Abstract
In NC machining of planar contours, the cutter path is frequently approximated by linear interpolation to a contour curve as target. In this situation, the interpolation error should be measured along the normal direction of the contour curve, and it is desired that only the fewest linear segments are needed with respect to the specified accuracy. However, how to determine the parameters of segment to match these requirements has not been completely solved. This paper presents a new linear interpolation approach for this problem. The approach is named as the optimal linear interpolation method and has three features which include: (1) the interpolation error along the normal direction of object curve satisfies the specified accuracy, and the number of the required segments is the fewest simultaneously; (2) the connection between any two adjacent segments on generated cutter path is natural and smooth, and the generation of extremely short segment is avoided as much as possible; (3) the algorithm is simple and with a high computation efficiency. The effectiveness of the proposed approach has been sufficiently confirmed by applying it to two interpolation examples of planar cam contours. At the same time, the reduction effect of interpolation data has also been verified through comparing the segment number required by the proposed method and Nishioka’s method, which is specially developed to precision machining of planar cam contour, relative to the same interpolation conditions.
Highlights
In science and engineering fields such as numerical analysis, image processing and numerical function generator design, a complex curve defined in a real interval [xL, xR], which is described as y = f (x), is often fitted with piecewise linear approximation, in order to improve the calculation efficiency and save the memories
From the result of Nishioka’s method, 90 segments were required with the specified interpolation accuracy value of 0.0005 mm, while the number of segments generated by the optimal linear interpolation method is 87 with the same accuracy value of ±0.00025 mm
3.4 Relationship between Interpolation Error Extremum Value and Segment Number In order to examine the relationship between the obtained interpolation error value 2e and the number of required segment n, the results shown in Table 1 and Table 3, the latter is based on each single concave-convex portion, are collated and s√ummarized in Fig. 9, where the horizontal axis is set to 1/ 2e and the vertical axis is set to n
Summary
In science and engineering fields such as numerical analysis, image processing and numerical function generator design, a complex curve defined in a real interval [xL, xR], which is described as y = f (x), is often fitted with piecewise linear approximation, in order to improve the calculation efficiency and save the memories. In such situation, the interpolation error for a linear segment of y = ai x + bi is usually expressed as | f (x) − (ai x + bi)|x=x∗ , x∗ ∈ [xL, xR].
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