Computer-Aided Design. The feedrate scheduling of NURBS interpolator for CNC machine tools

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1 Computer-Aded Desgn 43 (011) Contents lsts avalable at ScenceDrect Computer-Aded Desgn journal homepage: The feedrate schedulng of NURBS nterpolator for CNC machne tools An-Chen Lee a, Mng-Tzong Ln b,, Y-Ren Pan a, Wen-Yu Ln a a Department of Mechancal Engneerng, Natonal Chao Tung Unversty, Hsnchu 30010, Tawan, ROC b Department of Mechancal Desgn Engneerng, Natonal Formosa Unversty, Yunln 6301, Tawan, ROC a r t c l e n f o a b s t r a c t Artcle hstory: Receved 18 Aprl 010 Accepted 6 February 011 Keywords: NURBS Feedrate schedulng Curve splttng Off-lne scannng Ths paper proposes an off-lne feedrate schedulng method of CNC machnes constraned by chord tolerance, acceleraton and jerk lmtatons. The off-lne process for curve scannng and feedrate schedulng s realzed as a pre-processor, whch releases the computatonal burden n real-tme task. The proposed method frst scans a non-unform ratonal B-splne (NURBS) curve and fnds out the crucal ponts wth large curvature (named as crtcal pont) or G 0 contnuty (named as breakpont). Then, the NURBS curve s dvded nto several NURBS sub-curves usng curve splttng method whch guarantees the convergence of predctor corrector nterpolaton (PCI) algorthm. The sutable feedrate at crtcal pont s adjusted accordng to the lmts of chord error, centrpetal acceleraton and jerk, and at breakpont s adjusted based on the formulaton of velocty varaton. The feedrate profle correspondng to each NURBS block s constructed accordng to the block length and the gven lmts of acceleraton and jerk. In addton, feedrate compensaton method for short NURBS blocks s performed to make the jerk-lmted feedrate profle more contnuous and precse. Because the feedrate profle s establshed n off-lne, the calculaton of NURBS nterpolaton s extremely effcent for CNC hgh-speed machnng. Fnally, smulatons and experments wth two free-form NURBS curves are conducted to verfy the feasblty and applcablty of the proposed method. 011 Elsever Ltd. All rghts reserved. 1. Introducton In order to acheve hgh-speed and hgh-accuracy machnng, many scholars devote to nvestgate the felds such as parametrc curve nterpolaton, feedrate profle schedulng and servo-loop control technques. Snce 1950s, parametrc curves lke Bezer, B-splne and NURBS have been developed. Because of the benefts of NURBS [1], NURBS even becomes the standard format of freeform curve and surface n Recently, STEP complant NC programmng, STEP-NC has been specfed as a new NC data model [,3]. NURBS s adopted by STEP-NC and becomes the standard nterface for data exchange between CAD/CAM and CNC systems. How to desgn a relable and effcent NURBS nterpolator s crtcal for developng the next generaton ntellgent CNC machne tool. Shptaln et al. [4] frst proposed parametrc curve nterpolaton and realzed the parametrc nterpolator n CNC machne, whch accepts parametrc curve codes drectly from CAD/CAM. For generatng more precse moton trajectory, Yang and Kong [5] appled Taylor s expanson method to develop the frst-order and second-order nterpolaton algorthms wth constant feedrate. Correspondng author. Tel.: ; fax: E-mal addresses: aclee@mal.nctu.edu.tw (A.-C. Lee), mtln@nfu.edu.tw, ercfshxp@gmal.com (M.-T. Ln). Nevertheless, because of hgh-order truncaton errors [6], Taylor seres nterpolator mght not generate accurate feedrate command along hgh-curvature tool paths for hgh-speed machnng. Tsa and Cheng [7] proposed a closed-loop predctor corrector nterpolator (PCI) algorthm to replace Taylor s expanson method and provded the convergent condton of corrector. The advantage of PCI method s that the feedrate fluctuaton can be controlled through settng tolerance of feedrate error for ether gven constant or varable feedrate command. Erkorkmaz and Altntas [8] proposed a quntc splne nterpolaton method to mnmze feedrate fluctuaton. Ths s done by ether approxmatng the relaton between the arc length and splne parameter usng a feed correcton polynomal or by solvng the exact parameter usng an teratve nterpolaton method. Otherwse, Le et al. [9] proposed a fast NURBS nterpolaton method whch generated nverse length functons (ILF) for each parameter subnterval n off-lne. The new settng path parameter was calculated drectly by usng the ILF wthout the need for any tme-consumng computaton of NURBS dervatves and teratons n real tme. However, most of the proposed methods attempt to mantan constant feedrate wthout consderng chord error and acceleraton/deceleraton (ACC/DEC). Yeh and Hsu [10] frst proposed an adaptve-feedrate nterpolator to adjust curve speed accordng to chord tolerance. Zhmng et al. [11] presented a curvature-based nterpolaton algorthm based on curvature of curves. Yang and Narayanaswam [1] proposed an off-lne algorthm to detect feedrate senstve corners and /$ see front matter 011 Elsever Ltd. All rghts reserved. do: /j.cad

