Abstract
This paper presents a novel matrix-based approach for planar four-bar mechanisms, addressing type and dimensional synthesis under various constraints. Unlike conventional motion generation research that primarily focuses on object positions, we also consider velocity, acceleration, and pivot location restrictions. We introduce a unified design equation that directly incorporates kinematic and geometric constraints through mechanism parameters, solving them as a system of linear equations using singular value decomposition (SVD) and a quartic polynomial. Our approach computes both mechanism type and dimension configuration without prior type selection. It offers an integrated algorithm generating up to six distinct four-bar linkage mechanisms by capitalizing on a null space dimension of three for five constraints. Solving a quartic polynomial yields up to four real solutions representing constituent dyads. Additionally, our method highlights that kinematic mapping and quaternions, often used in motion synthesis, are not essential for the unified equation formulation. Demonstrated with practical examples, our approach proves effective in addressing complex design challenges.
