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Accurate Real-time CNC Curve Interpolators Based Upon Richardson Extrapolation

机译：基于Richardson推断的准确实时CNC曲线内插器

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摘要

Real-time CNC interpolators achieving a constant or variable feedrate V along a parametric curve r(xi) are usually based on truncated Taylor series expansions defining the time-dependence of the curve parameter xi. Since the feedrate should be specified as a function of a physically meaningful variable (such as time t, path arc length s, or curvature kappa ) rather than xi, successive applications of the differentiation chain rule are necessary to determine Taylor series coefficients beyond the linear term. The closed-form expressions for the higher-order coefficients are increasingly cumbersome to derive and implement, and consequently error-prone. To address this issue, the use of Richardson extrapolation as a simple means to compute rapidly convergent approximations to the higher-order coefficients is investigated herein. The methodology is demonstrated in the context of (1) an arc-length-dependent feedrate for cornering motions; (2) direct real-time offset curve interpolation; and (3) a curvature-dependent feedrate. All of these examples admit simple implementations that circumvent the need for tedious symbolic calculations of higher-order coefficients, and are compatible with real-time controllers with millisecond sampling intervals. (C) 2021 Elsevier Ltd. All rights reserved.
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