Development of a CNC interpolation scheme for CNC controller based on Runge-Kutta method. Abdullahil Azeem*, Syed Mithun Ali and Sanjoy Kumar Paul

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1 Int. J. Computer Aded Engneerng and Technology, Vol. 4, No. 5, Development of a CNC nterpolaton scheme for CNC controller based on Runge-Kutta method Bddut Bhattacharjee Unversty of Brtsh Columba, 3333 Unversty Way, Kelowna, BC, Canada E-mal: bddut@nterchange.ubc.ca Abdullahl Azeem*, Syed Mthun Al and Sanjoy Kumar Paul Department of Industral and Producton Engneerng, Bangladesh Unversty of Engneerng and Technology (BUET), Dhaka 1000, Bangladesh E-mal: azeem@pe.buet.ac.bd E-mal: mthun@pe.buet.ac.bd E-mal: sanjoy@pe.buet.ac.bd *Correspondng author Abstract: The parametrc nterpolators of modern CNC machnes use Taylor s seres approxmaton to generate successve parameter values for the calculaton of x, y, z coordnates of tool postons. In order to acheve greater accuracy, hgher order dervatves are requred at every samplng perod whch complcates the calculaton for contours represented by NURBS curve. In addton, ths method calculates the chordal error n a gven segment through estmaton of the curvature neglectng a fracton of the error. In order to avod calculatng hgher dervatves and make the calculatons smpler, ths paper proposes the classcal fourth-order Runge-Kutta (RK) method for the determnaton of successve tool postons requrng the calculaton of the frst dervatves only. Furthermore, a method of estmatng the chordal error on the average value of parameters at the end ponts of a gven curve segment s proposed here that does not requre the calculaton of curvature at every segment. Fnally, a varable feedrate nterpolaton scheme s desgned combnng the RK method of parameter calculaton and the proposed method of chordal error calculaton. Results show that reduced chordal error and feedrate fluctuatons are achevable wth the proposed nterpolator compared to the conventonal nterpolator based on Taylor s approxmaton wth hgher order terms. Keywords: CNC nterpolaton; CNC controller; Taylor s approxmaton; chordal error. Reference to ths paper should be made as follows: Bhattacharjee, B., Azeem, A., Al, S.M. and Paul, S.K. (01) Development of a CNC nterpolaton scheme for CNC controller based on Runge-Kutta method, Int. J. Computer Aded Engneerng and Technology, Vol. 4, No. 5, pp Bographcal notes: Bddut Bhattacharjee s a PhD student n the Advanced Control and Intellgent Systems Laboratory, Unversty of Brtsh Columba, Canada. He obtaned hs MSc n Engneerng from Industral and Producton Copyrght 01 Inderscence Enterprses Ltd.

2 446 B. Bhattacharjee et al. Engneerng Department of BUET, Bangladesh. Hs research nterest s n control systems. Currently, he s contnung research towards the development of feedback control system for a dgtal mcrofludc system. Abdullahl Azeem s a Professor n the Department of Industral and Producton Engneerng n BUET, Bangladesh. He obtaned hs PhD from the Unversty of Western Ontaro, Canada. Hs research nterests nclude three-axs and fve-axs CNC control mechansm, cuttng force optmsaton n CNC machnng and applcaton of artfcal ntellgence n ndustral engneerng. Syed Mthun Al s an Assstant Professor n the Department of Industral and Producton Engneerng n Bangladesh Unversty of Engneerng and Technology, BUET, Bangladesh. He obtaned hs MSc n Industral and Producton Engneerng from BUET. Hs research nterest ncludes applcaton of artfcal neural network, computer aded desgn and manufacturng and optmsaton. Sanjoy Kumar Paul s an Assstant Professor n the Department of Industral and Producton Engneerng n Bangladesh Unversty of Engneerng and Technology, BUET, Bangladesh. He obtaned hs MSc n Industral and Producton Engneerng from BUET. Hs research nterest ncludes applcaton of artfcal ntellgence n ndustral engneerng. 1 Introducton Free-form or sculptured surfaces are used n a varety of applcatons n automotve, aerospace, shp buldng ndustres. Des and moulds of automotve body, propeller and mpeller blades, and arcraft wngs etc. are typcal examples that nvolve free-form surface desgn and machnng. Modern CAD/CAM systems provde a desgner wth tools for defnng such free-form surfaces represented by 3-D parametrc curves and surfaces. Mult-axs computer numercal control (CNC) machnng s mostly used for machnng such complex free-form surfaces. When a free-form surface s machned usng a CNC machne, the path of the cuttng tool can be consdered as a 3-D splne curve. Therefore surface machnng nvolves four man phases: 1 optmsaton of machnng process parameters tool path plannng 3 tool path nterpolaton 4 Servo control. Phase 1 and Phase deals wth automatc feature recognton, the selecton of cutter and ts speed, feedrate and depth of cut etc. and the optmsaton of tool path. Phase 3 s concerned wth the converson of tool paths obtaned from a tool path plannng system nto tme-dependent commands for drvng the servo control system of a CNC machne mantanng the parameters resultng from Phase 1. Phase 4 manly deals wth closed loop control ssues such as adaptve control strateges, feedback, stablty etc. The desgners frst develop the model of the part n the CAD system. The part geometry s transferred to the CNC system by means of a part program (G-codes, M-codes) consstng manly of

