## 4. Stability Analysis
\begin{split}\dot{V}_{2}+\dot{V}_{4}&\leq-\frac{ \pi}{\beta T_{c}}\Big{(}\frac{1}{2}\mathbf{z}_{2}^{T}\mathbf{z}_{2}\Big{)}^{1- \frac{\beta}{2}}-\frac{\pi}{\beta T_{c}}\Big{(}\frac{1}{2}\mathbf{z}_{2}^{T} \mathbf{z}_{2}\Big{)}^{1+\frac{\beta}{2}}-\|\mathbf{z}_{2}\|\sum\limits_{i=0}^ {2}\dot{l}_{i}\ \|\ \mathbf{v}\ \|^{i}+\sum\limits_{i=0}^{2}\frac{1}{\gamma_{i}}(\dot{l}_{i}-l_{i}) \dot{\bar{l}}_{i}\\ &\leq-\frac{\pi}{\beta T_{c}}\Big{(}\vec{z}_{2}^{T}\frac{m}{2m} \frac{M_{a}}{m_{2}}\mathbf{z}_{2}\Big{)}^{1-\frac{\beta}{2}}-\frac{\pi}{\beta T _{c}}\Big{(}\vec{z}_{2}^{T}\frac{m}{2m_{a}}\frac{M_{a}}{m}\mathbf{z}_{2}\Big{)} ^{1+\frac{\beta}{2}}-\sum\limits_{i=0}^{2}-C_{\text{min}}(\dot{l}_{i}-l_{i}) \dot{l}_{i}^{1-\beta}-\sum\limits_{i=0}^{2}C_{\text{min}}(\dot{l}_{i}-l_{i}) \dot{l}_{i}^{1+\beta}\\ &\leq-\frac{\pi}{\beta T_{c}}\mathbf{v}_{2}^{1-\frac{\beta}{2}}- \frac{\pi}{\beta T_{c}}\mathbf{v}_{2}^{1+\frac{\beta}{2}}-\sum\limits_{i=0}^{ 2}C_{\text{min}}\big{(}\dot{l}_{i}^{2-\beta}-l_{i}^{2-\beta}\big{)}-\sum\limits_ {i=0}^{2}C_{\text{min}}\big{(}\dot{l}_{i}^{2+\beta}-l_{i}^{2+\beta}\big{)} \end{split}  \tag{66}\]  
By Lemma 4 and substituting d, we obtain  
\[ \begin{split}&-\sum\limits_{i=0}^{2}C_{\text{min}}\Big{(}\dot{l }_{i}^{2-\beta}-l_{i}^{2-\beta}\Big{)}-\sum\limits_{i=0}^{2}C_{\text{min}} \Big{(}\dot{l}_{i}^{2+\beta}-l_{i}^{2+\beta}\Big{)}\leq-\sum\limits_{i=0}^{2}C _{5}\Big{(}\widetilde{l}_{i}^{2}\Big{)}^{1-\frac{\beta}{2}}-\sum\limits_{i=0}^{ 2}C_{6}\Big{(}\widetilde{l}_{i}^{2}\Big{)}^{1+\frac{\beta}{2}}\\ &\leq-\sum\limits_{i=0}^{2}C_{5}\Big{(}\widetilde{l}_{i}^{2} \Big{)}^{1-\frac{\beta}{2}}-\sum\limits_{i=0}^{2}C_{6}\Big{(}\widetilde{l}_{i}^ {2}\Big{)}^{1+\frac{\beta}{2}}\leq-\frac{\pi}{\beta T_{c}}\bigg{(}\frac{1}{2} \sum\limits_{i=0}^{2}\frac{1}{\gamma_{i}}\widetilde{l}_{i}^{2}\bigg{)}^{1+ \frac{\beta}{2}}-\frac{\pi}{\beta T_{c}}\bigg{(}\frac{1}{2}\sum\limits_{i=0}^{2} \frac{1}{\gamma_{i}}\widetilde{l}_{i}^{2}\bigg{)}^{1-\frac{\beta}{2}}\\ &\leq-\frac{\pi}{\beta T_{c}}V_{4}^{1+\frac{\beta}{2}}-\frac{\pi}{ \beta T_{c}}V_{4}^{1-\frac{\beta}{2}}\end{split}  \tag{67}\]  
By combining (65) and (67), we obtain  
\[\begin{split}\dot{V}_{5}&\leq-\frac{\pi}{\beta T_{c}}V_{ 1}^{1+\frac{\beta}{2}}-\frac{\pi}{\beta T_{c}}V_{1}^{1-\frac{\beta}{2}}-\frac{ \pi}{\beta T_{c}}V_{2}^{1+\frac{\beta}{2}}-\frac{\pi}{\beta T_{c}}V_{2}^{1+ \frac{\beta}{2}}-\frac{\pi}{\beta T_{c}}V_{2}^{1-\frac{\beta}{2}}-\frac{\pi}{ \beta T_{c}}V_{4}^{1+\frac{\beta}{2}}-\frac{\pi}{\beta T_{c}}Vthe system is analyzed, proving that the network weights of the neural network will ultimately converge to a stable value. This convergence guarantees the stability and control accuracy of the closed-loop system, thereby achieving effective compensation for the underwater robot control system\". Generating predefined-time adaptive control laws without model parameters under systems containing unmodeled parts is a difficult task. Thus, decoupling the model into actuated and unactuated subsystems therefore leads the model to be decoupled into driven and non-driven subsystems, and a self-adaptation law is constructed to offset the effect of the unmodeled part of the system, while the radial-based neural network is designed to compensate for the effect of external perturbations on the system to avoid the appearance of steady-state error.