The speed-controlled interpolator for machining parametric curves

Transcription

1 COMPUTER-AIDED DESIGN Computer-Aie Design 31 (1999) The spee-controlle interpolator for machining parametric curves S.-S. Yeh, P.-L. Hsu* Department of Electrical an Control Engineering, National Chiao Tung University, Hsinchu, 300 Taiwan Receive 30 September 1998; receive in revise form 0 February 1999; accepte 1 March 1999 Abstract Moern CNC systems are esigne with the function of machining arbitrary parametric curves to save massive ata communication between CAD/CAM an CNC systems an improve their machining quality. Although available CNC interpolators for parametric curves generally achieve contouring position accuracy, the specifie feerate, which ominates the quality of the machining processes, is not guarantee ring motion. Recently, some approximation results concerning motion spee have been reporte [Shpitalni M, Koren Y, Lo CC. Computer-Aie Design 1994;6(11):83 838; Bei S, Ali I, Quan N. Trans ASME J Engng Instry 1993;115:39 336; Chou JJ, Yang DCH. Trans. ASME J Engng Instry 1991;113: ; Chou JJ, Yang DCH. Trans ASME J Engng Instry 199;114:15 ]. In this paper, by eriving a compensatory parameter, the propose interpolation algorithm has significantly improve curve spee accuracy. In real applications, the propose algorithm results in: (1) constant spee; an () specifie acceleration an eceleration (ACC/DEC) to meet the feerate commans. The motion accuracy an feasibility of the present interpolator have been verifie with a provie non-uniform rational B-spline (NURBS) example Elsevier Science Lt. All rights reserve. Keywors: Parametric curve; CNC interpolators; Spee accuracy; NURBS 1. Introction In moern CAD/CAM systems, profiles for parts like ies, vanes, aircraft moels, an car moels are usually represente in parametric forms. As conventional CNC machines only provie linear an circular arc interpolators, the CAD/CAM systems have to segment a curve into a huge number of small, linearize segments an sen them to CNC systems. Such linearize-segmente contours processe on traitional CNC systems are unesirable in real applications as follows: the transmission error between CAD/CAM an CNC systems for a huge number of ata may easily occur, i.e. lost ata an noise perturbation; the iscontinuity of segmentation eteriorates surface accuracy; the motion spee becomes unsmooth because of the linearization of the curve in each segment, especially in acceleration an eceleration. As the generate curves or profiles may be in a parametric form, only parametric curve information is require to be efficiently transferre among CAD/CAM/CNC systems as * Corresponing author. Tel.: ; fax: aress: plhsu@cc.nctu.e.tw (P.L. Hsu) shown in Fig. 1. Shpitalni et al. [1] propose the curve segments transfer between CAD an CNC systems an Bei et al. [] propose the B-spline curve an B-spline surface interpolation algorithm to obtain both accurate curves an gouge free surface. Huang an Yang [3] evelope a generalize interpolation algorithm for ifferent parametric curves with improve spee fluctuation. Moreover, Yang an Kong [4] stuie both linear an parametric interpolators for machining processes. In these CNC systems, parametric curves are profiles in ifferent formats like the Bezier curve, B-spline, cubic spline, an NURBS (non-uniform rational B-spline). The general parameter iteration metho use is u i1 ˆ u i D u i where u i is the present parameter, u i1 is the next parameter, an D u i is the incremental value. The interpolate points are calculate by substituting u i into the corresponing mathematical moel to recover the originally esigne curves. As the cutter moves straight between contiguous interpolate points, two position errors may occur as: (a) raial error; an (b) chor error [5] ring motion for a parametric curve as shown in Fig.. The raial error is the perpenicular istance between the interpolate points an the parametric curve. Basically, the raial error is cause by the rouning error of computer /99/$ - see front matter 1999 Elsevier Science Lt. All rights reserve. PII: S (99)

