## 1 Introduction  
The increasing number of people with motor deficiencies has been a crucial factor in developing rehabilitation devices. Motor injuries frequently occur in the upper and lower extremities due to degenerative diseases, neuronal disorders, sports injuries, traffic, and work accidents. Presently, rehabilitation devices for upper and lower limbs have been developed [1, 2]. Stationary rehabilitation devices are used more frequently in the rehabilitation of the patient. However, these devices are generally expensive because the control of several degrees of freedom is required to generate a rehabilitation routine. Rehabilitation systems based on closed-chain mechanisms are a low-cost alternative for health care [3]. These systems can provide a pre-established rehabilitation routine that controls a degree of freedom. Therefore, these devices can be considered for rehabilitation at home. In the design of closed-chain mechanisms, a synthesis process of the mechanism is carried out [4]. In this process, the length of links is determined to achieve the desired movements for rehabilitation. The graphic, analytic, and optimal methods [5] are the used techniques for the synthesis of closed-chain mechanisms. Graphic synthesis of mechanisms is performed by a visual representation of the mechanism, where Euclidean geometry concepts are used to obtain the design solutions. This method is characterized by providing quick solutions with a low precision rate. The analytical synthesis is based on getting algebraic expressions that describe the movement of the mechanism. This method offers solutions with a high precision rate. However, the number of precision points is limited by the number of independent variables of the closed-chain mechanism because it results in an overdetermined system of equations [6] requiring a numeric solution to solve them. The optimal synthesis of mechanisms is performed through the optimization problem solution. This synthesis method supports multiple-precision points, but the precision rate can decrease caused by the numerical optimization technique used. Currently, the last method is frequently used for the synthesis of mechanisms. In the optimal synthesis of closed-chain mechanisms for rehabilitation, indirect and direct search methods (numerical optimization techniques) [7] have been used to find suitable solutions. In the indirect search methods, derivatives of the objective function and constraints are required. For instance, [PERSON] and [PERSON] [8] designed a thumb prosthesis using a nine-bar spatial mechanism. For the solution of the synthesis problem, the function \"findminimum\" of Mathematica(r) was used. [PERSON] and [PERSON] [9] designed a lower limb rehabilitation system using a four-bar linkage. In the syntheses process, the method \"lsqnonlin\" of MATLAB(r) was implemented. [PERSON] et al. [10] designed a rehabilitation system using a four-bar linkage mechanism in the synthesis process. In that work, the search method \"fmincon\" of MATLAB(r) was implemented. [PERSON] et al. [11] designed a rehabilitation system using a ten-bar linkage mechanism. The synthesis problem was solved by Bertini(tm) software. [PERSON] et al. [12] designed a rehabilitation system using a Stephenson III linkage mechanism. The synthesis problem was solved using algorithms, such as Fletcher-Reeves, Newton, conjugate gradient, interior point, and Quasi-Newton. On the other hand, direct search methods do not require such derivatives, and the objective function is considered in its original form to find a solution. The use of metaheuristic algorithms as the direct search method is gaining more attention in the solution of mechanism synthesis problems [13] because they do not depend on the problem features (nonlinear, discontinuous, etc. ), they can endow different operators to find the most promising solution, and they are easy to implement.