2 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) planned ACC/DEC accordngly so that chord errors are bounded. Sun et al. [13] developed a gude splne-based feedrate schedulng method for machnng along curvlnear paths wth constrants of chord error and ACC/DEC. However, the jerk mght be out of lmt for machne f the curvature of a curve changes abruptly. To obtan a smooth jerk-lmted feedrate profle wth chord error constrant, many nterpolaton algorthms were developed n [14 17]. La et al. [17] proposed a more complete process to dentfy the segment ponts whose acceleraton changed across zero, as the bass of feedrate schedulng. The algorthm embedded n look-ahead module can plan jerk-lmted feedrate profle under varous contnuty condtons for composte curves. Nevertheless, most of the algorthms dd not nclude dynamcs effects of machne tool; thus the trackng or contourng error mght not keep wthn desred accuracy for real machnng. Lu et al. [15] consdered machnng dynamcs by utlzng notchng flterng or tme spacng based on FFT analyss to elmnate the components contanng hgh frequences or frequences matchng machne natural ones n the nterpolated acceleraton profle. However, they dd not provde the dynamcs model of machne tool and consder the effects of servo dynamcs. Ln et al. [18] and Tsa et al. [19] consdered the effect of servo dynamcs and appended servo dynamcs model to look-ahead functon. The experment results demonstrate that the trackng and contourng performance were mproved sgnfcantly. Dong and Stor [0] proposed a tme-optmal algorthm based on dynamcs of machne tool and capabltes of ndvdual moton axs. A mnmum-tme feedrate profle subject to the constrants of velocty, acceleraton, bandwdth and contourng error can be generated for a complex trajectory. Otherwse, Tkhon et al. [1] and Cho et al. [] proposed the crteron of feedrate schedulng n accordance wth materal removal rate and surface roughness, respectvely. Flesg and Spence [3] extended splne curve nterpolaton algorthm from three-axs to fve-axs machnng. Mohan et al. [4] presented a revew of varous parametrc nterpolaton methods for NURBS and dscussed the salent features, problems and solutons. Recent approaches on varable feedrate nterpolaton, parameter compensaton were also revewed and research trends were addressed. Feedrate schedulng s worthy of gong deep nto research snce t s one of the most mportant factors for achevng the effcency and qualty of machnng. Based on the archtecture of CNC controller, feedrate schedulng can be performed n on-lne or n off-lne mode. In partcular, real-tme nterpolaton algorthms wth look-ahead functon are proposed n [15,17 19]. However, whle complcated NURBS nterpolaton algorthms are performed n real tme, the large number of backtrackng process may cause tme-consumng computatons or the buffer could be used up for storng pre-nterpolated data. Falure of the nterpolator mght ncur tool chatter or breakage, even damage machne tool. To solve the problems of CNC machnng occurred n realty and to acheve the goal of hgh-speed and hgh-accuracy machnng, ths paper proposes an off-lne feedrate schedulng method for CNC machnes. The factors that affect machnng precson such as chord error, feedrate fluctuaton, and machne knematcs constrants are consdered smultaneously. The method frst fnds out crucal ponts as the bass of feedrate schedulng and splts a NURBS curve at breakponts nto several NURBS sub-curves. Therefore, the dvergence of PCI nterpolaton method can be avoded. Accordngly, the unque feedrate profle constraned by chord tolerance, acceleraton and jerk lmtatons for a NURBS curve s constructed. In the real-tme process, t only needs to perform PCI algorthm for updatng path parameter and the Cox de Boor algorthm for generatng nterpolaton pont. Takng the advantage of the proposed method, the complcated and heavy calculaton of backtrackng process n real tme can be avoded whle mantanng the desred precson wthn machne knematcs constrants.. NURBS curves and nterpolaton algorthms A NURBS curve C(u) can be expressed as follows [1]: C(u) = n N,p (u)w P =0 (1) n N,p (u)w =0 where P s the control pont, w s the correspondng weght of P, (n + 1) s the number of control ponts, and p s the degree of a NURBS curve. N.p (u) s the pth-degree B-splne bass functon defned on the non-unform knot vector U = {u 0, u 1,..., u n+p+1 }. The pth-degree B-splne bass functon s recursvely defned as follows 1 f u u < u N,0 (u) = +1 () 0 otherwse N,p (u) = u u N,p 1 (u) + u +p+1 u N +1,p 1 (u) u +p u u +p+1 u +1 = 0, 1,..., n. (3) For generatng a moton trajectory of parametrc curve C(u), the frst step s to determne the curve parameter u. Taylor seres expanson method s adopted n most NURBS nterpolaton algorthms. By employng Taylor s expansons of u(t) at t = t and neglectng hgh-order terms, the second-order Taylor nterpolaton algorthm s gven as [6]: u +1 = u + V(u ) T s C (u ) + 1 C (u ) A(u ) C (u ) C (u ) C (u ) 3 V(u ) T s where V(u ), A(u ), T s, C (u ) and C (u ) are the feedrate, acceleraton, samplng tme, frst and second dervatves of a NURBS curve, respectvely. For further reducng feedrate fluctuaton, the PCI method [7] s chosen to update the parameter u n ths paper. In the predctor stage, the nterpolaton command at the next samplng tme s estmated usng the equaton: u +1 = 3u 3u 1 + u. (5) In the correct stage, the followng equatons are utlzed to repeatedly update u +1 wthn samplng perod untl the specfed feedrate accuracy s satsfed. u (j) +1 = α(u u (j 1) +1 ) + u j 1 (6) α = β(v V (j 1) = V V (j 1) ) + V ) +1 C(u ) C(u (j 1) (4) (7) T s (8) where u s the value of parameter u at tme t, u (j) +1 s the value of parameter u after j teraton step at tme t +1, β s the correctonal coeffcent, V s the desred feedrate command at u and V (j 1) s the current feedrate command computed at u after (j 1) teraton, respectvely. The termnaton condton for the corrector s chosen as V V (j 1) V ε PCI (9) where ε PCI s the tolerance of feedrate command error.

3 614 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) Fg. 1. The flowchart of feedrate schedulng method. 3. Off-lne curve scannng In order to plan the feedrate profle of any gven NURBS curve, an off-lne process for curve scannng and feedrate schedulng s developed as a pre-processor, whch releases the computatonal burden n real-tme task. The flowchart of the proposed feedrate schedulng method s shown n Fg. 1. In the stage of curve scannng, the breakponts wth G 0 contnuty are detected by checkng the multplcty of knots n the knot vector of a NURBS curve, and the curve s splt nto several NURBS sub-curves. The adaptve-feedrate wth curvature-based feedrate nterpolaton algorthm [18] s utlzed to scan a NURBS curve for determnng crtcal ponts wth large curvatures. Furthermore, the sutable feedrate V at each breakpont s adjusted accordng to the lmts of velocty varaton on moton axes. The sutable feedrate V at each crtcal pont s evaluated wth the constrants of chord error, centrpetal acceleraton and jerk lmtatons. Consequently, the NURBS curve s dvded nto small NURBS blocks after determnng the breakponts and crtcal ponts, whch are called crucal ponts. The curve parameters u of crucal ponts are recoded smultaneously. The length S of each NURBS block between two adjacent crucal ponts s estmated by the adaptve Lobatto quadrature method [5]. Fnally, the scannng data (u, V, S ) for each NURBS sub-curve and block are obtaned and ready for the next stage of feedrate schedulng presented n Secton Knematc constrants To generate smooth tool path for NURBS curves, jerk-lmted feedrate profle should be planned n terms of acceleraton and jerk lmts. Here, A max and J max are denoted as acceleraton and jerk lmts on each axs for CNC machne, respectvely. The lmts