2 350 S.-S. Yeh, P.-L. Hsu / Computer-Aie Design 31 (1999) Fig. 1. The machining systems with parameters transmission. systems. With the rapi evelopment of microprocessors with higher precision, the raial error is no longer a major concern in present applications. In aition, the chor error is the maximum istance between the secant CD an the secant arc AB _.A small curvature raius with a fast feerate comman may cause the chor error. Thus, an aaptive feerate is require to keep the chor error within an acceptable range. However, cutting spee ominates the quality of the machining process. To achieve the specifie feerate for parametric curves is usually ifficult. The unesirable motion spee may eteriorate the quality of the machine surface. Several researchers have evelope ifferent interpolation algorithms for parametric curves to improve motion spee accuracy. Bei et al. [] set D(u i )as a constant to form the uniform interpolation algorithm which is the simplest metho an its chor error an curve spee however are not guarantee. Accoring to the analysis of CNC machine kinematics an cutter path geometry, an improve interpolation algorithm in position, velocity, an acceleration was propose by Chou an Yang [6] if the CNC machine kinematics moel is known exactly. Further, Houng an Yang [3] evelope the cubic spline parametric curve interpolator by using the Euler metho. Shpitalni et al. [1] erive the same interpolation algorithm by using Taylor s expansion. Lo an Chung [7] propose the contouring error compensation interpolation algorithm which contains real-time contouring error calculations an a simplifie parameter iteration metho to achieve satisfactory motion accuracy. In acceleration an eceleration, Kim [8] obtaine a simple metho for parametric curves while its position an spee errors are significant. The present spee-controlle interpolation algorithm is propose by eriving a suitable compensatory parameter for the first-orer approximation [1,6,8] to obtain esirable motion spee. Then, the propose interpolator is applie to the constant-spee moe an the acceleration/eceleration moe to achieve constant feerate an the specifie spee profiles, respectively. Thus, the present CNC interpolators result in stable motion an smoothly change spee to avoi mechanical shock or vibration in machining processes. The propose spee-controlle interpolators have also been successfully applie to a NURBS comman on a personal computer to achieve high motion precision.. Parametric curve formulation Suppose C(u) is the parametric curve representation function an the time function u is the curve parameter as u t i ˆu i an u t i1 ˆu i1 By using Taylor s expansion, the approximation up to the secon erivative is u i1 ˆ u i HOT t i1 t i 1 tˆti u As the curve spee V(u i ) can be represente as V u i ˆ ˆ i1 t i tˆti t tˆti the first erivative of u with t is obtaine as tˆti ˆ V u i By taking the erivative of Eq. (), the secon erivative of 1 Fig.. The raial error an the chor error.

3 S.-S. Yeh, P.-L. Hsu / Computer-Aie Design 31 (1999) u with t is u tˆtiˆ 1 6V u i 4 where ˆ uˆu ˆ i V u i tˆti By substituting Eq. (4) into Eq. (3), the secon erivative of u is rewritten as u tˆtiˆ V u i uˆu i 5 3 where ˆ C u By substituting Eq. (6) into Eq. (5), the secon erivative of u becomes! u V u i C u 7 tˆtiˆ 4 Let the sampling time in interpolation be secons an t i1 t i ˆ The first- an secon-orer approximation interpolation algorithms are obtaine by substituting Eqs. () an (7) into Eq. (1), respectively. By neglecting the higher orer term, the interpolation algorithms in Eq. (1) can be processe as follows: The first-orer approximation interpolation algorithm [1,6,9] u i1 ˆ u i V u i The secon-orer approximation interpolation algorithm [6,9] u i1 ˆ u i V u i! V u i Ts C u 4 where V(u i ) can be either the feerate comman, the specifie spee profiles of ACC/DEC, or any esire spee in a general machining process. 