A METHOD OF REAL-TIME NURBS INTERPOLATION WITH CONFINED CHORD ERROR FOR CNC SYSTEMS

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1 Vietnam Journal of Science and Technology 55 (5) (017) DOI: /55-518/55/5/906 A METHOD OF REAL-TIME NURBS INTERPOLATION WITH CONFINED CHORD ERROR FOR CNC SYSTEMS Nguyen Huu Quang *, Banh Tien Long Hanoi Univerity of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Ha Noi, Viet Nam * quang.nguyenhuu@hut.edu.vn Received: 8 February 017; Accepted for publication: 10 April 017 ABSTRACT Thi paper preent a method of real-time CNC interpolation for complex curved profile which can be decribed with NURBS equation. The real-time interpolation algorithm i baed on econd order Taylor expanion with the principle part being a formula for updating parametric value u after each ampling period. With the updated value of u, a new interpolated point i calculated baed on the DeBoor algorithm. Chord error caued by the interpolation algorithm depend on two main factor which are machining feedrate and radiu of curvature of the NURBS curve. In thi paper, an efficient method of limiting chord error i alo preented with the baic idea of reducing machining feedrate at poition with a radiu of curvature maller than a critical value. Simulation reult on Matlab oftware verify effectivene of the propoed method. Keyword: CNC interpolation; CAD/CAM; NURBS; DeBoor algorithm. 1. INTRODUCTION Nowaday parametric curve uch a Bezier, B-pline, NURBS, etc. are being widely applied in well-developed CAD ytem to deign part with complex curved profile. On the other hand, conventional CNC machine typically only upport motion along traight line (G01) and circular arc (G0,G03). To bridge the gap between CAD ytem and CNC machine there are two main approache: In the firt one, original parametric curve i approximated by a equence of hort linear egment with the ue of CAM oftware; And in another one, a new parametric interpolator i uggeted to be developed for CNC ytem o that it ha the ability of calculating interpolated poition in real-time baed on the parametric equation of the curve [1 5]. In the linear approximation method (the currently popular method), contour error depend on the number of linear egment. In order to achieve the deired tolerance (typically being 10 µ m ), the number of egment i uually very huge and the length of each egment i very hort, leading to the large ize of G-code file the deviation of machining feedrate, a well a the reduction of machining productivity and part urface quality, etc. In the real-time method, the interpolation algorithm tae account of the commanded feedrate in calculation of the tool

2 A method of real-time NURBS interpolation with confined chord error for CNC ytem poition for next cycle from the current poition. Therefore, the maching feedrate will be more table in the real-time method. Additionally, the ize of G-code file will be much maller, becaue the geometric parameter of the curve (e.g., control point coordinate weight etc.) will be paed to CNC ytem intead of end point coordinate of linear egment [1 5]. Thi paper preent a real-time interpolator for NURBS curve. NURBS (Non-Uniform Rational B-Spline) are generalization of non-rational and rational Bezier, a well a nonrational B-pline curve. They offer one common mathematical form for both tandard analytical hape (e.g., circular arc ellipe etc.) and free-form one with the great flexibility to deign a large variety of geometrie through adapting control point weight and not value. Alo, they can be evaluated reaonably fat by numerically table and accurate algorithm. With many advantage lited above and more, NURBS ha been widely ued in CAD ytem today. In that trend, real-time NURBS interpolator will promiingly have expanion application and alo promote the integration of CAD, CAM and CNC technologie to improve productivity and quality in mechanical production. In thi paper, the problem of chord error appeared in the realtime interpolation proce i alo conidered, and a imple but efficient formula to adapt machining feedrate i propoed for the purpoe of limiting thi ind of error.. PARAMETRIC NURBS CURVES A pth-degree NURBS curve i defined a follow [6]: where, [ ] n x( u) Ni, p ( u) wi Pi i= 0 C ( u) = y( u) =, a u b n z( u) Ni, p ( u) wi T i= 0 { P = X, Y, Z } (0 i n) are control point { w } are the correponding weight of i i i i { P i }. { Ni, p( u )} are pth-degree B-pline bai function defined on the not vector U = { a,..., a, u p+ 1,..., um p 1, b,..., b}. The pth-degree B-pline bai function are recurively p+ 1 p+ 1 defined a follow: N i,0 1, u [ ui, ui+ 1) ( u) = 0, u [ ui, ui+ 1) u u u u N ( u) N ( u) N ( u) i i+ p+ 1 i, p = i, p 1 + i+ 1, p 1 ui+ p ui ui+ p+ 1 ui+ 1 One of the mot important algorithm for NURBS i the DeBoor algorithm [6]. Thi algorithm i a generalization of the decateljau' algorithm for Bezier curve. It provide a fat and numerically table way to find a point on a NURBS curve, given a u value in the domain. Some important geometric propertie of NURBS curve (e.g., tangent, curvature, etc.) concern high order derivative of them. The th derivative of a NURBS curve C ( u), denoted by ( C ) ( u), i defined a following: ( ) ( ) d C C u = u = [ x ( u), y ( u), z ( u) ] T (4) du i () (3) 651

