## 1 Introduction  
A planar four-bar linkage is an important mechanism that is widely employed in industrial applications. For example, path-generating four-bar linkages (or four-bar path generators) are applied in thread pick-up mechanisms in sewing machines [1], agricultural rice seeding transplanting mechanisms [2], human gait rehabilitation systems [3] and knee exoskeletons [4], underwater manipulators [5], and in robotic fingers [6]. The synthesis problems of linkages can be categorized into three types [7]: (1) type synthesis, (2) number synthesis, and (3) dimensional synthesis. The dimensional synthesis problem involves the determination of the dimensional or geometric size of mechanisms, such as lengths, angles, initial motion position, and coordinates of the pivot, to ensure that the designed mechanisms can meet the required motion. The dimensional synthesis problems of mechanisms can be classified into function generation, path generation, and motion generation. The dimensional synthesis of path-generating four-bar linkages (termed as path synthesis problem in this work) involves the design of the parameters of a mechanism to ensure that the coupler point traces the desired path defined by a number of precision points, which is the central work of this research, and it is also a classical problem that has been investigated by numerous researchers for several decades, and some papers have reviewed the path synthesis problems [8, 9, 10]. Traditionally, the synthesis of such problems is solved usinggraphical methods [11], analytical methods [7], numerical methods [12,13], and a combined numerical and analytical method [14]. However, owing to the limitations of the number of precision points (the maximum precision points for the synthesis of four-bar path generation problem is nine), the solutions of path synthesis problems should address complex non-linear equation under constraints, which cannot be easily solved using traditional methods. With the rapid development of diverse optimization techniques, determining the fittest value through efficient numerical calculations has emerged as a major trend. This is because numerical calculations can rapidly search for the best design parameters within the design constraints, thus effectively reducing the calculation time and the converging rate to optimal solutions. Commonly used optimization methods can be divided into two main categories. The first category includes traditional gradient-based methods that require the determination of the derivative of the objective function, such as the Guass-Newton method [15], the gradient method [16, 17, 18, 19, 20], tabu-gradient method [21], ant-gradient method [22, 23, 24], and the sequential quadratic programming (SQP) method [25]. However, these methods cannot easily determine the function derivative when dealing with complex problems with nonlinear implicit objective functions. The second category includes free-derivative and population-based metaheuristic global optimization methods, which consist of two important groups. The first group includes the swarm intelligence-based algorithms inspired by biological swarm behavior, including the well-known particle swarm optimization (PSO) algorithm [26], cuckoo search (CS) algorithm [27], krill herd (KH) algorithm [28], and their improved variants. The second group includes the evolutionary-based algorithms inspired by biological evolution, including the well-known genetic algorithm (GA) [29], differential evolution (DE) algorithm [30], imperialist competitive algorithm (ICA) [31], teaching learning-based optimization (TLBO) algorithm [32], and their improved variants. For a comparative study comparing other optimal solutions reported in previous literature, here, we briefly review the metaheuristic global optimization methods and other methods applied to the optimal path synthesis problems. (1) Swarm intelligence-based algorithms:  
These methods include PSO, HPSO, MKH, CS, APT-FPSO, BAS, etc. [PERSON] et al.