3 Development of a CNC nterpolaton scheme for CNC controller 447 moton commands. These moton commands are translated n real tme by the CNC nterpolator nto a specal form, contanng the coordnates of successve tool postons, before beng sent to the control loops of machne tool drves. As most CNC nterpolators can process only straght lne and crcular arc moton commands, the parametrc curve representng the tool path s dscretsed nto a set of lne segments whch approxmate the orgnal curve to a desred accuracy. Researchers have developed varous technques of parametrc nterpolaton for the CNC machne tool s controller. The man objectve of all these research works was to reduce the chordal error and feedrate fluctuaton. To machne parts havng sculptured surfaces the cuttng tool has to traverse along a 3-D splne curve of varyng curvature at dfferent segments. Segments of the curve havng small radus of curvature are the sources of hgher chordal error f the tool traverses wth a constant feedrate. Agan, the surface qualty and machnng tme depend on the optmum feedrate. Hence, researchers tred to fnd out ways to resolve ths conflct. Chou and Yang (1991) developed a formulaton for the command generaton of parametrc curve machnng takng nto consderaton the dynamcs of the 3-axs CNC machne tool and the machnng process. The geometrcal propertes of a cutter path, ncludng poston, tangent, curvature and torson are analytcally related to the moton dynamcs of a machne tool ncludng feedrate, acceleraton, jerk and drvng force whch are needed for command generaton and machned surface qualty analyss. Based on ther work, Huang and Yang (199) developed a generalsed varable feedrate parametrc nterpolator by usng the Euler approxmaton method whch yelded satsfactory results compared to those from usng lnear, crcular and cubc splne nterpolators as long as the curvature of the curve for machnng was small. Bed and Quan (199) mplemented the nterpolatng B-splne curve through the data ponts lyng on the product rather than approxmatng those ponts. Later, Bed et al. (1993) extended the B-splne curve nterpolator to a surface nterpolator by utlsng the greater computatonal capacty of a network of computers called transputers. These transputers when nterfaced wth the controller of the machne tool can avod problems such as communcaton errors, jerky moton, and gougng. Wang and Yang (1993) developed a method of nterpolatng a set of dscrete data ponts to form a composte quntc splne for applcaton n precson machnng. The algorthm starts wth fttng a set of gven data wth a cubc splne and then on the bass of ths curve a composte quntc splne s generated wth nearly arc-length parametersaton, nd order contnuty and no hgh-order oscllatons. Koren et al. (1993) developed a real-tme constant feedrate nterpolaton algorthm for parametrc curves based on the frst- and second-order Taylor s seres approxmaton method. Shptaln et al. (1994) developed the same nterpolator for -D mplct curves and -D parametrc curves and acheved sgnfcant mprovements n terms of poston accuracy and feedrate devatons. Yang and Kong (1994) proposed a parametrc nterpolator for varable feedrate based on the frst- and second-order Taylor s expanson and compared lnear and parametrc nterpolators based on several aspects such as memory sze, feedrate fluctuaton and CPU tme. Krtss (1994) presented an ncremental step algorthm for general 3D and D parametrc curve nterpolaton, whch, for a gven poston of the tp of the cuttng tool, calculates the next poston as closely as possble to the curve by fulfllng two crtera. Ths algorthm requres many complex calculatons whch restrct ts use for real-tme applcaton. Feedback nterpolators were developed by Lo (1997) for both -D mplct curves and parametrc curves utlsng feedback acton to

