外文翻譯--加工參數(shù)曲線的速度控制插補器

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1、　　附錄A：英文資料　　The speed-controlled interpolator for machining parametric curves　　S.-S. Yeh, P.-L. Hsu 　　Abstract:

2、;　　Modern CNC systems are designed with the function of machining arbitrary parametric curves to save massive data communication between CAD/CAM and CNC systems and improve their machining quality. A

3、lthough available CNC interpolators for parametric curves generally achieve contouring position accuracy, the specified federate, with dominates the quality of the machining processes, is not guaranteed during motion. Re

4、cently, some approximation results concerning motion speed have been reported [Shpit　　Keywords:　　Parametric curve; CNC interpolators; Speed accuracy; NURBS<

5、/p>　　1. Introduction:　　In modern CAD/CAM systems, profiles for parts like dies, vanes, aircraft models are usually represented in parametric forms. As conventional CNC machines only pro

6、vide linear and circular arc interpolators, the CAD/CAM systems have to segment a curve into a huge number of small, linearized segments and send them to CNC systems. Such linearized-segmented contours processed on tradi

7、tional CNC systems are undesirable in real applications as follows:　　● the transmission error between CAD/CAM and CNC systems for a huge number of data may easily occur, i.e. lost data and noise pert

8、urbation;　　● the discontinuity of segmentation deteriorates surface accuracy;　　● the motion speed becomes unsmooth because of the linearization of the curve in each segment, espe

9、cially in acceleration and deceleration.　　As the generated curves or profiles may be in a parametric form, only parametric curve 　　information is required to be efficiently trans

10、ferred among CAD/CAM/CNC systems as shown in Fig. I. Shpitalni et al. [1] proposed the curve segments transfer between CAD and CNC systems and Bedi et al. [2] proposed the B-spline curve and B-spline surface interpolatio

11、n algorithm to obtain both accurate curves and gouge free surface. Huang and Yang [3] developed a generalized interpolation algorithm for different parametric curves with improved speed fluctuation. Moreover, Yang and Ko

12、ng [4] studie　　In these CNC systems, parametric curves are profiles in different formats like the Bezier curve, B-spline, cubic spline, and NURBS (non-uniform rational B-spline). The general paramete

13、r iteration method used is　　ui+1=ui+△(ui)　　where ui is the present parameter, ui+1 is the next parameter, and △(ui) is the incremental value. The interpolated points are calculat

14、ed by substituting ui into the corresponding mathematical model to recover the originally designed curves. As the cutter moves straight between contiguous interpolated points, two position errors may occur as: (a) radial

15、 error; and (b) chord error [5] during motion for a parametric curve as shown in Fig.2.　　CAD CNC　　Fig.1. The machining systems with parameters transm

16、ission　　Parametric curve 　　Cutter path ● D　　Radial error A Chord error　　Interpolated&

17、lt;p>　　B ●C　　Fig. 2. The radial error and the chord error　　The radial error is the perpendicular distance between the interpolated points and the parametric

18、 curve. Basically, the radial error is caused by the rounding error of computer systerms. With the rapid development of microprocessors with higher precision, the radial error is no longer a major concern in present appl

19、ications. In addition, the chord error is the maximum distance between the secant CD and the secant arc AB. A small curvature radius with a fast feedrate command may cause the chord e　　However, cutti

20、ng speed dominates the quality of the machining process. To achieve the specified feedrate for parametric curves is usually difficult. The undesirable motion speed may deteriorate the quality of the machined surface. Sev

21、eral researchers have developed different interpolation algorithms for parametric curves to improve motion speed accuracy. Bedi et al. [2] set △(ui) as a constant to form the uniform interpolation algorithm which is the

22、simplest method and its chord error and curve 　　The present speed-controlled interpolation algorithm is proposed by deriving a suitable compensatory parameter for the first-order approximation [1, 6,

23、 8] to obtain desirable motion speed. Then, the proposed interpolator is applied to the constant-speed mode and the acceleration/deceleration mode to achieve constant feedrate and the specified speed profiles, respective

24、ly. Thus, the present CNC interpolators result in stable motion and smoothly changed speed to avoid mechanical shock or vibration　　2.Parametric curve formulation: 　　Suppose C(u)

