## 3 Finite-Time ANFTSMC Design
### Controller Design Procedure
#### 3.2.2 Adaptive Algorithm Design  
An adaptive algorithm is developed to estimate the unknown upper bound of lumped disturbances. \(\hat{a}_{0i}\), \(\hat{a}_{1i}\), and \(\hat{a}_{2i}\) are the estimate values of \(a_{0i}\), \(a_{1i}\), and \(a_{2i}\): \(\hat{a}_{0i}=a_{0i}-\hat{a}_{0i}\), \(\hat{a}_{1i}=a_{1i}-\hat{a}_{1i}\), and \(\hat{a}_{2i}=a_{2i}-\hat{a}_{2i}\) are defined as the adaptive parameter estimation errors. In order to estimate the upper bound of lumped uncertainty online in real time, the adaptive law is designed as follows:  
Adaptive law of rotation subsystem:  
\[\begin{array}{l}\dot{\hat{a}}_{01}=\mu_{01}|s_{1}||e_{2}^{\left|\beta_{1}-1 \right.},\dot{\hat{a}}_{03}=\mu_{03}|s_{3}||e_{4}|^{\left|\beta_{3}-1\right.}, \dot{\hat{a}}_{05}=\mu_{05}|s_{5}||e_{6}|^{\beta_{5}-1}\\ \dot{\hat{a}}_{11}=\mu_{11}|s_{1}||e_{1}||e_{2}|^{\beta_{1}-1},\dot{\hat{a}}_{ 13}=\mu_{13}|s_{3}||e_{3}||e_{4}|^{\beta_{3}-1},\dot{\hat{a}}_{15}=\mu_{15}|s _{5}||e_{6}|^{\beta_{5}-1}\\ \dot{\hat{a}}_{21}=\mu_{21}|s_{1}||e_{2}|^{\beta_{1}},\dot{\hat{a}}_{23}=\mu_{ 23}|s_{3}||e_{4}|^{\beta_{3}},\dot{\hat{a}}_{25}=\mu_{25}|s_{5}||e_{6}|^{\beta _{5}}\end{array} \tag{25}\]  
Adaptive law of translation subsystem:  
\[\begin{array}{l}\dot{\hat{a}}_{07}=\mu_{07}|s_{7}||e_{8}|^{\left|\beta_{7}-1 \right.},\dot{\hat{a}}_{09}=\mu_{09}|s_{9}||e_{10}|^{\beta_{9}-1},\dot{\hat{a} }_{011}=\mu_{011}|s_{11}||e_{12}|^{\beta_{11}-1}\\ \dot{\hat{a}}_{17}=\mu_{17}|s_{7}||e_{7}||e_{8}|^{\beta_{7}-1},\dot{\hat{a}}_{ 19}=\mu_{19}|s_{9}||e_{9}||e_{10}|^{\beta_{9}-1},\dot{\hat{a}}_{111}=\mu_{11} |s_{11}||e_{11}||e_{12}|^{\beta_{11}-1}\\ \dot{\hat{a}}_{27}=\mu_{27}|s_{7}||e_{8}|^{\beta_{7}},\dot{\hat{a}}_{29}=\mu_ {29}|s_{9}||e_{10}|^{\beta_{9}},\dot{\hat{a}}_{211}=\mu_{211}|s_{11}||e_{12}|^{ \beta_{11}}\end{array} \tag{26}\]  
where \(\mu_{0i}\), \(\mu_{1i}\), and \(\mu_{2i}(i=1,3,5,\cdots,11)\) are positive constants.  
During the actual voyage of the vehicle, due to the influence of measurement noises, environmental disturbances, and parameter perturbations, the system state is usually not strictly fixed on the sliding-mode plane, so the adaptive parameter may increase infinitely, leading to system divergence. An easy way to overcome this shortcoming is to improve the adaptive law by using the dead zone technique, as follows [36]:  
\[\dot{\hat{a}}_{0i}=\left\{\begin{array}{l}\mu_{0i}|s_{i}||e_{i+1}|^{\beta_{ 1}-1},|s_{i}|>\varepsilon\\ 0,|s_{i}|\leq\varepsilon\end{array}\right.,\dot{\hat{a}}_{1i}=\left\{\begin{array} []{l}\mu_{1i}|s_{i}||e_{i}||e_{i+1}|^{\beta_{1}-1},|s_{i}|>\varepsilon\\ 0,|s_{i}|\leq\varepsilon\end{array}\right.,\dot{\hat{a}}_{2i}=\left\{\begin{array} []{l}\mu_{2i}|s_{i}||e_{i+1}|^{\beta_{i}},|s_{i}|>\varepsilon\\ 0,|s_{i}|\leq\varepsilon\end{array}\right. \tag{27}\]  
**Lemma 1** ([37; 38]).: _Consider a system (28a) if the Lyapunov function selected for its system state satisfies the following inequality (28b):_  
\[\dot{\xi}=f(\xi),f(0)=0,\xi(0)=\xi_{0},\xi\in\mathbb{R}^{n} \tag{28a}\] \[\dot{V}(\xi)\leq-\kappa V^{\gamma}(\xi) \tag{28b}\]  
_where \(f(\xi)=[f_{1}(\xi),f_{2}(\xi),\cdots,f_{n}(\xi)]^{\mathrm{T}}\) is a continuous vector field. \(\kappa\) is the positive constant, \(0<\gamma<1\), and \(\xi\) is the state of the system. The system can be considered finite-time stable. Further, we can calculate the convergence time of \(V(\xi)\) from the initial value \(V(0)\) to be 0, and the settling time of the system is determined as follows:_  
\[t=\frac{V^{1-\gamma}(0)}{\kappa(1-\gamma)} \tag{29}\]