## 2 Preliminaries and Problem Formulation
### The Finite-Time Attitude Constraint Controller Design
\begin{split}\dot{V}_{4}\leq&-k_{\varphi}\psi_{e}^{ 2}-k_{\varphi 1}|\psi_{e}|^{\frac{3}{2}}-(a_{r}-1)\zeta_{r}^{2}-a_{r1}\left| \zeta_{r}\right|^{\frac{3}{2}}+\psi_{e}r_{e}\\ &-\frac{y_{e}L\phi_{\dagger}\psi_{e}}{k_{y}^{2}-y_{e}^{2}}+r_{e} \left(\tau_{r}+d_{r}-m_{33}\dot{a}_{r}\right)\end{split}  \tag{42}\]  
Next, a finite-time disturbance observer is designed in order to estimate the compound disturbance \(d_{r}\):  
\[ \begin{cases}\dot{r}=\frac{1}{m_{33}}\left[\tau_{r}-b_{1}\bar{r}-b_{2}|\bar{r }|^{\gamma_{1}}\operatorname{sgn}(\bar{r})-\bar{d}_{r}\operatorname{sgn}( \bar{r})\right]\\ \hat{d}_{r}=m_{33}\dot{r}-\tau_{r}=-b_{1}\bar{r}-b_{2}|\bar{r}|^{\gamma_{1}} \operatorname{sgn}(\bar{r})-\bar{d}_{r}\operatorname{sgn}(\bar{r})\end{cases}  \tag{43}\]  
where \(\hat{d}_{r}\) is the observation value of the disturbance, \(0<\gamma_{1}<1\), \(b_{1}\), \(b_{2}\) are positive design constants and define the estimation error as \(\bar{r}=\dot{r}-r\). Then, taking the time derivative of \(\bar{r}\) results in  
\[\dot{\bar{r}}=\frac{1}{m_{33}}\left[-b_{1}\bar{r}-b_{2}|\bar{r}|^{\gamma_{1}} \operatorname{sgn}(\bar{r})-\bar{d}_{r}\operatorname{sgn}(\bar{r})-d_{r}\right] \tag{44}\]  
**Theorem 1**. : _For the mathematical model of an underactuated MSV (4) and (5), considering the compound disturbance \(d_{r}\), a finite-time disturbance observer is designed as (43), then the disturbance observation error \(\bar{d}_{r}\) can converge to 0 in finite time._  
Proof. : Consider a Lyapunov function as follows:  
\[V_{d_{r}}=\frac{1}{2}\dot{r}^{2} \tag{45}\]  
Taking the time derivative of \(V_{d_{r}}\) results in  
\[ \begin{split}\dot{V}_{d_{r}}&=\bar{r}\frac{1}{m_{33}} \left[-b_{1}\bar{r}-b_{2}|\bar{r}|^{\gamma_{1}}\operatorname{sgn}(\bar{r})- \bar{d}_{r}\operatorname{sgn}(\bar{r})-d_{r}\right]\\ &\leq-b_{1}\frac{1}{m_{33}}\dot{r}^{2}-b_{2}\frac{1}{m_{33}}|\bar{ r}|^{\gamma_{1}+1}\\ &=-2\frac{b_{1}}{m_{33}}V_{d_{r}}-2^{(\gamma_{1}+1)/2}\frac{b_{2} }{m_{33}}V_{d_{r}}^{(\gamma_{1}+1)/2}\\ &=-\chi_{1}V_{d_{r}}-\chi_{2}V_{d_{r}}^{(\gamma_{1}+1)/2}\end{split}  \tag{46}\]where \(\chi_{1}=2\frac{b_{1}}{m_{33}}\) and \(\chi_{2}=2^{(\gamma_{1}+1)/2}\frac{b_{2}}{m_{33}}\). According to the Lemma 1, the estimation error \(\tilde{r}\) converges to 0 in finite time, and the convergence time is: \(T_{1}\leq\frac{\ln\left(1+\frac{\chi_{1}}{\chi_{2}}J_{r}^{1-(\gamma_{1}+1)/2} \right)}{\chi_{1}(1-(\gamma_{1}+1)/2)}\). The observation error of the compound disturbance \(d_{r}\) is  
\[ \begin{split}\tilde{d}_{r}&=\hat{d}_{r}-d_{r}=m_{ 33}\dot{r}-\tau_{r}-(m_{33}\dot{r}-\tau_{r})\\ &=m_{33}\dot{r}\end{split}  \tag{47}\]  
Because it is proven that the estimation error \(\tilde{r}\) converges to zero in finite time, the disturbance observation error \(\tilde{d}_{r}\) can converge to 0 in finite time, that is \(\tilde{d}_{r}=0\), \(\forall t\geq T_{1}\). The Theorem 1 has been proven. To handle the actuator saturation (6), a finite-time attitude input saturation compensator is designed as follows:  
\[\hat{\chi}_{r}=-k_{\chi_{r1}}\chi_{r}-k_{\chi_{r2}}S_{\chi_{r}}+\Delta\tau_{r} \tag{48}\]  
where \(\chi_{r}\) is the output of the finite-time attitude input saturation compensator, \(k_{\chi_{r1}}>0\), \(k_{\chi_{r2}}>0\), \(\Delta\tau_{r}=\tau_{r}-\tau_{rc}\). \(S_{\chi_{r}}\) is designed as follows:  
\[S_{\chi_{r}}= \begin{cases}|\chi_{r}|^{\frac{1}{2}}\operatorname{sgn}(\chi_{r }),&\text{if }|\chi_{r}|\geq\varepsilon_{\chi_{r}}\\ \frac{3}{2}\varepsilon_{\chi_{r}}^{-\frac{1}{2}}\chi_{r}-\frac{1}{2}\varepsilon _{\chi_{r}}^{-\frac{3}{2}}\chi_{r}^{2},&\text{else}\end{cases}  \tag{49}\]  
The finite-time attitude tracking constraint controller can be designed as follows:  
\[\tau_{rc}=m_{33}k_{r}-k_{r}r_{e}-k_{r1}|r_{e}|^{\frac{1}{2}}-\psi_{e}-\tilde{ d}_{r}+k_{\chi_{r}}\chi_{r} \tag{50}\]  
where \(k_{r}\), \(k_{r1}\) and \(k_{\chi_{r}}\) are positive parameters. The Lyapunov function is defined as follows:  
\[V_{r}=V_{4}+\frac{1}{2}\chi_{r}^{2} \tag{51}\]  
Taking the time derivative of \(V_{r}\) along (48)-(50), we have  
\[