 Develope of Machable llpses by URBS Curves Yua L. La, Ja H. Che, a Ju P. Hug Absrac Owg o he hgh-spee fee rae a ulra sple spee have bee use oer ache ools, he ool-pah geerao plays a ey role he successful applcao of a Hgh-Spee Machg (HSM syse. Because of s porace boh hgh-spee achg a ool-pah geerao, approxag a coour by URBS fora s a poeal fuco CAD/CAM/CC syses. I s uch ore covee o represe a ellpse by paraerc for ha o coec pos laborously eere a CC syse. A ew approxag eho base o opu processes a URBS curves of ay egree o he ellpses s presee hs suy. Such operaos ca be he fouao of ool-raus copesao erpolaor of URBS curves CC syse. All operag processes for a CAD ool s presee a eosrae by praccal oels. Keywors llpse, Approxao, URBS, Opu. I. ITRODUCTIO U o he wesprea use of URBS-base curve D represeao a oo corol, he ea of Hgh-Spee Machg (HSM s creasg. I he applcao of HSM, oe of he pora copoes whch are ofe eglece s he ool-pah fore sraegy. The successful applcao of HSM ca oly be realze assocag wh proper CAM fucos. There are ay ehos of rawg a ellpse, clug foc, rael, cocerc-crcle, oblque-crcle, parallelogra [], wo-crcle a four-ceer [] ehos. The ellpses ay be regare as a soro of a crcle by uequal scale facors wo recos. I s sple bu o effce for raoal ools o eal wh, le scale a offse. I s uch ore covee o represe a ellpse by paraerc for ha o coec pos laborously eere a CC syse. Recely, Ros [3] has ae a pressve survey a coparso o four-arc approxao o ellpses. Qas [4] propose a aalycal fuco o f opu arcs o approxae a ellpse by axu error fuco. I has bee wely use CAD fel for ay years a s graually apple CAM area wh he prevalece of URBS Yua L. La s wh he Depare of Mechacal a Auoao geerg, Da-yeh Uversy, Chaghua, Tawa, R.O.C. (phoe: ; fax: , e-al: ylla@al.yu.eu.w Ja H. Che s wh he School of Physcal Therapy, Chug Sha Mecal Uversy, Tachug, Tawa, R.O.C. Ju P. Hug s wh he Depare of Auoao geerg, aoal Ch-Y Uversy of Techology, Tachug, Tawa, R.O.C. erpolaors equppe CC corollers. Wh URBS erpolao, he achg of sculpure surfaces s carre ou o a seres of URBS curves raher ha a large uber of shor les or arcs, whch s he case of CC corollers equppe wh lear a crcular erpolaors oly. Ths resuls ore flu ool pahs, aes he ool chage s ovg reco ore soohly a aas a hgher average fee rae. Alhough approxaos o crcular arcs a full crcles by paraerc fors have bee vesgae [5-9], here has bee very lle wor publshe o ellpses approxao usg URBS fora. A real ellpc par ca be ae by he correspog cug pahs gue by ool-raus copesao erpolaor. I ao, hese operaos base o paraerc curves offse algorhs [-6]. Fg. llusraes he cug as of URBS-base CC achg [7]. I oer CAD/CAM syses, ore a ore profles are represee paraerc fors o ee he requree of HSM. May researchers have evelope real e paraerc erpolaors for curve geerao usg URBS bass fuco [8-]. Soe aor coercal coroller aufacurers have brough URBS operaos o usry, such as Fauc 5M/6M [3] a Sees 84D [4]. Oce he sple fora s se o he coroller, he processor recly erpolaes he seges a exreely y ervals. The archecure of he coroller has loo-ahea feaures ha wll chage he fee-rae yacally o aap he sple o rap chages reco. Ths repor ells a eho base o opu processes a URBS curves of ay egree o represe ellpses. By pleeg he propose algorh o approxaos for ellpses a ellpses offse, he resuls show accurae perforaces evely. I ca be ae as coprehesve as oe ca age, he larger he uber of corol pos s, he hgher he qualy of effecveess s. The orgazao of he chaper s as follows. I seco he proceures for acqurg al coos are escrbe. Seco 3 scusses a buls global error fucos a seco 4 opzes locaos a weghs of all corol pos. xaples a coparsos are llusrae seco 5. I seco 6, he geerae ool pahs are verfe hrough URBS achg a close hs suy. 9

