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.

In this paper, we present an algorithm that automatically creates the linkage loop equations for planar one degree of freedom, 1DOF, linkages of any topology with revolute joints, demonstrated up to 8 bar. The algorithm derives the linkage loop equations from the linkage adjacency graph by establishing a rooted 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 nine unique 6-bar linkages with ground-connected inputs that can be constructed from the five distinct 6-bar mechanisms, Watt I–II and Stephenson I–III. Results also automatically produced the loop equations for all 153 unique linkages with a ground-connected input that can be constructed from the 71 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 1DOF linkages with revolute joints up to 8 bar. The methodology also provides a foundation for automated configuration analysis of 10-bar and higher linkages.

In this paper we present an algorithm applicable to the Watt I and Stephenson I six-bar linkage designs which determines if a candidate linkage moves smoothly through a desired range of input angles. Intended for use with a synthesis routine our algorithm uses the Jacobian of the linkage to determine if a linkage moves smoothly by identifying the continuous existence of a desired branch of a single circuit for all input angles within a bounded range. With the constraint that the input angle must be contained within the four-bar loop, the determinant of the Jacobian is factored into components that represent the individual linkage loops. The algorithm starts with a linkage of a known configuration which reaches one of the desired task positions to establish the set of signs for these determinants and this set is tracked for consistency throughout the bounded range of input angles. Linkages with defects that exist in a narrow range of input angles are addressed by numerically identifying the input angles which correspond to the minimal value of the Jacobian determinants. We verify that at these input angles the linkage remains on the same branch of the same circuit. A Watt I and Stephenson I six-bar linkage that passes this test will move smoothly through the desired range of input angles. Examples using Mathematica demonstrate the application of the algorithm on both the Watt I and Stephenson I linkage types.

In this paper we present an algorithm to determine if a six-bar linkage that has been designed as a constrained 3R chain using Burmester Theory is usable. A usable six-bar linkage is one that moves a workpiece smoothly through a given set of task positions with actuation by one joint parameter.Our algorithm is a two-step process. First the linkage is assembled in each of the task positions and is verified to have the same assembly configuration. Next, a numerical solution of the linkage is tracked between each task position and the assembly is verified to lie on the same branch of one coupler curve. A six-bar linkage that passes this two-part test is usable.An example using Mathematica demonstrates that this computation can be used to automatically evaluate a large number of design candidates.

No abstract