25、is the parametric curve representation function and the time function u is the curve parameter as 　　u (t i) = u i and u (t i+1) = u i+1　　By using Taylor’s expansion,

26、the approximation up to the second derivative is 　　u i+1 = u i +(du/dt│t=ti ) (ti+1–ti) + 1/2 (d2u/dt2│t=ti ) (ti+1–ti) 2+HOT (1)　　As the curve speed V (ui) can be represented a

27、s　　V(ui) = ‖dC(u)/dt‖u=ui = (‖dC(u)/du‖u=ui) (du/dt│t=ti ) 　　the first derivative of u with t is obtained as　　du/dt│t=ti = V(ui) / (‖dC(u)/dt‖u=ui) (2)

28、　　By taking the derivative of Eq. (2) , the second derivative of u with t is　　d²u/dt²│t=ti = –1/ (‖dC(u)/dt‖2u=ui){V(ui) [d(‖dC(u)/dt‖)│u=ui]/dt} (3)<p&

29、gt;　　where　　[d(‖dC(u)/dt‖)│u=ui]/dt={[d(‖dC(u)/du‖)│u=ui]/dt }(du/dt│t=ti )　　={[d(‖dC(u)/dt‖)│u=ui]/dt}{V(ui)/[‖dC(u)/du‖u=ui]} (4)　　B

30、y substituting Eq. (4) into Eq. (3), the second derivative of u is rewritten as　　d2u/dt2│t=ti = – V2 (ui) / (‖dC(u)/du‖3u=ui){[d(‖dC(u)/du‖)│u=ui]/du} (5)　　where</b&g

31、t;　　[d(‖dC(u)/du‖)│u=ui]/du = (dC(u)/du)(d2C(u)/du2)/(‖dC(u)/du‖u=ui) (6)　　By substituting Eq. (6) into Eq. (5), the second derivative of u becames　　d2u/dt2│

32、t=ti= – [V2 (ui)(dC(u)/du)(d2C(u)/du2)]/(‖dC(u)/du‖4u=ui) (7)　　Let the sampling time in interpolation be Ts seconds and 　　t i+1－t i = Ts　　The first- and

33、second-order approximation interpolation algorithms are obtained by substituting Eqs. (2) and (7) into Eq. (1) ,respectively. By neglecting the higher order term, the interpolation algorithms in Eq. (1) can be processed

34、as follows:　　The first-order approximation interpolation algorithm [1, 6, 9]　　u i+1 = u i + V(ui) Ts /(‖dC(u)/du‖u=ui) (8)　　The second-order approxima

35、tion interpolation algorithm [6, 9]　　u i+1 = u i + V(ui) Ts /(‖dC(u)/du‖u=ui) –{V2 (ui) T2s [(dC(u)/du)(d2C(u)/du2)]│u=ui}/ [2(‖dC(u)/du‖u=ui)] (9)　　where V (ui) can be eit

36、her the feedrate command, the specified speed profiles of ACC/DEC, or any desired speed in a general machining process.　　3.The speed-controlled interpolation algorithm:　　3.1. The

37、 compensatory parameter　　In Eqs. (8) and (9), the first-order and the second-order interpolation algorithms are approximated results by neglecting the higher order term as in Eq. (1) . Therefore, th

38、e approximation error for those methods is unavoidable and an interpolation algorithm concerning the curve speed is proposed in this paper. The present interpolation algorithm is based on the first-order approximation in

39、terpolation algorithm with a compensatory valueε(ui) as　　ui+1 = u’i+1 +ε(ui)　　whereε(ui) is the compensatory value and　　u’i+1 = ui +[ ( V(ui) Ts)/(‖dC(u)/du‖

40、u=ui)]　　To precisely calculate the compensatory valueε(ui), Taylor’s expansion for the curve and its speed must be concerned. Suppose　　Cx(u)<p&g

41、t;　　C(u) = [ ]　　Cy(u)　　is the parametric curve representation function for the X and Y axe, the interpolated points Cx(ui+1) and Cy(ui+1) are approxim

42、ated as follows:　　Cx(ui+1) = Cx(u’i+1) +(d Cx(u’i+1)/du)ε(ui) (10)　　Cy(ui+1) = Cy(u’i+1) +(d Cy(u’i+1)/du)ε(ui) (11)　　There are two approximation technique

