【文章內(nèi)容簡介】 U = X’ (u’i+1) 2 + Y’ (u’i+1) 2 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 Cx(u’i+1) / du Y’ (u’i+1) = d Cy(u’i+1) / du The pensatory valueε (ui) can be directly obtained as ε 1,2(ui) = [–Z177?！?( Z2 – 4UW )]/ 2U = {–[DXX’ (u’i+1) +DYY’ (u’i+1)] 177?！?[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 pensatory parameters 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 D = [ ] DY X’ (u’i+1) C’= [ ] Y’ (u’i+1) Eqs. (13) can be rewritten as ε 1,2(ui) = {– (DC’) 177。√ [‖ C’ ‖ 2 (V (ui) Ts) 2 –│ C’ D│ 2]}/‖ C’ ‖ 2 (14) where the geometrical relationship between vectors D and C correspond to parameters among which the parameters ui, u’i+1, ui+1 are shown in Fig. 3. θ is the angle between the difference vector D and the differential vector C’, and ‖ C’ ‖ 2 (V (ui) Ts) 2 –│ C’ D│ 2 = ‖ C’ ‖ 2 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 ＞ 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: (i) ε 1,2 (ui) are two different real numbers if [V178。 (ui) ＞ (‖ D‖ / Ts ) 2 sin2θ ] (ii) ε 1,2 (ui) are the same real numbers if [V178。 (ui) ＝ (‖ D‖ / Ts ) 2 sin2θ ] (iii) ε 1,2 (ui) are a pair of plex conjugate numbers if [V178。 (ui) ＜ (‖ D‖ / Ts ) 2 sin2θ ] a ui b Fig. 4. Illustrativem condition (ii) and (iii). Compared with (‖ D‖ / Ts ) and the desired speed V (ui), we conclude that the sign of {[V2 (ui) － (‖ D‖ / Ts ) 2 sin2θ ]} is dominated by angle θ . Conditions (ii) and (iii), which may produce the same real roots or plex conjugate roots of the quadratic equation, are shown in Fig. 4. 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 ] may occur. In physical meaning, the multiple real roots and the plex conjugate roots exist where the curvature is relatively large. In general applications, the curvature should be small to achieve precise 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 to Eq. (14) ε 1,2(ui) ={ – (DC’ /‖ C’ ‖ ) 177。√ [ (V (ui) Ts) 2 –│ C’ /‖ C’ ‖ D│ 2 ]} /‖ C’ ‖ ={– (‖ D‖ cosθ ) 177。√ [ (V (ui) Ts) 2 –‖ D‖ 2 sin2θ ]} / ‖ C’ ‖ 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 ε 1≈μ / [ 2‖ C’ ‖‖ D‖ cosθ ] ε 2≈ [ –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 pensation and forward motion during the interpolation process., the small pensatory parameter is preferable is as ε 1,2(ui) = [–Z177。√ ( Z2 – 4UW )]/ 2U = {–[DXX’ (u’i+1) +DYY’ (u’i+1)] 177。√ [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 speedcontrolled interpolation algorithm incorporates the first approximation interpolation algorithm and a suitable pensatory value which corrects the curve speed error, the obtained curve speed almost equals the specified speed V (ui) during the interpolating process. The roots condition {‖ C’ ‖ 2 ( V(ui) Ts) 2 –│ C’ D│ 2} can be examined before calculating the pensatory value. When the undesirable condition may occur, the pensatory value is set to be zero to avoid the plex 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 curve speed almost the same as the given constant feedrate mand during the machining process. The ACC/DEC mode makes the curve speed in smooth profiles with the specified speed for machining parametric curves. Y axis (mm) 150 100 50 0 –50 –100 –150 –150 –100 –50 0 50 100 150 X axis (mm) Fig. 5. The example of NURBS. : . The example of NURBS In this simulation, the interpolator is written by Turbo and is executed on a personal puter 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, and knot vector of NURBS are assigned as follows: ● The ordinal control points are 0 –150 –150 0 150 150 0 [ ], [ ], [ ], [ ], [ ], [ ], and [ ] (mm). 0 –150 150 0 –150 150 0 ● The weight vector is W = [1 25 25 1 25 25 1]. ● 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 = s and ● the feedrate mand is F = 200 mm/s = 12 m/min. In many recent applications, the machining is in a high speed like highspeed machining, . high –speed milling [1114], 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