2 , f < +,(, ( oherwse +, ( + +, ( (, [ ], ~ ( +, +, s he o vecor. The ervave of a B-sple curve s: C (, (, ( P (3 (,, ( + + +, ( + + +, + ( + +, ( Fg. Machg syse for URBS curves II. STABLISH TH IITIAL CODITIOS A. quvale Approxag Curve A ses ope shape coss of ( + corol pos, hus ( + corol pos ca for a B-sple curve whch s efe by he followg equao: C (, ( A URBS curve s efe by he followg equao: C( R, (, ( w P, ( w,, ( w P ( w R, ( P Where P are vecors copose of x a y cooraes of he corol pos, w are weghs for each po,, ( are B-sple bass fucos. ( p +, p s he egree of bass fuco. For exaple, whe orer s 3, he ra he efo of URBS s,.e. he URBS expresso s expresse by, a cosa. s he orer of a B-sple curve. The bass fuco s expresse by e Boor-Cox as follows:, ( The ervave of a URBS curve s: C ( R, ( P, w(, ( w, A ( R, (, R w( (, A ( w (4, w( A ( w ( A ( ( w( If a seres of corol pos exs, oe ca refe he shape of such a curve he followg ffere ways. They offer a we rage of ools o esg a aalyze shape eforao. They are: Movg of he corol pos P. Icreasg or ecreasg he uber of corol pos. Mulple locag of he corol pos. Chagg he orer of he bass fucos. Mofyg he weghs w for each corol po. Replacg he bass fuco. (close ufor, ope ufor, o-ufor Rearragg he o vecor [ ]. Usg ulple o values o vecor. Chagg he relave spacg of he os. A ellpse ca be represee paraercally by he equaos x a cosθ (5 y bsθ Where a a b are aor a or axes, a > b. x a y are he recagular cooraes of ay po o he ellpse, a he paraeer θ s he agle a he ceer easure fro he x-axs aclocwse. Frs, he al po-sequece curve C ( s saple base o equao (5. The propose eho s show he followg seps. By usg hs eho, oe ca f he al curve corol pos. 3

3 The proceure usg he propose algorh o oba he al coo s as follows:. For a arx equao o represe he curve usg he URBS fora. D (, (, ( 3, (., ( P D,, 3,., P, D3 3, 3, 3 3, 3., 3 P D,, 3,., P + [ D ] [ ][ P] (6. Fro sep, f [] s a Square Marx he [ P] [ ] [ D] (7 3. If [] s a o-square Marx he [ D ] [ ][ P] [P]: uow, [D]: ow. T T [ P] [[ ] [ ]] [ ] [ D] (8 4. I orer o ge [], eere he paraeers, ax s l s l s s D D s s s D D D D s D D s s s s ax, l 5. Solve he above equaos o f [P], l (9 Repea he above process, wo URBS curves C a ( a C b ( are obae by specfyg wo ffere ubers of corol pos. I orer o acqure a precse approxag curve, a bgger corol po uber s prove o C a (. The oher oe s prove a reasoable corol po uber o acqure a pleeg curve C b (, whch s expece o sore fewer corol pos. Ths corol po uber epe o he aor/or rao. Two URBS curves are obae o be he al coos. If here s o pressg reaso for og oherwse, here B-Sple shoul be efe as: o ulple corol pos. Ope ufor o vecor. The fora ca be accepe by every CAD/CAM/CC syse presely use. All efaul weghs w are assge oe o each corol po of he URBS curve, whch s he sae as B-sple curve. By usg he evaluag bou error fuco, he pleeg curve C b ( wll be ofe o opu coos seco 3. Fg. Illusrao of age a oral vecors of C ( B. Approxag Offse Curve Seco, he al offse po-sequece curve s saple base o oral reco of he orgal curve, as show Fg.. Saple pos ca be obae by C ( C( + ( ( where ( s he u oral vecor of C (, s he offse