4 448 B. Bhattacharjee et al. correct the feedrate error. A real-tme non-unform ratonal B-splne (NURBS) nterpolator was proposed by Zhang and Greenway (1998) for an artculated robot. In a new approach to CNC tool path generaton, Lo (1998) developed three dfferent methods of lnear, curve and surface nterpolator where the CL velocty s derved resultng n the moton of the cuttng edge wth the specfed feedrate. Yeh and Hsu (1999) developed a speed-controlled nterpolaton algorthm by dervng a compensatory parameter whch mproved the curve speed accuracy. Ln (000) proposed a real-tme surface nterpolator for 3-D parametrc surface machnng that can read surface g-codes, contanng geometrc nformaton such as the coeffcents of the parametrc surface, as well as cuttng condtons. A varable feedrate nterpolaton scheme for planar mplct curves was presented by Xu et al. (001) whch relates the geometrc propertes lke tangent, curvature etc to the knematcs of the tool along the curve. The expermental results proved that machnng effcency can be ncreased whle satsfyng a predefned tolerance of the contour error of the curve. Flesg and Spence (001) presented a new algorthm for the off-lne nterpolaton of data ponts and real-tme axs command generaton whch results n constant feedrate and reduced angular acceleraton. Lartgue et al. (001) presented an accurate method to generate a CNC tool path for a smooth free-form surface n terms of planar cubc B-splne curves whch can be fed nto the curve nterpolator. Results verfy the method for better machnng accuracy and lower memory requrements. Farouk and Tsa (001) proposed a systematc dervaton of the proper Taylor seres coeffcents for varable feedrate nterpolators, usng smple notatons and recursve formulae suted to real-tme computatons. Bahr et al. (001) presented a method of CNC nterpolaton based on modfed fnte forward dfferencng algorthm for fast evaluaton of ponts on a cubc parametrc curve. The results proved the method to be accurate, adaptve to curvature of the curve and reduces jerk. Zhmng et al. (00) presented a real-tme nterpolaton algorthm for NURBS curves based on the second-order Taylor s approxmaton whch resulted n smaller contour errors and feedrate fluctuatons. However, ther calculaton of contour error was based on crcular approxmaton for estmatng curvature of a segment whch can not gve the real value of the error. A method for lnear and angular feedrate nterpolaton for planar mplct curves was proposed by Xu (003) for mult-axs CNC machnng relatng the angular feedrate nterpolaton to the lnear feedrate nterpolaton. Nam and Yang (004) proposed a method that results n exact feedrate trajectory through jerk-lmted acceleraton profles for parametrc curves. The recursve trajectory generaton method estmates an admssble path ncrement and determnes the ntaton of the fnal deceleraton stage accordng to the dstance left to travel at every samplng tme. Cheng and Tsa (004) developed the NURBS nterpolator consderng acceleraton/deceleraton plannng before the nterpolaton. The expermental results suggest that nterpolator wth bell shape acceleraton/deceleraton plannng s satsfactory for the realsaton of acceleraton/deceleraton. In all the research works the nterpolaton for parametrc curves or surfaces s based on the Taylor s seres approxmaton for generatng successve values of the parameter. Consderng the accuracy of results frst three terms of the seres are used, requrng the second dervatve of the parametrc curve to be calculated at every samplng tme. Dervatves of curves lke B-splne and NURBS are qute complex and requres several teratons. Ths paper proposes the fourth-order Runge-Kutta (RK) method to fnd the parameter values requrng only the frst dervatves of parametrc curve. The RK method

