## 5 Conclusions  
This article proposed an improvement of a finger rehabilitation device based on a real trajectory. The slider-crank mechanism is one of the most studied in the theory of mechanisms due to its simplicity, efficiency, and suitability for many applications in which it can be used. The significance of the slider-crank mechanism presented in this article is that the mono-actuator action is applied toward a medical solution. The presented mechanism is a novel finger flexion-extension rehabilitation device.  
Its synthesis requires a careful analysis of the natural flexion-extension trajectory of the fingers. The focus of the optimization problem was the mono-objective formulation to minimize the error between the real trajectory and the mechanism's trajectory. This optimal solution allows the mechanism to follow a real trajectory more closely. The optimal dimensions of the finger rehabilitation prototype presented in this article allow for the optimization of the therapist's workspace without compromising the end-effector trajectory's performance.  
An experimental setup was performed to obtain real trajectories using a motion capture system and healthy volunteers. The real fingertip trajectory was used as an input to the synthesis problem of the slider-crank mechanism. The presented method is still available for all other subjects and fingers and is also based on the recorded real trajectories.  
The kinematic equations were reformulated to be included in the GA method. Using other synthesis methods for this application would mean omitting the design's original purpose: finger rehabilitation. The GA fitness function calculates the distance that the slider must travel to generate a flexion-extension path in the end-effector. This characteristic is a peculiarity of this function because GA proposes the optimal distance between the maximal flexion position and maximal extension position and does not need to be defined by the designer.  
Future work will focus on a new topology of mechanisms with more degrees of freedom, which could describe more complex shapes of trajectories. In the next work, the study of transmitted forces on the fingers will also be examined. The idea of studying the behavior of the exchange forces between the patient and the device is to develop a more appropriate control strategy that could address the differences in the fingers' capabilities.  
This is an important point because, at this stage, the mechanism can only reproduce the flexion-extension trajectory but cannot deal with situations such as the pain experienced by the patient when trying to complete the movements. A suitable rehabilitation device must consider the relationship between the force provided by the mechanism and finger movement.  
Conceptualization, M.A.L., E.C.C. and A.Z.G.; Methodology, A.Z.G. and M.A.L.; Software, M.A.L. and A.Z.G.; Validation, A.Z.G. and M.A.L.; Formal analysis, A.Z.G., M.A.L., and E.C.C.; Investigation, A.Z.G., M.A.L., and E.C.C.; Resources, M.A.L.; Data curation: A.Z.G., M.A.L., and E.C.C.; Writing-original draft preparation, A.Z.G.; Writing-review and editing: M.A.L., E.C.C., and A.Z.G.; Visualization, M.A.L., E.C.C., and A.Z.G.; Supervision, M.A.L. and E.C.C.; Project administration, M.A.L. and E.C.C.; Funding acquisition, M.A.L. and E.C.C. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Ethical review and approval were waived for this study due to non-invasive experimentation being performed.  
Verbal informed consent was obtained from all subjects involved in the study due to non-invasive experimentation being performed.  
Data sharing is not applicable to this article.  
The authors are grateful to the University of Poitiers, France and the Instituto Politecnico Nacional (IPN), Mexico. As well as the Consejo Nacional de Ciencia y Tecnologia (CONACYT), Mexico, for the facilities provided for this research.  
The authors declare no conflict of interest.