## 5 Conclusions  
This study provides a comprehensive theoretical and experimental analysis of the application of the standard Proportional-Integral-Derivative (PID) controller to bilateral systems consisting of robot manipulators with time-varying delays. Through a detailed closed-loop stability proof, it is demonstrated that PID control ensures the convergence of both manipulators to the desired position specified by the human operator on either side of the system. The stability proof provided does not rely on traditional Lyapunov theory or passivity assumptions, but instead employs \(\mathcal{L}_{2}\) and \(\mathcal{L}_{\infty}\) norms alongside [PERSON]'s Lemma. While it is assumed that human dynamics can be approximated by a simple PI control law, experimental results confirm that this assumption does not limit practical implementation. The derived tuning rules for the PID controllers yield consistent and robust performance, even when a human operator directly controls either of the two robots. Additionally, a comparison with the well-known PD + gravity compensation scheme demonstrates that PID controllers achieve comparable performance, making them a viable option for bilateral systems with time-varying delays. Experimental scenarios, including free motion, environmental contact, and force feedback, highlight the robustness and reliability of the PID control strategy, as well as its ability to maintain system stability and achieve a good degree of transparency. The main advantage of employing a standard PID controller in this context lies in its simplicity and robustness. Unlike advanced control techniques, which often require extensive computational resources or detailed system modeling, the PID controller provides a practical and reliable solution for systems where achieving stability under time-varying delays is more critical than attaining extreme precision. This study rigorously validates the application of a standard PID controller in bilateral teleoperation systems with time-varying delays, demonstrating that, for applications where simplicity, reliability, and robustness are prioritized, PID controllers represent an effective and practical choice. As illustrated in Figure 12, the standard PID controller achieves comparable performance to more complex control schemes, showcasing its viability as a practical and robust solution for teleoperated systems with time-varying delays. Figure 12: RMSE comparison for position tracking. Emulated dynamics human interaction PD + gravity compensation constrained motion and force reflection  
The primary contribution of this work lies in validating the applicability of standard PID control for systems where high precision or computational complexity is not required. In such contexts, the simplicity and inherent robustness of PID controllers make them the preferred choice, as they can handle delays and challenging environments without significant modifications to the controller architecture. Finally, while it is acknowledged that PID control cannot compete with more advanced techniques in terms of maximum performance, this work demonstrates that its application in bilateral systems with time-varying delays constitutes a practical and effective solution. This makes standard PID controllers a valuable tool for industrial and research applications in teleoperated robotics, especially where simplicity, reliability, and robustness are prioritized. However, for scenarios requiring extreme performance, the use of advanced control methodologies remains essential. While the results presented demonstrate the robustness and stability of the proposed standard PID control strategy in bilateral teleoperation systems with time-varying delays, certain limitations must be acknowledged. First, the theoretical stability proof assumes idealized conditions, such as bounded time delays and linearized system dynamics, which may not fully capture the complexities of highly nonlinear or uncertain real-world environments.