5 Development of a CNC nterpolaton scheme for CNC controller 449 s computatonally easer to mplement due to ts recursve nature. Ths research also proposes a dfferent method of calculatng the chordal error avodng the calculaton of curvature at every samplng tme. Based on the calculated error the algorthm utlses a correctve acton to fnd the new parameter value for whch the chordal error wll be wthn tolerance. Present method of CNC nterpolaton.1 Taylor s seres approxmaton The general form of a three dmensonal parametrc curve of nth order can be expressed as Pu ( ) = xu ( ) ˆ+ yu ( ) ˆj + zuk ( ) ˆ, and n n 1 n, x n 1, x 1, x 0, x xu ( ) = a u + a u + + a u+ a n n 1 n, y n 1, y 1, y 0, y yu ( ) = a u + a u + + a u+ a n n 1 n, z n 1, z 1, z 0, z zu ( ) = a u + a u + + a u+ a (1) where 0 u u max. The coordnates (x, y, z) of ponts on the curve are obtaned by substtutng the values of the parameter, u, n the above equatons. In a CNC machne tool, the tp of the cuttng tool moves from one such pont to the next and from there to the next, and so on. To fnd the successve values of the parameter, u, the followng teraton s used: ( ) u + = u +Δ u 1 where u s the parameter at tme t, u +1 s the parameter at tme t +1 and Δ(u ) s the ncremental value. The smplest approach would be to ncrement u unformly,.e., n equal small ncrements u, and calculate the correspondng x +1 = x(u +1 ), y +1 = y(u +1 ) and z +1 = z(u +1 ) at tme t +1. The samplng perod, T = t +1 t, s constant for a machne s controller. Ths approach has the followng problems: Dependng on the curvature the chordal length, 1/ L = ( x+ 1 x) + ( y+ 1 y) + ( z+ 1 z ), between ponts (x +1, y +1, z +1 ) and (x, y, z ) s not equal. But the tool traverses ths dstance n equal tme ntervals T. So the feedrate s not constant, rather t vares from one segment to another along the curve. The optmal sze of the ncrement Δu s not known. If Δu s too small, the resultant L s are too small as well and machnng slows down. If Δu s too bg, the resultant L s are too bg and machnng accuracy s not mantaned. That s why real-tme parametrc nterpolator cannot be based on unform segmentaton of the curve accordng to equal ncrements of the parameter, u. Segmentaton of the curve

6 450 B. Bhattacharjee et al. should be such that the chordal lengths, L, of each segment are equal so that the feedrate does not vary along the curve. Researchers have developed the method for determnng successve values of u so that the chordal lengths L (machned at each samplng perod T) are constants, resultng n a constant feedrate. The feedrate, V(u) along the curve s defned by or, ds ds du Vu ( ) = = dt du u = u dt t = t du V V t = t = = dt ds dx( u) dy( u) dz( u) du + + u = u du u = u du u = u du u = u () (3) The soluton of the above equaton for a constant V results n the requred value of the parameter, u, at any tme t. As analytcal soluton of the above equaton s dffcult and nfeasble to mplement, a numercal method s used to fnd the soluton. The functon u s expanded at the neghbourhood n tme t by usng the Taylor s seres approxmaton. ( + ) ( ) ( + ) ( t ) + 1 t du d u u t = u t + t t + + HgherOrderTerms dt t = t dt t = t du T d u u = u + T + + HgherOrderTerms + 1 dt t = t dt t = t (4) Fnally, performng a few steps of dfferentaton and neglectng the Hgher-order terms n (4), u + 1 = u + ( VT ) VT dx( u) dy( u) dz( u) + + du du du u = u u = u u = u dx( u) d x( u) dy( u) d y( u) + du u = u du du u u u = u du = u = u dz( u) d z( u) + du u = u du u = u dx( u) dy( u) dz( u) + + du u = u du u = u du u = u The above equaton s the second order Taylor s Seres approxmaton to fnd the value of parameter, u +1, at tme t +1, calculated from the current value, u, and the dervatves of (5)

7 Development of a CNC nterpolaton scheme for CNC controller 451 the current tool poston (x, y, z ) wth respect to u. Ths new value, u +1, s substtuted n equaton (1) to fnd the next reference tool poston (x +1, y +1, z +1 ). Although ths nterpolaton algorthm s supposed to provde constant feedrate machnng, there are varatons n actual cuttng speed due to the approxmaton of Taylor s Seres. Neglectng the hgher order terms (HOTs) of the seres results n a value of u +1 dfferent from that whch would be obtaned f the HOT were used. As a result, the chordal dstance, L, between (x, y, z ) and (x +1, y +1, z +1 ), traveled by the cuttng tool n T seconds, s such that the actual feedrate of the tool between ths two postons s not exactly V, the specfed feedrate. The actual speed, V, at the th segment s calculated from the followng equaton: 1 L 1 = = ( + 1 ) + ( + 1 ) + ( + 1 ) T T V x x y y z z (6). Determnaton of chordal error As the cutter moves straght between contguous nterpolated ponts, two poston errors may occur as: a radal error b chord error, durng moton for a parametrc curve as shown n Fgure 1. Fgure 1 The radal and chordal errors Parametrc curve Radal error D Cutter path A Chord error B Interpolated pont C The radal error s the perpendcular dstance between the nterpolated ponts and the parametrc curve. Bascally, the radal error s caused by the roundng error of computer systems. Wth the rapd development of mcroprocessors wth hgher accuracy n computng, the radal error s no longer a major concern n present applcatons. The chord error s the maxmum dstance between the nterpolated trajectory CD and the arc AB on the desred curve. As the samplng tme, T, for a CNC machne s controller s constant, the chordal error depends on the machnng feedrate and the radus of curvature of the parametrc curve. It can be understood from Fgure 1 that chordal error s hgher f