4 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) Table 1 Parameters of a hat curve. Parameters Fg.. The hat curve. Items Control ponts:p 9 3 (0, 0, 0); ( 150, 50, 0); ( 50, 50, 0); (0, 150, 0); (150, 150, 0); (150, 0, 0); (50, 50, 0); (50, 150, 0); (0, 0, 0) Knot vector:u 1 1 0, 0, 0, 1/6, 1/3, 1/3, /3, /3, 5/6, 1, 1, 1 Weghts:w 1 9 1,, 3, 1, , 1, 3,, 1 Degree:p of acceleraton and jerk n the normal and tangent drectons are set as A n = A t = A max J n = J t = J max (10) where A n, A t, J n and J t are the centrpetal acceleraton, tangent acceleraton, centrpetal jerk and tangent jerk, respectvely. It demonstrates that f the feedrate profle s planned wth the constrants of maxmum centrpetal and tangent acceleratons, the acceleraton on each axs would not exceed the lmtaton and the same token for jerk. Therefore, tool chatterng or system vbraton due to hgh jerk can be avoded when adoptng the settngs. 3.. NURBS curve splttng on breakponts The functon of curve splttng plays an mportant role n the off-lne curve scannng method. Snce G 1 contnuty s necessary for calculatng curve length accurately by numercal ntegraton method, f a pth-degree NURBS curve has p repeated knots n the knot vector except frst and last (p + 1) knots, the curve should be splt nto at least two sub-curves. If a knot has multplcty r = p, a breakpont wth G 0 contnuty may occur. After nsertng (p r +1) knots at the knot wth multplcty r, an orgnal NURBS curve can be splt nto two NURBS sub-curves usng the recursve algorthm of knot refnement [1]. A hat curve s provded here to llustrate the concept of curve splttng. The curve s a degree two NURBS curve wth 9 control ponts shown n Fg.. The parameters of the curve are lsted n Table 1 unless stated otherwse. Snce ts knot vector U = {0, 0, 0, 1/6, 1/3, 1/3, /3, 5/6, 1, 1, 1} has two knots wth multplcty at the parameter u.e., 1/3 or /3, two breakponts represented by D and E are detected. Therefore, the curve s splt nto three new NURBS sub-curves such as AD, DE and EA shown n Fg.. In ths paper, the values of two knots and two breakponts are recorded n a curve scannng algorthm. The scannng data wll be used to adjust the feedrate profle and to estmate the length of a NURBS curve n Sectons 3.3 and 3.5, respectvely. Fg. 3. Velocty varatons of two axes across a breakpont Feedrate adjustng at breakponts wth G 0 contnuty Whle the tool moves across breakponts wth G 0 contnuty, t may result n volent change of acceleraton or jerk on each axs. To mantan the contnuty of acceleraton at the breakponts, t s neffcent that the feedrate n one of the axes should approach to zero. The method n [6] for lmtng velocty varaton at breakponts s adopted to solve ths problem. As shown n Fg. 3, P 1, P and P +1 are the current poston, breakpont and next poston, F s the feedrate command across the breakpont, θ 1 s the angle between the lne P 1 P and x-axs, θ s the angle between the lne P P +1 and x-axs. Therefore, the velocty varaton on x-axs and y-axs are gven as V x = F cos θ cos θ 1 V y = F sn θ sn θ 1. (11) If the velocty varaton on any axs s larger than the lmt of velocty varaton V max, t s necessary to adjust the feedrate command F to meet the constrants of velocty varaton, acceleraton and jerk usng the followng equatons: V max = mn A max T s, J max T s / R v = max(v x, V y )/V max V = F mn A max T s, J max T = / s R v max( cos θ cos θ 1, sn θ sn θ 1 ) (1) where A max, J max and R v are the maxmum acceleraton and jerk on each axs, and scalng factor, respectvely. Here, V max s set as mn A max T s, J max T s /. Eq. (1) llustrates that the acceleraton or jerk on any axs caused by velocty varaton should never exceed the lmts of acceleraton and jerk wthn one samplng tme. The curve parameter u and sutable feedrate V at each breakpont are recorded and wll be used to plan the jerk-lmted feedrate profle n Secton Feedrate adjustng at crtcal ponts wth large curvatures Although feedrates at breakponts have been adjusted accordng to knematc property n Secton 3.3, sharp corners wth large curvatures n a NURBS curve stll could volate knematc property for hgh-speed machnng. To fnd out crtcal ponts whch are generally called sharp corners and determne ther correspondng feedrates, three constrants of chord error, centrpetal acceleraton and jerk are consdered smultaneously n Eq. (13). The crtcal curvature κ cr for dentfyng the crtcal ponts s gven as κ cr = mn 8δ (V max T s ) + 4δ, A n, Vmax J n V 3 max (13)

