NURBS curve method Taylor's launch type of interpolation arithmetic Wan-Jun Zhang1,2,3,a, Shan-Ping Gao1,b Xi-Yan Cheng 1,c &Feng Zhang2,d

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1 Advances n Engneerng Research volme 118 nd Internatonal Conference on Atomaton Mechancal Control and Comptatonal Engneerng (AMCCE 17) NURBS crve method Taylor's lanch type of nterpolaton arthmetc Wan-Jn Zhang13a Shan-Png Gao1b X-Yan Cheng 1c &Feng Zhangd 1 Qanzho Insttte of Informaton Engneerng 36 Chna School of Mechancal Engneerng Xan Jao tong Unversty Chna Lanzho Instry and Eqpment Co Ltd Lanzho 735 Chna a gszwj_4@163com b @qqcom c @qqcom d @qqcom Keywords: NURBS crve; velocty plannng nterpolaton algorthm; real-tme nterpolaton; smlaton Abstract The algorthm of NURBS nterpolaton was ntrocedt s the key for NURBS crve nterpolaton that formla of Taylor development of frst-order dervaton of two order n the nterpolaton cycle nder the conon of certan nterpolaton ncrement only and nterpolaton speed by changng the nterpolaton ncrements can acheve correcton crve of nterpolaton Ths paper ntroces a method sng DSP moton control card for NURBS crve nterpolaton methods ths method can not only shorten the nterpolaton tme also can be controlled by moton control card nterrpt tmer to adjst nterpolaton The last example shows that the nterpolaton algorthm not only meet the control reqrements bt also meets the reqrements of hgh speed and hgh precson nterpolaton 1 Introcton The traonal nmercal control machnng system ses the dscretzaton of the straght lne segment approxmaton Cased by the velocty profle crve dscontnty and flctaton and generate a lot of data ncrease the brden of the CNC system t s dffclt to meet the hgh speed hgh precson machnng needs [1] In order to solve ths problem the processng stablty hgh speed and hgh effcency are mproved the Shptaln and [] are calclated accordng to the frst order Taylor expanson approxmaton to calclate the nterpolaton ponts Yang and other [3] based on the se of two order Taylor expanson approxmaton to calclate the nterpolaton pont mprove the approxmaton error Althogh the above algorthm can obtan a constant speed wthot consderng the error control [4] For NURBS crve nterpolaton [5-8] crve nterpolaton algorthm and two step desgn expanson by Taylor these docments are abot NURBS nterpolaton crve and srface nterpolaton parameters recrsve Taylor expanson of one order and two order solton of complcated processng error Ths paper manly ntroces a NURBS crve nterpolaton algorthm of Taylor seres expanson one order and two order dervatve for NURBS crve nterpolaton and Taylor nterpolaton speed ncrement whch can be smplfed nto a dervatve nterpolaton; NURBS crve can be acheved n the real-tme nterpolaton of DSP moton control card can mprove the speed and precson of [9] nterpolaton operaton savng tme nterpolaton In modern CAD/CAM systems NURBS(Non-Unform Ratonal B-Splne) more and more wdely has sed n the CNC ma chne tools feld It has many character[1-3]: NURBS can gve a nfed mathematcal representaton for srfaces and crves NURBS can change the shape by modfyng weght vector and control pont etc Bt NURBS has a shortcomng: nterpolaton tme bgger and NURBS crve nterpolaton error are not easy changed Lanzho Instry and Eqpment Co Ltdresearchers [4-6] proposed an NURBS algorthms whch based on real-tme nterpolaton and adaptve nterpolaton Lteratre [7-9] gve some NURBS nterpolaton algorthms whch makes NC programmng complcated and nterpolaton calclate complcated Shptaln et al [9] derved the same nterpolaton algorthm by sng Taylor s expanson Hong and Yang [1] were gven Cbc splne crve nterpolator by sng Eler algorthm Lteratre [11] research on hgh-grade CNC machnes tools CNC system for B-Splne crve method of Hgh-speed Copyrght 17 the Athors Pblshed by Atlants Press Ths s an open access artcle nder the CC BY-NC lcense ( 53