3. The spee-controlle interpolation algorithm 3.1. The compensatory parameter In Eqs. (8) an (9), the first-orer an the secon-orer interpolation algorithms are approximate results by neglecting the higher orer term as in Eq. (1). Therefore, the approximation error for those methos is unavoiable an an interpolation algorithm concerning the curve spee is propose in this paper. The present interpolation algorithm is base on the first-orer approximation interpolation algorithm with a compensatory value e(u i )as u i1 ˆ u 0 i1 e u i where e(u i ) is the compensatory value an u 0 i1 ˆ u i V u i To precisely calculate the compensatory value e(u i ), Taylor s expansion for the curve an its spee must be concerne. Suppose C u ˆ Cx u # C y u is the parametric curve representation function for the X an Y axes, the interpolate points C x (u i1 ) an C y (u i1 ) are approximate as follows: C x u i1 ˆC x u 0 i1 C x u 0 i1 e u i 10 C y u i1 ˆC y u 0 i1 C y u 0 i1 e u i 11 There are two approximation techniques use in eriving the interpolation algorithm. The first one is the Taylor s expansion of the parameter u with respect to the time t to obtain the first- or secon-orer approximation interpolation algorithms [1,6,9], as in Eqs. (8) an(9). The secon one is the Taylor s expansion of curve C with respect to the 9

4 35 S.-S. Yeh, P.-L. Hsu / Computer-Aie Design 31 (1999) DX ˆ C x u 0 i1 C x u i DY ˆ C y u 0 i1 C y u i Fig. 3. Geometrical representation of parameters. X 0 u 0 i1 ˆ C x u 0 i1 Y 0 u 0 i1 ˆ C y u 0 i1 The compensatory value e(u i ) can be irectly obtaine as e 1; u i ˆ Z ^ p Z 4UW U ˆ q DX X 0 u 0 i1 DY Y 0 u 0 i1 ^ X 0 u 0 i1 Y 0 u 0 i1 Š V u i DY X 0 u 0 i1 DX Y 0 u 0 i1 Š X 0 u 0 i1 Y 0 u 0 i1 13 parameter u as in Eqs. (10) an (11). By comparing Eqs. (1), (8) an (9) with Eqs. (10) an (11), the higher-orer term of the Taylor s expansion in the first approximation is estimate as the compensatory value e(u i ) by applying the secon approximation. Eqs. (10) an (11) implies that the ifference between position # 3 C x u i1 C x u 0 an 4 i1 5 C y u i1 C y u 0 i1 is etermine by the slope of curve with the ajustment gain e(u i ). Although V(u i ) is the instantaneous velocity at u ˆ u i in the interpolation algorithm, it is usually assigne as the constant feerate comman F in real interpolation applications. The following equation is provie to accurately achieve a linear motion from # # C x u i C x u i1 to C y u i C y u i1 with the esire spee V(u i ): q C x u i1 C x u i Š C y u i1 C y u i Š ˆ V u T i s 1 then, a quaratic equation for the compensatory parameter is erive as Ue Ze W ˆ 0 where U ˆ X 0 u 0 i1 Y 0 u 0 i1 Z ˆ DX X 0 u i1 DY Y 0 u i1 Š W ˆ DX DY V u i 3.. Selection of compensatory parameters As the two values in Eq. (13) are the roots of a quaratic equation, characteristics of roots have to be iscusse in real applications. Define two vectors as ~D ˆ DX # DY ~C 0 ˆ X 0 u 0 # i1 Y 0 u 0 i1 Eq. (13) can be rewritten as q e 1; u i ˆ ~D ~C 0 ^ ~C 0 V u i ~C 0 ~D ~C 0 14 where the geometrical relationship between vectors ~D an ~C 0 correspon to parameters among which the parameters u i, u 0 i1, u i1 are shown in Fig. 3. u is the angle between the ifference vector ~D an the ifferential vector ~C 0, an ~C 0 V u i ~C 0 ~D ˆ ~C 0 Ts V u i # ~C 0 ~D ˆ ~C 0 Ts 4V u i ˆ ~C 0 Ts V ~D 4 u i ~C 0 ~C 0 D ~ sin u5 ~C 0 T s As ~C 0 0 an Ts 0, { ~C 0 V u i ~C 0 ~D } has the same sign as ( V D u i ~ #) sin u

