Intelligent NURBS interpolator based on the adaptive feedrate control
Chinese
Journal of
Aeronautics
Chinese Journal of Aeronautics 20(2007) 469-474
https://www.sodocs.net/doc/f711461180.html,/locate/cja
Intelligent NURBS Interpolator Based on the Adaptive
Feedrate Control
Zhang Deli *, Zhou Laishui
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Received 19 December 2006; accepted 21 March 2007
Abstract
To satisfy the need for high-speed and high-accuracy machining of NURBS curve. Firstly the form of NURBS curve is analyzed and Talor’s expansion of the parameter u with respect to time t is used to obtain the algorithm of the first order approximation interpola-tion. Secondly, based on the algorithm of the controlled chord error interpolator, an intelligent interpolation algorithm of the adaptive feedrate control is proposed. According to the actual machining capacity of machine tools, this algorithm uses look-ahead method, which dispenses with the complicated computation of the end point estimation of NURBS curve, to analyze the curve segment required by the maximum deceleration distance. Thus, the feedrate could decrease in advance and vary with the curvature and the variation ratio of cur-vature, which makes machining motion quite smooth. Not only could high accuracy and fine surface quality be achieved during high-speed machining, but also the overload of cutter tools is avoided on corners. Finally, in order to facilitate the calculation of interpo-lation, the dynamic matrix representation and efficient algorithm of curvature computation of the NURBS curve are presented. Keywords:CNC; NURBS curve; interpolation; high-speed machining
1 Introduction In aeronautical and astronautical devices, high- accuracy integral components with complex surface and profile are widely used, which are usually pro-duced by milling, for example, in case of manufac-turing integral impellers. The milling process of those integral components needs high performance CNC system. However, the traditional milling method, which uses piecewise linear segment to ap- proximately approach the profile curve segment to be milled, results in low efficiency and poor surface quality. Thanks to advanced CNC system and en-hanced calculating speed of computers, an NURBS curve generated by free form surface in CAD sys-
*Corresponding author. Tel.: +86-25-84604619.
E-mail address : nuaazdl@https://www.sodocs.net/doc/f711461180.html,
Foundation item: National Excellent Young Teacher Encouragement Plan
of China
tems can be directly machined. Hence the following questions are resolved: (1) the memory and trans-mission load of a large number of data of NC code due to piecewise line segment approximation ap-proach; (2) bad surface quality of parts [1-3].
The simplest interpolation algorithm is that the feedrate remains constant except at start and end points. It means that chord error varies with the curvature of curve and the machining accuracy can not be guaranteed. A controlling chord error method was proposed that the feedrate varies with the cur-vature of curve. But the tangent and normal direc-tion acceleration of machine tools is limited by the capacity of electrical and mechanical system.
Therefore, it is concluded that the feedrate can not be rapidly decreased to the one that makes the chord error within the specified tolerance at sec-tional curve where the variation ratio of curvature
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becomes very big; and furthermore, the feedrate which satisfies the need of chord error may not be responded to machine tools because of the limited normal acceleration of machine tools when the in-stantaneous radius of curvature is very small. Based on the algorithm of the controlling chord error in-terpolator, an intelligent interpolation algorithm of the adaptive feedrate control is proposed. Depend-ing on the actual machining capacity of machine tools, this algorithm uses look-ahead method to analyze the sectional curve that the maximum deceleration distance requires, which dispenses with the complicated calculation of the end point estima-tion of NURBS curve. Thus, the feedrate could de-crease in advance and varies with the curvature and the variation ratio of curvature making machining motion quite smooth.
2 Real-time Adaptive Interpolation of 3D NURBS
2.1 Def inition of NURBS
A NURBS curve is a vector-valued rational polynomial defined by the following form
,0,0
()n
i i i k i n
i i k i w d N u C u w N u
|| (1)
where d i ,i = 0, ···, n , is control point, w i ,i = 0, ···, n ,is weight fact, N i,k (u ) is the normalized B-spline basis function of order k with the following nota-tions:
1
,01,,11,1111when 0others 0
define 00i i i i i k i k i k i k i k i i k i u u u N u u u u u N u N u N u u u u u ?- ?°
ˉ°
° °
?
