An analytical curvature-continuous Bézier transition algorithm for high-speed machining of a linear tool path

Abstract

An analytical curvature-continuous path-smoothing algorithm is developed for the high speed machining of a linear tool path. The algorithm can be used in a post-processing stage or NC unit. Every segment junction of the linear tool path, which is the point of tangent discontinuity, is blended by inserting two cubic Bézier spiral curves. A tool path, which is composed of cubic Bézier curves and lines, is then obtained to replace the linear tool path. The new tool path is everywhere curvature-continuous, and both the tangent and curvature discontinuities are avoided. The feed motion will be more stable since the discontinuities are the most important sources of feed fluctuation. In the blending algorithm, the approximation error at the segment junction is accurately guaranteed and the control points of the two cubic Bézier transition curves are all analytically computed. The maximal curvature in every Bézier transition pair, which is critical for velocity planning, is also analytically computed. The analytical expressions provide a way to optimize the curvature radii of the transition curves to pursue the high feed speed. The path-smoothing methods for the post-processing stage and NC unit are both developed. The computational examples confirm the validity of the algorithm. The transition algorithm has been integrated into an open NC system. Cutting experiments show that the curvature-continuous tool path generates smoother feed and consumes shorter machining time than the original linear tool path.