## 4 Results and Discussion
### RTHS Performance  
RTHS introduces errors due to factors inherent to its methodology, such as discrepancies between the numerical simulation (NS) model and the physical structure, electronic noise in the experimental signals of the experimental substructure (ES), and the dynamics of actuators within the transfer systems, among others [86; 87]. Consequently, various indices have been proposed to assess the accuracy of RTHS, accounting for potential deviations between the reference and experimental signals. Many of these evaluation metrics emphasize the local response during hybrid simulations (HS), particularly focusing on the physical component and its interaction with its numerical counterpart. The evaluation of local performance was based on the synchronization of the boundary conditions between the numerical response (\(u_{i,d}^{*}\)) sent to the actuator and the displacement achieved in the physical component (\(u_{i,m}^{*}\)). The normalized root mean square in the experiment (NRMSE) can be used, according to Equation (14), to obtain a single value representing the difference between the actuator signals in the time domain. In addition, frequency domain comparison, Equations (15)-(19) are used, corresponding to the frequency evaluation index (FEI), which compares the fast Fourier transform (FFT) of the desired signal \(u_{i,d}^{*}\) with that of the measured \(u_{i,m}^{*}\) at the actuator, the dominant frequency during RTHS \(f^{eq}\), the generalized amplitude \(A_{0}\), and the delay \(\delta\), respectively [88].  
\[NRMS_{error}=\sqrt{\frac{\sum_{j-1}^{N}\left[u_{i,d}^{*}(j)-u_{i,m}^{*}(j) \right]^{2}}{\sum_{j-1}^{N}\left[u_{i,d}^{*}(j)\right]^{2}}} \tag{14}\]  
\[FEI=\sum_{j=1}^{N}\left(\frac{u_{i,m}^{*}(j)}{u_{i,d}^{*}(j)}.\frac{\left\|u_ {i,d}^{*}(j)\right\|^{l}}{\sum_{j=1}^{P}\left\|u_{i,d}^{*}(j)\right\|^{l}}\right) \tag{15}\]  
\[f^{eq}=\frac{\sum_{j-1}^{N}\left(\left\|u_{i,d}^{*}(j)\right\|^{l}.f_{i} \right)}{\sum_{j-1}^{N}\left\|u_{i,d}^{*}(j)\right\|^{l}} \tag{16}\]  
\[A_{0}=\left\|FEI\right\| \tag{17}\]  
\[\varnothing=\arctan\left[\frac{\text{Im}(FEI)}{Re(FEI)}\right] \tag{18}\]  
\[\delta=-\frac{\varnothing}{2\cdot\!\tau\cdot f^{eq}} \tag{19}\]In addition, the global indices consider the interaction between the numerical and experimental substructures, and they were evaluated in the time domain, quantifying the largest difference between the calculated and measured displacements using \(e_{DM}\) and \(e_{MD}\), as shown in Equations (20) and (21) [85].  
\[e_{DM}=\frac{\left|\left|u_{i,d}^{*}\right|^{max}-\left|u_{i,m}^{*}\right|^{ max}\right|^{max}}{\left|u_{i,m}^{*}\right|^{max}} \tag{20}\]  
\[e_{MD}=\frac{\left|u_{i,d}^{*}-u_{i,m}^{*}\right|^{max}}{\left|u_{i,m}^{*} \right|^{max}} \tag{21}\]  
All indexes were calculated for all test subjects and all gait velocities evaluated. The mean values per each velocity are listed in Table 3. All tests developed a high level of horizontal tracking, with a generalized amplitude \(A_{0}\) around 1.0 and global errors lower than 2.95%, which allowed us to infer that the vertical and lateral transfer system, integrated in HSI-MTE, has high performance and allows performing RTHS with a high level of accuracy, which demonstrates its optimal performance during the RTHS to simulate general HSI effects.