Graphical singularity analysis of planar parallel manipulators

This paper introduces a new approach to identify singularities of planar parallel manipulators (PPMs). This method is based on Maxwell´s reciprocal figure theory which establishes a duality between self stressed frameworks and reciprocal figures, which are abstract dual representations of frameworks. We use line geometry tools to introduce a new graphical construction called the mechanism´s line of action graph (MLG). The MLG is introduced in order to implement Maxwell´s reciprocal figure theory to mechanisms. We show that the configurations where the MLG has a connected reciprocal figure imply a singularity in the mechanism. This singularity analysis tool is also used to trace the singularity loci of the PPM. Finally, we provide detailed examples of the singularity analysis of two common PPMs; one consists of three limbs with a passive revolute joint, actuated prismatic joint and another passive revolute joint (3-RPR), the other consists of three limbs with three revolute joints, where only the first is actuated (3-RRR)