8 45 B. Bhattacharjee et al. the radus of curvature of arc AB s smaller. So f the tool traverses along the curve wth a constant feedrate, V, there wll be hgh machnng error on the segments of the curve where the rad of curvature are small. Agan for a gven curvature f the feedrate s hgh then the chordal dstance between two adjacent tool postons s also hgh, whch n turn results n hgh chordal error. To estmate the chordal error, t s assumed that the curve segment between Pu ( ) and Pu ( + 1) s a crcular arc wth radus ρ at parameter u, as shown n Fgure. The radus ρ at parameter u s found from the curvature, κ, at parameter u whch s calculated by the followng equaton: Fgure Estmaton of chordal error Curve Crcle L A B E C O ρ D κ = P u P u ( ) ( ) P ( ) 3 u (7) The chordal dstance, L, s approxmated by P( u ) P( u ) + 1. So the chordal error, or, E = OB OA= OB OD AD L 1 1 VT 1 VTκ 1 1 E = ρ ρ = = κ κ κ Ths approach of calculatng chordal error neglects BC. Hence the actual error wll be greater than that calculated from equaton (8) (8) 3 Proposed method of CNC nterpolaton As was explaned n the prevous secton, due the omsson of HOTs of the Taylor s Seres the actual feedrate between successve tool postons vares from the

9 Development of a CNC nterpolaton scheme for CNC controller 453 predetermned feedrate. Hence for constant feedrate nterpolaton, the feedrate varaton s unacceptable. To reduce ths varaton the fourth-order RK method s proposed for fndng the parameter values. 3.1 RK method The Fourth-order RK method s explaned here. For an ordnary dfferental equaton of the followng form dy f ( x, y) dx = the soluton of the above accordng to RK method s: y 1 = y +Φ ( x, y, h) h (9) + where h = x +1 + x, and Φ(x, y, h) s called the ncrement functon, whch can be nterpreted as a representatve slope over the nterval. The ncrement functon can be wrtten n general form as Φ= ak + a K + + a K 1 1 n n where the a s are constants and the K s are (, ) (, ) (, ) K1 = f x y K = f x + p1h y + q11k1h K3 = f x + ph y + q1k1h+ qkh K = f x + p h y + q K h+ q K h + q K h ( 1, 1,1 1 1, 1, 1 1 ) n n n n n n n where the p s and q s are constants. For fourth order the value of n s equal to 4. Accordng to the Fourth-order RK method equaton (9) s wrtten as 1 y+ 1 = y + ( K1 K K3 K4) h, where (, ) K1 = f x y 1 1 K = f x + h, y + K1h 1 1 K3 = f x + h, y + Kh K = f x + h y + K h (, ) 4 3 Thus, accordng to the Fourth-order RK method, the soluton to equaton (3) s

10 454 B. Bhattacharjee et al. where and 1 u u ( K K K K ) T = (10) K 1 3 ( ) = f u = dx( u) dy( u) dz( u) + + du u = u du u = u du u = u KT 1 V K = f u + = dx( u) dy( u) KT + 1 KT 1 du u = u+ du u = u + K = dz( u) + KT 1 du u = u + V V dx( u) dy( u) dz( u) KT + KT + KT du u = u + du u = u + du u = u + K 4 = V dx( u) dy( u) dz( u) + + du du du u = u + K3T u = u + K3T u = u + K3T The RK method acheves the accuracy of a Taylor seres approach wthout requrng the calculaton of hgher dervatves. To acheve hgh accuracy from the Taylor s Seres the hgher dervatves are requred. Current methods utlse the seres up to the nd dervatve. Computaton of the dervatves makes the nterpolator slower, whch becomes sgnfcant n case of curves represented by B-Splne and NURBS. The RK method s also effcent for computer mplementaton because of the recurrence relatonshps of the K s. That s why the proposed method of nterpolaton s faster and generates more accurate parameter values compared to that generated from Taylor s Seres Approxmaton. As a result, the actual feedrate between successve tool postons remans very close to the predetermned feedrate wth neglgble varatons. 3. Determnaton of chordal error The proposed nterpolaton utlses a new method for fndng the chordal error. The exact value of chordal error s the maxmum dstance between the straght lne connectng two adjacent tool postons and the arc between them as shown n Fgure 3.

