## 5 Numerical Simulations  
### Simulation Results  
To validate the effectiveness of the predefined-time backstepping controller, its performance in perturbation estimation and convergence characteristics was evaluated through MATLAB2022a simulations. The parameters of the underactuated underwater vehicle are provided in Table 1 and refer to Table 2 for parameter settings. [5]. The initial position was set as \(\boldsymbol{\eta}=x,y,z,\theta,\psi^{T}=[14,6,4,0,0]^{T}\). Considering the operational conditions, the unknown perturbation was modeled as \(\boldsymbol{f}=[\exp(-7/5t)+\exp(-6*7t)+3,0,0,\exp(-3/2t)+5,\exp(-t)+4]^{T}\). The desired trajectory was designed to follow: \(\boldsymbol{\eta}_{1d}=[10\sin(0.06t)+10\cos(0.03t),10\sin(0.03t)+10\cos(0.06t ),0.1t]^{T}\). The system parameters of the underwater robot are presented in Table 1. For the control strategy, the predefined-time virtual control law was implemented as provided in Equation (43), while the backstepping controller was applied according to Equation (55). The adaptive control law was detailed in Equation (48), and the iterative neural network strategy was outlined in Equation (54). The constrained parameters of the predefined-time controller were selected as \(\tau_{Lt}=-300\), \(\tau_{Lt}=\tau_{Lr}=-500\), \(\tau_{Ht}=500\), \(\tau_{Ht}=\tau_{Hr}=700\). The RBFNN consisted of 12 neurons, each employing a Gaussian radial basis function. The center of each Gaussian function was randomly selected within the range (0, 10) for each of the three dimensions, and the variance for all the Gaussian functions was set to 0.1. Figures 3 and 4 illustrate the three-dimensional trajectory tracking results and the tracking performance in three directions for the underactuated AUV, respectively. The experimental results indicate that, considering both unmodeled dynamics and time-varying external disturbances, the underactuated AUV is capable of accurately tracking the given three-dimensional trajectory, with tracking errors converging within 10 s. The significant deviation between the initial position and the desired trajectory is the primary cause of large control errors in the early stages of underwater robot operation. During this process, the control system requires a certain amount of time to adjust the robot's position and orientation, gradually reducing the initial error through corrective actions, ultimately steering the robot towards the desired trajectory. Figure 5 further validates the rationality of the predefined-time controller design, showing that no chattering phenomena occurred during the entire control process. Figure 6 compares the external disturbances with the output of the RBF neural network (RBFNN). The results demonstrate that after 5 s of iteration, the RBFNN output \(\boldsymbol{\hat{f}}_{NN}\) closely approximated the assumed time-varying disturbance \(\boldsymbol{f}_{a}\). Figure 7 depicts the dynamic evolution of the adaptive laws, showing that the adaptive signals exhibited smooth and bounded behavior throughout the control process. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline \(M\) & \(183\) & \(M_{\hat{q}}\) & \(-86\) \\ \(I_{xx}\) & \(3\) & \(X_{u|u|}\) & \(-16\) \\ \(I_{zz}\) & \(95\) & \(Y_{v|v|}\) & \(-542\) \\ \(X_{u}\) & \(-49\) & \(Z_{w|w|}\) & \(-422\) \\ \(Y_{v}\) & \(-243\) & \(M_{q|u|}\) & \(-62\) \\ \(Z_{w}\) & \(-230\) & \(N_{r|r|}\) & \(-78\) \\ \(M_{q}\) & \(-140\) & \(N_{r}\) & \(-86\) \\ \(N_{r}\) & \(-160\) & \(Z_{w}\) & \(-230\) \\ \(X_{u}\) & \(-13\) & \(Y_{v}\) & \(-243\) \\ \hline \end{tabular}
\end{table} 
Table 1: Underactuated autonomous underwater vehicle parameters.