Abstract
This paper was performed using geometry and interactive motion (GIM) software, a learning and research software designed to facilitate the kinematic analysis and synthesis of planar mechanisms. This study presents an analytical and geometric approach to the synthesis of planar four-bar linkages for precise path generation using sixth order connecting coupler curves. Emphasis is placed on defining linkage configurations that enable a coupler point to pass through a set of three, four, or five user-defined precision points. The methodology integrates classical kinematic principles, Roberts' theorem, and focal circle theory to identify double points and optimize the trajectory of the coupler. Using complex number representations and algebraic formulations, the equations governing coupler motion are derived and applied to various linkage types, including crank-rocker, double-rocker, and limit mechanisms. Parallelogram constructions and focal circles are used to detect critical configurations and analyze the continuity and self-intersections of coupler paths. Computational simulations validate the synthesis approach across different configurations, highlighting trade-offs between accuracy and mechanical feasibility. The results demonstrate the applicability of the method in advanced motion control tasks such as robotic arms, dwell mechanisms, and function generation. This work contributes to the field of kinematic synthesis by offering a rigorous framework for linkage design with enhanced precision, reliability, and versatility.