## 1 Introduction
[50] considered the tracking error of a mechanism, the deviation of its transmission angle from 90\({}^{\circ}\), and the maximum angular velocity ratio as the multi-objectives of an optimization problem, and employed a hybrid Pareto GA with a built-in adaptive local search to optimally design a path-generating four-bar linkage. [PERSON] et al. [51] proposed a modified error function and employed a DE algorithm to solve the optimal path synthesis problems. [PERSON] et al. [52] proposed a GA-based method, named the NSGA-II method, for the optimal coupler-curve synthesis of a planar four-bar linkage. [PERSON] et al. [53] proposed a DE-based algorithm, called Ingenieria Mecanica Malaga (IMMa) optimization algorithm with self-adaptive control parameters technique, IOA*at, to solve the dimensional synthesis problems of five cases of four-bar and one case of six-bar path-generating mechanisms. [PERSON] [54] proposed a two-phase synthesis method in which the first phase involves shape optimization followed by a scale-rotation-translation (SRT) synthesis optimization for the synthesis of path-generating four-bar mechanisms. Next, [PERSON] employed a GA-DE evolutionary algorithm to solve the optimal synthesis problem, and two representative cases were investigated to verify the higher accuracy of the solutions achieved by the proposed method. [PERSON] et al. [55] utilized an imperialist competitive algorithm (ICA) along with a parallel simulated annealing (SA) algorithm, called the ICA-SA method, for the dimensional synthesis of path-generating four-bar mechanisms with and without prescribed timing, and compared the performance of the algorithms to those of some other heuristic algorithms (GA, DE, and PSO). [PERSON] et al. [56] proposed a one-phase synthesis method without using the input angles as a design variable for optimum path synthesis problems, and employed a DE algorithm with an SRT technique (DE-SRT) to improve the solution reliability and accuracy. [PERSON] et al. [57] included two-swim operators to the chemotaxis process of the modified bacterial foraging optimization algorithm (TS-MBFOA), an evolutionary-based algorithm, to solve the optimal synthesis problems of path-generating four-bar mechanisms. [PERSON] et al. [58] proposed a self-adaptive population size TLBO (SAP-TLBO) algorithm for the optimal path synthesis of a four-bar linkage. Two traditional path generation test problems were conducted, and the optimum results revealed the superiority of the SAP-TLBO algorithm compared to original TLBO method. [PERSON] et al. [59] simultaneously considered three conflicting objective functions, which are the tracking error, the deviation of the transmission angle from 90\({}^{\circ}\), and the maximum angular velocity ratio, as a multi-objective function for the optimal synthesis of four-bar mechanisms using an ICA coupled with a GA (ICA-GA). [PERSON] et al. [60] proposed a combined mutation differential evolution (CMDE) algorithm for the optimum synthesis of path-generating four-bar mechanisms. The combined strategies of the mutations: are DE/best/1, DE/current-to-best/1, and DE/rand/1. The enhanced population diversity and search effectiveness of the CMDE algorithm compared to DE was demonstrated using five representative problems. [PERSON] et al. [61] introduced an objective function called circular proximity function (CPF), which exhibits the lowest number of optimization variables, and DE, called the CPF-DE method, to solve the optimal path synthesis problems. [PERSON] et al. [62] proposed a TLBO algorithm to solve the optimal synthesis problems of human knee exoskeletons and a crank-rocker four-bar path generating linkage. [PERSON] et al. [63] presented a one-phase synthesis method that can reduce the number of design parameters for the optimal synthesis of a four-bar mechanism generating open/closed paths with prescribed timing. Three cases were investigated using a DE algorithm to validate the effectiveness of the method. [PERSON] et al.