Singularity Distance Computations for 3-RPR Manipulators using Extrinsic Metrics

It is well-known that parallel manipulators are prone to singularities. There is still a lack of distance evaluation functions, referred to as metrics, for computing the distance between two 3-RPR configurations. The presented extrinsic metrics take the combinatorial structure of the manipulator into account as well as different design options. Using these extrinsic metrics, we formulate constrained optimization problems, which aim to find the closest singular configurations for a given non-singular configuration. The solution of the corresponding system of polynomial equations relies on algorithms from numerical algebraic geometry implemented in the software package Bertini. Moreover, we developed a computational pipeline for computing the singularity distance along a 1-parametric motion of the manipulator. To facilitate these computations for the user, an open-source interface is developed between software packages Maple, Bertini, and Paramotopy. The presented approach is demonstrated based on a numerical example.