2 Advances n Engneerng Research volme 118 real-tme nterpolaton arthmetclo and Chng[1-13] proposed the error nterpolaton algorthm whch error calclatons changed by crve chord On the bass of the research above a velocty plannng nterpolaton algorthm based on NURBS crve s presented n ths paper Frthermorevelocty plannng nterpolaton algorthm can meet wth NURBS crve nterpolaton We can se velocty plannng nterpolaton algorthm that calclate x y z Smlaton reslts show that the proposed NURBS crve nterpolator meet the hgh-speed and hgh-accracy nterpolaton reqrements of CNC systems ( ) Interpolaton algorthm of NURBS crve 1 Mathematcal representaton of NURBS crves In ths paper NURBS crve s sed to represent a parametrc of a Improved algorthm adaptve of NURBS crve and t s ntroced frst Spposed p can be represented a Improved algorthm adaptve of NURBS crve Whle NURBS [3] are parametrcally mathematcal defnton by the followng Eq(1): n p ω d n ω N k n k p N k (1) Where s cbc tme a Improved algorthm adaptve of NURBS crve each parameter k the order of a Improved algorthm adaptve of NURBS crve p s the control ponts ω s the weght vector N k s the blendng fncton 1 1 N k other N N k k 1 k 1 N k k 1 k 1 1 defne () By sbstttng U U c c d d m p 1 p 1 p 1 p 1 An algorthm for real tme nterpolaton of NURBS crve Taylor expanson 1 d 1 T H OT t t T t t H O T Where denotes the hgher order terms 1)Parametrc eqatons are expressed by Taylor's formla: An approxmate formla for the frst order expanson of Taylor's formla s obtaned 1 dp (4) (5) (6) Set NURBS/B crve p s expressed as T t 1 t when the nterpolaton cycle T yo can meet 1 the Taylor formla one order expanson nterpolaton approxmate teraton 54

3 Advances n Engneerng Research volme 118 (7) d p V T 1 dp dp By the formla (8) we can know that there R s a ras of crvatre of any pont R 1 dp d p p dp d p (8) p When R Where 1 dp (9) If the ras of crvatre s small the Taylor formla satsfes the two order teraton A dervatve of formla (4) and Nka Msakshkko (1) t s common type: 3 1 dp (1) NURBS crve method Taylor's lanch type of nterpolaton arthmetc By the formla (1) a recrsve approxmaton s almost eqal to the contor error psh and the two order approxmate recrsve calclaton bt the two order approxmate recrsve volme so ths paper ses an order recrsve nterpolaton algorthm By sng frst order recrsve nterpolaton the frst dervatve: x p x y z V dp p p (11) y z dx dy dz x y z 1 Calcated x y z d p x x d p y y d p z z Spposed 13 where V dp V 14 x y z In CNC system the nterpolaton perod s always constant as T t 1 t then 1 55

4 Advances n Engneerng Research volme 118 dp 1 x y z 15 From Eq(1) Eq(13) Eq(14) Eq(15)we shold get Eq(16) ω x N k ω N x ( ) ω N ω x N k k ω N k ω N k k 16 Where N k 1 N 1 k 1 N k k k k 1 1 ω N ω y k y ( ) N ω N 17 ω y k k ω N k ω N k ω N ω z N k z ( ) ω N 18 ω z N k k ω N k ω N k N k k 19 From Eq(16) Eq(17) Eq(19) we shold get ( x y z ) NURBS crve method Taylor's lanch type of nterpolaton arthmetc shold be fnshed 4 NURBS crve method Taylor's lanch type of nterpolaton arthmetc flow chart NURBS crve method Taylor's lanch type of nterpolaton arthmetc based on NURBS crve s explaned as follow: Step1: Inpt date are from NURBS crve parameters model sch as NURBS crve control ponts weght vector and so on Step: By sng two order of Taylor s expanson s Calcated 1 Step3: Calcated x y z By sng Eq(7) Eq(1) and Eq(13) the NURBS crve poston ( x y z ) can be calclated by sng 1 By sbstttng 1 dp 1 x y z Step4: NURBS crve nterpolaton s fnshed 56

5 Advances n Engneerng Research volme 118 Start Inpt NURBS crve paramter NURBS crve Calclate Taylor s expanson Calclate ( x y z ) dp 1 y z x Get calclate Reslt ( x y z ) NURBS crve method Taylor's lanch type of nterpolaton arthmetc fnshed? End Fg 1: Flowchart of algorthm 5 Experment smlaton and analyss 51 Interpolaton example In ths smlaton ths nterpolaton scheme s realzed on the moton controller developed by or own lab based on DSP TMSC543Development envronment s a PC wth AMD Sempron 81Ghz CPUGB RAM and man freqency s 144MHzmachne tool s machne center Machnng parameters and dynamcs parameters are shown n Table 1 In the paper velocty plannng nterpolaton algorthm based on NURBS crve parameter the control ponts weght vector and knot vector of NURBS for the provded example are assgned as follows: T p (x y z ) Some of the data nterpolaton calclaton as shown n Tab1 Tab 1 Some of the data nterpolaton calclaton Table I 1 3 x y z 1 x y z J 3mm s the maxmm vale of the chord The allowable acceleraton and jerk 15μm F 1mm s -1 error δ max the command nterpolaton federate The SIEMENS 81D CNC system has a wde and large share n the CNC system market and they make NC parameters as NC program command CNC machne tool Velocty plannng nterpolaton algorthm based on NURBS crve G code G6: NURBS crvetxt RT ' n the CNC machne tools complete nterpolaton G6 K4 U X1 Y Z W1 U X Y1 Z1 W1/5 U X-1 Y1 Z1 W1/5 U X- Y Z W1 U 57