3 Nguyen Huu Quang, Banh Tien Long ( 1) ( ) dc ( u) C ( u) =, (5) du According to [6], the derivative of a pth degree NURBS curve i a (p-1)th degree NURBS curve. Therefore, the DeBoor algorithm can alo be applied to calculate high order derivative of NURBS curve. 3. DESIGN OF A REAL-TIME NURBS INTERPOLATOR Baed on input data including: (a) parametric equation of the NURBS path; (b) machining feedrate V ; (c) interpolation period T; and (d) the current poition of the tool, the real-time NURBS interpolator deigned in thi ection will calculate the target poition of the tool for next interpolation cycle in conideration of the chord error tolerance A NURBS interpolation algorithm baed on the econd order Taylor expanion According to differential geometry, the feedrate calculated a following: and, V = = = u From (6) we can deduce: du V = dt C ( u) d u dt V along the NURBS path can be dc( u) dc( u) du du C ( ) (6) dt du dt dt () V C ( u), C ( u) = 4 C ( u) where, denote calar product of two vector and denote magnitude of a vector. Define u = u( T ), V, = V ( T ). By uing Taylor expanion, we have the approximation up to the econd derivative a following: du T d u + 1 t= T t= T u = u + T + + ε (9) dt dt In Equation (9), ε i truncation error which i determined in requirement. Subtituting (7) and (8) into (9) we have the following econd order Taylor approximation: u V T C ( u ), C ( u ) () V T + 1 = u + 4 C ( u ) C ( u ) The Equation (10) ha a recurive form and can be ued to update the parametric value u after each interpolation period. With the new value of u, the next target tool poition can be obtained by applying the parametric equation of the NURBS curve and the DeBoor algorithm: (8) (10) 65

4 A method of real-time NURBS interpolation with confined chord error for CNC ytem ( x, y, z ) = C ( u ) (11) = Equation (10) and (11) form the bai of a NURBS interpolation algorithm. They can be implemented in real-time by digital ignal proceing ytem. 3.. Chord error calculation In an interpolation period, the tool move linearly on the chord connecting C ( u ) and C ( u + 1) and caue the chord error which i the maximum deviation of the chord from the correponding curve egment. To evaluate thi ind of error, we conider a circular arc which i tangent with and ha the ame radiu of curvature a the NURBS curve at C ( u ), a demontrated in Figure 1. If the tool didn t move along the NURBS curve but the circular arc in one interpolation period with feedrate V, then it would reach a poition P ( u + 1) intead of C ( u + 1). Let ρ denote the radiu of curvature of the NURBS curve at C ( u ), and define l = P( u + 1) P( u ). Becaue the value of u and u + 1 are very cloe together, the chord error in movement along the NURBS path from C ( u ) to C ( u + 1) can be etimated by the chord error, e, in movement along the circular path from P ( u ) to P ( u + 1) (with P( u ) C ( u ) ). From geometric relationhip we can infer: l e = ρ ρ (1) Figure 1. Etimation of chord error. Additionally, l approximate to the path length which the tool move in one period T, o: l = V T (13) From and we deduce: e V T V T V = ρ ρ = 4 V 8 T ρ 4 ρ + ρ 4 T (14) 653

5 Nguyen Huu Quang, Banh Tien Long Equation (14) i an etimation of the chord error. It how that the chord error i dependent on two major variable which are the feedrate, V, and the radiu of curvature of the curve, ρ. The radiu of curvature of the NURBS curve, ρ, can be calculated a following [6]: ρ = C C ( u ) ( u ) C ( u ) 3 () where, ymbol denote vector product of two vector. (15) 3.3. Confining the chord error Let e max denote the chord error tolerance. We have the following contraint: In (16), e V e V e = (16) 8 max c max ρ T ρc V c i the commanded feedrate and chord error i propoed a following: Vc T ρc =. The method of confining the 8 e If ρ ρc then the machining feedrate i maintained a commanded feedrate: V, = Vc. If ρ < ρc then the machining feedrate in current interpolation cycle i reduced a below: V ρ = Vc. (17) ρc max 4. SIMULATION AND DISCUSSION Simulation method i ued to evaluate the real-time NURBS interpolation algorithm in two cae: without and with conideration of the chord error tolerance. The tet NURBS curve i choen a Figure. Figure. The tet NURBS curve (olid line). Dahed line repreent it control polygon. A,B,C mar high curvature zone. Figure 3. Curvature plot of the tet NURBS curve. A,B,C mar correponding curvature of A,B,C poition on Figurre. A hown in Figure 3, it curvature varie coniderably along the curve. In general, thi characteritic mae the tet curve a difficult toolpath for CNC machining [7, 8]. Geometric 654

