Automated generation of linkage loop equations for planar 1-DOF linkages, demonstrated up to 8-bar

In this paper we present an algorithm that automatically creates the linkage loop equations for planar 1-DoF linkages of any topology with rotating joints, demonstrated up to 8-bars. The algorithm derives the linkage loop equations from the linkage graph by establishing a cycle basis through a single common edge. Divergent and convergent loops are identified and used to establish the fixed angles of the ternary and higher links. Results demonstrate the automated generation of the linkage loop equations for the five distinct 6-bar mechanisms, Watt I-II and Stephenson I-III, as well as the seventy one distinct 8-bar mechanisms. The resulting loop equations enable the automatic derivation of the Dixon determinant for linkage kinematic analysis of the position of every possible assembly configuration. The loop equations also enable the automatic derivation of the Jacobian for singularity evaluation and tracking of a particular assembly configuration over the desired range of input angles. The methodology provides the foundation for the automated configuration analysis of every topology and every assembly configuration of 1-DoF linkages with rotating joints up to 8-bar. The methodology also provides a foundation for automated configuration analysis of 10-bar and higher linkages