Accurate Real-time CNC Curve Interpolators Based Upon Richardson Extrapolation

Abstract

Real-time CNC interpolators achieving a constant or variable feedrate V along a parametric curve r(ξ) are usually based on truncated Taylor series expansions defining the time-dependence of the curve parameter ξ. Since the feedrate should be specified as a function of a physically meaningful variable (such as time t, path arc length s, or curvature κ) rather than ξ, 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.