43、s used in deriving the interpolation algorithm. The first one 　　C’ θ u’i+1　　u i+1 　　D　　u i

44、　　Fig. 3. Geometrical representation of parameters.　　is the Taylor’s expansion of the parameter u with respect to the time t to obtain the first-or second-order approximation int

45、erpolation algorithms [1, 6, 9], as in Eqs. (8) and (9). The second one is the Taylor’s expansion of curve C with respect to the parameter u as in Eqs. (10) and (11). By comparing Eqs. (1), (8) and (9) with Eqs. (10) and

46、 (11), the higher-order term of the Taylor’s expansion in the first approximation is estimated as the compensatory valueε(ui) by applying the second approx　　Cx(ui+1) Cx(u’i+1)</p

47、>　　[ ] and [ ]　　Cy(ui+1) Cy(u’i+1)　　is determined by the slope of curve with the adjustment gainε(ui).<p

48、>　　Although V (ui) is the instantaneous velocity at u = ui in the interpolation algorithm, it is usually assigned as the constant feedrate command F in real interpolation applications. The following equation is provid

49、ed to accurately achieve a linear motion from　　Cx(ui) Cx(ui+1)　　[ ] to [ ]　　Cy(ui) Cy(ui+1)<

50、;/p>　　with the desired speed V (ui):　　{√{[Cx(ui+1) – Cx(ui)] 2 + [Cy(ui+1) – Cy(ui)] 2}}/ Ts = V (ui) (12) 　　then, a

51、 quadratic equation for the compensatory parameter is derived as 　　Uε2 + Zε+ W = 0　　where 　　U = X’ (u’i+1) 2 + Y’ (u’i+1) 2<

52、;p>　　Z = 2[DXX’ (u’i+1) +DYY’ (u’i+1)]　　W=DX2 + DY2 – (V (ui) Ts) 2　　DX = Cx(u’i+1) – Cx(ui)　　DY = Cy(u’i+1) – Cy(ui)　　X’ (u’i+1) = d

53、 Cx(u’i+1) / du　　Y’ (u’i+1) = d Cy(u’i+1) / du　　The compensatory valueε(ui) can be directly obtained as　　ε1,2(ui) = [–Z±√( Z2 – 4UW )]/ 2U = {–[DXX’ (u’

54、i+1) +DYY’ (u’i+1)]　　±√[X’ (u’i+1) 2 + Y’ (u’i+1) 2] ( V(ui) Ts) 2 – [DY X’ (u’i+1) – DX Y’ (u’i+1)] 2}/ [X’ (u’i+1) 2 + Y’ (u’i+1) 2] (13)　　Selection of compensatory param

55、eters　　As the two values in Eq. (13) are the roots of a quadratic equation, characteristics of roots have to be discussed in real applications. Define two vectors as　　DX

56、　　D = [ ]　　DY　　X’ (u’i+1) 　　C’= [ ]　　Y’ (u’i+1)　　Eqs.

57、 (13) can be rewritten as　　ε1,2(ui) = {– (DC’) ±√[‖C’‖2 (V (ui) Ts) 2 –│C’×D│2]}/‖C’‖2 (14)　　where the geometrical relationship between vectors D and C correspond to p

58、arameters among which the parameters ui, u’i+1, ui+1 are shown in Fig. 3. θis the angle between the difference vector D and the differential vector C’, and 　　‖C’‖2 (V (ui) Ts) 2 –│C’×D│2 = ‖C’‖2

59、 Ts2 [ V2 (ui) – (│C’×D│2/‖C’‖2Ts2 ) ]　　= ‖C’‖2Ts2 [ V2 (ui) –│(C’ /‖C’‖) ×(D/ Ts)│2]　　=‖C’‖2 Ts2 [ V2 (ui) –‖D / Ts‖2sin2θ]　　As ‖C’‖2 ＞ 0 and Ts2

60、＞ 0, {‖C’‖2 (V (ui) Ts) 2 –│C’×D│2 } has the same sign as　　{[ V2 (ui) –‖D / Ts‖2sin2θ] }　　The solution of Eq. (13) can be in the following three categories:<p

61、>　　(i) ε1,2 (ui) are two different real numbers if　　[V² (ui) ＞ (‖D‖/ Ts ) 2 sin2θ]　　(ii) ε1,2 (ui) are the same real numbers if　　[V² (ui) ＝ (‖

