Kinematic synthesis of Stephenson III six-bar function generators

• The kinematic synthesis of Stephenson III function generators that can pass through 11 accuracy points is formulated.
• The synthesis equations are reduced to a polynomial system with a multihomogeneous degree of 55,050,240.
• Polynomial homotopy was used to find 834,441 nonsingular solutions to a numerically general synthesis system, establishing efficient parameter homotopies.
• Linkage solutions are sorted by cognates and analyzed to verify their performance.
• The theory is employed to produce a desired torque-angle profile, forming a nonlinear, negative spring useful for a dynamic wrist brace.

This paper presents a direct solution of the kinematic synthesis equations for Stephenson III six-bar function generators to achieve as many as 11 accuracy points. The approach is similar to that used to design Stephenson II function generators, except additional algebraic manipulations reduce the system to a multihomogeneous degree of 55,050,240. A numerically general multihomogeneous homotopy was used to obtain 834,441 nonsingular solutions, which were then used to construct an efficient parameter homotopy for specific tasks consisting of 11 accuracy points. The thousands of linkage solutions found by this parameter homotopy are sorted and analyzed to verify nonbranching movement through the specified task positions. An example is presented of a function generator that creates a specified torque-angle profile for a dynamic wrist splint that cancels the effects of spasticity in the wrists of stroke survivors.

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