## 3 Results and Discussion  
To validate the digital twin, the designed trajectory in the virtual model was compared with the one executed by the physical prototype using a knife-handle type trajectory. Inverse kinematic data and actuator readings were analyzed using trajectory graphs, position and angular velocity charts, and statistical tests. The trajectory was set for a clockwise path of six seconds with a cycloidal profile, capturing results at 25 frames per second. Figures 11 and 12 confirm that the positions and velocities are within permissible limits, and the sampling time ensures data transmission. Figure 11: Angular position. When analyzing two datasets to determine if one has greater variability than the other, an equality of variances test, a key statistical tool, was used. This test evaluates whether the variability within the datasets is similar, using a significance level of 5%, which implies operating with 95% confidence, a standard recommended by various authors for result validation. According to the results presented in Tables 1 and 2, the obtained \(p\)-values are greater than the 5% significance level, and the confidence interval includes the number one, specified in the null hypothesis of equal variances. A high \(p\)-value suggests strong evidence in favor of not rejecting the null hypothesis, implying no statistically significant difference in the variability between the data of the trajectory designed in the virtual model and the data obtained from the trajectory executed by the physical prototype. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Actuator Positions** & **Actuator Velocities** \\ \hline Null hypothesis & \(H_{0}:\frac{\sigma_{\psi}^{2}}{\sigma_{\psi}^{2}}=1\) & \(H_{0}:\frac{\sigma_{\psi}^{2}}{\sigma_{\psi}^{2}}=1\) \\ Alternative hypothesis & \(H_{\text{A}}:\frac{\sigma_{\psi}^{2}}{\sigma_{\psi}^{2}}\
eq 1\) & \(H_{\text{A}}:\frac{\sigma_{\psi}^{2}}{\sigma_{\psi}^{2}}\
eq 1\) \\ Confidence interval & [0.740; 1.408] & [0.809; 1.538] \\ \(p\)-value & 0.897 & 0.503 \\ Significance level & 0.050 & 0.050 \\ Result & The null hypothesis is not rejected & The null hypothesis is not rejected \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of variances. Figure 12: Angular velocity. Figure 13 illustrates the trajectory traced by the prototype's laser compared to the trajectory calculated by the virtual model. The high similarity between the two trajectories confirms that the digital twin accurately reproduces the behavior of the physical prototype. This validation highlights the system's excellent performance, demonstrating precise alignment between the virtual and physical environments. The integration of a digital twin with a physical prototype demonstrated its effectiveness as a tool for trajectory planning and control in robotic manipulators, aligning with the growing emphasis on digital twins in Industry 4.0. The comparison of trajectories in the virtual and physical environments revealed a high level of fidelity, as evidenced by statistical tests confirming the equivalence of variances and means for both actuator positions and velocities. This outcome underscores the reliability of the digital twin in replicating real-world behavior, validating its capability for precise synchronization and control. Furthermore, the use of cycloidal profiles in trajectory generation ensured smooth \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Actuator Positions** & **Actuator Velocities** \\ \hline Null hypothesis & \(H_{0}:\mu_{x}-\mu_{y}=0\) & \(H_{0}:\mu_{x}-\mu_{y}=0\) \\ Alternative hypothesis & \(H_{\Lambda}:\mu_{x}-\mu_{y}\
eq 0\) & \(H_{\Lambda}:\mu_{x}-\mu_{y}\
eq 0\) \\ Confidence interval & \([-2.14;1.984]\) & \([-0.513;3.226]\) \\ \(p\)-value & \(0.817\) & \(0.154\) \\ Significance level & \(0.050\) & \(0.050\) \\ Result & The null hypothesis is not rejected & The null hypothesis is not rejected \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of means. Figure 13: Trajectory traced.