## 3 Methodology
### Asymptotic Stability Analysis  
**Theorem 4**.: _Assume that the trajectory tracking of the underactuated AUV is described by the differential model Equation (9) and Assumptions 1-2 are fulfilled. If the disturbances are estimated by Equations (12) and (13), the command integral filter is designed as Equation (23), and the control strategy is obtained from Equations (37) and (38) and Equations (55)-(57). The system (9) is asymptotically stable._  
**Proof.** In order to analyze the asymptotic stability analysis of the system (9), a Lyapunov function is chosen:  
\[U=V+W=\frac{1}{2}yy^{\rm T}+\frac{1}{2}z_{c}z_{c}^{\rm T}. \tag{66}\]  
Additionally, the derivative of (66) is  
\[\dot{U}=\frac{\partial V}{\partial y}f(y,z_{s})+\frac{\partial W}{\partial z_ {c}}g(y,z)+\frac{\partial V}{\partial y}[f(y,z)-f(y,z_{s})]+\bigg{[}\frac{ \partial W}{\partial y}-\frac{\partial W}{\partial z_{c}}\frac{\partial z_{c} }{\partial y}\bigg{]}f(y,z)-\frac{\partial W}{\partial z_{s}}\frac{\partial z _{s}}{\partial t}. \tag{67}\]  
In (67), because \(-\ddot{\beta}+\ddot{\gamma}_{d}\) is usually very small in practice, \(-\dot{\beta}+\dot{\gamma}_{d}\) can be treated as a constant and \(\frac{\partial W}{\partial z_{s}}\frac{\partial z_{s}}{\partial t}\) could be negligible.  
\[\frac{\partial V}{\partial y}f(y,z_{s})=\dot{V}_{2}\leq-n_{2}V_{2}+\delta_{2} =-n_{2}\|y\|^{2}+\delta_{2}, \tag{68}\]  
\[\frac{\partial W}{\partial z_{c}}g(y,z)=\dot{V}_{4}\leq-n_{4}V_{4}+\delta_{4} =-n_{4}\|z_{c}\|^{2}+\delta_{4}, \tag{69}\]  
\[\frac{\partial V}{\partial y}[f(y,z)-f(y,z_{s})]\leq|r_{c}||\gamma_{c}|+M_{1} (-X_{u}+k_{2})|v_{c}||u_{e}|\leq\lambda_{1}\|y\|\|z_{c}\|, \tag{70}\]  
where \(\lambda_{1}=[1+M_{1}(k_{2}-X_{u})]/B_{2}B_{4}\).  
\[\bigg{[}\frac{\partial W}{\partial y}-\frac{\partial W}{\partial z_{c}}\frac{ \partial z_{s}}{\partial y}\bigg{]}f(y,z)=-k_{1}\bigg{[}\frac{a_{13}}{Y_{v}} uv_{c}-v_{c}\bigg{]}(r_{c}-k_{1}\gamma_{e})\leq(\lambda_{2}\|y\|+\lambda_{3}\|z_{c} \|)\|z_{c}\|, \tag{71}\]  
where \(\lambda_{2}=k_{1}^{2}(1+|a_{13}/Y_{v}||u|_{\max})/B_{2}B_{4}\lambda_{3}=(k_{ 1}+|a_{13}/Y_{v}||u|_{\max})/B_{4}^{2}\).  
Substituting conditions (68)-(71) into (67) can yield the inequality.  
\[\begin{array}{ll}\dot{U}&=\frac{\partial V}{\partial y}f(y,z_{s})+\frac{ \partial W}{\partial z_{c}}g(y,z)+\frac{\partial V}{\partial y}[f(y,z)-f(y,z_ {s})]+\bigg{[}\frac{\partial W}{\partial y}-\frac{\partial W}{\partial z_{c} }\frac{\partial z_{c}}{\partial y}\bigg{]}f(y,z)\\ &\leq-n_{2}\|y\|^{2}+\delta_{2}-n_{4}\|z_{c}\|^{2}+\delta_{4}+\lambda_{1}\|y \|\|z_{c}\|+(\lambda_{2}\|y\|+\lambda_{3}\|z_{c}\|)\|z_{c}\|\\ &\leq-\overline{n}U\end{array}. \tag{72}\]  
Inequality (72) is satisfied with \(\overline{n}=\min\{n_{2}(1-\theta_{1}),n_{4}(1-\theta_{2})\}\) for \(\forall\|y\|\geq\mu_{1}=\left(\frac{\lambda_{1}B_{4}+\delta_{2}/B_{2}}{n_{2} \theta_{1}}\right)>0\), \(\forall\|z_{c}\|\geq\mu_{2}=\left(\frac{\lambda_{2}B_{2}+\lambda_{3}B_{4}+ \delta_{4}/B_{4}}{n_{4}\theta_{2}}\right)>0\), and \(0<\theta_{1}<1\), \(0<\theta_{2}<1\).  
According to the converse Lyapunov theorem, the origin of the full system is asymptotically stable if inequality (72) holds. Namely, the compact sets \(\Omega_{2}\) and \(\Omega_{4}\) provide a mathematical bound on the design parameters. \(\square\)