## 4 Simulation Results  
To verify the effectiveness of the fixed-time adaptive fault-tolerant control (FT-AFTC) scheme proposed in this paper, the mathematical model of Cybership II [25] is selected to carry out simulation experiments. The detailed model parameters are listed in Table 1[26]. The external time-varying disturbance is set as \(d=[0.5\sin(0.2t),0.6\cos(0.4t),0.3\sin(0.5t)]^{T}\). For the thruster faults model, we set \(\theta=diag\{0.6+0.2\sin(0.5t),0.7+0.15\sin(0.4t)\}\), \(0.8+0.1\sin(0.3t)\}\) and \(\delta=[0.15\sin(1.5t),0.2\sin(2t),0.25\sin(0.5t)]^{T}\). The controller parameters are selected as \(k_{1}=2\), \(k_{2}=2\), \(p=\frac{7}{9}\), \(q=\frac{11}{9}\), \(\gamma_{1}=1\), \(\gamma_{2}=1\), and \(\sigma=0.01\). The initial states and the desired position are given in the following two cases: Case 1: \(\eta_{1}=[-2,2,\pi/18]^{T}\), \(\
u_{1}=[0.1,0.1,0.1]^{T}\), and \(\eta_{d1}=[0,0,0]^{T}\). Case 2: \(\eta_{2}=[2,-3,\pi/9]^{T}\), \(\
u_{2}=[0.5,0.5,0.2]^{T}\), and \(\eta_{d2}=[-1,1,\pi/6]^{T}\). Under the same conditions, the adaptive backstepping sliding mode control (ABSMC) strategy in the literature [27] and the nonlinear fault-tolerant controller (NFTC) in [28] are used for simulation comparison. The simulation results in Case 1 are shown in Figures 1-4. Figure 1 shows the tracking trajectory curve under the three control strategies. The FT-AFTC control scheme proposed in this paper can quickly converge to the desired position in a fixed time, even in the case of actuator failure. However, the ABSMC control strategy in reference [27] significantly overshoots the tracking trajectory and has a significantly slower convergence time than the FT-AFTC method. In addition, the convergence speed of the system under the NFTC control scheme in [28] is significantly slower than that of the FT-AFTC control method. Figure 2 shows the velocity response curves under the three controllers. Figure 3 provides the tracking error response curves under the action of the FT-AFTC control scheme, the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** & **Parameters** & **Values** \\ \hline \(m\) & 23.8 & \(Y_{r}\) & 0.1079 & \(Y_{rr}\) & 2 \\ \(I_{z}\) & 1.76 & \(N_{p}\) & 0.1052 & \(Y_{rr}\) & 1 \\ \(x_{g}\) & 0.046 & \(N_{vv}\) & 5.0437 & \(Y_{rr}\) & 3 \\ \(X_{u}\) & \(-0.7225\) & \(X_{du}\) & \(-2\) & \(N_{vp}\) & 5 \\ \(X_{uu}\) & \(-1.3274\) & \(Y_{dr}\) & \(-10\) & \(N_{r}\) & 4 \\ \(X_{uuu}\) & \(-5.8664\) & \(Y_{dr}\) & 0 & \(N_{rr}\) & 0.5 \\ \(Y_{vp}\) & \(-0.8612\) & \(N_{dr}\) & 0 & \(N_{rr}\) & 0.8 \\ \(Y_{vv}\) & \(-36.2823\) & \(N_{dr}\) & \(-1\) & \(-\) & \(-\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hydrodynamic parameters of the Cybership II. ABSMC approach in [27], and the NFTC strategy in [28]. Compared with the ABSMC control strategy and the NFTC control scheme, the FT-AFTC algorithm can quickly converge the tracking error to the equilibrium point within a fixed time and have a minor overshoot. Additionally, the control input response curves are displayed in Figure 4. According to Figure 4, the controller proposed in this paper has less control over energy consumption. Similarly, Figures 5-8 display the simulation results under the three control schemes in Case 2. Figures 5 and 6, respectively, show the trajectory and velocity response curves of the dynamic positioning ship under the control schemes of FT-AFTC, NFTC, and ABSMC. Figure 7 displays the convergence curves of tracking errors under three different controllers. Compared with the ABSMC and NFTC approaches, the proposed FT-AFTC control scheme has faster convergence speed and better transient performance. Figure 8 shows the control input curves of three control methods. While ensuring high-precision control performance, the FT-AFTC control scheme significantly reduces control energy consumption compared to the ABSMC and NFTC schemes.