11 Development of a CNC nterpolaton scheme for CNC controller 455 Fgure 3 Estmaton of chordal error on the pont of average parameter L E The curve segment between ponts Pu ( ) and Pu ( + 1) can be approxmated by a crcular arc. The pont on the curve segment correspondng to the average value of the parameters u and u +1 can be consdered as approxmately the most dstant pont from the straght lne Pu ( + 1) Pu ( ). Therefore, the chordal error, E, s the perpendcular u+ 1 u dstance from pont P + to the lne P( u+ 1) P( u). Mathematcally, u + u+ 1 P( u+ 1 ) P( u ) E = P P( u) ( 1 ) P u P( u) + 1 = ( Pm P ) ( P + 1 P ) (11) L ( ym y)( z+ 1 z) ( zm z)( y+ 1 y) 1 ( ym y)( z+ 1 z) ( zm z)( y+ 1 y) = + L + ( ym y)( z+ 1 z) ( zm z)( y+ 1 y) 3.3 Varable feed rate nterpolaton Ths secton presents the proposed varable feedrate nterpolaton scheme for CNC machne tool s controller. As was ponted out earler that constant feedrate nterpolaton does not guarantee that the chordal error wll be wthn the specfed tolerance. Chordal error may exceed the specfed tolerance f the radus of curvature of a curve segment s smaller. Therefore, an nterpolaton algorthm s necessary whch generates varable feedrate to keep the chordal error wthn predetermned tolerance. The algorthm should check the magntude of chordal error frst, then dependng on whether t s wthn tolerance the next tool poston wll be generated mantanng some relatonshp whch s explaned below. Fgure 4 shows how the chord length of a crcle vares n relaton to the dstance between the mdpont of that chord and that of the correspondng arc.

12 456 B. Bhattacharjee et al. Fgure 4 Varaton of chordal error wth chord length P T C Q R S B O A Let the radus of the crcle be r, then ( ) ( ( /) ) ( ( /) ) QR = OR OQ = OR OA AQ = OR OA AP H = r r AP (1) Smlarly, t can be shown that, ( ( /) ) ST = r r AC H (13) Now, consderng the arc segment of the splne curve between successve tool postons as a crcular arc, n Fgure 4, A s the current tool poston, P or C s the next tool poston, AP or AC s the tool trajectory and QR or ST s the chordal error. From equatons (1) and (13), the general form can be expressed as: ( L ) ' E = ρ ρ, where ρ s the radus of curvature of the th segment, s the chordal error, E L s the chord length. or ( 8 ρ 4 ) ' L = E E (14) If the current tool poston s A and the next poston, generated from the nterpolaton algorthm usng the RK method, s such that the calculated chordal error s greater than the tolerance then the next tool poston wll not be that generated from RK method. To mantan the chordal error wthn tolerance the next poston wll be derved by usng the relaton of equaton (14), where E wll be replaced by tolerance. The resultng L s the chord length between the corrected next tool poston and the current poston. To fnd the parameter u + 1 that wll gve the corrected next tool poston, the algorthm uses

13 Development of a CNC nterpolaton scheme for CNC controller 457 V = L / T. Equaton (11) needs the radus of curvature for the correspondng curve segment and requres several computaton steps. In order to fnd a smple relatonshp between L and E, L s plotted for dfferent values of ρ where the range of E s 0 E ρ. The chord error resultng from usng the Taylor s Seres Approxmaton or RK method hardly exceeds 0.1 mm, but the error tolerance can be much smaller than 0.1 mm. Now Fgure 5 shows that for a gven value of ρ, the relatonshp between L and E can be farly approxmated by a lnear relatonshp for 0 E 0.1 mm. Therefore, n ths range of E, or, ' = L E L Tolerance ( ) ' L = L E * Tolerance (15) And the feedrate to be used by the RK method to fnd the next parameter u +1 s (( )* ) V = L E Tolerance T Fgure 5 Relaton between chord length and chord error for dfferent values of radus of curvature L (mm) E (mm) 4 Results and dscussons The proposed varable feedrate nterpolaton employs the fourth-order RK method to generate the successve values of the parameter.

