Computational Design of Stephenson II Six-Bar Function Generators for 11 Accuracy Points

This paper presents a design methodology for Stephenson II six-bar function generators that coordinate 11 input and output angles. A complex number formulation of the loop equations yields 70 quadratic equations in 70 unknowns, which is reduced to a system of ten eighth degree polynomial equations of total degree $810=1.07×109$⁠. These equations have a monomial structure which yields a multihomogeneous degree of 264,241,152. A sequence of polynomial homotopies was used to solve these equations and obtain 1,521,037 nonsingular solutions. Contained in these solutions are linkage design candidates which are evaluated to identify cognates, and then analyzed to determine their input–output angles in each assembly. The result is a set of feasible linkage designs that reach the required accuracy points in a single assembly. As an example, three Stephenson II function generators are designed, which provide the input–output functions for the hip, knee, and ankle of a humanoid walking gait.