Automated Generation of Linkage Loop Equations for Planar 1-DoF Linkages, Demonstrated up to 8-Bar

Abstract

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.

Synthesis of eight-bar linkages by constraining a 6R loop

“…This is done using the analysis algorithm developed by Parrish et al (2014), which uses the adjacency matrix of a linkage graph to characterize the design, Tsai (2000).…”

“…We use the automated system developed by Parrish et al (2014) for the analysis of eight-bar linkages. This algorithm reads the location of the pivots of the eight-bar linkage in the first task position and the adjacency matrix of the linkage graph to formulate the three loop equations of the eight-bar linkage.…”

A Design System for Six-Bar Linkages Integrated With a Solid Modeler

“…This linkage graph and the input crank are used by an analysis algorithm developed by Parrish et al [10].…”

Synthesis of Useful Eight-Bar Linkages as Constrained 6R Loops

“…The design procedure uses an analysis algorithm described in Parrish and McCarthy [15,16] for forward kinematics of multiloop linkage systems. The algorithm uses an automated loop generation technique based on Graph theory.…”

“…For that we use the procedure developed by Parrish et al(2013) [15,16]. Plecnik and McCarthy (2012) [17] demonstrated the effectiveness of random variation of the task positions within tolerance zones in order to increase the number of candidate linkages, for evaluation in the design of a 5SS spatial platform.…”