14 458 B. Bhattacharjee et al. Fgure 6 Flowchart of the varable feedrate nterpolaton algorthm usng the 4th order RK method START (u = 0) u = u V = V V = V gven Fourth-order Runge-Kutta Method u = u +1 V = V gven u +1 u +1 > u max? No Calculate (u + u +1 ) / Calculate E at u = (u + u +1 ) / E > Tolerance? Yes Calculate L = (L / E ) * Tolerance Calculate V = L / T Yes No u +1 = u max Calculate V, L, E Calculate x(u), y(u), z(u) STOP Calculate V Calculate x(u), y(u), z(u) If the chordal error, calculated at the average value of the current and next parameter, s less than the specfed tolerance, the next tool poston s found by usng the next parameter and the tool traverses at the specfed feedrate. If the chordal error exceeds the tolerance, the chord length that would result n error wthn tolerance s calculated and the resultng feedrate s utlsed by the RK method to generate the next corrected parameter value. The varable feedrate nterpolaton scheme s shown n Fgure 6. Ths algorthm s tested for feasblty and performance by wrtng the nterpolator n MATLAB 6.5 and performng smulatons on a personal computer wth.41 GHz CPU for a 3-D cubc splne curve. Ths secton dscusses the smulaton results of dfferent nterpolators n terms of the generated tool postons, chordal error and feedrate for the parametrc curve mentoned n

15 Development of a CNC nterpolaton scheme for CNC controller 459 the prevous secton. Frst, the chordal error and feedrate are compared between the nterpolaton usng the Taylor s approxmaton and that usng the RK method for constant feedrate mode. Then the chordal error, feedrate and tool postons generated from the varable feedrate nterpolaton are shown. Fgure 7 shows the cubc splne curve, for whch the control ponts are gven n Table 1 and the mathematcal expresson s: 3 x = 50u 100u + 50u 3 y = 1u + 35u 55 u, 0 u 1. 3 z = 10u + 1u 4u Table 1 Control ponts of the cubc splne curve No x y z Fgure 7 3-D vew of the cubc splne curve (see onlne verson for colours) Smulaton results for the cubc splne are shown n Fgure 8, Fgure 9, Fgure 10, Fgure 11 and Fgure 1. For constant feedrate nterpolaton, the Taylor s seres approxmaton results n larger chordal error compared to that of RK method for the cubc splne curve as shown n Fgure 8. Specfcally, the RK method reduces the chordal error by 6% of that resultng from Taylor s approxmaton. Error ncreases followng the same pattern for both methods. The curvature of the curve starts ncreasng for parameter values greater than 0. and maxmum curvature corresponds to parameter values between 0.5 and 0.6. The maxmum error results from the curve segment where the curvature s

16 460 B. Bhattacharjee et al. the maxmum. It s noteworthy that snce n Taylor s approxmaton, the error s calculated based on the estmaton of the curvature for a gven segment neglectng a small porton of the error, the actual error s greater than that shown n Fgure 8. For a gven segment the maxmum chordal error s very dffcult to calculate analytcally and would be nfeasble to mplement nto the real-tme nterpolator. That s why to better estmate the error, RK method employs the calculaton of error at the average value of the parameters at the end-ponts of a gven segment. It s worth mentonng here that the samplng frequency of the nterpolator, whch s an electronc controller, has drect nfluence on the chordal error and hence, on the accuracy of machned surfaces. Wth the ncrease of samplng frequency the dstance travelled by the tool tp s reduced. Thus, the chordal error n the regon of hgher curvature wll be less than that resultng from usng an nterpolator wth lower samplng frequency (Farouk and Tsa, 001). The RK method of nterpolaton also results n better feedrate profle compared to that from the Taylor s approx. for constant feedrate mode as shown n Fgure 9. The fluctuaton of feedrate s greater for latter method where the maxmum devaton s about 0.3%. For the RK method, the maxmum feedrate devaton s 0.15%. Also, the RK method results n a 34% reducton of the feedrate error. The reason for the varaton n actual feedrate can be explaned by consderng that both the methods utlses an approxmaton to generate successve parameter values and the tool travels the dstance between two adjacent postons n a samplng tme, whch s constant for a machne s controller. So the chordal length between the generated next poston and the current poston vares dependng on whether the tool s passng through a segment of hgh curvature or low curvature. As RK method gves better accuracy of generatng successve parameters, the varaton n chordal length s less than that of the Taylor s approxmaton method. Fgure 8 Chordal error from constant feedrate (00 mm/sec) nterpolaton for the cubc splne (see onlne verson for colours)