°
° °
°?
?? (2)
2.2 The algorithm of controlled chord error
interpolation [4-5]
Let the function of parameter u with respect to
t be u (t i ) = u i ,u (t i +1) = u i +1, using Talor’s expansion, the approximation up to the second derivative is 22
1
112
d d ()()
d d H.O.T
i i i i t t i i t t i i u u u u t t t t t t (3)
the feedrate is defined by
d d d d d d s s u V u t u t
(4) where
d d d ,,d d d s x y x'y'u u u
d .d z z'u
Let the sampling time interval be T s seconds and T i +1–T i =T s . Then the first-order interpolation algorithm is obtained by substituting Eq.(4) into Eq.(3), and neglecting the high order term, Eq.(3) can be processed as follows
s 1d d i
i i i u u V u T u u s u
Supposing the sectional curve of u ?[u i ,u i +1] is approximated by circle arc ( shown in Fig.1), where r is the radius of curvature when parameter u =u i ,C (u i ) and C (u i +1) are the interpolated points on curve when u = u i ,u
= u i +1, respectively.
Fig.1 Estimation of next interpolated point.
Given chord length L = || C (u i +1) – C (u i )||, the
approximated curve speed V (u i ) is defined as
s
i L
V u T
and the derived chord error
r H Then the obtained V (u i ) in function of chord error ?is
Zhang Deli et al. / Chinese Journal of Aeronautics 20(2007) 469-474 · 471 ·
s
2i V u T
(5) If the chord error ? and sampling time T s are con-stant,V (u i ) varies with the radius r of curvature.
But the tangent and normal acceleration of machine tools are limited by the capacity of electri-cal and mechanical system. Hence, the required normal acceleration that causes the chord error within the specified tolerance at curve section where the curvature is very big would surpass the permit-ted value a Nmax , and the required tangent accelera-tion that causes the chord error within the specified tolerance at curve section where the variation ratio of curvature is large would surpass the permitted value a max , so the mechanical characteristics of ma-chine tools must be considered when the feedrate is adaptively adjusted. Therefore, the adaptive algo-rithm of feedrate adjustment is introduced below. 2.3 The interpolation algorithm of controlled
normal acceleration [1,6]The relationship between normal acceleration a ,feedrate V (u i ) and the radius r of curve curvature can be expressed as follows
2()i V u a r
The maximum permitted speed is
max V (6)
The adaptively adjusted feedrate is derived from
Eq.(5) and Eq.(6),
max max max
s
()when
2V u V V V T
-!°°?
°°ˉ(7)
But the actual tangent acceleration may not
satisfy the adjusted feedrate calculated by Eq.(7) at curve section where the variation ratio of curvature is very big. That means the feedrate can not decline to a very small one at short time which leads to ap-pearance of over-cutting. These problems can be resolved by the following algorithm.
2.4 The interpolation algorithm of controlled tangent acceleration
Let the maximum granted acceleration be a max ,and the corresponding feedrate be V max , on condi-tion actual deceleration be equal to
a max , then the decelerating distance s m from V max to V = 0 is as follows
s m =2
max V /(2a max )
Therefore, several sampling interpolated seg-ments on curve must be analyzed together in ad-vance in order to ensure that the total length of these sampling interpolated segments is at least equal to s m , and only in this way, the current machining segment is immune from feedrate smoothing of the latter sampling interpolated segments.
As shown in F ig.2, suppose each length of sampling interpolated segment is s 0,s 1, ···, s j , re-spectively and the feedrate is defined as V 0,V 1, ···, V j ,V j
+1, respectively.
F ig.2 Tracks of segment.
If s = (2j V –21j V )/2a max >s j in the j th sampling interpolated segment, V j must be adjusted.
If s i +1+ ··· +s j < s < s i +s i +1 + ··· +s j , the feedrate must be adjusted through to the (i +1)th point V i +1.Thus, adjusted feedrate 1i V
c can be obtaine
d as fol-lows
1 i V c
correspondingly,
2i V c
until j V c , feedrate adjustment men-tioned above only satisfies cutting condition from the (i +1)th to j th sampling interpolated segments, but the adjustment of the feedrate on the (i +1)th sampling interpolated segment is liable to affect the (i –1)th, (i –2)th sampling interpolated segments and so on. Hence, the speed of the corresponding sam-
· 472 · Zhang Deli et al. / Chinese Journal of Aeronautics 20(2007) 469-474
pling interpolated segments must be adjusted until the total length of these sampling interpolated seg-ments equals at least to s m , and current segment can be processed accordingly.
