## 3 Finite-Time ANFTSMC Design
### Controller Design Procedure  
#### 3.2.1 NFTSMC Design  
Below, we first define tracking errors as follows:  
\[ \begin{bmatrix}\begin{array}{ccc}e_{1}&e_{3}&e_{5}\end{array}\end{bmatrix}  ^{\text{T}}= \begin{bmatrix}\begin{array}{ccc}\phi-\phi_{d}&\theta-\theta_{d} &\psi-\psi_{d}\end{array}\end{bmatrix} ^{\text{T}} \tag{17}\] \[ \begin{bmatrix}\begin{array}{ccc}e_{2}&e_{4}&e_{6}\end{array} \end{bmatrix} ^{\text{T}}= \begin{bmatrix}\begin{array}{ccc}\dot{\phi}-\dot{ \phi}_{d}&\dot{\theta}-\dot{\theta}_{d}&\dot{\psi}-\dot{\psi}_{d}\end{array} \end{bmatrix} ^{\text{T}} \tag{18}\]  
where \( \begin{bmatrix}x_{d}&y_{d}&z_{d}\end{bmatrix} ^{\text{T}}\) and \( \begin{bmatrix}\phi_{d}&\theta_{d}&\psi_{d}\end{bmatrix} ^{\text{T}}\) are the desired position and attitude angle, respectively. The sliding surfaces of the translation subsystem and the rotation subsystem are introduced as \( \begin{bmatrix}S_{A}=\begin{bmatrix}S_{1}&S_{3}&S_{5}\end{bmatrix}^{\text{T}} \end{bmatrix} \) and \( \begin{bmatrix}S_{P}=\begin{bmatrix}S_{7}&S_{9}&S_{11}\end{bmatrix}^{\text{T}}\end{bmatrix} \)[32; 33; 34]. \[ \begin{array}{l}s_{1}=e_{1}+b_{1}|e_{1}|^{s_{1}}sign(e_{1})+b_{2}|e_{2}|^{ \beta_{1}}sign(e_{2})\\ s_{3}=e_{3}+b_{3}|e_{3}|^{s_{3}}sign(e_{3})+b_{4}|e_{4}|^{\beta_{3}}sign(e_{4}) \\ s_{5}=e_{5}+b_{5}|e_{5}|^{s_{5}}sign(e_{5})+b_{6}|e_{6}|^{\beta_{5}}sign(e_{6}) \end{array}  \tag{19a}\]  
Figure 3: Block diagram of the coaxial HAUV’s control structure. \[ \begin{array}{l}s_{7}=e_{7}+b_{7}|e_{7}|^{\alpha_{7}}sign(e_{7})+b_{8}|e_{8}|^{ \beta_{7}}sign(e_{8})\\ s_{9}=e_{9}+b_{9}|e_{9}|^{\alpha_{9}}sign(e_{9})+b_{10}|e_{10}|^{\beta_{9}}sign( e_{10})\\ s_{11}=e_{11}+b_{11}|e_{11}|^{\alpha_{11}}sign(e_{11})+b_{12}|e_{12}|^{\beta_{11} }sign(e_{12})\end{array}  \tag{19b}\]  
Here, \(b_{i},b_{i+1}(i=1,3,5,7,9,11)\) are positive constants, and \(1<\beta_{i}<2\), \(\alpha_{i}>\beta_{i}\). The time derivative of the sliding surfaces can be estimated as follows:  
\[\dot{s}_{i}=e_{i+1}+\alpha_{i}b_{i}|e_{i}|^{\alpha_{i}-1}e_{i+1}+\beta_{i}b_{i+ 1}|e_{i+1}|^{\beta_{i}-1}\dot{e}_{i+1} \tag{20}\]  
By setting \(\dot{s}_{i}=0\), an equivalent control term can be obtained. Then, the equivalent controller equations of attitude and position are shown in Equations (21a) and (21b), respectively:  
\[ \begin{array}{l}U_{2 eq}&=\left(-x_{4}x_{6}\left(I_{yyw}-I_{zw}\right)+\left( k_{8}k_{9w}+k_{8}k_{9w}\right)|x_{2}|x_{2}\right)\\ &+I_{zw}\left(\ddot{\theta}_{d}-\frac{1}{\beta_{10}^{2}}|e_{2}|^{2-\beta_{1}} \left(1+\alpha_{1}b_{1}|e_{1}|^{\alpha_{1}-1}\right)sign(e_{2})\right)\\ U_{3 eq}&=\left(-x_{2}x_{6}\left(I_{zw}-I_{zw}\right)+\left(k_{8}k_{9w}+k_{8} k_{9a}\right)|x_{4}|x_{4}\right)\\ &+I_{yw}\left(\ddot{\theta}_{d}-\frac{1}{\beta_{3}^{2}}|e_{4}|^{2-\beta_{3}} \left(1+\alpha_{3}b_{3}|e_{3}|^{\alpha_{3}-1}\right)sign(e_{4})\right)\\ U_{4 eq}&=\left(-x_{2}x_{4}\left(I_{zw}-I_{zw}\right)+\left(k_{8}k_{9w}+k_{8} k_{9w}\right)|x_{6}|x_{6}\right)\\ &+I_{zw}\left(\ddot{\theta}_{d}-\frac{1}{\beta_{3}^{2}}|e_{6}|^{2-\beta_{5}} \left(1+\alpha_{5}b_{5}|e_{5}|^{\alpha_{5}-1}\right)sign(e_{6})\right)\end{array}  \tag{21a}\] \[ \begin{array}{l}v_{xeq}=\left(k_{8}k_{xw}+k_{8}k_{xx}\right)|x_{8}|x_{8} /m_{w}+\ddot{x}_{d}-\frac{1}{\beta_{5}^{2}\beta_{18}}|e_{8}|^{2-\beta_{7}} \left(1+\alpha_{7}b_{7}|e_{7}|^{\alpha_{7}-1}\right)sign(e_{8})\\ v_{yeq}=\left(k_{8}k_{yw}+k_{8}k_{9a}\right)|x_{10}|x_{10}/m_{w}+\ddot{y}_{d}- \frac{1}{\beta_{9}\lambda_{10}}|e_{10}|^{2-\beta_{9}}\left(1+\alpha_{9}b_{9}| e_{9}|^{\alpha_{9}-1}\right)sign(e_{10})\\ v_{zeq}=\left(k_{8}k_{2w}+k_{8}k_{2a}\right)|x_{12}|x_{12}/m_{w}+\ddot{z}_{d}- \frac{1}{\beta_{11}\lambda_{12}}|e_{12}|^{2-\beta_{11}}\left(1+\alpha_{11}b_{1 1}|e_{11}|^{\alpha_{11}-1}\right)sign(e_{12})\end{array}