62、D‖/ Ts ) 2 sin2θ]　　(iii) ε1,2 (ui) are a pair of complex conjugate numbers if　　[V² (ui) ＜(‖D‖/ Ts ) 2 sin2θ]　　a ui<

63、b>　　b　　Fig. 4. Illustrativem condition (ii) and (iii).　　Compared with (‖D‖/ Ts ) and the desired speed V (ui), we conclude that the sign of

64、{[V2 (ui) － (‖D‖/ Ts ) 2 sin2θ]}　　is dominated by angle θ. Conditions (ii) and (iii), which may produce the same real roots or complex conjugate roots of the quadratic equation, are shown in Fig. 4.

65、As the vector D and the differential vector C’ are almost perpendicular and parameter ui+1 is near points a or b as shown in Fig. 4, sin2θ≈1 and the conditions of　　[V2 (ui) ≦ (‖D‖/ Ts ) 2 sin2θ]</

66、p>　　may occur. In physical meaning, the multiple real roots and the complex conjugate roots exist where the curvature is relatively large. In general applications, the curvature should be small to achieve p

67、recise interpolation. Thus, conditions (ii) and (iii) are not allowed in real applications and only the two different real roots as in condition (i) are concerned in the present algorithm.　　According

68、 to Eq. (14)　　ε1,2(ui) ={ – (DC’ /‖C’‖) ±√[ (V (ui) Ts) 2 –│C’ /‖C’‖×D│2 ]} /‖C’‖　　={– (‖D‖cosθ) ±√[ (V (ui) Ts) 2 –‖D‖2 sin2θ]} / ‖C’‖<b&g

69、t;　　Let　　(V (ui) Ts) 2 –‖D‖2 = μ　　By applying Taylor’s expansion, roots of the quadratic equation have the approximated values in simple forms as

70、ε1≈μ/ [ 2‖C’‖‖D‖cosθ]　　ε2≈[ –2‖D‖cosθ] /‖C’‖　　As μ is small, the first root is near zero and the other root is negative and relatively large. To achieve reliable compensation and

71、 forward motion during the interpolation process., the small compensatory parameter is preferable is as　　ε1,2(ui) = [–Z±√( Z2 – 4UW )]/ 2U = {–[DXX’ (u’i+1) +DYY’ (u’i+1)]

72、±√[X’ (u’i+1) 2 + Y’ (u’i+1) 2] ( V(ui) Ts) 2 – [DY X’ (u’i+1) – DX Y’ (u’i+1)] 2}/ [X’ (u’i+1) 2 + Y’ (u’i+1) 2]　　As the present speed-controlled interpolation algorithm incorporates the first

73、approximation interpolation algorithm and a suitable compensatory value which corrects the curve speed error, the obtained curve speed almost equals the specified speed V (ui) during the interpolating process. The roots

74、condition　　{‖C’‖2 ( V(ui) Ts) 2 –│C’ ×D│2}　　can be examined before calculating the compensatory value. When the undesirable condition may occur, the compensatory value is se

75、t to be zero to avoid the complex conjugate roots.　　In real machining processes, the present interpolator achieves (1) a constant speed and (2) specified ACC/DEC. The constant speed mode keeps the cu

76、rve speed almost the same as the given constant feedrate command during the machining process. The ACC/DEC mode makes the curve speed in smooth profiles with the specified speed for machining parametric curves.

77、　　Y axis (mm)　　150　　100　　50　　0<p&g

78、t;　　–50　　–100　　–150　　–150 –100 –50 0 50 100 150 X axis (mm)&

79、lt;p>　　Fig. 5. The example of NURBS.　　4.Applications:　　4.1. The example of NURBS　　In this simulation, the interpolator is written by Turbo C2.0 and is exe

80、cuted on a personal computer with both 80 and 200 MHZ CPU. The present interpolator is applied to a NURBS [10] parametric curve with two degrees as shown in Fig. 5.　　The control points, weight vector

81、, and knot vector of NURBS are assigned as follows: 　　● The ordinal control points are　　0 –150 –150 0 150 150 0　　[ ], [

82、 ], [ ], [ ], [ ], [ ], and [ ] (mm).　　0 –150 150 0 –150 150 0　　● The weight vector is W = [1 25 25 1 25 25 1]