As for acceleration sampling interpolated seg-ment, shown in F ig.5, if V 1 must be adjusted by 2V (8) otherwise, V 2 will exert influence on the next feedrate analysis. When the end point of curve with u = 1 is reached, the feedrate must decrease to V = 0, and also must be adjusted with Eq.(8). Obviously, the proposed algorithm obviates the need for estimating the end point of machining curve. But in Ref.[7], a large amount of dynamic calculation of the curve length is conducted because of the end point estima-tion of curve which influences the real-time re-sponse. 3 Dynamic Matrix Representation of NURBS Curve Let i =1i = u i +1–u i , 2 i = u i +2–u i , 3i = u i +3– u i , ···, especially, 00i , to knot vector U , yield d u =^`01,,,n k , to u ?[u i ,u i +1], define +1()/()()/i i i i i t u u u u u u , then t ?[0,1]. If NURBS curve is the third order, K = 3, then the i th section of curve can be represented by matrix as follows 3322231132231[1]()[1]i i i i i i i i i i i i i i i w d w d t t t w d w d p t w w t t t w w ao??? ????????? ao??? ?????????M M where, 0?t ?1,i =3, 4, ···, n 11 12131421 2223243132333441 42 43 44i m m m m m m m m m m m m m m m m ao???? ?? ???? M 22 111132323 1211111 11232311 2 11213323 1133442211 114344 2323 11033303303i i i i i i i i i i i i i i i i i i i m m m m m m m m m m m m m m ao ?? ?? ? ? ?? ?? ?? ? ? ?? ?? ?? a?? ?? ?? ?? o?? ??? ????? ? Expand every term of this matrix and define a =m 11w i –3d i –3+m 12w i –2d i –2+m 13w i –1d i –1+m 14w i d i , b = m 21w i –3d i –3+m 22w i –2d i –2+m 23w i –1d i –1+m 24w i d i ,c =m 31w i –3d i –3+m 32w i –2d i –2+m 33w i –1d i –1+m 34w i d i ,e =m 41w i –3d i –3+m 42w i –2d i –2+m 43w i –1d i –1+m 44w i d i ,a 1= m 11w i –3+ m 12w i –2+ m 13w i –1+m 14w i ,b 1= m 21w i –3+ m 22w i –2+ m 23w i –1+m 24w i ,c 1= m 31w i –3+ m 32w i –2+ m 33w i –1+m 34w i ,e 1= m 41w i –3+ m 42w i –2+ m 43w i –1+m 44w i ,then 2323 1111i a bt ct et p t a b t c t e t As the control points d i and weights w i are known, matrix M i is only related to knot vector U and can be calculated before the interpolation of the i th section of curve. Hence, only the incremental value ?t of parameter t needs to be calculated during interpolation, which increases the calculation speed greatly. 4 The Software Realization of the Proposed Algorithm At the start portion of curve, trapezoidal or S-curve profile acceleration method and the pro-posed algorithm are applied separately. Comparing the speeds calculated by the two methods, the minimum speed of them is used, the proposed algo-rithm is applied alone until the later speed is great Zhang Deli et al. / Chinese Journal of Aeronautics 20(2007) 469-474 · 473 · than the former one. To avoid too much time spent on adjusting feedrate by Eq.(8), in which frequent and tedious calculation of square roots is demanded, the feedrate square is memorized other than the feedrate for pre-interpolation segments ?feedrate adjustment can be processed by the following equation 2221max 1 2V V a s (9) Only when the interpolation segment is to be machined is the square root of feedrate square cal-culated. Thus, every adjustment of feedrate of a segment is processed only four times with one addi-tion or subtraction (2a max can be treated as one pa-rameter). In the process of recursive calculation, only the value of feedrate is changed while the others remain unchanged until the interpolation segment is to be machined. Current pre-interpolation point is re-placed by the actual interpolation point, as shown in Fig.3, the 1st, 4th and 6th points are pre-calculated according to the chord error and the maximum feedrate, the 2nd, 3rd and 5th points are the actual interpolation points which are calculated by the al-gorithm of controlled tangent and normal accelera- tion. Fig.3 The interpolated points of NURBS curve. Because the feedrate between any two adjacent pre-interpolation points must satisfy the following inequality 1max s i i V V a T ?The feedrate between any two adjacent actual interpolated points must also satisfy the above re-quirement. The feedrate of the 2nd, 3rd and 5th points can be derived as follows V 2= (V 1+ V 4)/2V 3= (V 2+ V 4)/2V 5= (V 4+ V 6)/2 Finally, the incremental value d u of parameter U and the actual interpolated points are calculated. 