sace. ( ( y (, x ( x ( + y (, ( x (, y ( (, ( Repea he sae proceures as escrbe prevous seco, a approxag offse curve C ( ca be obae. A correspog URBS curve C e ( s fore by C (. Ths offse URBS curve also ca be acqure by equao (4. Sae as above, by usg he evaluag bou error fuco, he pleeg curve C e ( wll be ofe o opu coos seco 3. I sees he sples way s o se paraeer ] sprea he curve regularly. [ III. BUILD VALUATIG BOUD RROR FUCTIOS quvale Approxag Curves There were wo popular approxag error easures a fe saple pos alog C ( by copug equaos (3 a (5 earler sues. ( Cb( ( Mze axu uclea sace: ax( ( ax( Cb( Ca ( (3 or ( Cb( (4 Mze axu squares sace: ax( ( ax( C ( C ( (5 b a The above easurg errors represe local varaos. They ee o be rasfore o easure global varaos hroughou 3

4 he curve ( ( b C a. Assug wo paraerc curves ( C a C wh slar oreao o vecors ], oe ca oralze hese wo o vecors [ ] a ] o oe o vecor,.e. [ ], ax[ ], [ ] a ax[ ], he., [,] xe he uclea sace fuco (3 a (5 bewee C a ( a C b ( o global fferece fucos: b a ( Cb( ( ( Cb( (6 b a ( Cb ( ( ( Cb( (7 where s he saplg uber. The above global error fucos are perhaps he sples C b o se ways o easure error for he approxae curves (. Afer coparg hese wo error fucos, perfors beer ha. Ths arcle uses as he B fuco. B. Approxag Offse Curves I hs seco, a very ffere eho o prevous oes s use o offse a URBS curve. The followg are he aor seps:. Bul a valuag Bou rror fuco.. Saple a offse po-sequece curve base o he orgal curve s frs ervaves. 3. Use saplg pos a pre-specfe paraeers (such as uber of corol pos, egree erao uber o for a ew offse curve as escrbe he prevous seco. 4. Relocae he frs a las corol pos o he exac posos,.e., cosras of epo C couy. 5. Durg each erao, opze he oher corol pos by usg a web-le search algorh. I orer o f a beer soluo, bul a esag fuco s spesable o hs wor o easure errors. arler sues easure approxae errors, geeral, a fe saple pos alog C a ( by copug: ( abs( ax( ( ax( abs( C ( C ( (8 [ e a where ( s he offse sace varao. lber a Cohe [5] propose aoher easurg crero o prove he offse perforace by copug: [ a ( abs( ax( ( ax( abs( C ( C ( (9 e a The above easurg error represes local varaos. They ee o be rasfore o easure global varaos hroughou he curve C a (. Assug wo paraerc curves C a a C e wh slar oreao o vecors [ ], oe ca oralze hese wo o vecors [ ] a [ ] o oe o vecor,.e. [ ], ax[ ], [ ] a ax[ ], he., [,] xe he offse sace varao fuco o area fferece fuco: ( abs( Ca ( b ( ( abs( a ( where s he saplg uber. ( abs( b ( ( abs( a ( The above area error fucos are perhaps he sples ways o easure error for he approxae offse curve C e ( o se, a copare saces bewee pos o equal paraeers value wh he offse sace. However, he po C e ( ay o le o he oral reco a he po C a ( [6]. I orer o coquer hs recoal error-proe, hs paper, we propose a ew crero o evaluae varao hroughou he curve. Afer coparg hese wo error fucos, perfor beer he. Ths arcle uses as our error fuco. Ths valuag Bou rror (B fuco cosss of offse sace a agle varao δ, was efe as: a b θ ( cos ( a b a C a (, b C ( C ( e a π δ abs( θ (, ( or b Ce (, δ ( abs( θ ( f ( cos( δ ( 3 + f ( ( where s he saplg uber. The average valuag Bou rror per u legh perceage s efe as: 3 ( B % (3 3

