## 2 Preliminaries and Problem Formulation
### The Finite-Time Attitude Constraint Controller Design  
Step 1. (Kinematics controller). Define the yaw tracking error as follows:  
\[\psi_{e}=\psi-\psi_{d} \tag{25}\]  
Define the Lyapunov function \(V_{2}\) as follows:  
\[V_{2}=\frac{1}{2}\psi_{e}^{2} \tag{26}\]  
Taking the time derivative of \(V_{2}\), we obtain  
\[\dot{V}_{2}=\psi_{e}\left(r-\dot{\psi}_{d}\right) \tag{27}\]  
Step 2. (Dynamics controller). Define the yaw angle rate error as follows:  
\[r_{e}=r-\alpha_{r} \tag{28}\]  
where \(\alpha_{r}\) is the virtual control law. Substituting (28) into (27) yields  
\[\dot{V}_{2}=\psi_{e}\left(r_{e}+\alpha_{r}-\dot{\psi}_{d}\right) \tag{29}\]  
In order to avoid the complicated calculation of the derivative of \(\alpha_{r}\), the proposed command filter is shown in Figure 3. Furthermore, an auxiliary system is presented to compensate for the estimation error. A second-order command filter is expressed as follows:  
\[ \begin{cases}\dot{\rho}_{1}=\rho_{2}\\ \dot{\rho}_{2}=-2\xi_{n}\omega_{n}\left(\rho_{2}+\frac{\omega_{n}^{2}}{2\xi_{n} \omega_{n}}(\rho_{1}-\alpha_{rc})\right)\end{cases}  \tag{30}\]  
where \(\xi_{n}\) and \(\omega_{n}\) are positive design parameters; \(\alpha_{rc}\) is the input signal of the second-order command filter; \(\rho_{1}\) and \(\rho_{2}\) denote the estimations of \(\alpha_{r}\) and \(\dot{\alpha}_{r}\), respectively. Define the estimation error as follow:  
\[\Delta\alpha_{r}=\alpha_{r}-\alpha_{rc} \tag{31}\]  
Substituting (31) into (29) yields  
\[\dot{V}_{2}=\psi_{e}\left(r_{e}+\alpha_{rc}+\Delta\alpha_{r}-\dot{\psi}_{d}\right) \tag{32}\]  
Therefore, design a nominal virtual yaw rate \(\alpha_{rc}\) for (32) as follows:  
\[\alpha_{rc}=-k_{\psi}\psi_{e}-k_{\psi 1}|\psi_{e}|^{\frac{1}{2}}+\dot{\psi}_{d }+c_{r}\xi_{r}-\frac{c_{r}^{2}}{2}\psi_{e}-\frac{y_{e}U\phi_{1}}{k_{y}^{2}-y_ {e}^{2}} \tag{33}\]  
where \(k_{\psi}\), \(k_{\psi 1}\) and \(c_{r}\) denote the positive design constants. To compensate the estimation error \(\Delta\alpha_{r}\) of the virtual yaw rate, the auxiliary system is presented as follows:  
\[\dot{\zeta}_{r}=\left\{ \begin{array}{cc}0,&|\zeta_{r}|\leq\zeta_{r}^{*}\\ -a_{r}\zeta_{r}-a_{r1}|\zeta_{r}|^{\frac{1}{2}}+b_{r}\Delta\alpha_{r}-\left( \frac{1}{2}b_{r}^{2}\Delta\alpha_{r}^{2}+\psi_{e}\Delta\alpha_{r}\right)\zeta _{r}^{-1},&|\zeta_{r}|>\zeta_{r}^{*}\end{array} \right. \tag{34}\]  
where \(a_{r}>1\), \(a_{r1}>0\) and \(b_{r}>0\); \(\zeta_{r}^{*}\) is a positive parameter. The Lyapunov function is defined as follows:  
\[V_{3}=V_{2}+\frac{1}{2}\zeta_{r}^{2} \tag{35}\]  
Invoking (33) and (34), the time derivative of \(V_{3}\) is given by  
\[\dot{V}_{3}= \dot{V}_{2}+\zeta_{r}\dot{\zeta}_{r} \tag{36}\] \[= \psi_{e}r_{e}-k_{\psi}\psi_{e}^{2}-k_{\psi 1}|\psi_{e}|^{\frac{3}{2}}+c_{r} \psi_{e}\zeta_{r}-\frac{1}{2}c_{r}^{2}\psi_{e}^{2}\ -a_{r}\zeta_{r}^{2}-a_{r1} \left|\zeta_{r}\right|^{\frac{3}{2}}\] \[+b_{r}\zeta_{r}\Delta\alpha_{r}-\frac{1}{2}b_{r}^{2}\Delta{\alpha _{r}}^{2}-\frac{y_{e}U\phi_{1}\psi_{e}}{k_{y}^{2}-y_{e}^{2}}\]  
Based on the Young's inequality, we can obtain  
\[c_{r}\psi_{e}\zeta_{r}\leq\frac{1}{2}c_{r}^{2}\psi_{e}^{2}+\frac{1}{2}\zeta_{ r}^{2} \tag{37}\]  
Figure 3: Configuration of the command filter. \[b_{r}\zeta_{r}\Delta\alpha r\leq\frac{1}{2}b_{r}^{2}\Delta a_{r}^{2}+\frac{1}{2} \zeta_{r}^{2} \tag{38}\]  
Substituting (37) and (38) into (36), we have  
\[ \begin{split}\dot{V}_{3}\leq&-k_{\varphi}\psi_{e}^{ 2}-k_{\psi 1}|\psi_{e}|^{\frac{3}{2}}-(a_{r}-1)\zeta_{r}^{2}-a_{r1}\left|\zeta_{r} \right|^{\frac{3}{2}}\\ &+\psi_{e}r_{e}-\frac{y_{e}L\phi_{\dagger}\psi_{e}}{k_{y}^{2}-y_{ e}^{2}}\end{split}  \tag{39}\]  
Taking the time derivative of \(r_{e}\) along (5), we have  
\[\dot{r}_{e}=\dot{r}-\dot{a}_{r}=\frac{1}{m_{33}}\left(\tau_{r}+d_{r}\right)- \dot{a}_{r} \tag{40}\]  
The Lyapunov function is defined as follows:  
\[V_{4}=V_{3}+\frac{1}{2}m_{33}r_{e}^{2} \tag{41}\]  
Taking the time derivative of \(V_{4}\) along (39) and (40), we have  
\[