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# Asymmetrical Pythagorean-hodograph spline-based $${{\mathrm{C}}}^{4}$$ C 4 continuous local corner smoothing method with jerk-continuous feedrate scheduling along linear toolpath

The International Journal of Advanced Manufacturing Technology（2022）

In computer numerical control systems, linear segments 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, the inserted spline will be micro to avoid overlap. This will increase the curvature extreme of the spline and reduce the feedrate on it. In this article, the corners are smoothed by a $${C}^{4}$$ 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 constrain 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 condition 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 $$7.27$$ to $$75.11\%$$ compared to $${G}^{3}$$ and $${G}^{4}$$ methods.

Local corner smoothing, Pythagorean-hodograph spline, Continuity, Jerk-continuous feedrate scheduling
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