6 A method of real-time NURBS interpolation with confined chord error for CNC ytem parameter of the tet NURBS curve are lited in Table 1. The imulation program i developed on the Matlab oftware with the interpolation period T choen a 1 m. Table 1. Geometric parameter of the tet NURBS curve. Degree p=3 Knot vector (0,0,0,0,0.5,0.5,0.75,1,1,1,1) P 0 =(10,0,0), w 0 =1 P 1 =(0,,0), w 1 =1 P =(1,8,0), w =1 Control point and weight P 3 =(10,0,0), w 3 =1 P 4 =(8,8,0), w 4 =1 P 5 =(0,,0), w 5 =1 P 6 =(10,0,0), w 6 = Evaluation of the NURBS interpolator without confined chord error The NURBS interpolation algorithm without confined chord error (Section 3.1) i imulated with the tet NURBS curve. With different feedrate which are 100 mm/ 60 mm/ and 30 mm/ the chord error are hown in Figure 4. It can be een from Figurre 4 the trend of increaing chord error when the feedrate get higher value. With the feedrate of 100 mm/ (6 m/min), the chord error can be larger than 10 µm which i quite large. The chord error decreae to be lower than 5 µm with the feedrate of 60 mm/ and continue decreaing to about 1 µm with the feedrate of 30 mm/. Thee reult confirm the idea of reducing chord error by reducing machining feedrate. For another remar, we can ee that the range of machining feedrate below 30 mm/ in which the chord error i about 1 µm, in fact, i uitable with almot normal CNC machining application. So, the problem of limiting chord error only get more important in high-peed machining. Figure 1. Chord error caued by NURBS interpolator without confined chord error. 655

7 Nguyen Huu Quang, Banh Tien Long 4.. Evaluation of the NURBS interpolator with confined chord error The NURBS interpolator with confined chord error i imulated with emax = 1µ m. When the commanded feedrate i et to be 100 mm/ Figure 5 (a) how that the chord error i reduced ignificantly in comparion with the previou imulation reult in Figure 4. At everal poition the chord error are till larger than 1 µm, becaue the feedrate adapting formula i only baed on an approximate etimation of the chord error. 656 Figure. Simulation reult of the NURBS interpolator with confined chord error: (a),(b) correponding to the feedrate of 100 mm/; (c),(d) correponding to the feedrate of 00 mm/. Figure 5(b) repreent the machining feedrate. It can be een that the feedrate i reduced at ome poition (mared a A,B,C), a a reult of the error limiting algorithm. At other poition the machining feedrate i maintained almot the ame a commanded feedrate. When the commanded feedrate i doubled to 00 mm/ the chord error i hardly greater, a hown in Figure 5(c). Thi reult confirm the effectivene of the error limiting algorithm. However, the number of poition with feedrate reduction increae, a hown in Figure 5(d). The reaon i that the critical radiu of curvature, ρ c, get larger when the commanded feedrate get higher, leading to the expanion of the curve egment at which machining feedrate need to adapt. Thi effect will decreae the average machining feedrate, a well a increae the machining time. 5. CONCLUSION In thi paper, a real-time interpolation method i preented for parametric NURBS curve. The interpolation algorithm i baed on the econd order Taylor expanion. Thi method can

8 A method of real-time NURBS interpolation with confined chord error for CNC ytem maintain a mall feedrate fluctuation, but the chord error can be large at poition on the curve which have a high curvature (i.e., ha a mall radiu of curvature). The problem of chord error can be overcome by monitoring the radiu of curvature along the NURBS curve during interpolation time, and adapting the machining feedrate uch that it get properly maller value at poition which have a radiu of curvature below a critical value, ρ c. Simulation reult on the Matlab oftware how that, although the feedrate adapting formula i quite imple, it i very ueful in repecting the chord error tolerance. Acnowledgement. Thi reearch i funded by the Hanoi Univerity of Science and Technology (HUST) under project number T016-PC-061. REFERENCES 1. Koren Y., Lo C. C., Shpitalni M. - CNC interpolator: algorithm and analyi Ame Prod Eng Div Publ Ped, Ame, New Yor, USA 64 (C) (1993) Shpitalni M., Koren Y., Lo C. C. - Realtime curve interpolator Comput. De. 6 (11) (1994) Zhiming X., Jincheng C., Zhengjin F. - Performance evaluation of a real-time interpolation algorithm for NURBS curve Int. J. Adv. Manuf. Technol. 0 (4) (00) Hong Tzong Yau, Ming Tzong Lin, Meng Shiun Tai - Real-time NURBS interpolation uing FPGA for high peed motion control, CAD Comput. Aided De. 38 (10) (006) Banh Tien Long, Nguyen Huu Quang - A method of real-time NURBS interpolation for CNC ytem Journal of Science & Technology Technical Univeritie 114 (016) (in Vietnamee). 6. Le Piegl, Wayne Tiller, The NURBS boo, nd ed., Springer, New Yor, Banh Tien Long, Bui Ngoc Tuyen - Surface machining theory and application in mechanical engineering, Vietnam Education Publihing Houe, Ha Noi, 01 (in Vietnamee). 8. Banh Tien Long, Tran The Luc, Nguyen Chi Quang - Theory of cutting tool geometry and machining, Science and Technic Publihing Houe, Ha Noi, 006 (in Vietnamee). 657