For 11 positions
Function generation refers to the use of a linkage to coordinate angles. This has many useful applications and function generators are often installed as elements of other linkages, such as path generators. Designing function generators is a particular challenge because of the large degree of their design equations.
We have solved the design equations for Stephenson II and Stephenson III six-bar linkages. The total degree for the design equations for each case is 1.07×109. However, we show a more in depth solution count limits the number of roots to 264,241,152 and 55,050,240. Furthermore, a numerical reduction by homotopy results in approx. 1,500,000 and 800,000 solutions for each case. The solution sets we approximated during these numerical reductions can be used as starting points with the homotopy solver Bertini to solve these large systems for specific cases in a couple of hours.
Numerical reductions for the Stephenson II case were completed over 311 hours on 256×2.2GHz cores of the UCI HPC. Following solves take 2 hours on a single 64 core node. Numerical reductions for the Stephenson III case were completed over 40 hours on 512×2.6GHz cores of the San Diego Supercomputing Center. Following solves take 1 hour on a single 64 core node.
For 8 positions
Function generation is a practical and commonly studied motion requirement. Function generation refers to the coordination of input and output joint parameters of a mechanism. For example, in the figure of the four-bar below, function generation involves the coordination of the angles φ and ψ.
The addition of two more bars provides for much more exotic and accurate functions. The six-bars that are most useful for function generation are Watt II, Stephenson II, and Stephenson III.
Six-bars are capable of producing 11 precision positions. The numerical complexity of these systems requires more study. A simpler approach is design for 8 precision positions. The design equations for these systems are highly nonlinear but tractable, which allows for the generation of many candidate designs. For example, when synthesizing for the parabola and ψ=φ2/90°, the results yielded 86, 19, and 73 defect-free linkages for the Watt II, Stephenson II, and Stephenson III designs.
M. Plecnik and J. M. McCarthy, 2014. “Numerical synthesis of six-bar linkages for mechanical computation,” Journal of Mechanisms and Robotics.