5 616 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) a b c d Fg. 4. The geometry and knematc propertes of a hat contour (a) curvature profle of hat curve; (b) crtcal ponts n Cases I and VI; (c) crtcal ponts n Cases II and III; (d) crtcal ponts and breakponts n Case II. where δ, T s and V max are chord tolerance, samplng tme and maxmum feedrate, respectvely. In Eq. (13), the frst condton s derved from the adaptve-feedrate nterpolaton algorthm [10]; the second and thrd condtons are derved from the equatons of centrpetal acceleraton and jerk n [17]. Any nterpolaton pont wth ts curvature κ beng larger than the crtcal curvature s defned as a canddate pont snce t exceeds one of three constrants. The pont whch has the local maxmum curvature among the canddate ponts s defned as a crtcal pont and ts sutable feedrate V s modfed accordng to the followng equaton. V = mn 1 T s κ 1 A n J δ,, 3 n (14) κ κ κ where κ s the curvature of the crtcal pont. The hat curve s provded to further demonstrate the concept. The contour and curvature profles of the hat curve are shown n Fgs. and 4(a), respectvely. In general, one could expect that four ponts marked as B, C, F and G should be dentfed as crtcal ponts because the curvatures at these ponts are local maxmum. In fact, the ponts C and F mght satsfy three constrants under some geometry and knematc condtons, only the ponts B and G need to be regarded as crtcal ponts. In order to demonstrate how many crtcal ponts are detected under dfferent condtons, four cases are tested and the correspondng parameters are lsted n Table. Case I ncludes the followng default values such as δ = 1 µm, V max = 100 mm/s, A n = 800 mm/s and J n = mm/s. In Case I, two crtcal ponts of B and G are detected as shown n Fg. 4(a) and (b). The crtcal curvature (κ cr = 0.08) s determned by the second condton n Eq. (13), and the sutable feedrates at the crtcal ponts are constraned by centrpetal acceleraton. When the gven feedrate ncreases to 50 mm/s n Case II, four crtcal ponts of B, C, F and G are obtaned as shown n Fg. 4(c). The crtcal curvature (κ cr = 0.018) s stll determned by the second condton. As compared wth Case I, f the chord tolerance s reduced from 1 µm to 0.1 µm n Case III, four crtcal ponts of B, C, F and G are detected. The crtcal curvature (κ cr = 0.0) s determned by the frst condton. The sutable feedrates are lmted by chord tolerance. As compared wth Case I, f the centrpetal acceleraton ncreases to 000 mm/s n Case VI, only two crtcal ponts marked as B and G are detected. The crtcal curvature (κ cr = 0.016) s determned by the thrd condton. The sutable feedrates are constraned by centrpetal jerk. Fnally, the curve parameters u and sutable feedrates V at the crtcal ponts are stored and wll be utlzed n Sectons 3.5 and The length estmaton of NURBS blocks After detectng all breakponts and crtcal ponts wthn a NURBS curve, ther parameters u and sutable feedrate V are obtaned; the curve s dvded nto small NURBS blocks correspondng to the scannng data (u, V ). For example, the hat curve s dvded nto seven NURBS blocks AB, BC, CD, DE, EF, FG and GA shown n Fg. 4(d). The length of each NURBS block s requred for feedrate schedulng. However, t s dffcult to calculate the length by ntegratng NURBS parametrc equaton. The adaptve Lobatto quadrature method [5] s adopted to estmate the length of each NURBS block accurately. Gven a tolerance of length error ε, the length of each block S s calculated over the parameter nterval

6 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) Table The parameters of four test cases for the hat curve (fxed J n = mm/s 3 ). Chord tolerance δ (µm) Feedrate V max (mm/mn) Centrpetal acceleraton A n (mm/s ) Crtcal curvature κ cr (1/mm) Case I mn(0.0, 0.08, 0.16) Case II mn(0.03, 0.018, 0.041) Case III mn(0.0, 0.08, 0.16) Case VI mn(0.199, 0.0, 0.16) Fg. 5. Feedrate schedulng for a NURBS block. u s, ue usng the followng equaton: S u e u s C (u)du ± ε (15) where s the ndex of each NURBS block and u s /ue are the start/end curve parameters of th block. After performng the procedure of curve scannng, the scannng data (u, V, S ) for each NURBS block wll be provded to the feedrate schedulng algorthm whch s presented n the next secton. 4. Feedrate schedulng algorthm When the process of curve scannng s fnshed, the scannng data (u, V, S ) are obtaned and stored. In the stage of feedrate schedulng, the goal s to construct a jerk-lmted feedrate profle for each NURBS block through the scannng data. Its concept s shown n Fg. 5, where C(u ) and C(u +1 ) are the start and end ponts n each NURBS clock, V and V +1 are the correspondng feedrates at the start/end ponts, S s the length of each NURBS block. V s, V e and V fs denote the start, end, and constant feedrate after feedrate schedulng, respectvely. N a, N c and N d are the number of samplng tme n the ACC, constant feedrate (CF) and DEC sectons of feedrate profle. Fnally, the feedrate profle and correspondng parameters (V s, V fs, V e, N a, N c, N d ) of a NURBS curve are obtaned by connectng those feedrate profles for each NURBS block Sne-curve velocty profle Snce sne-curve velocty profle shown n Fg. 6 s more contnuous than trapezod and blended splne velocty profle, t s chosen to generate the feedrate profle for each NURBS block n ths paper and ts velocty equaton s gven as [7]: V tan (t) [ V fs V s t t0 sn π 1 T a = V fs t s t < t c [ V fs V e t tc sn π 3 T d ] V s, t 0 t < t s (16) ] V e, t c t < t e Fg. 6. Sne-curve velocty profle. where T a = N a T s and T d = N d T s. Dfferentatng Eq. (16) yelds the acceleraton equaton, V fs V s π t t0 cos π 1, t 0 t < t s T a T a A tan (t) = 0 t s t < t c (17) V fs V e π t tc cos π 3, t c t < t e. T a T a Dfferentatng Eq. (17), one obtans the jerk equaton, J tan (t) V fs V s π t t0 sn π 1 t 0 t < t s T a T a = 0 t s t < t c (18) V fs V e π t tc sn π 3, t c t < t e. T a T a The lmts of tangent acceleraton and jerk can be determned by settng the feedrate V fs = V max : V A tan (t) = fs V s π t t0 cos π 1 T a T a V fs V s π A t T a V fs V s π t t0 J tan (t) = sn π 1 (19) T a T a V fs V s π J t. T a