17 Development of a CNC nterpolaton scheme for CNC controller 461 Fgure 9 Actual feedrate from constant feedrate (00 mm/sec) nterpolaton for the cubc splne curve (see onlne verson for colours) Fgure 10 Chordal error from varable feedrate nterpolaton wth specfed feedrate 00 mm/sec, tolerance 1 μm, for the cubc splne curve (see onlne verson for colours)

18 46 B. Bhattacharjee et al. Fgure 11 Actual feedrate from varable feedrate nterpolaton wth specfed feedrate 00 mm/sec, tolerance 1 μm, for the cubc splne curve Fgure 1 Tool postons generated from varable feedrate nterpolaton wth specfed feedrate 00 mm/sec, tolerance 1 μm, for the cubc splne curve (see onlne verson for colours)

19 Development of a CNC nterpolaton scheme for CNC controller 463 Fgure 10, Fgure 11 and Fgure 1 shows the results of proposed varable feedrate nterpolaton for the cubc splne curve. The chordal error remans wthn the tolerance of 1μm. There are two postons where the values of chordal error are approxmately equal to the prescrbed tolerance. The reason for these s that the algorthm generates the next poston command f the chordal error s less than 1 μm, otherwse t recalculates the next poston. The segments of the curve havng low curvature result n chordal error close to the tolerance. The feedrate profle resultng from the varable feedrate nterpolaton s as expected. The porton of the curve havng low curvature results error less than 1 μm and so the tool moves wth the specfed feedrate of 00 mm/sec. For the segments of the curve resultng error greater than tolerance, the nterpolator adjusts the feedrate so that chordal lengths are very small and the tool traverses these small dstances at constant tme (.e., samplng tme). The varaton n feedrate s also smooth resultng n smooth acceleraton and deceleraton of the tool. 5 Conclusons In ths paper a varable feedrate nterpolaton algorthm s suggested whch changes the feedrate dependng on chordal error. In ths method, the chordal error s calculated at the average value of the parameters of the end ponts of the gven curve segment. Thus, the calculaton of curvature at each segment s avoded. The nterpolator employs the fourthorder RK method to generate successve values of the parameter. The RK method only requres the frst dervatves of parametrc curve but acheves the accuracy of Taylor s approxmaton wth hgher order dervatves. Thus, the proposed method smplfes the nterpolaton process. The algorthm works as follows: the RK method generates the next value of parameter whch s used to calculate the coordnates of the next tool poston only f the chordal error s wthn tolerance. If the error exceeds the tolerance the nterpolator calculates the value of feedrate that would result n a next parameter for whch error wll be wthn the tolerance. The proposed nterpolaton s compared wth the tradtonal method by performng smulatons for a cubc splne curve. The results show that the proposed method of nterpolaton usng the RK method generates the parameter values for whch the feedrate profle s smoother than that from usng the Taylor s approxmaton n constant feedrate mode. The varable feedrate nterpolaton results n chordal error wthn the tolerance and the feedrate remans constant at the specfed value as long as error s wthn lmt. The feedrate s decreased when necessary to mantan the error wthn tolerance. Hence, the proposed varable feedrate nterpolaton results n much smoother fluctuaton of feedrate whle mantanng the chordal error wthn tolerance. References Bahr, B., Xao, X. and Krshnan, K. (001) A real-tme scheme of cubc parametrc curve nterpolatons for CNC systems, Computers n Industry, Vol. 45, No. 3, pp Bed, S. and Quan, N. (199) Splne nterpolaton technque for NC machnes, Computers n Industry, Vol. 18, No. 3, pp Bed, S., Al, I. and Quan, N. (1993) Advanced nterpolaton technques for NC machnes, Transactons of ASME, Journal of Engneerng for Industry, Vol. 115, No. 3, pp

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