For the trajectory synthesis problem of four-bar linkages, the optimization method offers advantages in applicability, speed, and solution accuracy. However, traditional optimization models typically treat all 10 independent parameters of four-bar linkages as variables, resulting in complex processes. Moreover, such methods are limited to specific tasks and lack robustness to geometric transformations. Therefore, this paper proposes a reduced-dimensional optimization method for four-bar linkage trajectory synthesis based on Fourier harmonic characteristic parameters. It employs a Fourier series to represent the linkage angle function and curve, extracts the harmonic components, and normalizes them to derive these parameters. Four bar-length parameters are identified as optimization variables by establishing relationships between the harmonic characteristic parameters, along with multiple constraints and three distance-based objective functions. The method is independent of discrete point counts and reduces optimization variables for four-bar linkage trajectory synthesis to four. Compared to traditional direct designs, which use all 10 unknown parameters as variables, this approach substantially reduces the optimization dimension. Finally, the correctness and effectiveness of the proposed method for reduced-dimensional trajectory optimization of four-bar linkages are verified through four numerical examples.
Yu et al. (Wed,) studied this question.