5 Experiments In order to authenticate the proposed method, the feedrate is measured. The whole process is com-pleted by DSP-based motion control board with the following machining parameters: the maximum fee- drate is 6 m/min; the starting feedrate is 0.5 m/min; the acceleration in the feed direction, a max , is 9.8 m/s 2; the normal acceleration is 0.98 m/s 2; the in-terpolation time interval T s is 2 ms; the cutter radius is 3 mm; the contour error is 2 P m. Fig.4 shows the NURBS curve and its control points and Fig. 5 shows its curvature radius. Fig.4 The NURBS curve and its control points. Fig.5 The curvature radius of the NURBS curve. As shown in Fig.6, at some segments, accelera-tion or deceleration exceeds the given value. But after the adjustment of the interpolation algorithm of controlled tangent and normal acceleration (shown in Fig.7), the variation of feedrate and feed acceleration or deceleration all fall in the rang of CNC system parameters. Fig.8 shows the machined NURBS surface and F ig.9 shows the cross section · 474 · Zhang Deli et al. / Chinese Journal of Aeronautics 20(2007) 469-474 of the machined NURBS surface. Fig.6 The feedrate after the adjustment of the interpolation algorithm of the controlled chord error. Fig.7 The feedrate after the adjustment of controlled tan- gent and normal acceleration. Fig.8 The machined NURBS surface. Fig.9 The cross section of the machined NURBS surface. 6 Conclusions Of great importance to high-speed and high- accuracy machining system, the interpolation of NURBS curve has drawn close attention in this realm. Although the time spent by the proposed al-gorithm is much more than that by the constant feedrate algorithm, still, it has such advantages as follows: (1) Taking the actual capacity of machine tools in full consideration. (2) Being able to control contour errors. (3) Making the motion of machine tools quite smooth. (4) Improving the surface quality of work-pieces.References [1] Cheng M Y , Tsai M C, Kuo J C. Real-time NURBS command generators for CNC servo controllers. International Journal of Machine Tools and Manufacture 2002; 42: 801-803. [2] Lartigue C, Thiebaut F, Maekawa T. CNC tool path in terms of B-spline curves. Computer-Aided Design 2001; 33(4): 307-319. [3] Bahr B, Xiao X, Krishnan K. A real time scheme of cubic para-metric curve interpolations for CNC systems. Computers in In-dustry 2001; 45: 309-317. [4] Yeh S S, Su P L H. Adaptive feedrate interpolation for parametric curves with a confined chord error. Computer-Aided Design 2002; 34(3): 229-237. [5] Farouki R T, Tsai Y F. Exact Taylor series coefficient for variable feedrate CNC curve interpolators. Computer-Aided Design 2001; 33(2): 155-165. [6]Farouki R T, Manjunathaiah J, Nicholas D, et al. Variable feedrate CNC interpolators for constant material removal rates along Py-thagorean-hodograph curves. Computer-Aided Design 1998; 30 (8): 631-640. [7]Krishnan K K, Kappen J, Bahr B. Calculation of variable feedrate and machining. Transactions of NAMRI/SME 2001; 24: 429-435. Biography: Zhang Deli Born in 1973, he is a post-doctoral fellow. He focuses his re-search interest on CAD/CAM and CNC. E-mail: nuaazdl@https://www.sodocs.net/doc/f711461180.html,