5 B equals o zero f orhogoaly a exac sace are preserve. Oe purpose of hs B fuco res o aa a cosa sace bewee offse a base curves. The oher ore sgfca purpose ca recfy hese wo curves approachg he sae reco. Fg. 3 Regularze he paraeers of a The average valuag Bou rror equals o zero f B orhogoaly a exac sace are preserve. Due o he coveece of coparg boh base a offse curves, hs suy regularzes he paraeers uforly a fe seges. The llusrao of relae curve s pheoeo s show Fg. 3. There are wo eos of hs error fuco, as show Fg. 3, oe purpose of hs B fuco res o aa a cosa sace bewee offse a base curves. The oher ore sgfca purpose ca recfy hese wo curves approachg sae reco. IV. LOCATIOS AD WIGHTS OPTIMIZATIO Ths suy use a raag web-le search algorh o eec opu posos. I hs eho, opu paraeers shoul be efe frs,.e. raus sep r sep, agle sep θ sep a covergece. A progra algorh s offere below o help curre corol pos o f he curre opu locaos urg each erao Beg Copue al Se curre Se Searchg_sep sep r ow sep Do ul sep < Do o _ag ( π _ ag θ sep Se ew locao for curre corol po p p x p x + r ow cos( θ, p y p y + r ow s( θ Copue If or ow > he ow a p ow p Loop If curre > he ow curre ow a r ow row + a sep p curre p ow lse row row a sep sep sep / 5 f Loop The efaul al search-raus sep r sep. Fro he above algorh, he s geg saller a saller, whe sep s less ha covergece,.e., r sep <, he search process wll sop o hs corol po. Durg each evaluag erao, we perurb he corol pos p ~ p o evaluae he area error. ach corol po chage wll affec he GB fuco a fluece he ex searchg erao sage. Thus ay chage he fal covergece suao of he opu soluo. I orer o ee G couy, he frs a las corol pos were fxe a he exac posos. The seco a h ( corol pos are resrce o locae a he age vecors of he sar a e pos of a ellpse, hus esure URBS curves ca ee he requree of G couy. A progra algorh s offere below o help curre corol pos o f he curre opu weghs urg each erao. Ths subroue ees o be reae wce, oe s posve reco, a he oher s egave reco Beg Copue al error Se o_ax 5, o_err Se curre Se Searchg_sep sep Do ul o_err > o_ax w w + sep ow Copue If ow < curre he curre ow o_err lse w w sep sep sep /5 o_erro_err+ f Loop

6 Fg. 4 (a True ellpse wh a5 a b3. (b Approxao by 96 sragh les. (c Approxao by crcular arcs. ( Approxao by a URBS curve of egree 4 a 48 corol pos. (e Approxao by a URBS curve of egree 4 a corol pos. (f Approxao by a URBS curve of egree 4 a corol pos. (g Approxao by a URBS curve of egree 5 a corol pos. (h Approxao by a URBS curve of egree 6 a corol pos V. ILLUSTRATIO AD APPLICATIOS Fg. 4(a shows a rue ellpse wh aor raus a5 a or raus b3. Fg. 4(b shows a approxao o Fg. 4(a by 96 sragh les. Fg. 4(c shows a approxao o Fg. 4(a by crcular arcs. Fg. 4( shows a approxao o Fg. 4(a by a URBS curve of egree 4 a 48 corol pos whou opu processes. Fg. 4(e shows a approxao o Fg. 4(a by a URBS curve of egree 4, ufor o vecor a corol pos. Fg. 4(f shows a approxao o Fg. 4(a by a URBS curve of egree 4, ufor o vecor a corol pos. Fg. 4(g shows a approxao o Fg. 4(a by a URBS curve of egree 5, ufor o vecor a corol pos. Fg. 4(h shows a approxao o Fg. 4(a by a URBS curve of egree 6, ufor o vecor a corol pos. Fg. 4(e-(g were obae wh URBS curves a reae by he propose opu processes. A error coparso s show Fg. 5. The error whe usg corol pos s less ha.6. Whe usg corol pos, all errors ca corol uer.. The expere resuls show ha o aer wha egree he URBS curve s ore corol pos perfors beer ha fewer corol pos..3.5 A approxao o he above ellpse offse wh a sace of s show Fg. 6. I shows a approxao o Fg. 4(a by a URBS curve of egree 5, ufor o vecor a corol pos. A error coparso s show Fg. 7. The error afer usg