Approximate Differentiators have been extensively used in the estimation of joint velocity measurements of industrial manipulators. In this letter, we exploit key differences between the continuous-time model of the first-order Approximate Differentiator, commonly known as the Dirty-derivatives Filter (DF), and its discrete-time model. We show that the discrete-time filter shares the characteristics of an exponentially weighted moving average; in particular, the filter smooths the derivative of its input. We numerically and experimentally demonstrate how the filter estimation performance follows a convex trend in function of the filter bandwidth. We further demonstrate how the bandwidth at which the filter achieves “optimal” performance varies with the frequency of the filter's input. Inspired by the latter, we propose an Approximate Differentiator with Varying Bandwidth (ADVB) where the filter bandwidth varies based on the magnitude of the position tracking error. We illustrate the superiority of the proposed ADVB over various differentiation schemes both numerically and experimentally.

中文翻译：

---

用于控制跟踪应用的具有不同带宽的近似微分器


近似微分器已广泛用于估计工业机械手的关节速度测量。在这封信中，我们利用一阶近似微分器（通常称为脏导数滤波器 (DF)）的连续时间模型与其离散时间模型之间的关键差异。我们证明了离散时间滤波器具有指数加权移动平均线的特征；特别是，滤波器平滑其输入的导数。我们通过数值和实验证明了滤波器估计性能如何遵循滤波器带宽函数的凸趋势。我们进一步演示了滤波器实现“最佳”性能的带宽如何随滤波器输入频率的变化而变化。受后者的启发，我们提出了一种可变带宽近似微分器（ADVB），其中滤波器带宽根据位置跟踪误差的大小而变化。我们通过数值和实验证明了所提出的 ADVB 相对于各种微分方案的优越性。