## 4 Experimental Results
### Experiment 5: Force Reflection  
The previous experiment demonstrated the robust performance of PID controllers in constrained motion scenarios. Nevertheless, the results presented in Figure 8 reveal a lack of force matching, which limits the system's transparency. To overcome this limitation, the concept of delayed kinematic correspondence, as outlined in [35], is utilized. Force feedback is incorporated using controllers (64)-(68) with the parameter \(\mathbf{K}_{\text{fi}}=1350\mathbf{I}\). This control strategy relies on torque measurements, derived through the relationship \(\mathbf{\tau}_{i}=\mathbf{J}_{i}^{\text{T}}(\mathbf{q}_{i})\mathbf{F}_{i}\)[30], where \(\mathbf{J}_{i}(\mathbf{q}_{i})\) denotes the geometric Jacobian, and \(\mathbf{F}_{i}\) corresponds to the measured forces. The principle of delayed kinematic correspondence ensures minimal position errors on the local side and accurate force tracking errors on the remote side. This mechanism allows the operator to experience the remote environment as if it were directly reproduced locally. Unlike Experiment 4, the operator maneuvers the local end-effector to bring the remote end-effector into contact with the book. Subsequently, a circular motion is performed on the book, followed by contact with the aluminum plate, where a linear motion is executed. Finally, the end-effector is returned to its initial position. Figure 10 presents the force tracking results, demonstrating acceptable performance during contact with the book (from \(t\approx 7\) s to \(t\approx 24\) s) and the aluminum plate (from \(t\approx 29\) s to \(t\approx 44\) s). These results indicate that the remote manipulator accurately applies the forces commanded by the operator. The RMSE values for force tracking are presented in Table 7, demonstrating significant improvements compared to Table 5. These results highlight the effectiveness of incorporating force feedback in enhancing system performance. The force tracking RMSE values in Table 5 reveal that the addition of the force feedback term enhances transparency during teleoperation. The lower RMSE in the \(x\) and \(z\) directions highlights the effectiveness of force control in providing the human operator with a more accurate perception of the remote environment. This improvement demonstrates the ability of the PID controller with force feedback to handle dynamic interactions while preservingstability and robustness. Position tracking performance during force reflection is shown in Figure 11.  
The RMSE values, provided in Table 8, confirm that the PID controller maintains precise tracking throughout the experiment, even under varying force conditions. The position tracking RMSE values in this table show a slight improvement compared to constrained motion experiments without force feedback. This enhancement reflects the synergistic effect of force feedback on position control, as the human operator benefits from increased transparency when manipulating the local end-effector. These results validate the proposed PID controller's adaptability and effectiveness in scenarios requiring both position and force control. This level of position tracking performance is particularly critical during constrained motion scenarios. Since both robots share identical kinematic configurations, the operator perceives the remote environment as being directly reproduced locally [35]. The round trip time (RTT), with a mean value of 1.3 s, remains a key metric for evaluating bilateral system performance. As noted in [42], this RTT value represents superior performance for the standard PID controller compared to most alternative algorithms, as detailed in Table 3 of the reference. In conclusion, the standard PID controller demonstrates acceptable performance across all test scenarios. For further performance enhancements, more advanced control methodologies may be considered. To summarize the results of the experiments presented, Figure 12 provides a comparative analysis of the RMSE values for position tracking across different scenarios. This