opu processes ca corol uer Fg. 6 A approxao o a ellpse offse Ial Locaos Op. Weghs Op Fg. 5 A error coparso 4-p 4-p 5-p 6-p Fg. 7 A error coparso The applcao of curre eho o egeerg probles s frsly show Fg. 8, where ouer a er profles are approxae by URBS curves. Fg. 9 shows a achg process of ellpses by a ache ceer wh FAUC 6M coroller, URBS-base ool pah, hgh precso coour corol (HPCC a lear oors [3]. Fg. shows a ellpc box (aor aeer s 8 a or aeer s was acqure by asseblg hree peces of ellpc MDF-boars (Meu Desy Fberboar. Ths suy preses a opu proceure o acqure a URBS curve o approxae a ellpse. The URBS-fora achg coes of FAUC corollers, lse Table I, are aope hs 34

7 suy, where coe G5 P a coe G5 P are use o acvae a ur off he HPCC fuco, respecvely. I ao, G6. s he cug sruco, P s he orer of URBS, K represes os a R represes her assocae weghs. Fg. A MDF-boar ellpc box VI. COCLUSIO Fg. 8 Tool-pah geerao of ellpses Fg. 9 MDF-boar (Meu Desy Fberboar achg of ellpses TABL I MACHIIG CODS FOR TH LLIPSS USIG URBS Ouer URBS ool-pah Ier URBS ool-pah G5 P G5 P P5 K X. Y-66.5 R. F3 P5 K X. Y R. F3 K X-7.76 Y-66. R. K X Y R. K X Y R.47 K X-36.7 Y R.68 K X-3.74 Y-4.73 R.9 K X Y R.8 K X Y4.486 R.7 K X-86.9 Y R.39 K X-9.37 Y68.57 R.7 K X-4.36 Y55.54 R.38 K X9.33 Y68.46 R.7 K X4.9 Y55.53 R.39 K3 X Y4.499 R.7 K3 X Y34.58 R.38 K4 X3.5 Y R.9 K4 X9.476 Y R.7 K5 X43.7 Y R.47 K5 X36.9 Y-49.6 R.66 K6 X7.694 Y-66.3 R. K6 X5.56 Y R. K7 X. Y-66.5 R. K7 X. Y R. G5 P G5 P Ths suy rouces a accurae algorh for ellpse approxao, a algorh base o opu processes a URBS curves of ay egree. I such a way, he orgal aa use o efe URBS curves, whch clues corol pos, weghs a os, wll be rase o corollers as G-coes. The ere curve whch was oce escrbe by ay blocs of shor les or arcs ow requres a sgle progra bloc oly. As a resul, he sze of C progras ca be grealy reuce. The sze reuco s up o / o / relave o a lear-erpolao par progra. Ths parally solve oe of he savaages of applyg lear a crcular erpolaos HSM - large sze of C progras for coplcae geoeres. URBS erpolao s oe ype of curve erpolao. To ae avaage of he capably, oe requree s a CAM syse capable of oupug URBS ool pahs. Ths eho s servceable egeerg applcaos. The propose eho eecs opu locaos for each corol po by perurbg he wh a raag web-le search algorh. These corol pos for a accurae ellpse wh a ufor o vecor whch ca be accepe by ay oer CAD/CAM/CC syse. By pleeg he slar processes bu ffere error fucos, a offse curve ca be obae wh URBS represeao. ex sage of hs research, by refg he searchg correlao bewee corol po posos a correspog weghs, he propose operag oel possbly wor o approxao o geeral curves wh URBS curves. Such operaos ca be he fouao of ool-raus copesao erpolaor of URBS curves CC syse. RFRCS [] F.. Gesece, A. Mchell, H.C. Specer, I.L. Hll, R.O. Lovg, J.T. Dygo, geerg Graphcs, Maclla, 985. [] C. Jese, a J.D. Helsel, geerg Drawg a Desg, McGraw-Hll, 985. [3] P.L. Ros, A survey a coparso of raoal pecewse crcular approxaos o he ellpse, Copuer Ae Geoerc Desg, 6, 999, pp [4] W.H. Qa, a K. Qa, Opsg he four-arc approxao o ellpses, Copuer Ae Geoerc Desg, 8,, pp.-9. [5] C. Blac, a C. Schlc, Accurae Paraerzao of Cocs by URBS, I Copuer Graphcs a Applcaos, 6(3, 996, pp [6] J.M. Carcer,. Maar, J.M. Pea, Represeg crcles wh fve corol pos, Copuer Ae Geoerc Desg,, 3, pp.5-5. [7] J.J. Chou, Hgher orer Bezer crcles, Copuer-Ae Desg, 7, 995, pp

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