Real-time CNC interpolators achieving a constant or variable feedrate $V$ along a parametric curve $\mathbf{r}\left(\xi \right)$ 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.

实时CNC插补器实现恒定或可变进给率 $V$ 沿参数曲线 $\mathbf{[R}（\xi ）$ 通常基于截断的泰勒级数展开式，该展开式定义了曲线参数的时间依赖性 $\xi$。由于应将进给率指定为物理上有意义的变量（例如时间）的函数$Ť$，路径弧长 $s$或曲率 $\kappa$） 而不是 $\xi$，微分链规则的连续应用对于确定超出线性项的泰勒级数系数是必要的。高阶系数的闭式表达式越来越难以推导和实现，因此容易出错。为了解决这个问题，使用Richardson外推法在本文中，作为计算快速收敛到高阶系数的简单方法的简单方法，本文进行了研究。在以下方面证明了该方法：（1）转弯运动的弧长相关进给率；（2）直接实时偏移曲线插补；（3）曲率相关的进给率。所有这些示例都采用了简单的实现方式，从而避免了对高阶系数进行繁琐的符号计算的需求，并且与具有毫秒采样间隔的实时控制器兼容。