## 4 Discussion  
The shape and amplitude of the flexion-extension trajectory had to be considered to design a better mechanism. This is because the mechanism must be able to adjust the range of motion for different flexion-extension amplitudes. Since the optimized mechanism's configuration is an end-effector, it allows for the extrapolation of the result obtained for use with different subjects. The flexion-extension movement is not affected since the trace is at the fingertip. Using the PCA technique, it was possible to transform the real flexion-extension movement from the _xyz_ space to the _xz_ plane. This simplification of the workspace simplifies the design of the mechanism and does not compromise the trajectory's shape. The use of PCA to derive postural synergies has mainly been used to study the hand's different kinematic configurations during the grasping and manipulation processes [48; 49; 50; 51]. This article did not use PCA to reduce kinematic configurations during grasping. It consisted of verifying that the fingers' flexion-extension movement in space could be simplified to a movement in a plane, resulting in a design simplification of the mechanisms that tried to reproduce this trajectory. The rehabilitation mechanism presented did not seek to emulate the hand's grip functions, but sought to intervene during the first stage of rehabilitation where the physiotherap repeatedly and independently mobilized the patient's fingers to reduce stiffness and expand the range of motion. Regarding GA performance, the low measure of population diversity ensured that the best individuals were close to the optimal solution. The number of generations and the population size used also contributed to finding the best solution. The algorithm converges due to the average distance between individuals in terms of fitness value decreasing as generations passed. The capability to converge to an optimal solution means that the objective function was successfully minimized based on the error between the real finger trajectory and the mechanism's trajectory. The error between the original mechanism's trajectories and the real finger increased as the flexion-extension movement was performed. In contrast, this error decreased with the dimensions of the mechanism obtained through the GA. The amplitude of the trajectory \(\gamma\) of the optimal design had a difference of almost 2.5 degrees, compared with the real trajectory considered for the optimization process. It is assumed that the difference between trajectories is representative of the mechanism's performance. As can be observed in Figure 23, the mechanism trajectory followed the real trajectory very closely. This can be an indicator of mobility recovery in the patient because as the patient has an improvement in the rehabilitation process, the amplitude of the trajectory will increase. It is important to point out that clinical tests are required to evaluate the device's feasibility and potential benefits. Figure 25: Four mechanisms from the index to small fingers. Even though the design results for a slider-crank-based rehabilitation device are promising, this type of mechanism cannot reproduce more closed trajectories. This means that for a certain number of patients, the device cannot meet successful rehabilitation criteria. One of the issues addressed and successfully handled in this work is that the design vector includes the geometric parameters of the slider-crank mechanism and the distance that the slider needs to displace to generate the desired trajectory. This point is significant because this distance is the input of the mechanism and traditionally considered as known. For example, this distance was fixed in the existing prototype as a consideration of the designer at 125 mm [34]. The mechanism design was limited only to physical therapy because it only helps the patient improve flexion-extension movements.