7 618 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) Feedrate profle generaton for each NURBS block In ths secton, the procedure of feedrate schedulng for each NURBS block s shown n Fg. 7 and the detals are llustrated schematcally as follows. Step I. In the frst step, t needs to calculate the number of samplng tme N a and N d n the ACC/DEC sectons of sne-curve profle. For gven start feedrate V s, allowable feedrate V fs = V max and lmts of tangent acceleraton and jerk, the number N a can be derved from Eq. (19), N a = max V fs V s π V fs V s 1 π,. (0) A t T s J t T s The process for calculatng N d s smlar to that of calculatng N a. The areas under the ACC/DEC sectons of sne-curve profle are derved as ts [ Vmax V s t t0 S a = sn π 1 ] V s dt T a t 0 = (V s + V max ) N a T s te [ Vmax V e t tc S d = sn π 3 ] V e dt T d t c = (V e + V max ) N d T s. (1) Therefore, the number of samplng tme n the CF secton of sne-curve profle s obtaned: N c = S S a S d V max T s. () If N c > 0, t means that the length of NURBS block S s long enough to plan the CF secton wth maxmum feedrate V max, and the feedrate profle s planned as Fg. 7(a). Otherwse, the process goes to Step II. Step II. For the feedrate profle wthout CF secton, t s necessary to reduce the feedrate from maxmum feedrate V max to allowable feedrate V fs. It s represented as Na 1 [ (V fs V s ) j sn π + 1] T s + V s N a T s j=0 + Nd 1 j=0 1 N a [ (V fs V e ) j sn π 3 N d + 1] T s + V e N d T s + V fs N c T s = S. (3) V fs = Smplfyng Eq. (3), one obtans the allowable feedrate V fs as S T s + 1 (av s + bv e ) V s N a V e N d (4) (a+b) + N c where a = N a 1 j j=0 [sn π( N a 1 ) + 1] and b = N d 1 j j=0 [sn π( N d 3 ) + 1]. If the V fs s greater than max (V s, V e ), the process goes to feedrate schedulng wthout CF secton as shown n Fg. 7(b). In ths case, N a and N d are calculated usng Eq. (0), and N c s equal to zero. There are three cases accordng to the relaton between V s and V e (.e., V s > V e or V s < V e or V s = V e ). Otherwse, the acceleraton or jerk may exceed the lmtaton, so that feedrate schedulng must be performed further and the process goes to Step III for the case of V fs max(v s, V e ). Step III. If the allowable feedrate V fs s greater than mn (V s, V e ), the feedrate profle wth only ACC or DEC secton s consdered as shown n Fg. 7(c). In ths case, the V fs s set as max (V s, V e ), and the parameter N a or N d s evaluated from Eq. (0). Eq. (4) s used to calculate the V fs. Note that f V s > V e, the parameters N a, N c and a n Eq. (4) are set to zero; otherwse, the parameters N d, N c and b are set to zero. If the V fs s smaller than mn (V s, V e ) or equal to mn (V s, V e ), the acceleraton or jerk stll could exceed the lmtaton. Therefore, t needs to perform feedrate schedulng further and goes to Step IV. Step IV. For the case of V fs mn(v s, V e ), the feedrate profle wth only CF secton s adopted as shown n Fg. 7(d). The feedrates V s and V e are all set as mn (V s, V e ). Therefore, the length of NURBS block S s the only condton for feedrate schedulng. The V fs s determned through Eq. (4) where the parameters N a, N d, a and b are set to zero, and the process of feedrate schedulng for each NURBS block s fnshed. In summary, the concept of feedrate schedulng for each NURBS block s summarzed as follows: 1. By the constrants of tangent acceleraton and jerk, the numbers of samplng tme n the ACC/DEC secton of sne-curve velocty profle can be evaluated from Eq. (0).. Accordng to the length of NURBS block, the allowable feedrate V fs after feedrate schedulng s evaluated from Eq. (4). 3. Fnally, the feedrate schedulng data (V s, V fs, V e, N a, N c, N d ) are determned by the feedrate schedulng algorthm Short NURBS block determnaton and pror handlng In the process of feedrate schedulng proposed n the last secton, the boundary condton V s or V e may be changed n some stuatons. Ths problem may cause jump and dscontnuty at the juncton of two feedrate profles correspondng to two adjacent NURBS blocks. Furthermore, the acceleraton and jerk could exceed the lmtaton. For example, f the length of NURBS block s too short, ts feedrate profle could be changed from the form of Fg. 7(b) to the form of Fg. 7(c) or (d) n the process of feedrate schedulng, so V s or V e s changed, too. Therefore, one hope to determne how short NURBS block the boundary condton may be changed. As shown n Fg. 8, the crteron length of NURBS block satsfed V fs = max(v s, V e ) can be derved as 1 (V s + V e )N a T s + V e N d T s, V s < V e S std = 1 (V (5) s + V e )N d T s + V s N a T s, V s > V e (N a + N d )V s T s V s = V e. Through Eq. (5), f S S std, the boundary condton s not changed after feedrate schedulng process and the correspondng NURBS block s called a long NURBS block. Otherwse, the NURBS block s called a short NURBS block. For avodng the jump and dscontnuty at the juncton of two feedrate profles, t s necessary to check f the next NURBS block s short or not before schedulng the feedrate profle for th NURBS block. If so, searchng all of the followng short NURBS blocks s performed n prorty, e.g., ( + 1)th, ( + )th,..., ( + k)th block are short. After feedrate schedulng for those short NURBS blocks and th NURBS block s performed n the opposte drecton, e.g. ( + k)th, ( + k 1)th,..., ( + 1)th, th, the work of feedrate schedulng s moved to (+k)th block and goes on. The concept of feedrate reschedulng for short NURBS blocks s llustrated n Fg. 9.

8 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) Fg. 7. The flowchart of feedrate profle generaton for each NURBS block. Fg. 8. The crteron of short NURBS block Termnal error compensaton of NURBS block The work of off-lne feedrate schedulng for the whole NURBS curve s over up to here. Snce the length S of each NURBS block s estmated by the adaptve Lobatto quadrature method, estmaton error exsts between actual movng length and estmated length, and t may cause a problem of termnal error after nterpolaton. The actual movng length S tr after nterpolaton and the estmaton error S are calculated usng the followng equatons: N tr S tr = C(uj+1 ) C(u j ) (6) j=1 S = S tr S (7) where s the ndex of NURBS block and N tr s the number of nterpolaton ponts wthn the th block. To solve the termnal error problem, the error dstance s compensated by desgnng a compensated feedrate curve for dstrbutng t n the last N com

