## 1 Introduction
Five study cases of dimensional synthesis for path generation with and without prescribed timing were solved and comparatively studied with the well-known vector loop-based synthesis method, and were solved by metaheuristic algorithms. [PERSON] et al. [76] proposed a repellency evolutionary algorithm (REA) to solve the dimensional synthesis of path-generating four-bar mechanisms. The REA consists of two repulsive mutation behaviors, which exhibit two common characteristics: (i) the population no longer learns from the current global optimum to prevent the population from falling into the local optimum region; (ii) the offspring searches any direction except the location of the parent. [PERSON] et al. [77] proposed a hybrid-combined differential evolution and Jaya (HCDJ) algorithm, which uses a modified initialization, a hybrid-combined mutation between the DE and Jaya algorithm, and an elitist selection for the dimensional synthesis of path-generating four-bar mechanisms with symmetrical motions. Five representative problems were tested, and the HCDJ algorithm was confirmed to provide an improved optimal solution than the original DE and Jaya methods and some algorithms presented in the literature. [PERSON] et al. [78] developed a new constraint-handling method called the individual repairing (IR) method as an alternative to the penalty function method and applied three metaheuristic methods (GA, DE, and PSO) for the optimal synthesis of path-generating four-bar linkages. Two cases with prescribed timing and one case without prescribed timing were tested to verify the reliability of the IR method, and the results revealed that the optimal solutions of the IR method are superior to those of the algorithms proposed by [PERSON] et al. [46]. (3) Path-based synthesis methods:  
These methods include the Fourier descriptors (FDs) method and Haar wavelet transform method. [PERSON] [79] employed the harmonic analysis method and normalization technique to create a library of path generation mechanisms of variable types to be generated and stored in terms of the normalized Fourier coefficients (NFCs) of the path they generate. [PERSON] et al. [80] proposed an objective function based on FDs that evaluates only the shape difference between desired and candidate coupler curves for an optimal synthesis problem. The FDs method has the ability to decouple the nine design variables involved in path generation problem, and [PERSON] et al. [80] first applied the FDs method to reduce the dimension of the search space from nine to five. [PERSON] et al. [81] presented a case-based approach using neural networks to synthesize the dimensions of planar four-bar path generators, in which the shape of the desired path was represented using normalized FDs. [PERSON] et al. [82] proposed a refined numerical representation (RNR) method to describe the shape character of path curves and define the objective function of the optimum path synthesis problems. In addition, they determined the possible solutions using the artificial immune network global searching (AIS) method. [PERSON] [83] utilized NFCs to represent a closed curve, and employed GA, named the GA-FC method, for the synthesis of a four-bar path generator. [PERSON] et al. [84] developed a wavelet-based neural network method to describe the close path shape and optimally synthesize crank-rocker path-generating linkages. [PERSON] et al. [85] utilized the NFCs of a curve curvature as a curve description for the optimal synthesis of a path mechanism. In their work, the distance norm between two curves was considered as the distance between their NFCs, which was used as the objective function, and GA was applied for the optimal path synthesis problems. [PERSON] et al. [86] proposed an approach where artificial neural networks (ANNs) were used to establish a relationship between the FDs of a desired coupler curve and the corresponding dimensions of a four-bar linkage. [PERSON] et al.