## 4 Stability Analysis  
**Theorem 3**. : _Consider the whole control system of the mathematical model (4) and (5) with external disturbances, model parametric uncertainties, and input saturation, and suppose that Assumptions 1 and 2 are satisfied. The update law and the desired heading angle are presented as (19) and (20), the auxiliary system is developed as (34), the finite-time disturbance observers are designed as (43) and (61), the finite-time tracking constraint controllers are designed as (50) and (68), and the finite-time input saturation compensators are designed as (48) and (66). Select the appropriate controller parameters: \(k_{r}>\frac{k_{rx}}{2}+\frac{1}{2}\), \(k_{x_{t1}}>\frac{k_{rx}}{2}+\frac{1}{2}\), \(k_{u}>\frac{k_{xu}}{2}+\frac{1}{2}\) and \(k_{\chi_{u1}}>\frac{k_{xu}}{2}+\frac{1}{2}\), then the following conclusions are valid:_  
1. _All the error signals of the whole control system are uniformly ultimately bounded. Additionally, position errors meet the constraint requirements, that is_ \(|x_{e}|<k_{x}(t)\) _and_ \(|y_{e}|<k_{y}(t)\)_._
2. _The tracking error of an underactuated_ \(MSV\) _can converge into a small neighborhood around zero in finite time._  
Proof. : Construct the Lyapunov function of the whole control system as follows:  
\[\begin{split} V=& V_{1}+V_{r}+V_{u}\\ =&\frac{1}{2}\log\frac{k_{x}^{2}}{k_{x}^{2}-x_{e}^{ 2}}+\frac{1}{2}\log\frac{k_{y}^{2}}{k_{y}^{2}-y_{e}^{2}}+\frac{1}{2}\psi_{e} ^  
where \(K_{1}=\min\left\{\begin{aligned} &\frac{2k_{1}}{k_{x}^{2}-x_{e}^{2}}, \frac{2k_{3}U}{(k_{y}^{2}-y_{e}^{2})\sqrt{\delta_{y}^{2}+\Delta^{2}}},2k_{\psi },2(a_{r}-1),\frac{2}{m_{33}}(k_{r}-\frac{k_{\chi_{r}}}{2}-\frac{1}{2}),\\ &\min\left\{ \begin{aligned} &\frac{2^{3}k_{2}}{k_{x}^{2}-x_{e}^{2}}, \frac{2^{3}k_{4}U}{(k_{y}^{2}-y_{e}^{2})\sqrt{\delta_{y}^{2}+\Delta^{2}}},2^{ 3}k_{\psi 1},2^{3}a_{r1},\\ &\left(\frac{2}{m_{33}}\right)^{\frac{3}{3}}\!\!k_{r1},2^{\frac{3 }{3}}\!\!k_{\chi_{r2}},\left(\frac{2}{m_{11}}\right)^{\frac{3}{3}}\!\!k_{u1}, 2^{\frac{3}{3}}\!\!k_{\chi_{u2}}\end{aligned} \right\}\text{ and }K_{3}=\Delta\tau_{r}^{2}+\Delta\tau_{u}^{2}\). 1. According to (77), we have \[\dot{V}\leq-K_{1}V-K_{2}V^{\frac{3}{4}}+K_{3}\leq-K_{1}V+K_{3}\] (78) Because \(\frac{2k_{1}}{k_{x}^{2}-x_{e}^{2}}\geq\epsilon\), \(\frac{2k_{3}U}{(k_{y}^{2}-y_{e}^{2})\sqrt{\delta_{y}^{2}+\Delta^{2}}}\geq\kappa\), similar to [15] and [20], solving the inequality (78) yields \[0\leq V\leq\Gamma=\left(V(0)-\frac{K_{3}}{K_{1}}\right)e^{-K_{1}t}+\frac{K_{3 }}{K_{1}}\] (79)  
Obviously, the BLF (14) is bounded. Therefore, it can be seen intuitively obtained that \(x_{e}\), \(y_{e}\), \(\psi_{e}\), \(\zeta_{r}\), \(u_{e}\), \(r_{e}\), \(\chi_{u}\) and \(\chi_{r}\) are uniformly ultimately bounded. In addition, the BLF (14) satisfies the following inequalities:  
\[ \begin{array}{l}\frac{1}{2}\log\frac{k_{x}^{2}}{k_{x}^{2}-x_{e}^{2}}\leq V \leq\Gamma\\ \frac{1}{2}\log\frac{k_{y}^{2}}{k_{y}^{2}-y_{e}^{2}}\leq V\leq\Gamma\end{array}  \tag{80}\]  
Then, by solving [80], we obtain:  
\[ \begin{array}{l}|x_{e}|\leq k_{x}\sqrt{1-e^{-2}}<k_{x}\\ |y_{e}|\leq k_{y}\sqrt{1-e^{-2}}<k_{y}\end{array}  \tag{81}\]  
Therefore, this means that the position errors of the underactuated MSV meet the constraint requirements--i.e., \(|x_{e}|<k_{x}\), \(|y_{e}|<k_{y}\). 2. When \(V\geq\frac{K_{3}}{\Pi K_{1}}\), \(K_{3}\leq\Pi K_{1}V\), where \(0<\Pi<1\). Therefore, from (77), we can see: \[ \begin{array}{l}\dot{V}\leq-K_{1}V-K_{2}V^{\frac{3}{4}}+K_{3}\\ \leq-(1-\Pi)K_{1}V-K_{2}V^{\frac{3}{4}}\end{array} \] (82)  
From Lemma 1, we can infer that \(V\) can converge to region \(\left\{\Omega:\Omega\leq\frac{K_{3}}{\Pi K_{1}}\right\}\) in finite time. The convergence time is: \(T=\frac{4}{(1-\Pi)K_{1}}\ln\frac{(1-\Pi)K_{1}V^{\frac{1}{2}}(0)+K_{2}}{K_{2}}\). This means that the position tracking errors \(|x_{e}|<k_{x}\), \(|y_{e}|<k_{y}\) in \(T\).