## 4 Experimental Results
### Experiment 1: Emulated Human Dynamics  
In the first experiment, the interaction of a human operator with the local robot is emulated using a PI control law, as described in (9)-(11). Standard PID controllers are implemented on both robots in accordance with (6)-(8) and the tuning rules outlined in Section 4.1. Both robots begin with the same initial conditions: \(\dot{\boldsymbol{q}}_{i}(0)=\boldsymbol{0}[^{\circ}/\text{s}]\), \(\boldsymbol{q}_{i}(0)=[0^{\circ},30^{\circ},-90^{\circ}]^{\text{T}}\).  
The desired position is defined as:  
\[\boldsymbol{q}_{\text{d}}=\big{[}\begin{array}{ccc}50^{\circ}&15^{\circ}&-4 0^{\circ}\end{array}\big{]}^{\text{T}}. \tag{73}\]  
The induced RTT for this experiment is depicted in Figure 3. Figure 4 illustrates the tracking error \(\boldsymbol{\delta}_{1}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Value** & **Unit** \\ \hline \(\lambda_{\text{hl}}\) & \(5.75\times 10^{-4}\) & N\(\text{s}/\text{m}\) \\ \hline \(\lambda_{\text{br}}\) & \(5.42\times 10^{-4}\) & N\(\text{s}/\text{m}\) \\ \hline \(\lambda_{\text{Hl}}\) & 0.0070 & N\(\text{s}/\text{m}\) \\ \hline \(\lambda_{\text{Hr}}\) & 0.0050 & N\(\text{s}/\text{m}\) \\ \hline \(k_{\text{cl}}\) & 0.0031 & N\(/\text{m}\) \\ \hline \(k_{\text{cr}}\) & 0.0021 & N\(/\text{m}\) \\ \hline \(k_{\text{gl}}\) & 0.1086 & N\(/\text{m}\) \\ \hline \(k_{\text{gr}}\) & 0.1317 & N\(/\text{m}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: System parameter values and their respective units.  
Despite the considerable RTT, the tracking error converges to zero, indicating that both manipulators achieve the desired positions. It is worth noting that, although the tracking error for the first joint becomes zero at approximately \(t\approx 4\) s, it does not remain constant but increases slightly before converging to zero again.  
This transient behavior occurs because the remote robot's position continues to change, affecting the local manipulator's desired position. Ultimately, however, all tracking errors converge to zero and remain stable, as expected.  
To quantify the system's performance, the Root Mean Square Errors (RMSEs) were calculated using the following metric:  
\[\text{RMSE}(e)=\sqrt{\frac{1}{m}\sum_{k=1}^{m}e(k)^{2}}, \tag{74}\]  
where \(m\) denotes the sample size and \(e\) represents the tracking error of interest. Table 2 presents the performance indices obtained for this experiment.  
The RMSE values in Table 2 demonstrate the effectiveness of the PID controllers under emulated human dynamics. Joint 3 shows the highest RMSE, which is likely due to its larger range of motion and sensitivity to communication delays.  
These results highlight the need for precise gain tuning to address time delay effects and ensure stable performance. Overall, the error values validate the theoretical tuning guidelines provided earlier.  
These results experimentally validate the tuning rules provided in Theorem 1. However, they do not confirm the feasibility of replacing the PI control law with an actual human operator. Addressing this question is the focus of the second experiment.  
Remark 7: _Rather than directly supplying the desired position \(\mathbf{q}_{\text{d}}\) to the PI control law, fifth-order polynomials were programmed. This approach better emulates the behavior of a human operator while protecting the experimental testbed._  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Joint** & **RMSE**\([^{\circ}]\) \\ \hline \(1\) & 8.4560 \\ \hline \(2\) & 4.3047 \\ \hline \(3\) & 9.4689 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experiment 1: RMSE for position tracking \(\mathbf{\xi}_{1}\) under emulated human dynamics.