## 4 Results and Discussion
### Monte Carlo Simulation Analysis against Different Communication Delays  
Figure 22 shows randomized communication delay samples based on the measured communication delay samples. Here, we used mean square error as a comparison criterion as follows:  
\[\text{RMSE}=\frac{1}{N}\sum_{k=1}^{N}\sqrt{\frac{1}{L}\sum_{i=1}^{L}(\theta_{ k}^{i}-\hat{\theta}_{k}^{i})^{2}}, \tag{27}\]  
where \(L\) is a sequence of trials for the simulation, \(N\) is the number of samples, and \(\hat{\theta}\) is the compensated outlier sample. As for the comparison, the following parameters were used in the simulation (\(L:100,N=N_{s}\), where \(N_{s}\) refers to the number of samples). In  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Index** & **Raw Samples** & **Predictor** & **Outlier Compensated Predictor** \\ \hline Mean error (deg) & \(3.58\times 10^{-4}\) & \(9.58\times 10^{-3}\) & \(1.08\times 10^{-4}\) \\ Standard deviation (deg) & \(7.7\times 10^{-2}\) & \(3.54\times 10^{0}\) & \(4.72\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison result of mean error and standard deviation between raw samples, predictor-based framework, and proposed outlier compensated predictor.  
Figure 21: Comparison of estimated error between each method and non-delayed signal samples. The diamond markers represent the estimated outlier among the groups.  
Figure 20: Example scene of outlier compensation predictor-based framework and original predictor-based framework.  
addition, for different communication delay conditions, artificially generated delay samples \(\widehat{T}_{\rm c}\) were used as follows:  
\[\widehat{T}_{\rm c}=\sigma T_{\rm c}+\mu, \tag{28}\]  
where \(\sigma\) is a scaler for the deviations, and \(\mu\) is a scaler for the average communication delay.  
Figure 23 shows a comparison of the RMSE for each method, including the proposed approach. From the result, it can be seen that our proposed method can resist the communication delay outlier compared to the predictor-based approach in terms of RMSE comparison.  
The results underscore the significant ramifications of communication delay outliers in teleoperation, particularly in critical situations like emergencies and collision avoidance scenarios. These outliers can exacerbate challenges in maintaining situational awareness, potentially leading to confusion or misinterpretation of the remote environment, thus compromising decision-making accuracy.  
To effectively address these implications, robust strategies for detecting and mitigating communication delay outliers in teleoperation systems are imperative. We advocate for the adoption of the proposed method for detecting communication delay outliers, which can complement existing approaches aimed at bolstering the resilience of teleoperation control signals.  
This recommendation extends beyond vehicular applications to encompass various frameworks utilizing teleoperation, such as surgical robots, unmanned ground vehicles  
Figure 23: Comparison of RMSE for each method using the Monte Carlo simulation with differing \(\mu\) (0 s, 0.1 s, 0.2 s) and \(\sigma\) (0, 0.05, 0.10).  
Figure 22: Randomized communication delay samples based on the measured communication delay samples.  
(UGVs), and drones. By implementing such strategies across these diverse domains, we can enhance the reliability and safety of teleoperation systems, ultimately benefiting a wide range of applications and industries.