83、.　　● The knot vector is U = [0 0 0 1/4 1/2 1/2 3/4 1 1 1].　　The interpolating processes are as follows:　　● the sampling time in interpolation is Ts

84、=0.002 s and　　● the feedrate command is F = 200 mm/s = 12 m/min.　　In many recent applications, the machining is in a high speed like high-speed machining, e.g. high –speed millin

85、g [11-14], machining by linear motor [15, 16], and laser machining [17]. In this example, the provided weight vector which results in sharp corners is used to exam the speed deviation of different interpolation algorithm

86、s under the feedrate command F = 200 mm/s = 12 m/min.　　4.2. The constant-speed mode　　The curve speed fluctuations for different interpolation algorithms are compared as below: (a

87、) the uniform, (b) the first-order approximation [1, 6, 9] (c) the second-order approximation [6, 9] and (d) the proposed speed-controlled mode. Simulation results for different interpolation algorithms are shown in Fig.

88、 6-8 and are summarized in Table 1. The curve speed fluctuation ratio for each intermediate point is calculated by ηi = (F－Vi)/F where Vi is the curve speed from the interpolated point C(　　m

89、m/sec　　206　　204　　202　　200<

90、b>　　198　　196　　194　　0 1 2 3 4 5 6 Time (sec)　　Fig. 6. S

91、imulation results of the first-order approximation.　　mm/sec　　200.15　　200.1<b

92、>　　200.05　　200　　199.95　　199.9　　199.85

93、　　0 1 2 3 4 5 6 Time (sec)　　Fig. 7. Simulation result of the second-order approximation　　By applying the present algorithm, the ro

94、ots of the quadratic for the compensatory value are also shown in Fig. 9. Fig. 9 indicates that one root is near zero which is adopted here while the other is negative and relatively large. The curve speed accuracy of th

95、e interpolator by applying the uniform interpolation algorithm is the worst cast because the undefined real map operation of curve and parameters is not uniform . Although the present interpolation algorithm takes a long

96、er period to comput　　4.3. The ACC/DEC mode　　In order to avoid shock or vibration of mechanical systems when staring and slowing down the axial travel, ACC/DEC algorithms are requ

97、ired during the machining process. The conventional ACC/DEC algorithms for parametric curves usually result in larger radial and chord errors.　　mm/sec

98、　　200.005 　　200.004　　200.003　　200.002　　200.001<

99、/p>　　200　　199.999　　199.998　　199.997

100、199.996　　0 1 2 3 4 5 6 Time (Sec) 　　Fig. 8. Simulation results of the present constant-speed interpolation.　　Magnitu

101、de (×10ˉ3)　　0　　–0.5　　–1　　–1.5<

102、b>　　–2　　–2.5　　–3 　　0 1 2 3 4 5 6 Time (sec)　　Fi

103、g. 9. Roots of quadratic equationε1,2(ui).　　Kim [8] proposed the parametric ACC/DEC algorithm. As the mapping operation between the curve and the parameter is not uniform in nature, the ACC/DEC in th

104、e uniform parameter interpolation algorithm cannot obtain the desirable acceleration and deceleration.　　The curve speed errors in different acceleration profiles－linear, parabolic, and exponential pr

105、ofiles with different interpolations as (a) the first-order approximation, (b) the second-order approximation, and (c) the speed-controlled interpolation－are compared. The curve speed errors　　by appl

106、ying the parabolic ACC/DEC method are shown in Figs. 10-12 and are also summarized　　Speed deviation ( mm/sec )　　0.4　　0.35</b&g

107、t;　　0.3　　0.25　　0.2　　0.15　　0.1&

108、lt;/b>　　0.05　　0　　0 0.005 0.01 0.015 0.02 0.025 0.03　　Time (sec) </p

109、>　　Fig. 10. Speed error of the first-order approximation during acceleration.　　Table 1　　Simulation results for differdent interpolation algorithm

110、s　　Table 2　　Measurement of the maximum curve speed error　　in Table 2. Results indicate that the largest speed error is caused by applying

111、the exponential ACC/DEC method and the smallest curve speed error is obtained by applying the parabolic ACC/DEC method. According to the simulation results, the curve speed is also changed parabolically during the accele

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外文翻譯--加工參數(shù)曲線的速度控制插補器

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