## 4 Experimental Results
### Experiment 4: Constrained Motion  
One of the key strengths of the PID control law is its robustness, which makes it particularly suitable for applications involving constrained motion in bilateral systems. \begin{table}
\begin{tabular}{c c} \hline \hline
**Joint** & **RMSE** [\({}^{\circ}\)] \\ \hline
1 & 2.8800 \\ \hline
2 & 1.9506 \\ \hline
3 & 1.1187 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Experiment 3: position tracking RMSE with human interaction using PD + gravity compensation. While the stability proof in Section 3 focuses on free motion, it is essential to test the system's robustness when the remote robot interacts with its environment. In this experiment, the remote robot contacts a book and an aluminum plate, as shown in Figure 7.  
The book, being slightly deformable and subject to friction effects, poses a modeling challenge. Conversely, the aluminum plate is rigid, with lower but measurable friction. This complexity makes the experiment a robust test of the controller's capabilities. Force sensors attached to both end-effectors measure the forces applied during contact. The experiment involves moving the remote end-effector to exert force on the book along the \(z\)-axis and later on the aluminum plate along the \(y\)-axis. Force measurements are displayed in Figure 8.  
Force measurements on the local side reflect the corresponding delays, as observed on the remote side. The results show that forces on both sides are not directly related, indicating that the human operator applies higher forces when the local end-effector is unconstrained than  
Figure 8: Experiment 4: constrained motion. Force measurements: \(\mathbf{F}_{\mathrm{e}}(t)\)(- - -), \(\mathbf{F}_{\mathrm{h}}(t-T_{\mathrm{I}}(t))\)(—). Figure 7: Experiment 4: surfaces used for constrained motion testing. when motion is restricted. For instance, this can be observed before contact with the book at \(t\approx 9\) s.  
Figure 9 shows the position tracking \(\mathbf{q}_{1}(t)\) versus \(\mathbf{q}_{\text{r}}(t-T_{\text{r}}(t))\). During intervals when the remote robot contacts its environment, the local position becomes constant but no longer matches the remote position, as occurs during free motion. This behavior indicates that the operator cannot move the local end-effector in the direction constrained by the environment. Despite these challenges, the PID controller maintains the stability of the closed-loop system under perturbations. The force tracking errors remain bounded, as indicated in Table 5.  
The RMSE values in this table demonstrate that the force tracking errors remain bounded during constrained motion scenarios. The higher RMSE values in the \(y\) and \(z\) directions are consistent with the increased difficulty of maintaining precise force control when interacting with deformable and rigid surfaces, such as the book and aluminum plate used in the experiment. Despite these challenges, the PID controller effectively handles the dynamic perturbations introduced by environmental constraints. In addition, the PID controller effectively ensures position tracking stability during constrained motion, as evidenced by the position tracking RMSE values in Table 6. \begin{table}
\begin{tabular}{c c} \hline \hline
**Joint** & **RMSE** [\({}^{\circ}\)] \\ \hline \(1\) & 3.1200 \\ \hline \(2\) & 3.9702 \\ \hline \(3\) & 1.4177 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Experiment 4: constrained motion. Position tracking RMSE. \begin{table}
\begin{tabular}{c c} \hline \hline
**Component** & **RMSE** [N] \\ \hline \(x\) & 0.3449 \\ \hline \(y\) & 1.7693 \\ \hline \(z\) & 1.8941 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Experiment 4: constrained motion. Force tracking RMSE. Figure 9: Experiment 4: constrained motion. Position tracking: \(\mathbf{q}_{1}(t)\) (—), \(\mathbf{q}_{\text{r}}(t-T_{\text{r}}(t))\) (- - -). The position tracking RMSE values indicate that the PID controller maintains stability and acceptable performance even under constrained motion conditions.