5 S.-S. Yeh, P.-L. Hsu / Computer-Aie Design 31 (1999) Fig. 4. Illustrative conitions (ii) an (iii). The solutions of Eq. (13) can be in the following three categories: (i) e 1, (u i ) are two ifferent real numbers if V u i! # ~D sin u (ii) e 1, (u i ) are the same real numbers if V u i ˆ! # ~D sin u an only the two ifferent real roots as in conition (i) are concerne in the present algorithm. Accoring to Eq. (14)! v ~C ~D 0 u ^ t V u ~C 0 i T ~C 0 s ~C 0 ~D e 1; u i ˆ ~C 0 q ˆ ~D cosu ^ V u i ~D sin u ~C 0 ˆ ~D cosu ^ Let V u i ~D ˆ m q V u i ~D ~D cos u ~C 0 By applying Taylor s expansion, roots of the quaratic equation have the approximate values in simple forms as m e 1 ~C 0 ~D cosu (iii) e 1, (u i ) are a pair of complex conjugate numbers if V u i! # ~D sin u Compare with ~D= an the esire spee V(u i ), we e u i ˆ DX X 0 u 0 i1 DY Y 0 u 0 i1 Š e ~D cosu ~C 0 As m is small, the first root is near zero an the other root is negative an relatively large. To achieve reliable compensation an forwar motion ring the interpolation process, the small compensatory parameter is preferable as q X 0 u 0 i1 Y 0 u 0 i1 Š V u i DY X 0 u 0 i1 DX Y 0 u 0 i1 Š X 0 u 0 i1 Y 0 u 0 i1 Š conclue that the sign of ( V D u i ~ #) sin u is ominate by angle u. Conitions (ii) an (iii), which may proce the same real roots or complex conjugate roots of the quaratic equation, are shown in Fig. 4. As the vector ~D an the ifferential vector ~C 0 are almost perpenicular an parameter u i1 is near points a or b as shown in Fig. 4, sin u 1 an the conitions of V u i! # ~D sin u may occur. In physical meaning, the multiple real roots an the complex conjugate roots exist where the curvature is relatively large. In general applications, the curvature shoul be small to achieve precise interpolation. Thus, conitions (ii) an (iii) are not allowe in real applications As the present spee-controlle interpolation algorithm incorporates the first approximation interpolation algorithm an a suitable compensatory value which corrects the curve spee error, the obtaine curve spee almost equals the specifie spee V(u i ) ring the interpolating process. The roots conition {~C 0 V u i ~C 0 ~D } can be examine before calculating the compensatory value. When the unesirable conition may occur, the compensatory value is set to be zero to avoi the complex conjugate roots. In real machining processes, the present interpolator achieves (1) a constant spee an () specifie ACC/DEC. The constant spee moe keeps the curve spee almost the same as the given constant feerate comman ring the machining process. The ACC/DEC moe makes the curve spee in smooth profiles with the specifie spee for machining parametric curves.

6 354 S.-S. Yeh, P.-L. Hsu / Computer-Aie Design 31 (1999) Fig. 5. The example of NURBS. 4. Applications 4.1. The example of NURBS In this simulation, the interpolator is written by Turbo C.0 an is execute on a personal computer with both 80 an 00 MHz CPU. The present interpolator is applie to a NURBS [10] parametric curve with two egrees as shown in Fig. 5. The control points, weight vector, an knot vector of NURBS are assigne as follows: The orinal control points are # # # # ; ; ; ; # # # ; ; an mm : The constant-spee moe The curve spee fluctuations for ifferent interpolation algorithms are compare as below: (a) the uniform, (b) the first-orer approximation [1,6,9] (c) the secon-orer approximation [6,9] an () the propose spee-controlle moe. Simulation results for ifferent interpolation algorithms are shown in Figs. 6 8 an are summarize in Table 1. The curve spee fluctuation ratio for each intermeiate point is calculate by h i ˆ F V i =F where V i is the curve spee from the interpolate point C(u i )toc(u i1 ) an F is the given feerate comman. Accoring to the simulation results, most interpolation algorithms except the uniform interpolation result in high contouring accuracy with relatively small chor errors. As shown in Figs. 6 8, The weight vector is W ˆ [ ]. The knot vector is U ˆ [ ]. The interpolating processes are as follows: the sampling time in interpolation is ˆ 0:00 s an the feerate comman is F ˆ 00 mm/s ˆ 1 m/min. In many recent applications, the machining is in a high spee like high-spee machining, e.g. high-spee milling [11 14], machining by linear motor [15,16], an laser machining [17]. In this example, the provie weight vector which results in sharp corners is use to exam the spee eviation of ifferent interpolation algorithms uner the feerate comman F ˆ 00 mm/s ˆ 1 m/min. Fig. 6. Simulation results of the first-orer approximation.

