## 4 Numerical Simulations and Analysis  
Building on the theoretical foundation, the subsequent section is dedicated to numerical simulations and subsequent analyses aimed at substantiating the efficacy of the proposed method. The primary objective of these simulations is to evaluate the practical viability and effectiveness of the Gaussian-process-based MPC approach in achieving precise motion control and obstacle avoidance for AUVs. A meticulous examination of the obtained results will yield valuable insights, further validating the method's suitability across various underwater environments. Such rigorous scrutiny is essential for establishing the method's suitability and drawing scientifically sound conclusions regarding its performance in various scenarios.  
Given the complexities of the underwater environment and the operational demands placed on AUVs, the attainment of robust motion control capabilities and effective obstacle avoidance capabilities remains of paramount importance. In this section, we rigorously assess the effectiveness of the proposed method in facilitating AUV obstacle avoidance through comprehensive simulations. The initial part of the analysis entails the realization of trajectory tracking via 6-DOF AUV control, providing a validation of the feasibility and control accuracy of the Gaussian-process-based MPC approach. Subsequently, in the second and third parts, the method's obstacle avoidance prowess is investigated in both static and dynamic obstacles, employing the underactuated AUV horizontal plane model. Finally, the fourth part explores obstacle avoidance simulations for the underactuated AUV model in a three-dimensional space.  
During the subsequent simulation experiments, Gaussian noise characterized by a variance of \(\sigma^{2}=10^{-5}\) is intentionally introduced into the dataset. To provide a comprehensive context, Table 1 furnishes us with precise values concerning the AUV's inertia and hydrodynamic parameters. These parameters play a pivotal role in our endeavor to meticulously model and govern the intricate motion dynamics of the AUV, ensuring our simulations remain rooted in reality.  
The incorporation of Gaussian noise, paired with the vital parameters from Table 1, sets the stage for our extensive investigation, encompassing several key facets of AUV control and obstacle avoidance. Our comprehensive analysis spans trajectory tracking, the avoidance of static obstacles on the horizontal plane, the handling of dynamic obstacles in the same plane, and finally, the challenging domain of obstacle avoidance in three-dimensional space. Each of these segments aims to scrutinize and validate the Gaussian-process-based MPC approach under various conditions, further bolstering our understanding of its practicality and effectiveness in the underwater realm.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline \(m\) & 30.48 kg & \(X_{ii}\) & \(-\)0.93 kg \\ \(I_{zz}\) & 3.45 kg\(\cdot\)m\({}^{2}\) & \(Y_{v}\) & \(-\)35.5 kg \\ \(Y_{\delta}\) & 9.64 kg/(m\(\cdot\)rad) & \(Y_{r}\) & 1.93 kg\(\cdot\)m/rad \\ \(N_{\delta}\) & \(-\)6.15 kg/rad & \(N_{r}\) & \(-\)4.88 kg\(\cdot\)m\({}^{2}\)/rad \\ \(X_{or}\) & 35.5 kg/rad & \(N_{v}\) & 1.93 kg\(\cdot\)m \\ \(X_{rr}\) & \(-\)1.93 kg\(\cdot\)m/rad & \(X_{u|u|}\) & \(-\)1.62 kg/m \\ \(Y_{ur}\) & 5.22 kg/rad & \(Y_{0|v|}\) & \(-\)1310 kg/m \\ \(Y_{uzv}\) & \(-\)28.6 kg/m & \(Y_{r|r}\) & 0.632 kg\(\cdot\)m/rad\({}^{2}\) \\ \(N_{ur}\) & \(-\)2 kg\(\cdot\)m/rad & \(N_{0|v|}\) & \(-\)3.18 kg \\ \(N_{uv}\) & \(-\)24 kg & \(N_{r|r}\) & \(-\)94 kg\(\cdot\)m\({}^{2}\)/rad\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specific values of the AUV’s inertia and hydrodynamic parameters.