## 1 Introduction
It is worth mentioning that the dimensional synthesis of a simple mechanism such as a four-bar mechanism is not a trivial task and can essentially be performed by graphical [19], algebraic [20], analytic [19], or optimization [21] methods. The first three methods are helpful for synthesizing four-bar mechanisms when the trajectories (paths) are relatively simple. As trajectories become more complicated, the optimization method turns out to be more effective, so its use is now widespread [22; 23; 24; 25; 26]. The optimization-based synthesis method involves formulating and solving an optimization or mathematical programming problem. This problem consists of finding the dimensional information of the links of a mechanism whose movement minimizes the differences between the trajectory produced by a point of interest in the mechanism and the desired path while at the same time satisfying a series of constraints related to manufacturing limitations, space conditions available for the operation of the mechanism, or even compliance with Grashof's laws, to mention a few examples. Hence, optimization problems related to the dimensional synthesis of four-bar Grashof-type mechanisms are complex, and classical optimizers such as Sequential Quadratic Programming or the least square optimization technique may have difficulties in finding appropriate solutions, as reported in the specialized literature [22]. It is difficult for practitioners to execute the classical optimizers because of the challenges in defining the initial conditions and the algorithms' reliance on gradient information. Fortunately, stochastic techniques known as metaheuristics can find good solutions to complex optimization problems, such as mechanism synthesis, at a reasonable computational cost [27]. Metaheuristics in engineering applications have grown considerably due to their high effectiveness [28]. This has led to the development of different alternatives to these techniques to solve more and more specific problems [28]. However, the operation of these alternatives takes as a starting point the operation of state-of-the-art metaheuristics [29]. Among these metaheuristic techniques, Differential Evolution (DE) [30] along with its various variants, are recurring options for solving mechanism synthesis problems [31; 32; 33]. The four-bar mechanism is one of the most widely used in various applications and complex machines such as sewing machines, round balers, suspension systems of automobiles, pumpacks, prosthetic knees, and others, because this mechanism can trace complex trajectories by appropriately setting link lengths without the need to use complex mechanisms, even though the latter can provide significant flexibility in the synthesis [34]. Thus, nowadays, researchers are continually coming up with new approaches to state and resolve the four-bar mechanism's synthesis problem [26; 35] to improve the quality of design objectives. Thus, this work developed a comparative study of the most-used variants of DE in search of the solution to the most common conceptualizations of the dimensional synthesis problem of four-bar Grashof-type mechanisms. The study is intended to serve as a tool for choosing the most suitable alternative to solve this type of problem and as a support for developing algorithms that can solve specific related cases more effectively. To illustrate the study's usefulness, a Simple and Improved version of DE (SIDE) was proposed that combines the functionality of the DE variants with the best results from the dimensional synthesis problem. This version of the DE enhances the results obtained with the original variants and also with respect to the cutting-edge methods employed in the four-bar mechanism synthesis problem. The obtained mechanism with the proposed SIDE generates paths more accurately with an increment in the computational effort of the algorithm. Therefore, the proposal's main value is based on engineering applications, where accuracy is one of the most important criteria to consider.