7 S.-S. Yeh, P.-L. Hsu / Computer-Aie Design 31 (1999) Fig. 7. Simulation results of the secon-orer approximation. all the maximum spee eviation occurs at the four sharp corners. Moreover, the spee-controlle interpolation algorithm provies the best curve spee accuracy ring the interpolating process. Simulation results show that the curve spee eviates within [0.004, 0.004] mm/s by applying the present spee-controlle interpolation algorithm. However, the secon-orer approximation achieves larger spee eviation within [0., 0.] mm/s an the firstorer approximation achieves the largest spee eviation within [5, 5] mm/s. By applying the present algorithm, the roots of the quaratic equation for the compensatory value are also shown in Fig. 9. Fig. 9 inicates that one root is near zero which is aopte here while the other is negative an relatively large. The curve spee accuracy of the Fig. 9. Roots of quaratic equation e 1, (u i ). interpolator by applying the uniform interpolation algorithm is the worst case because the unefine real map operation of curve an parameters is not uniform. Although the present interpolation algorithm takes a longer perio to compute the compensatory value, its processing time can be significantly improve with an upate CPU to achieve esirable performance The ACC/DEC moe In orer to avoi shock or vibration of mechanical systems when starting an slowing own the axial travel, ACC/DEC algorithms are require ring the machining process. The conventional ACC/DEC algorithms for parametric curves usually result in larger raial an chor errors. Fig. 8. Simulation results of the present constant-spee interpolation. Fig. 10. Spee error of the first-orer approximation ring acceleration.

8 356 S.-S. Yeh, P.-L. Hsu / Computer-Aie Design 31 (1999) Table 1 Simulation results for ifferent interpolation algorithms Type Measure Mean-square of spee error (mm/s) The maximum spee fluctuation ratio The maximum chor error (mm) Computation time with 80 MHz CPU (ms) Computation time with 00 MHz CPU (ms) Uniform First orer approximation [1,6,9] Secon orer approximation [1,6,9] Constant spee Table Measurement of the maximum curve spee error Interpolation ACC type Linear (mm/s) Parabolic (mm/s) Exponential (mm/s) First orer approximation Secon orer approximation Spee controlle Kim [8] propose the parametric ACC/DEC algorithm. As the mapping operation between the curve an the parameter is not uniform in nature, the ACC/DEC in the uniform parameter interpolation algorithm cannot obtain the esirable acceleration an eceleration. The curve spee errors in ifferent acceleration profiles linear, parabolic, an exponential profiles with ifferent interpolations as (a) the first-orer approximation, (b) the secon-orer approximation, an (c) the speecontrolle interpolation are compare. The curve spee errors by applying the parabolic ACC/DEC metho are shown in Figs an are also summarize in Table. Results inicate that the largest spee error is cause by applying the exponential ACC/DEC metho an the smallest curve spee error is obtaine by applying the parabolic ACC/DEC metho. Accoring to the simulation results, the curve spee is also change parabolically ring the acceleration process. Results also inicate that the present interpolation algorithm achieves the best spee accuracy with error magnitue 3 5 orer less than when other methos are use. In general, ACC/DEC functions are require in real Fig. 11. Spee error of the secon-orer approximation ring acceleration. Fig. 1. Spee error of the spee-controlle interpolation ring acceleration.

