## 1 Introduction
[87] presented a synthesis method for the open path generation of a four-bar mechanism using the Haar wavelet transform and normalization to extract the wavelet output feature parameters (WOFP) of an open path. [PERSON] et al. [88] presented a parametrization-invariant FD-based method for the optimal synthesis of a path-generating four-bar mechanism. In addition, they determined the optimal values of the design parameters using a back propagation (BP) neural network algorithm. [PERSON] et al. [89] proposed a non-uniform FD-based path synthesis algorithm with an optimal parameterization scheme for a path-generating four-bar mechanism and determined the optimal solution using the Nelder-Mead simplex optimization method. [PERSON] et al. [90] proposed a two-step optimization methodology for path-generating, crank-rocker four-bar linkages based on first-order and second-order derivations of a specified coupler curve, and a hybrid Taguchi-random coordinate search algorithm (HTRCA), which is an experimental-based and free-derivative method, was applied to minimize the objective function (the root mean square error) with the reference slope and its change in the angle of the slope. [PERSON] et al. [91] applied the HTRCA method to synthesize three cases of path-generating four-bar linkages with and without prescribed timing. They compared the optimal results of the HTRCA method with those of some evolutionary algorithms (GA-CSP, GA, PSO, DE, and GA-DE) to verify the robustness and efficiency of the HTRCA method. [PERSON] et al. [92] presented an analytical approach for four-bar path generation synthesis based on the FCs of the path, named the Anal-FC method. Based on the Anal-FC method, the synthesis problem was reduced to solving two polynomial equations without any limitation on the number of the precision point. Other optimization methods:  
[PERSON] et al. [93] applied a Stackelberg game-theoretic approach for the multi-objective optimal synthesis of path-generating four-bar mechanisms. The tracking error and the deviation of the transmission angle from 90\({}^{\circ}\) were considered as the bi-objective function. [PERSON] et al. [94] proposed an approach based on the synergy of cooperative game theory, reliability-based design optimization (RBDDO), and Monte Carlo simulation (MCS), called RBDO-MCS method, to solve the problem of the multi-objective optimal synthesis of path-generating four-bar mechanisms. In the approach, three performance criteria: tracking error, deviation of the transmission angle from 90\({}^{\circ}\), and probability of failure of Grashof's constraint, were considered as the three objective functions. Owing to the random search characteristics of the optimization methods, the efficiency (or effectiveness) and accuracy are significantly affected by the selected parameters of the algorithms, such as the number of populations, crossover ratio, scale factor, and iterative number used in the method. The obtained optimization results may differ at each time; thus, effectively determining the performance of an optimization algorithm is a critical problem. To find accurate and efficient global optimization algorithms for solving high dimensional synthesis problems, such as the synthesis of a six-bar or eight-bar path-generating linkage, is the intent of this preliminary study. In previous research on optimization methods, comparison tests were often performed based on some benchmark functions or representative problems, and the statistical mean value and standard deviation (SD) were applied to judge the merits of the methods.