Accurate Real-time CNC Curve Interpolators Based Upon Richardson Extrapolation. (June 2021) - Farouki, Rida T.
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. Highlights: Accurate real-time CNC interpolators based on Richardson extrapolation are introduced. The need to derive cumbersome symbolic expressions for higher-order derivatives is avoided. The methodology is simple to implement and compatible with millisecond sampling intervals. High-speed cornering, offset curve, and curvature-dependent feedrate examples are included.