9 S.-S. Yeh, P.-L. Hsu / Computer-Aie Design 31 (1999) machining with ifferent spee profiles, like linear, parabolic, an exponential types [5,8]. In this paper, the eceleration algorithm is the same as the acceleration algorithm except that the spee is ecreasing ring the process. Results inicate that the present spee-controlle interpolation algorithm achieves smooth acceleration an eceleration with the specifie ACC/DEC spee profiles. 5. Conclusions The propose spee-controlle interpolation algorithm has significantly improve curve spee accuracy for parametric curves by incluing a compensatory value. The constant-spee moe operates the contiguous interpolate points with a constant curve spee an the ACC/DEC moe generates interpolate points successively with smooth spee variation. The propose interpolation algorithm provies the best accuracy in both position an spee in real machining processes. Although the processing time of the propose interpolation algorithm is a little longer than the others, it has been successfully implemente with faster computers. References [1] Shpitalni M, Koren Y, Lo CC. Realtime curve interpolators. Computer-Aie Design 1994;6: [] Bei S, Ali I, Quan N. Avance techniques for CNC machines. Trans ASME J Engng Instry 1993;115: [3] Huang JT, Yang DCH. A generalize interpolator for comman generation of parametric curves in computer controlle machines. Japan/USA Symposium on Flexible Automation 199;1: [4] Yang DCH, Kong T. Parametric interpolator versus linear interpolator for precision CNC machining. Computer-Aie Design 1994;6(3): [5] Koren Y. Computer control of manufacturing systems. New York: McGraw-Hill, 198. [6] Chou JJ, Yang DCH. Comman generation for three-axis CNC machining. Trans ASME J Engng Instry 1991;113: [7] Lo CC, Chung CY. Curve generation an control for biaxial machine tools. J. CSME 1997;18: [8] Kim DI. Stuy on interpolation algorithms of CNC machine tools. IEEE Instry Applications Conference 1995;3: [9] Chou JJ, Yang DCH. On the generation of coorinate motion of fiveaxis CNC/CMM machines. Trans ASME J Engng Instry 199;114:15. [10] Piegl L, Tiller W. The NURBS book. Berlin: Springer, [11] Davies MA, Dutterer B, Pratt JR, Schaut AJ. On the ynamics of highspee milling with long, slener enmills. Cirp Annuals 1998;47: [1] Tung ED, Urushisaki Y, Tomizuka M. An algorithm for high-spee control of machine tools. IEEE Proc ACC 1993; [13] Tung ED, Tomizuka M, Urushisaki Y. High-spee en milling using a feeforwar control architecture. Trans ASME J Manufacturing Sci Engng 1996;118: [14] Schulz H, Moriwaki T. High-spee machining. Cirp Annuals 199;41/4(1): [15] Alter DM, Tsao TC. Optimal feeforwar tracking control of linear motors for machine tool rives. IEEE Proc. ACC. 1995; [16] Czajkowski S, Eielberg B, Heinemann G, Stollberger C. Linear motors: the future of high-performance machine tools. Am Machinist 1996;140: [17] Hermanns C. Making lasers more proctive. Manufacturing Engng 1997;119:4 50. Syh-Shiuh Yeh receive the BS egree in mechanical engineering an MS egree in electrical an control engineering from National Chiao Tung University, Taiwan, ROC in 1994 an 1996, respectively. He is currently a PhD caniate in the Institute of Electrical an Control Engineering at National Chiao Tung University, Taiwan, ROC. His research interests inclue motion system an control esign. Pau-Lo-Hsu (M 9) was born in a Taiwanese Christian pastor an missionary family. He receive the BS egree from National Cheng Kung University, The MS egree from University of Delaware, an the PhD egree from the University of Wisconsin-Maison, in 1978, 1984, an 1987, respectively, all of which were in mechanical engineering. After a twoyear military service in King-Men, he worke for San-Yang (Hona) Instry in 1980/1981 an Sanvik (Taiwan) in 1981/198. He then pursue his master s egree uner Prof. Tsu- Wei Chou an PhD egree uner Prof. Sam Wu. In 1988, he joine the Department of Electrical an Control Engineering, National Chiao Tung University as an Associate Professor an was promote to Professor in Since 1998, he is the Chairman of the epartment. His research interests inclue mechantronics, motion control, an fault iagnostic systems.