Five-axis Spline Interpolation Algorithm for Digital Manufacturing System
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1 3r Iteratoal Coferece o Mechatrocs, Robotcs a Atomato (ICMRA 25 Fve-axs Sple Iterpolato Algorthm for Dgtal Mafactrg System LI Hyg, a a CHE Lagj 2,b * Zhegzho Isttte of Aeroatcal Istry Maagemet, Cha 2 Zhegzho Isttte of Aeroatcal Istry Maagemet, Cha a lhyg.lhy@63.com, b chejaxclj@63.com * correspog athor Keywors: URBS crve, locato a feerate cotrollg, ctter locato ata Abstract. At preset, o-uform Ratoal B-Sple(URBS crves are mostly se to geerate arbtrary cotor of a complex shape the CAD fel. I ths artcle, a ew metho wth URBS techology s presete a realze a locato a feerate cotrollg system. I the metho, the tool paths a the ctter locatos(cl were represete to URBS crves base o the same ot vector. The above CL ata was frstly calclate a the trasforme to fve moto commas of fve axes of mache throgh the real-tme post-processo algorthm. The accelerato/ecelerato cotrollg metho s also presete to avo the mpact of mache. The propose 5-axs sple terpolato metho s realze a the expermetal reslt of machg shows that the metho s val. Itrocto Tll ow, most of complex procts sch as aeroatcal trbe blaes, mpellors, es, mols etc. are mache o 5-axs Compter mercal Cotrolle (CC maches. The core of the maches s the cotor cotrollg system whch eces machg effcecy a accracy of a part. However, the crret 5-axs CC system mostly prove oly le terpolators, that s, tool moto alog straght les s spporte. To mache a srface, the tool paths, whch are also ow as the ctter locato (CL paths, are typcally approxmate wth pecewse lear segmets by CAD/CAM systems. The approxmately machg metho has ts ow rawbacs avacg effcecy a accracy of a mache part. To overcome the savatages, t mst be tae to accot that the 5-axs cotrol system has tellget ablty to geerate the tool paths of the mache srfaces. o-uform Ratoal B-Sple (URBS, as the oly staar of ata-exchage of proct sactoe by ISO, has bee mostly se almost all CAD/CAM systems to escrbe wor-pece s srface. Base o the realty, the CC system shol be prove wth the ablty to rectly geerate 5-axs URBS crves or srfaces real-tme. Some URBS terpolato methos ha bee propose by several vestgators[2-6]. Cheg et al.[4] ha propose a real-tme URBS crve moto comma geerator for CC maches. Zhmg et al.[6] evelope a URBS crve terpolator for CC machg base o the geometrc propertes of the tool path. However, most of them ha cocetrate ther attetos o 3-axs ball-e machg, bt for 5-axs URBS srface terpolator, lttle has bee oe. Whe evelopg a realzg a 5-axs CC system wth fcto of URBS terpolato, relate sty mst be carre ot. I ths paper, a real-tme 5-axs URBS crve terpolato metho s gve a also ts format of G coe. URBS Metho A p-egree URBS crve wth parameter ca be efe as follows: 25. The athors - Pblshe by Atlats Press 286
2 C ( [ ( ( ( ] T x y z ( W P ( W where P s the th 3-D cotrol pot; W s the correspog weght factor of P ; (+ s the mber of cotrol pots;,p (, B-sple bass fcto wth egree of p, ca be calclate by the followg formla: Ï Ï ( < + Ô, Ì Ô Ó ( otherwse Ô - + p+ - Ì ( - ( + +, p-( Ô + p - + p+ -+ Ô Ô (prescrbe ÔÓ where [,..., + p + ] s the ot vector. More formato abot URBS ca be fo Ref.[]. ( (2 Locato Cotrollg Metho The tas of real-tme URBS sple terpolato s to calclate ext terpolato pero s CL ata that clg tool-tp locato coorato a tool-axs oretato a the trasfer the CL ata to mache tool s moto commas sch as X,Y,Z,A,C throgh post-processo. As show Fg., C ( a C ( s the ctter ceter pot sple a the oretato sple, respectvely. The two sples are costrcte the same ot vector. Assme that parameter s fcto of tme t, that s, (t. By sg Taylor s expaso of the parameter wth respect to tme t to obta the frst orer approxmato terpolato algorthm, the frst orer approxmato p to the frst ervatves s + + T s t t (3 t where T s s terpolato pero, a + are correspog parameters of crret a ext tme t a t +. z C ( C ( y C ( T + - e + T + + C ( + C ( + x C ( Fgre. Prcple of URBS terpolato metho The feerate of tool-tp pot alog the crve C ( s efe by 287