9 60 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) Table 3 Parameters of knematc constrants and feedrate schedulng method. Parameters Symbols Unts Samplng tme T s 0.00 s Chord error δ 1 µm Maxmum feedrate V max 50 mm/s Maxmum acceleraton A max 800 mm/s Maxmum jerk J max mm/s 3 Length estmaton error ε 0.1 µm Feedrate error of PCI ε PCI 0.1% The correctonal coeffcent of PCI β 0.9 The number of feedrate compensaton N com 50 samplng tme. The sne-curve nteror π, 3π s utlzed as the compensated feedrate curve shown at the bottom of Fg. 1. Its equaton s represented as V(j) = V [ com j sn π N 1 ] + 1. (8) Let the area under the compensated feedrate curve be equal to S, and represented as S = N com 1 j=0 V(j)T s (9) where N com s the number of samplng tme of the compensated feedrate curve. The parameter V com can be obtaned from Eqs. (8) and (9). If V com s postve, the actual movng length of NURBS block s larger than the estmated length,.e. S > 0; otherwse, the actual movng length s smaller than the estmated length,.e. S < 0. To calculate the error dstance, the nterpolaton should be performed for each NURBS block. The sum of the orgnal feedrate profle and compensated feedrate profle becomes the new feedrate command profle whch s used n the fnal N com tmes nterpolaton. It s noted that only N com samplng ponts need to backtrack, and the compensated feedrate curve generates new curve ponts n real-tme nterpolaton process. Fnally, the problem of termnal error s elmnated va the process of feedrate compensaton. 5. Smulaton and expermental verfcaton In the followng, analytcal smulatons and experments are performed to demonstrate the feasblty and applcablty of the proposed feedrate schedulng method. The performance evaluaton among frst-order Taylor, second-order Taylor and PCI nterpolaton algorthms are also performed to verfy the effcency of the proposed method Effcency and accuracy analyss The envronment of smulaton conssts of Intel Core GHz personal computer wth Wndows XP operatng system, and the proposed feedrate schedulng method s developed by MATLAB. The PCI nterpolaton algorthm n [7] s appled to generate curve parameter. Even though PCI has a dsadvantage of stablty problem that corrector may not converge at breakponts wth G 0 contnuty, t stll works n ths paper snce the NURBS curve whch contans the breakponts has been splt nto several NURBS sub-curves. In ths secton, two free-form curves are chosen to smulate and to compare the effcency and accuracy among the frst-order Taylor, second-order Taylor and PCI nterpolaton algorthms. The parameters of knematc constrants and feedrate schedulng method are presented n Table 3 unless stated otherwse. To llustrate the computng effcency, the number and total clock cycles of arthmetc operatons requred to realze varous nterpolaton technques are summarzed n Tables 4 and 5, respectvely. The degree of NURBS curve p, the number of moton axes N a and the teraton number of PCI N j are consdered as performance nfluencng factors. To mprove feedrate accuracy, PCI algorthm requres to perform Eqs. (6) (9) recursvely. Snce the Cox de Boor algorthm needs p(p+1) tmes of functon calls to generate an nterpolaton pont C(u), the total number of ADD, SUB, MUL and DIV operatons for PCI ncreases proportonal to the factor p(p+1) (N j 1)N a n [9]. In contrast to PCI, snce the Taylor seres methods only need to calculate the frst and second dervatves of a NURBS curve n Eq. (4) usng the Cox de Boor algorthm, the total number for the frst- and second-order Taylor methods ncreases p(p 1) proportonal to the factors N a and N a (p 1), respectvely. Therefore, the computaton of PCI algorthm consumes the most clock cycles as compared wth the others shown n Table 5. In the hat case wth p =, N a = 3, N j =, the total clock cycles consumed by PCI s about 4.08 and 6.63 tmes of that consumed by the frst- and second-order Taylor methods, respectvely. Although computng effcency of PCI algorthm s not better than the Taylor seres methods, t s stll worth to apply PCI method to reduce feedrate fluctuaton for hgh-speed machnng. Therefore, the hat and butterfly curves are tested to prove the accuracy and feasblty of the proposed feedrate schedulng algorthm wth PCI method. The adaptve-feedrate nterpolaton algorthm (ADA), adaptve-feedrate wth curvature-based feedrate nterpolaton algorthm (ADACB) and proposed feedrate schedulng method (FS) are adopted to plan the feedrate profle of a NURBS curve; the frstand second-order Taylor methods, and PCI methods are appled to generate an nterpolaton pont. The comparsons of feedrate errors among varous NURBS nterpolaton algorthms are summarzed n Table 6. It s clear that the PCI method acheves the best feedrate accuracy as compared wth the Taylor methods for the two curves. Furthermore, the FS wth PCI method can outperform all of the other algorthms both n maxmum and root mean square of feedrate accuracy. The correspondng feedrate error s wthn the tolerance 0.1%, and root mean square of feedrate error s only 0.04%. It s noted that the ADA algorthm wth frst-order Taylor method s not sutable for hgh-speed machnng snce the maxmum feedrate fluctuatons are over a hundred percent of gven feedrate command. In the followng sectons, smulatons are conducted to demonstrate the feasblty and applcablty of the proposed feedrate schedulng method. Table 4 Arthmetc operatons of varous NURBS nterpolaton methods. Arthmetc operaton Frst-order Taylor Second-order Taylor PCI ADD N a + N a p(p 1)/ N a (p 1) + N a + (N j 1) [N a p(p + 1)/ + N a + 1] + N a (p + p + 1) SUB 3N a p(p 1)/ 3N a (p 1) + 1 MUL N a (p p + 1) + 1 N a (p 1) + N a + 13 (N j 1) [N a p(p + 1)/ + N a + 3]+N a (3p + 3p+1)+ (N j 1) N a (p + p + 1) + + N a (p + p + 1) + DIV N a p(p 1)/ + 1 N a (p 1) + 4 (N j 1) [N a p(p + 1)/ + 3] + N a (p + p) + SQRT 1 1 N j Note. ADD: Addton, SUB: Subtracton, MUL: Multplcaton, DIV: Dvson, SQRT: Square Root, p: the degree of NURBS curve, N a : the number of moton axes, N j : the number of teratons for PCI.

