Five Position Synthesis of a Planar Four-Bar Linkage
This paper presents a tutorial on the formulation and solution of the five position synthesis equations for a four-bar linkage in a way that is convenient for implementation using polynomial homotopy software. Complex numbers are used to formulate the loop equations for the four-bar linkage with a coupler reference frame in each of five task positions. We use the loop equations for both cranks of the linkage and their conjugates, and the normalization conditions for each of the joint angles in four displaced positions relative to the initial configuration of the linkage, in order to obtain 8 quadratic equations in 8 unknowns. While there are other formulations for these synthesis equations, we show that this formulation is convenient for solution by polynomial homotopy continuation. An example code is provided as well as a sample calculation.
Keywordskinematic synthesis four-bar linkage geometric design
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