6 Advances n Engneerng Research volme 118 U5 U3 Fgres and Fgres 3 for CNC machne tool n velocty plannng nterpolaton algorthm based on NURBS crve Fg : Velocty plannng nterpolaton algorthm based on NURBS crve Fg 3 : Comparson fgre of nterpolaton error Fg 4 : Comparson fgre of nterpolaton error 5 Data analyss In order to show the sperorty of velocty plannng nterpolaton algorthm the nterpolaton accracy and the nterpolaton tme and the nterpolaton tme data table are made accordng to the graph of Fgre3 as shown n Table Tab Table analyss of nterpolaton of NURBS crve reslts Parameters algorthms NURBS crve nterpolaton algorthm Velocty plannng nterpolaton algorthm Interpolaton tme(s) Max nterpolaton error (mm) Mn nterpolaton error (mm)

7 Advances n Engneerng Research volme 118 As can be seen from Fg3 and table n the process of the nterpolaton nterpolaton tme reced Max nterpolaton error deceased Mnx step error vale s 798 whch meet the expected to velocty plannng nterpolaton algorthm e to rece nterpolaton tme and Max nterpolaton error To verfy the hgh effcency and relablty of ths velocty plannng nterpolaton algorthm are appled n the experments to make a comparson It can be seen that velocty plannng nterpolaton algorthm based on NURBS crve s feasble and effcent 6 Smmary In the paper velocty plannng nterpolaton algorthm based on NURBS crve s ntroced Frstly the second-order Taylor expanson s appled on the nmerator n NURBS crve representaton wth parameter crve Thenvelocty plannng nterpolaton algorthm can meet wth NURBS crve nterpolaton We can se velocty plannng nterpolaton algorthm that calclate ( x y z ) Smlaton reslts show that the proposed NURBS crve nterpolator meet the hgh-speed and hghaccracy nterpolaton reqrements of CNC systems The nterpolaton of NURBS crve shold be fnshed In adonnc machnng tme can be reced Implementaton on NC machne has proven the feasblty of a developed nterpolaton algorthm The smlaton reslts show that the algorthm s correct; t s consstent wth a NURBS crve nterpolaton reqrements Acknowledgements The athors thank the fnancal spports from Natonal Natral Scence Fondaton of Chna(Grant no ) and Scence and Technology Major Project of Hgh-grade NC Machne Tools and Basc Manfactrng Eqpment (1ZX41-181) References [1] Fang Y-Xang L Wen-Xe Based on the geometrc propertes of the cbc nform B-Splne crve strctre descrpton[j] Jornal of engneerng graphcs pp [] Zhang Wan-Jn H Ch-Bng Zhang Feng et al Honng machne moton control card three B splne crve method of nterpolaton arthmetc for CNC system [J]Chnese Jornal of Manfactrng Technology & Machne Tool 8(8) pp4-43agst 1 [3] Zhang Wan-Jn HU Ch-bng WU Za-xn etal Research on modfcaton algorthm of Three B Splne crve nterpolaton technology [J]Chnese Jornal of Manfactrng Technology & Machne Tool pp febrary 13 [4] Zhang Wan-Jn Zhang Feng Zhang Go-ha Research on a algorthm of adaptve nterpolaton for NURBS crve [J]Appled Mechancs and Materals Vol pp16163 December 14 [5] Zhang Wan-Jn Zhang Feng Zhang Go-ha Research on modfcaton algorthm of Cbc Bsplne crve nterpolaton technology [J]Appled Mechancs and Materals Vol pp December 14 [6] Zhang Wan-Jn Zhang Feng Zhang Wan-Lang Research on a NURBS crve of tmng / nterrpt nterpolaton algorthm for CNC system [J]Chnese Jornal of Manfactrng Technology & Machne Tool 4(4) pp aprl 15 [7] Kong Fan-Go Hao Shang-HaZhongYan-Zh NURBS crve nterpolaton algorthm to acheve the VC [J] Chna new technology and new procts 17 pp8-9 9 [8] Ye Bo-ShengYang Sh-Z CNC system n cbc B-Splne crve nterpolaton method [J ] Chna Mechancal Engneerng 9 ( 3) pp [9] L He-Cheng Wang Y-Png An nterpolaton based genetc algorthm for slovng nonlnear blevel programmng problemschnese Jornal of Compters 31(6) pp Jne 8 [1] Shptaln M Koren Y Lo CC Realtme crve nterpolators Compter- Aded Desgn 6 pp

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