10 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) Fg. 9. The reschedulng process for short NURBS blocks. Table 5 Total clock cycles of varous NURBS nterpolaton methods (fxed N a = 3, N j = ). NURBS curve Frst-order Taylor Second-order Taylor PCI Hat (p = ) Butterfly (p = 3) Note. The clock cycles for performng double-precson arthmetc operatons on Intel Core 7 chp are gven as ADD(3), SUB(3), MUL(5), DIV(4), SQRT(34) n [8]. 5.. Smulatons of hat curve The frst NURBS curve s a hat contour shown n Fg. and the parameters of smulatons are lsted n Table 3. The hat curve has nne control ponts and ts degree s two. Fg. 10(b) llustrates that the proposed algorthm detects three breakponts,.e., A, D, and E, and dvdes the curve nto three NURBS sub-curves (AD, DE and EA) usng the curve splttng method. Furthermore, the proposed method obtans four zones where the curvatures are larger than the crtcal curvature κ rc = as shown n Fg. 10(c). It also fnds out four crtcal ponts,.e., B, C, F, and G, and further dvdes the curve nto seven NURBS blocks such as AB, BC, CD, DE, EF, FG and GA based on the breakponts and crtcal ponts. Fnally, the sne-curve feedrate profle for the hat curve s planned as shown n Fg. 10(d). As shown n Fg. 11(a) (c), the profles of chord error, acceleraton and jerk generated by the proposed method are all constraned on the values of 1 µm, 800 mm/s and mm/s 3, respectvely. As shown n Fg. 11(d) (f), the velocty, acceleraton and jerk profles of x-axs and y-axs are also constraned on the gven lmts of 50 mm/s, 800 mm/s and mm/s 3, respectvely. From the smulaton results, t demonstrates that the precson and the knematc constrants of CNC machne are all satsfed. It s obvous that the curve conssts of two breakponts D and E wth G 0 contnuty wthn the curve. Snce the drectons of the moton path change nstantaneously whle crossng these two breakponts, the feedrate varaton would cause the ndvdual moton axs to exceed the constrants of acceleraton and jerk f the feedrates are not adjusted. Ths example shows that not only the proposed algorthm can correct the feedrates at two breakponts of D and E to sutable values usng Eq. (1), but also the acceleraton and jerk are bounded. Besdes, the feedrates at four crtcal ponts of B, C, F, and G are adjusted accordng to Eq. (14). Note that the condtons for the breakponts wth G 0 contnuty are strcter than those for crtcal ponts wth G 1 contnuty. That s why the feedrates at the breakponts of D and E are far less than those at the crtcal ponts of B, C, F, and G shown n Fg. 10(d) Smulatons of butterfly curve To further llustrate the merts of the proposed feedrate schedulng method, one consders the second NURBS curve whch s more complex than the hat curve. The butterfly curve has 51 control ponts and ts degree s three n [9]. Fg. 1(a) (d) show the butterfly contour, ts sub-curves and blocks, ts curvature and feedrate profle, respectvely. For the gven parameters lsted n Table 3, the crtcal curvature κ cr s obtaned as 0.018, the proposed method dentfes 31 crtcal ponts marked as C 1 to C 31 as shown n Fg. 1(b) (c), and the curve s dvded nto 3 NURBS blocks ncludng C 0 C 1, C 1 C, C C 3, C 3 C 4,..., C 30 C 31 and C 31 C 0. The algorthm further estmates the length of each block and plans the sne-curve feedrate profle for each block. It s noted that the curve detects ten short blocks such as C 4 C 5, C 8 C 9, C 9 C 10, C 13 C 14, C 14 C 15, C 15 C 16, C 16 C 17, C C 3, C 3 C 4 and C 8 C 9 utlzng the crteron of short NURBS block n Eq. (5). Therefore, the short blocks are combned wth the nearby long blocks through the feedrate reschedulng process as shown n Fg. 9. For example, snce the block C 4 C 5 s too short and ts end velocty V e s too low, ts deceleraton profle cannot be planned Table 6 Feedrate errors of varous NURBS nterpolaton algorthms (fxed V max = 50 mm/s). NURBS curve Hat Butterfly Interpolaton algorthm ADA ADACB FS (proposed) MAX/RMS (%) MAX/RMS (%) MAX/RMS (%) Frst-order Taylor / / /4.0 Second-order Taylor / / /4.10 PCI 0.10/ / /0.04 Frst-order Taylor 1.8/ / /.06 Second-order Taylor 3.14/ / /0.8 PCI 0.10/ / /0.04 Note. ADA: Adaptve-feedrate nterpolaton algorthm, ADACB: Adaptve-feedrate wth curvature-based feedrate nterpolaton algorthm, FS: The proposed feedrate schedulng method.

11 6 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) Table 7 Parameters of the servo control systems and moton controllers. Quantty Symbol Value Unts x-axs y-axs z-axs Servo system dynamcs a s b mm/volt Poston controller K pp /s Velocty controller K vp Volts/mm T /s Velocty feedforward controller K vff ω n Hz a b c d Fg. 10. The hat curve: (a) contour, (b) sub-curves and blocks, (c) curvature profle and (d) feedrate profle. under the gven jerk constrant. The proposed algorthm combnes the long block C 3 C 4 and the short block C 4 C 5 nto one long block C 3 C 5, and plans the deceleraton profle for the block as shown n Fg. 7(c.). Fnally, the algorthm reschedules the feedrate profle for each combned block shown n Fg. 1(d), and the number of NURBS blocks decreases from 3 to 6. Comparng Fg. 1(c) wth Fg. 1(d), t llustrates that the feedrates at crtcal ponts C 5, C 8, C 10, C 13, C 19, C, C 4 and C 7 wth larger curvatures are lower than those of other crtcal ponts wth smaller curvatures. Agan, one can check that the feedrate profle of the butterfly curve satsfes the constrants of chord error, acceleraton and jerk lmtatons shown n Fg. 13(a) (c). As shown n Fg. 13 (d) (f), the profles of velocty, acceleraton and jerk for x-axs and y-axs are also constraned on the gven lmts of 50 mm/s, 800 mm/s and mm/s 3, respectvely. From the smulaton results, t demonstrates that the proposed feedrate schedulng method acheves the desred precson and satsfes the knematc constrants of CNC machne for hgh-speed machnng Smulaton comparsons among dfferent NURBS nterpolaton algorthms In order to pont out the dfferences among the proposed method and prevous researches ncludng dynamc-based nterpolator wth real-tme look-ahead algorthm (DBLA) n [18] and adaptve-feedrate nterpolaton algorthm (ADA) n [10], some smulatons are performed as follows. The block dagram and specfcaton of AC servo control systems are referred to [18]. The parameters of servo control systems and moton controller are lsted n Table 7 unless stated otherwse. For the hat case, snce the DBLA and ADA algorthms cannot detect the breakponts D and E as shown n Fg., they wll not sutably decrease the feedrate and consequently deterorate

12 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) a d b e c f Fg. 11. Feedrate schedulng for the hat curve: (a) chord error profle, (b) acceleraton profle, (c) jerk profle, (d) x- and y-axes velocty profles, (e) x- and y-axes acceleraton profles and (f) x- and y-axes jerk profles. sharply n large chord, trackng and contourng errors around the breakponts D and E. However, snce the proposed algorthm can handle the problem of G 0 contnuty n the off-lne process, the performance can meet the specfcatons. The hat curve wth 1/4 scale and the feedrate command 1454 mm/s s adopted to demonstrate the merts of the proposed method. Smulaton results are shown n Fg. 14. Statstcal data are summarzed n Table 8 where the proposed method can reduce the maxmum contour error by 65.56% and 68.95% as compared to that of DBLA and ADA algorthms, respectvely. Furthermore, for both the hat and butterfly cases, f the maxmum feedrate command s gven as mm/mn, the samplng tme ms, the constrants of chord error, acceleraton and jerk 1 µm, 800 mm/s and mm/s 3, respectvely, the maxmum allowable feedrate F max for DBLA algorthm can be determned as F max = 1 ( T a) A max = A max J max = mm/ mn A max 4.4 mm/s where T a s the maxmum acceleraton tme of DBLA algorthm. The DBLA algorthm constraned the maxmum feedrate by mm/mn, far less than the admssble feedrate. Nevertheless, the proposed feedrate schedulng method can acheve the feedrate command mm/mn by plannng the sne velocty