3 C( V ( t Sce the crve spee C( V ( Ê C ( ˆÊ ˆ Á Á Ë Ë t t The frst ervatve of wth t s obtae as V ( t t t C( Therefore, the frst-orer terpolato algorthm s obtae by sbstttg Eq.(6 to Eq.(5, Eq. (5 ca be processe as follows: TV s ( + + (7 C( The frst a seco ervatve of C ( wth s obtae as ( W P ( W ( W P - 2 C( (8 p, ( W Ê ˆ Áp, ( W Ë where the geeral algorthm for st orer ervatve of,p ( s È (, p ( ( p Í - (9 ÍÎ + p- - + p - + C( + a C ( + are obtae by sbstttg the calclate + to C ( a C (. Sppose the tool-axs t vector st +, so C ( T C ( + ( C ( - C ( + + (4 (5 (6 Feerate Cotrollg Metho I ths paper, t assme that the feerate cotrollg law s lear varable wth accelerato a a ecelerato -a a the estato cotrollg feerate s v as show Fgre 2. V v a -a S D t Fgre 2. Moel of lear feerate cotrollg For the accelerato process, the followg metho ca be se V ( at For the accelerato process, the ey problem to be solve s evalato of ecelerato pot. I Fgre 2, S D, stace of ecelerato, ca be calclate as follows S D v 2 ( 2a 288
4 The stace of ecelerato, whch s also the legth betwee ecelerato pot a estato pot alog the sple crve, ca be calclate as follows C ( Ú S D D (2 where D s correspog parameter of the ecelerato pot. The followg ewto-rapso metho s se to calclate D. If f( C ( Ú - S D (3 we have f( D Assme the tal vale of D s, that s, f( -S D,the we have the followg eqato + - f( / f ( (4 where s teger. Eq.(4 ca be eqvalet to the followg formla f( + + C( (5 f( Eqs.(4 a (5 ca be obtae sg Smpso tegral metho. Realzato The propose 5-axs URBS terpolato metho has bee realze the evelopg 5-axs CC system. The type of the 5-axs mache s table tltg/rotatg. Fg.4 shows that the CC system s cotrollg cttg tool alog a URBS crve path to mache a mpeller. Fgre 3. Applcato tegral mpeller machg Smmary Thoroghly aalyzg the efects of the exstg 5-axs lear terpolato metho se the machg of sclptre srface, 5-axs URBS terpolato metho wth locato a feerate cotrollg s presete a realze a real-tme CC system. Dfferet from the tratoal machg metho of the 5-axs lear terpolato, the propose metho ca rectly obta a crve o a free-form srface. Coseqetly, the mber of the C coe for the same machg path ca be greatly ecrease a machg effcecy a accracy s also mprove. 289
5 Acowlegmet Ths research s wholly spporte by the project of atoal atral Scece Foato of Cha mbere wth a the program for Scece & Techology Iovato Talets Uverstes of Hea Provce mbere wth 3HASTIT36 throgh Zhegzho Isttte of Aeroatcal Istry Maagemet. Refereces [] L. Pegl, O URBS: a srvey, IEEE Compter Graphcs & Applcato, ( [2] M. Tho, et al, URBS terpolator for costat materal removal rate ope C mache tools, Iteratoal Joral of Mache Tools & Mafactre, 44( [3] B. Bahr, X. Xao, a K. Krsha, A real-tme scheme of cbc parametrc crve terpolatos for CC systems, Compters Istry, 45( [4] M. Cheg, M. Tsa, a J. Ko, Real-tme URBS comma geerators for CC servo cotrollers, Iteratoal Joral of Mache Tools & Mafactre, 42( [5] R. E. Sorace, V. S. Rehart, a S. A. Vagh, Hgh-spee gtal-to-rf coverter, U.S. Patet (997. [6] X. Zhmg, C. Jcheg, a F. Zhegj, Performace evalato of a real-tme terpolato algorthm for URBS crves, Iteratoal Joral of Avace Mafactrg Techology, 2( [7] G. Qy, R. Zhag, a B. Greeway, Developmet a mplemetato of a URBS crve moto terpolator, Robotcs a Compter-Itegrate Mafactrg, 4( [8] M. Tsa, C. Cheg, a M. Cheg, A real-tme URBS srface terpolator for precso three-axs CC machg, Iteratoal Joral of Mache Tools & Mafactre, 43(
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