1、PDF外文：http:/ A：英文资料 The speed-controlled interpolator for machining parametric curves S.-S. Yeh, P.-L. Hsu Abstract: Modern CNC systems are designed with the function of machining arbitrary parametric curves to save m
2、assive data communication between CAD/CAM and CNC systems and improve their machining quality. Although 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 d
3、uring motion. Recently, some approximation results concerning motion speed have been reported Shpitalni M, Koren Y, LoCC. Computer-Aided Design 1994；26(11):832-838； Bedi S, Ali I, Quan N. Trans ASME J Engng Industry 1993；115:329-336； Chou JJ, Yang DCH. Trans. ASME J Engng Industry 1991；113:305-310；
4、Chou JJ,Yang DCH. Trans ASME J Engng Industry 1992；114:15-22. In this paper, by deriving a compensatory parameter, the proposed interpolation algorithm has significantly improved curve speed accuracy. In real applications, the proposed algorithm results in: (1) constant speed； and (2) specified acce
5、leration and deceleration and deceleration (ACC/DEC) to meet the feedrate commands. The motion accuracy and feasibility of the present interpolator have been verified with a provided non-uniform rational B-spline (NURBS) example. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Param
6、etric curve； CNC interpolators； Speed accuracy； NURBS 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 provide linear and circular arc interpolators, the CAD/CAM syst
7、ems 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 traditional CNC systems are undesirable in real applications as follows: the transmission error between CAD/CAM and CNC systems for a huge
8、number of data may easily occur, i.e. lost data and noise perturbation； the discontinuity of segmentation deteriorates surface accuracy； the motion speed becomes unsmooth because of the linearization of the curve in each segment, especially in acceleration and deceleration. As the genera
9、ted curves or profiles may be in a parametric form, only parametric curve information is required to be efficiently transferred 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-spl
10、ine curve and B-spline surface interpolation 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 Kong 4 studied both linear and parametric
11、 interpolators for machining processes. 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 parameter iteration method used is 错误！未指定书签。 ui+1=ui+ (ui)
12、;where ui is the present parameter, ui+1 is the next parameter, and (ui) is the incremental value. The interpolated points are calculated by substituting ui into the corresponding mathematical model to recover the originally designed curves. As the cutter moves straight between contiguous interpolat
13、ed points, two position errors may occur as: (a) radial error； and (b) chord error 5 during motion for a parametric curve as shown in Fig.2. CAD &n
14、bsp; CNC Fig.1. The machining systems with parameters transmission Parametric curve &
15、nbsp; Cutter path D Radial error A &
16、nbsp; Chord error Interpolated B C Segmentation Parametric curve
17、represented model Parametric curve interpolator Tool motion Man-Machine interface device Fig. 2. The radial error and the chord error The radial error is the perpendicular distance between the interpolated points and the parametric curve. Basically, the radial
18、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 applications. In addition, the chord error is the maximum distance between the secant CD and the secant arc AB. A
19、small curvature radius with a fast feedrate command may cause the chord error. Thus, an adaptive feedrate is required to keep the chord error within an acceptable range. However, cutting speed dominates the quality of the machining process. To achieve the specified feedrate for p
20、arametric curves is usually difficult. The undesirable motion speed may deteriorate the quality of the machined surface. Several 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 unif
21、orm interpolation algorithm which is the simplest method and its chord error and curve speed however are not guaranteed. According to the analysis of CNC machine kinematics and cutter path geometry , an improved interpolation algorithm in position, velocity, and acceleration was proposed by Chou and
22、 Yang 6 if the CNC machine kinematics model is known exactly. Further, Houng and Yang 3 developed the cubic spline parametric curve interpolator by using the Euler method. Shpitalni et al. 1 derived the same interpolator algorithm by using Taylors expansion. Lo and Chung 7 proposed the contouring er
23、ror compensation interpolation algorithm which contains real-time contouring error calculations and a simplified parameter iteration method to achieve satisfactory motion accuracy. In acceleration and deceleration ,Kim 8 obtained a simple method for parametric curves while its position and speed err
24、ors are significant. The present speed-controlled interpolation algorithm is proposed by deriving a suitable compensatory parameter for the first-order approximation 1, 6, 8 to obtain desirable motion speed. Then, the proposed interpolator is applied to the constant-speed mode and the a
25、cceleration/deceleration mode to achieve constant feedrate and the specified speed profiles, respectively. Thus, the present CNC interpolators result in stable motion and smoothly changed speed to avoid mechanical shock or vibration in machining processes. The proposed speed-controlled interpolators have also been successfully applied to a NURBS command on a personal computer to achieve high motion precision. 2.Parametric curve formulation: Suppose C(u) is the parametric curve representation function and the time function u is the curve parameter as
