1、A CNC machine tool interpolator for surfaces of cross-sectional designSotiris L. Omiroua,_, Andreas C. NearchouAbstractA machining strategy for milling a particular set of surfaces, obtained by the technique of cross-sectional design is proposed. Thesurfaces considered are formed by sliding a Bezier
2、 curve (profile curve) along another Bezier curve (trajectory curve). The curves arelocated in perpendicular planes. The method employs a three-axis CNC milling machine equipped with suitable ball-end cutter and isbased on the locus-tracing concept. 1. IntroductionIn the automobile, aerospace and ap
3、pliances industry, a variety of functional or even aesthetic free-form surfaces are engaged by engineers and designers to achieve the desired performance of a product. The machining of such complex geometries is a basic problem in computer-aidedmanufacturing since the available NC machines are const
4、rained, by their software, to linear and circular motions. In this paper we deal with a set of surfaces obtained with this design technique. More particularly we use Bezier curves to define the shapes of both the profile and the trajectory. Bezier curves as free-form curves are a powerful designing
5、tool. They need only a few points to define a large number of shapes, hence their wide use in CAD systems. The principle for generating the considered surfaces is shown in Fig. 1. The curves are located in perpendicular planes. The upper end of the profile curve lies on the trajectory curve which is
6、 a plane contour. Fig. 2 shows a sample surface obtained by the above-mentioned technique. This paper, following the present intention of research engineers to take advantage of the hardware capabilities of modern CNC systems, proposes a real-time surface interpolator for machining the specified sur
7、faces onFig. 1. Surface is generated by sliding the profile curve along thetrajectory curve.Fig. 2. Sample surface obtained by cross-sectional designvertical three-axis CNC milling machine. However we keep in mind that whenever feasible, three-axis milling procedures are often preferred due to consi
8、derations of cost. For the considered surfaces, inaccessibility issues are directly dependent upon the form of the profile curve. So by controlling the form of theaccuracy are the main advantages of this manufacturing method.Finally, accuracy is obtained by applying the locus-tracing concept for dri
9、ving the tool along the Beziers offset. The concept is generally applicable in motion generation. In this paper, its application is illustrated in the context of motion generation along Beziers offset. Compared to the customary offset-modeling schemes, an additional advantage besides accuracy, is th
10、e fact that we avoid the complexity of using an exact analytic expression or a piecewise-analytic approximation for the offset. 2. Cross-sectional design with Bezier curvesMany commonly seen and useful surfaces are surfaces of cross-sectional design. For example a surface of revolution is produced u
11、nder this technique. The surface is generated by revolving a given curve about an axis. The given curve is a profile curve while the axis is the axis of revolution. This paper deals with a more complex type of surface which is an extension to the surfaces of revolution. We still need aprofile curve
12、that rotates about the axis of revolution, but the rotation is controlled by a trajectory curve. Now, the profile curve swings about the axis of revolution, guided by the trajectory curve. Both curves, profile and trajectory, are Bezier curves located in perpendicular planes. A Bezier curve of degre
13、e n is a polynomial interpolation curve defined by en t 1T points defining the Bezier control polygon. The interpolation basis functions used in Bezier interpolation are the Bernstein polynomials defined for degree n aswhere the binomial coefficients are given byThe parameter t is in the range 0,1 a
14、nd there are n t 1 polynomials defined for each i from 0 to n. The Beziercurve is therefore defined over the interval 0,1 aswhere bi are the control points defining the Bezier polygon. A recursive algorithm defined by de-Casteljau 3,5,12, calculates for a given control polygon the point that lies on
15、 the Bezier curve for any value of t, and can be used to evaluate and draw the Bezier curve simply, without using the Bernstein polynomials. The algorithm advances by creating in each step a polygon of degree one less than the one created in the previous step until there is only one point left, whic
16、h is the point on the curve. The polygon vertices for each step are defined by linear interpolation of two consecutive vertices of the polygon from the previous stepwith a value of t (the parameter):An interactive drawing tool based on the de-Casteljau algorithm, capable to design and manipulate Bezier curves supports the method proposed in this paper. Since the design process is very often iterative, the designer first lets
