Efficient high-speed cornering motions based on continuously-variable feedrates. I. Real-time interpolator algorithms

Abstract

The problem of high-speed traversal of sharp toolpath corners, within a prescribed geometrical tolerance 𝜖, is addressed. Each sharp corner is replaced by a quintic Pythagorean–hodograph (PH) curve that meets the incoming/outgoing path segments with G ² continuity, and deviates from the exact corner by no more than the prescribed tolerance 𝜖. The deviation and extremum curvature admit closed-form expressions in terms of the corner angle 𝜃 and side-length L, allowing precise control over these quantities. The PH curves also permit a smooth modulation of feedrate around the corner by analytic reduction of the interpolation integral. To demonstrate this, real-time interpolator algorithms are developed for three model feedrate functions. Specifying the feedrate as a quintic polynomial in the curve parameter accommodates precise acceleration continuity, but has no obvious geometrical interpretation. An inverse linear dependence on curvature offers a purely geometrical specification, but incurs slight initial and final tangential acceleration discontinuities. As an alternative, a hybrid form that incorporates the main advantages of these two approaches is proposed. In each case, the ratio $f=V_{\min }/V_{0}$ of the minimum and nominal feedrates is a free parameter, and the improved cornering time is analyzed. This paper develops the basic cornering algorithms—their implementation and performance analysis are described in detail in a companion paper.