It is well-known that parallel manipulators are prone to singularities. However, there is still a lack of distance evaluation functions, referred to as metrics, for computing the distance between two 3-RPR configurations. The proposed extrinsic metrics take the combinatorial structure of the manipulator into account as well as different design options. Utilizing these extrinsic metrics, we formulate constrained optimization problems. These problems are aimed at identifying the closest singular configurations for a given non-singular configuration. The solution to the associated system of polynomial equations relies on algorithms from numerical algebraic geometry implemented in the software package \texttt{Bertini}. Furthermore, we have developed a computational pipeline for determining the distance to singularity during a one-parametric motion of the manipulator. To facilitate these computations for the user, an open-source interface is developed between software packages \texttt{Maple}, \texttt{Bertini}, and \texttt{Paramotopy}. The effectiveness of the presented approach is demonstrated based on numerical examples and compared with existing indices evaluating the singularity closeness.

翻译：众所周知，并联机构易出现奇异位形。然而，目前仍缺乏用于计算两个3-RPR构型之间距离的距离评估函数（即度量）。本文提出的外在度量考虑了机构的组合结构以及不同的设计选项。利用这些外在度量，我们构建了约束优化问题，旨在识别给定非奇异构型下最近的奇异构型。相关多项式方程组的求解依赖于软件包\texttt{Bertini}中实现的数值代数几何算法。此外，我们开发了一套计算流程，用于确定机构在单参数运动过程中与奇异位形的距离。为方便用户执行这些计算，我们在软件包\texttt{Maple}、\texttt{Bertini}和\texttt{Paramotopy}之间开发了一个开源接口。通过数值算例验证了所提方法的有效性，并与现有的奇异接近度评估指标进行了比较。