Numerical Synthesis of Six-Bar Linkages for Mechanical Computation

This paper presents a design procedure for six-bar linkages that use eight accuracy points to approximate a specified input-output function. In the kinematic synthesis of linkages, this is known as the synthesis of a function generator to perform mechanical computation. Our formulation uses isotropic coordinates to define the loop equations of the Watt II, Stephenson II, and Stephenson III six-bar linkages. The result is 22 polynomial equations in 22 unknowns that are solved using the polynomial homotopy software BERTINI. The bilinear structure of the system yields a polynomial degree of 705,432. Our first run of BERTINI generated 92,736 nonsingular solutions, which were used as the basis of a parameter homotopy solution. The algorithm was tested on the design of the Watt II logarithmic function generator patented by Svoboda in 1944. Our algorithm yielded his linkage and 64 others in 129 min of parallel computation on a Mac Pro with 12 x 2.93 GHz processors. Three additional examples are provided as well.