Singularity Analysis of an Extensible Kinematic Architecture: Assur Class $N$⁠, Order $N−1$

This paper presents an analysis to create a general singularity condition for a mechanism that contains a deformable closed contour. This kinematic architecture is particularly relevant to rigid-link, shape-changing mechanisms. Closed contour shape-changing mechanisms will be shown to belong to the Assur classification because of the pattern of interconnections among the links. The general singularity equations are reduced to a condensed form, which allows geometric relationships to be readily detected. The analysis is repeated for alternative input links. A method for formulating the singularity condition for an Assur Class $N$⁠, knowing the condition for a Class $N−1$ mechanism, is given. This approach is illustrated with several examples.