Asymmetrical pythagorean-hodograph spline-based C4 continuous local corner smoothing method with jerk continuous feedrate scheduling along linear toolpath

In computer numerical control systems, linear segments, which are generated by computer-aided manufacturing software, are the most widely used toolpath format. Since the linear toolpath is discontinuous at the junction of two adjacent segments, the fluctuations on velocity, acceleration and jerk are inevitable. Local corner smoothing is widely used to address this problem. However, most existing methods use symmetrical splines to smooth the corners. When any one of the linear segments at the corner is short, to avoid overlap, the inserted spline will be micro, thereby increasing the curvature extreme of the spline and reducing the feedrate along it. In this article, the corners are smoothed by a 𝐶4 continuous asymmetric Pythagorean-hodograph (PH) spline. The curvature extreme of the proposed spline is investigated first, and 𝐾=2.5 is determined as the threshold to constarin the asymmetry of the spline. Then a two-step strategy is used to generate a blended toolpath composed of asymmetric PH splines and linear segments. In the first step, the PH splines at the corners are generated under the premise that the transition lengths do not exceed half of the length of the linear segments. In the second step, the splines at the corners are re-planned to reduce the curvature extremes, if the transition error does not reach the given threshold and there are extra linear trajectories on both sides of the spline trajectory. Finally, the bilinear interpolation method is applied to determine the critical points of the smoothed toolpath, and a jerk-continuous feedrate scheduling scheme is presented to interpolate the smoothed toolpath. Simulations show that, under the condition of not affecting the machining quality, the proposed method can improve the machining efficiency by 7.84% to 23.98% compared to 𝐺3 and 𝐺4 methods.

Possibilities and Advantages of Rational Envelope and Minkowski Pythagorean Hodograph Curves for Circle Skinning

Minkowski Pythagorean hodograph curves are widely studied in computer-aided geometric design, and several methods exist which construct Minkowski Pythagorean hodograph (MPH) curves by interpolating Hermite data in the R2,1 Minkowski space. Extending the class of MPH curves, a new class of Rational Envelope (RE) curve has been introduced. These are special curves in R2,1 that define rational boundaries for the corresponding domain. A method to use RE and MPH curves for skinning purposes, i.e., for circle-based modeling, has been developed recently. In this paper, we continue this study by proposing a new, more flexible way how these curves can be used for skinning a discrete set of circles. We give a thorough overview of our algorithm, and we show a significant advantage of using RE and MPH curves for skinning purposes: as opposed to traditional skinning methods, unintended intersections can be detected and eliminated efficiently.