13 64 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) a b c d Fg. 1. The butterfly curve: (a) contour, (b) sub-curves and blocks, (c) curvature profle and (d) feedrate profle. Table 8 Performance comparsons of smulatons among dfferent NURBS nterpolaton algorthms for hat contour. Feedrate command (mm/mn) 1454 Interpolaton algorthm Chord error (µm) Trackng error (µm) Contour error (µm) Machnng tme (s) MAX RMS MAX (x/y-axs) RMS (x/y-axs) MAX RMS PRE MAC Total ADA / / DBLA / / FS / / Note. MAX: maxmum, RMS: root mean square, PRE: pre-process, MAC: machnng. profle through Eqs. (16) (19); the process of feedrate schedulng s shown n Fg Expermental results In ths secton, experments are performed on a three-axs engravng machne wth Panasonc MINAS-A4 servo drvers and MSMD04S1S servo motors. The drvers of servo motors are set to torque mode. The off-lne feedrate schedulng algorthm s mplemented on MATLAB, and the three-axs engravng machne shown n Fg. 15(a) s controlled by a PC-based moton controller wth Intel Core GHz mcroprocessor. The hat and butterfly curves are tested under feedrate command of mm/mn. Snce the maxmum veloctes of servo drvers are lmted to 10 mm/s, the maxmum feedrate for real machnng s set to be 700 mm/mn. The feedrate schedulng method mplemented as a pre-process consumes about s and s for the hat and butterfly cases, respectvely. The real machnng test of the butterfly curve s shown n Fg. 15(b). Some experments are performed to do the comparatve analyss among the ADA, the FS wth and wthout real machnng for the butterfly curve. Snce the DBLA cannot meet the specfcaton of maxmum feedrate, only the ADA s used for comparson. Expermental results are shown n Fg. 16. Statstcal data are summarzed n Table 9. The ADA whch only consders chord errors cause hgh jerks, large trackng and contour errors at the crtcal ponts of C 1, C 5, C 8, C 10, C 13, C 19, C, C 4, C 7, C 31, as shown n Fg. 16(b) (d). The chord errors cannot be constraned by 1 µm usng the ADA for hgh-speed machnng as shown n Fg. 16(a). Nevertheless, the FS detects the crtcal ponts n advance, and plans a smoother feedrate profle under the gven constrants of chord error, acceleraton and jerk as shown n Fg. 16(f). Fg. 16(e) (h) demonstrate that the proposed FS can mprove the chord, trackng and contour performances smultaneously as compared wth the ADA. As shown n Table 9, the total machnng tmes for performng the FS wth and wthout real machnng are s and s, respectvely. The trackng and contour errors are almost the same as expected between smulaton and expermental results. The FS can reduce the maxmum contour error by 76.5% as compared wth the ADA, wth a penalty of 464.7% more machnng tme. The expermental results demonstrate the feasblty of the proposed off-lne feedrate schedulng method and the applcablty for real-tme mplementaton.

14 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) a d b e c f Fg. 13. Feedrate schedulng for the butterfly curve: (a) chord error profle, (b) acceleraton profle, (c) jerk profle, (d) x- and y-axes velocty profles, (e) x- and y-axes acceleraton profles and (f) x- and y-axes jerk profles. Table 9 Performance comparsons among dfferent NURBS nterpolaton algorthms for butterfly contour. Feedrate command (mm/mn) 700 Interpolaton algorthm Note. SIM: smulaton, EXP: experment. Chord error (µm) Trackng error (µm) Contour error (µm) Machnng tme (s) MAX RMS MAX (x/y-axs) RMS (x/y-axs) MAX RMS PRE MAC Total ADA / / FS (SIM) / / FS (EXP) / /

15 66 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) (a) ADA. (b) DBLA. (c) FS. (d) ADA. (e) DBLA. (f) FS. Fg. 14. Smulaton results of ADA, DBLA and FS nterpolaton algorthms for the hat curve. Fg. 15. The real machnng test (a) three-axs engravng machne, (b) the butterfly contour.

16 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) a e b f c g d h Fg. 16. Expermental results of ADA and FS nterpolaton algorthms for the butterfly curve.

17 68 A.-C. Lee et al. / Computer-Aded Desgn 43 (011) Conclusons In ths study, an off-lne feedrate schedulng method constraned by chord tolerance, acceleraton and jerk lmtatons s proposed to construct a jerk-lmted feedrate profle of NURBS curve. In off-lne process, scannng a NURBS curve n advance s performed to fnd out the breakponts wth G 0 contnuty, and the curve s splt nto several NURBS sub-curves. The proposed method also detects the crtcal ponts wth large curvatures, and the curve s further dvded nto small blocks. Furthermore, the sutable feedrates at breakponts and crtcal ponts are determned based on the constrants of chord tolerance, acceleraton and jerk lmtatons, and velocty varaton. The length of each NURBS block s estmated and stored as the scannng data. The feedrate profle for each NURBS block s constructed by utlzng the results of curve scannng. In addton, feedrate compensaton method for short NURBS block s establshed to make the whole feedrate profle more contnuous and precse. The proposed method s realzed as a preprocessor, whch avods the complcated and heavy calculatons n real tme. In real-tme process, t s smple and effcent to perform the nterpolaton algorthm and to generate the curve pont. Fnally, smulatons and experments are conducted to verfy the feasblty and applcablty of the proposed feedrate schedulng method. References [1] Pegl L, Tller W. The NURBS Books. second ed. Berln, Hedelberg: Sprnger; [] Xu XW, Newman ST. Makng CNC machne tools more open, nteroperable and ntellgent a revew of the technologes. Computers n Industry 006;57(): [3] Xu XW. Realzaton of STEP-NC enabled machnng. Robotcs and Computer- Integrated Manufacturng 006;(): [4] Shptaln M, Koren Y, Lo CC. Real-tme curve nterpolators. Computer-Aded Desgn 1994;6(11):83 8. [5] Yang DCH, Kong T. 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