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A CNC machine tool interpolator for surfaces of cross-sectional design
Sotiris L. Omiroua,_, Andreas C. Nearchou
Abstract
A machining strategy for milling a particular set of surfaces, obtained by the technique of cross-sectional design is proposed. The
surfaces considered are formed by sliding a Bezier curve (profile curve) along another Bezier curve (trajectory curve). The curves are
located in perpendicular planes. The method employs a three-axis CNC milling machine equipped with suitable ball-end cutter and is
based on the locus-tracing concept.
1. Introduction
In the automobile, aerospace and appliances 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-aided
manufacturing since the available NC machines are constrained, 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 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 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 surfaces on
Fig. 1. Surface is generated by sliding the profile curve along the
trajectory curve.
Fig. 2. Sample surface obtained by cross-sectional design
vertical three-axis CNC milling machine. However we keep in mind that whenever feasible, three-axis milling procedures are often preferred due to considerations 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 driving the tool along the Bezier’s offset. The concept is generally applicable in motion generation. In this paper, its application is illustrated in the context of motion generation along Bezier’s offset. Compared to the customary offset-modeling schemes, an additional advantage besides accuracy, is the 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 curves
Many commonly seen and useful surfaces are surfaces of cross-sectional design. For example a surface of revolution is produced under 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 a
profile curve 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 degree 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 as
where the binomial coefficients are given by
The parameter t is in the range [0,1] and there are n t 1 polynomials defined for each i from 0 to n. The Bezier
curve is therefore defined over the interval [0,1] as
where 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 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, which 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 step
with 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 the computer draw the Bezier curve defined by a given polygon. Next, checks whether the shape is acceptable (or optimal) based on various criteria, and, if necessary, adjusts the location and the number of the polygon vertices. The edit, add, move and delete operations of this drawing tool, presented in Figs. 3(a)–(d), respectively, were used to achieve the desired form for a profile curve. Once the forms of the profile and the trajectory curve are definitively accepted, the coordinates of their control points are advanced to the input of the CNC surface interpolator, constituting part of the geometric information required.
3. Offset tracing for a Bezier curve
An accurate machining of the considered surfaces requires accurate offset cutter paths along the trajectory and the profile curves. Since both of them are implemented in terms of Bezier curves our interest is focused on the motion generation along Bezier’s offset. The generation of an accurate motion along Bezier’s offset is treated as a
locus-tracing problem. The formulation of the interpolation algorithm demonstrates the versatility and effectiveness of the locus-tracing concept in this practical case of machining. The algorithm guides the tool-center through repeated application of two analytically implemented
construction operations, maintaining exact contact (within 1BLU1) along the entire path. In each iteration, the set of candidate steps is represented by the vector expression
assuming a unit of length equal to the step size. The number of possible steps in each point is 8 (Fig. 4). The last inequality excludes the combination of zero values for both dX, dY, which does not constitute a step. The optimal step is one, which maximizes the advance TidP (Fig. 5) along the local tangent Ti while, at the same time, it satisfies a criterion of proximity to the offset. Implementation of the proximity criterion requires the use of a proximity function which, in the neighborhood of Pi, provides a measure of closeness to the offset.
A suitable proximity function is derived from the fixed
distance property of the offset
where d is the radius of the cutting tool.
Notice that for P lying on the offset p