We present an efficient algorithm for computing the closest singular configuration to each non-singular pose of a 3-RPR planar manipulator performing a 1-parametric motion. By considering a 3-RPR manipulator as a planar framework, one can use methods from rigidity theory to compute the singularity distance with respect to an intrinsic metric. There are different design options as the platform/base can be seen as a triangular plate or as a pin-jointed triangular bar structure. Moreover, we also allow the additional possibility of pinning down the base/platform triangle to the fixed/moving system thus it cannot be deformed. For the resulting nine interpretations, we compute the corresponding intrinsic metrics based on the total elastic strain energy density of the framework using the physical concept of Green-Lagrange strain. The global optimization problem of finding the closest singular configuration with respect to these metrics is solved by using tools from numerical algebraic geometry. The proposed algorithm is demonstrated based on an example.

翻译：本文提出了一种高效算法，用于计算执行一参数运动的3-RPR平面机械臂在每个非奇异位姿下距离最近的奇异位形。通过将3-RPR机械臂视为平面框架，可利用刚度理论中的方法基于内在度量计算奇异性距离。由于平台/基座既可被视为三角形板也可被视为销接三角形杆结构，因此存在不同设计方案。此外，我们还允许将基座/平台三角形固定于固定/运动系统的附加可能性，从而使其不可变形。针对由此产生的九种解释，我们基于格林-拉格朗日应变的物理概念，利用框架的总弹性应变能密度计算对应的内在度量。通过数值代数几何工具求解了关于这些度量的全局优化问题——寻找最近奇异位形。最后通过实例验证了所提算法。