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 , 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 . 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.

Efficient high-speed cornering motions based on continuously-variable feedrates. II. Implementation and performance analysis

“…In the companion paper [5] of this two-part study, a family of novel real-time interpolator algorithms were described, for the execution of continuously-variable feedrates along G 2 continuous Pythagorean-hodograph (PH) quintic corner segments. These corner curves and feedrate functions form the basis of a comprehensive sharp-corner traversal strategy.…”

“…Complete details of the PH curve corner rounding constructions, and real-time interpolator algorithms for smooth feedrate suppression along them, may be found in [5]. The present study develops these basic utilities into a cohesive strategy for traversing piecewise-linear part programs that contain many sharp corners, and thoroughly investigates its practical performance.…”

“…The present study develops these basic utilities into a cohesive strategy for traversing piecewise-linear part programs that contain many sharp corners, and thoroughly investigates its practical performance. It is of interest to identify significant advantages or shortcomings (if any) of the three cornering feedrate functions proposed in [5], as well as the efficiency improvements that the proposed cornering strategy offers over the traditional "full stop" approach to negotiating sharp corners.…”

“…The special algebraic structure of the PH curves [3] allows a closed-form reduction of the interpolation integral for each of these feedrate variations, providing very accurate and efficient real-time interpolator algorithms. Further details on the PH quintic corner curves and associated cornering feedrate functions may be found in [5].…”

“…The problem of corner rounding in conjunction with adaptive feedrate scheduling has received relatively little attention, compared to individual treatments of these two topics. Prior work on corner rounding with PH curves (including some studies that employ variable feedrates) was briefly summarized in [5]. Industrial machine tool controllers must also provide strategies to deal with the kinematic difficulties of sharp corners.…”