Roberts-Chebyshev theorem and analytical synthesis of four-bar path generators

This paper presents an interactive analytical and optimization synthesis of four-bar path generators with a total number of parameters. It is established that for a crank-and-rocker linkage reproducing a closed-loop coupler curve, there are only two valid mechanisms tracing this trajectory. The third cognate mechanism received by the Roberts-Chebyschev theorem is unable to work, as it does not keep the assembly configuration during the motion cycle. For a double crank four-bar path generator we can always synthesize three cognate linkages.