## 4 Conclusions  
This study developed and validated a digital twin of the 2 RRR planar parallel manipulator, integrating CAD software (SolidWorks, 2024), real-time monitoring tools, and an interactive graphical interface for trajectory generation and system control. The digital twin allowed seamless communication with the physical prototype through a scalable electronic communication system that utilized microcontrollers, servomotors, and specialized algorithms. This integration enabled real-time trajectory execution, ensuring precision and adaptability in the manipulator's operations.  
The results of the statistical analysis confirmed a high level of concordance between the angular movements of the digital model and the physical prototype, despite minor differences. Hypothesis tests, conducted with a 95% confidence level, validated the statistical equivalence of position and velocity data between the virtual and physical systems. These findings underscore the reliability of the digital twin in replicating the behavior of the physical prototype. The observed differences in angular velocities were attributed to the dynamics of the system, including the limitations of the servomotors' precision and potential calibration errors, which may affect the exact reproduction of trajectories.  
The developed system demonstrates the potential of digital twin technology for improving the design, control, and validation of robotic systems. By enabling iterative testing and real-time adjustments, the digital twin enhances efficiency, reduces mechanical strain, and facilitates the optimization of robotic trajectories under various conditions. This work contributes to the growing body of knowledge on digital twins in robotics and highlights their role as a reliable and effective tool for trajectory planning, system validation, and performance improvement in industrial applications.  
Conceptualization, C.A.M.-M., N.I.M.P., K.D.O.-Q., G.A.H.-L., L.V.V.-U. and G.C.D.-P.; methodology, C.A.M.-M., N.I.M.P., K.D.O.-Q., G.A.H.-L., L.V.V.-U. and G.C.D.-P.; wildiation, C.A.M.-M., N.I.M.P., K.D.O.-Q., G.A.H.-L., L.V.V.-U., G.C.D.-P., E.A.A.-E. and D.V.; formal analysis, C.A.M.-M., N.I.M.P., K.D.O.-Q., G.A.H.-L., L.V.V.-U., G.C.D.-P., E.A.A.-E. and D.V.; investigation, C.A.M.-M., N.I.M.P., K.D.O.-Q., G.A.H.-L., L.V.V.-U. and G.C.D.-P.; resources, C.A.M.-M., N.I.M.P., K.D.O.-Q., G.A.H.-L., L.V.V.-U. and G.C.D.-P.; data curation, C.A.M.-M., G.C.D.-P., E.A.A.-E. and D.V.; writing--original draft preparation, C.A.M.-M., G.C.D.-P., E.A.A.-E. and D.V.; writing--review and editing, C.A.M.-M., G.C.D.-P., E.A.A.-E. and D.V.; visualization, C.A.M.-M., N.I.M.P., K.D.O.-Q., G.A.H.-L., L.V.V.-U., G.C.D.-P., E.A.A.-E. and D.V.; supervision, C.A.M.-M., G.A.H.-L., L.V.V.-U. and D.V.; project administration, C.A.M.-M., G.A.H.-L. and L.V.V.-U.; funding acquisition, C.A.M.-M., G.A.H.-L. and L.V.V.-U. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Universidad Tecnologica de Pereira, Colombia, through the Vicerrectoria de Investigacion, Innovacion y Extension (Research project \"8-23-1\") entitled \"Desarrollo de una Metodologia para la Implementacion de Gemelos Digitales en Procesos de Monitoreo y Gestion de la Integridad Mecanica de Activos\". Data Availability Statement: Data available upon request from the authors.  
The authors declare no conflicts of interest.