A Simple Proof of the Existence of the Collinearity Circle for a Force-Input Coupler-Driven Four-Bar

This research investigates analytical analysis and synthesis problems of force-input and coupler-driven four-bar linkages. Very little bibliography can be found of this type of mechanism in contrast to the conventional torque-input crank-driven four-bars. Traditional four-bar mechanisms use the adjacent links jointed to the ground as input/output links; whereas a coupler-driven four-bar mechanism is actuated by applying the force or torque directly to the coupler link, the member has no direct connection to the base. In this paper, the transmission performance indices, TI (Transmissibility Index) and MI (Manipulability Index) are reviewed, Collinearity Points are defined, where both the MI and TI are unity and therefore the optimal motion transmission performance can be achieved. The Collinearity Circle, which is the locus of all Collinearity Points, is proposed, a novel performance indicator which is used to monitor the effectiveness of the transmission in a force-input coupler-driven four-bar. The beauty of this Collinearity Circle lies not only in its convenient shape, which is a circle, but also in its derivation that can be shown to be merely geometry-dependent. Just like the famous Instant Center, which is also only geometry-dependent, this new Collinearity Circle will prove to be a handy addition to the kinematics toolbox for its power to enable speedy construction and ballpark estimations on the transmission properties of force-input coupler-driven mechanisms. Observations and applications are presented.