## 4 Experimental Results
### Gain Tuning  
The initial step involves tuning the PID controllers and the PI control law according to the conditions given in Theorem 1. As a starting point, the bounds \(\lambda_{\mathrm{hi}}\), \(\lambda_{\mathrm{Hi}}\), \(k_{\mathrm{ci}}\), and \(k_{\mathrm{gi}}\) must be obtained. To this end, an experimental approach was implemented, consisting of recording position and velocity data from both robots while performing random movements that span the entire task space. This information was saved in a data file. Once the data acquisition was completed, the recorded information was processed to calculate the bounds of the dynamics according to Properties 1, 3, and 4:  
\[\lambda_{\mathrm{hi}} =\min_{\mathbf{q}_{i}\in\mathbb{R}^{n}}\lambda_{\mathrm{min}}\Big{(} \mathbf{H}_{i}(\mathbf{q}_{i})\Big{)}, \tag{69}\] \[\lambda_{\mathrm{Hi}} =\max_{\mathbf{q}_{i}\in\mathbb{R}^{n}}\lambda_{\mathrm{max}}\Big{(} \mathbf{H}_{i}(\mathbf{q}_{i})\Big{)},\] (70) \[k_{\mathrm{ci}} =\max_{\mathbf{q}_{i},\mathbf{d}_{i}\in\mathbb{R}^{n}}\frac{\|\mathbf{C}_{ \mathbf{q}_{i}}\mathbf{q}_{i}\|}{\|\mathbf{q}_{i}\|},\] (71) \[k_{\mathrm{gi}} =\max_{\mathbf{q}_{i}\in\mathbb{R}^{n}}\|\mathbf{g}_{i}(\mathbf{q}_{i})\|. \tag{72}\]  
Figure 3: Typical round-trip time delay (RTT). Figure 2: Teleoperation system composed of two _Geomagic Touch_ haptic devices. It is important to highlight that having prior knowledge of the system model is a prerequisite. In this case, the model for both robots was obtained from [30]. The resulting parameter values are summarized in Table 1.  
These parameter values represent the lightweight and compact design of the robotic manipulators, evident from their low inertia, damping, and gravitational terms. The values were experimentally derived by analyzing the system dynamics across its operational workspace. The small values of \(k_{\text{cl}}\) and \(k_{\text{cr}}\) indicate minimal Coriolis and centrifugal effects, which is consistent with the manipulators' limited mass and low-speed operation. These parameters were essential to determine the appropriate control gains, ensuring stability and performance under time-varying delays. The controller gains were determined following the tuning guidelines established in the previous section. First, the derivatives of time delays were assumed to be bounded by \(T_{1}^{*}=T_{\text{r}}^{*}=1\). The parameters \(\alpha_{i}\) and \(\beta_{i}\) were set as \(\alpha_{1}=\alpha_{\text{r}}=\beta_{1}=\beta_{\text{r}}=1\), representing constant time delays. By defining \(\delta_{\text{el}}=\delta_{\text{er}}=1\), from (54) and (55), it was obtained that \(\lambda_{\text{min}}(\boldsymbol{\Lambda}_{1})\), \(\lambda_{\text{min}}(\boldsymbol{\Lambda}_{\text{r}})\geq 4\). Through trial and error, the following gains were selected: \(\boldsymbol{\Lambda}_{1}=\boldsymbol{\Lambda}_{\text{r}}=11\boldsymbol{I}\) to enhance damping and improve performance. Similarly, setting \(\delta_{\text{el}}\) and \(\delta_{\text{sl}}\) to \(\delta_{\text{el}}=\delta_{\text{er}}=\delta_{\text{sl}}=\delta_{\text{sr}}=1\) resulted in \(\lambda_{\text{min}}(\boldsymbol{K}_{\text{vr2}})\geq 4\) according to (56). Thus, a value of \(\boldsymbol{K}_{\text{vr2}}=19\boldsymbol{I}\) was chosen based on (57). This allowed for the definition of \(\boldsymbol{B}_{1}=20\boldsymbol{I}\) as per (58). By assigning \(\boldsymbol{K}_{\text{wl}}=\boldsymbol{K}_{\text{wrt}}=30\boldsymbol{I}\) and \(\epsilon=10\), the resulting gains were calculated as \(\boldsymbol{K}_{\text{vl1}}=245.22\boldsymbol{I}\) and \(\boldsymbol{K}_{\text{vr1}}=226.39\boldsymbol{I}\). Consequently, the overall gains are \(\boldsymbol{K}_{\text{vl}}=264.22\boldsymbol{I}\) and \(\boldsymbol{K}_{\text{vr}}=230.39\boldsymbol{I}\), following the computations in (46).