## 3 Predefined-Time Backstepping Control
### Control Law Design  
The design of the controller is mainly divided into two parts:  
* Constructing the Lyapunov function to design the virtual control law;
* Adopting the RBF neural network to estimate the external disturbances of the system and identifying the unmodeled part of the system by the adaptive law. The predefined-time adaptive control law is designed in detail as follows:  
**Step 1:** Designing the virtual control law \(\alpha\), substituting \(\upsilon_{e}\) into (39)\[\dot{\mathbf{z}}_{1}=-\mathbf{\omega}\mathbf{z}_{1}+\Lambda\mathbf{v}_{e}+\Lambda\mathbf{\alpha}+\mathbf{v }_{1}^{*}-J_{1}^{T}\dot{\mathbf{\eta}}_{g}+\dot{M} \tag{40}\]  
Constructing alternative Lyapunov functions  
\[V_{1}=\frac{1}{2}\mathbf{z}_{1}^{T}\mathbf{z}_{1} \tag{41}\]  
Compute the derivative with respect to time for \(V_{1}\)  
\[\dot{V}_{1}=\mathbf{z}_{1}^{T}\Big{(}-\mathbf{\omega}\mathbf{z}_{1}+\Lambda\mathbf{v}_{e}+ \Lambda\mathbf{\alpha}+\mathbf{v}_{1}^{*}-J_{1}^{T}\dot{\mathbf{\eta}}_{g}+\dot{M}\Big{)} \tag{42}\]  
Design the virtual control law \(\mathbf{\alpha}\) as  
\[\mathbf{\alpha}=\Lambda^{-1}(-C_{1}\mathbf{K}_{1}diag\Big{(}\left|\mathbf{z}_{1}\right|^{1 +\beta}\Big{)}\text{sgn}(\mathbf{z}_{1})-C_{2}\mathbf{K}_{2}diag\Big{(}\left|\mathbf{z}_{1 }\right|^{1-\beta}\Big{)}\text{sgn}(\mathbf{z}_{1})-\mathbf{v}_{1}^{*}+J_{1}^{T}\dot{ \mathbf{\eta}}_{g}-\dot{M}) \tag{43}\]  
where \(\mathbf{\alpha}=\big{[}\mathbf{a}_{u},\mathbf{a}_{q},\mathbf{\alpha}_{r}\big{]}^{T},\mathbf{a}_{u },\mathbf{a}_{q},\mathbf{\alpha}_{r}\) is the virtual control law in the direction of the three driving degrees of freedom. \(\mathbf{K}_{1},\mathbf{K}_{2}\) is a positive definite symmetric matrix defined as \(\mathbf{K}_{1}=\text{diag}([k_{11},k_{12},k_{13}]),\mathbf{K}_{2}=\text{diag}([k_{21},k_{22},k_{23}]),k_{11},k_{12},k_{13}\), \(\beta\in(0,1)\) is a predefined time control parameter, \(C_{1},C_{2}\) is a predefined time control parameter satisfying the following definitions \(C_{1}=\frac{\pi}{3\beta\beta T_{c}\Lambda\text{min}2^{1+\beta}},C_{2}=\frac{ \pi}{\beta T_{c}\Lambda\text{min}2^{1-\beta}}\), and \(T_{c}\) is a predefined time. **Step 2:** Design of drive subsystem controllers \(\mathbf{\tau}_{a}\)  
Extended alternative Lyapunov functions:  
\[V_{3}=V_{1}+V_{2}=\frac{1}{2}\mathbf{z}_{1}^{T}\mathbf{z}_{1}+\frac{1}{2m}\mathbf{z}_{2}^ {T}\mathbf{M}_{a}\mathbf{z}_{2} \tag{44}\]  
Here, \(\mathbf{z}_{2}=\mathbf{v}_{e},m_{1}\leq\left\|\mathbf{M}_{a}\right\|\leq m_{2},\,m>m_{2}\), which is derived by differentiating with respect to time relative to \(V_{3}\):  
\[ \begin{array}{rl}\dot{V}_{3}&=\mathbf{z}_{1}^{T}\Big{(}-\mathbf{\omega}\mathbf{z}_{1}+ \Lambda\mathbf{v}_{e}+\Lambda\mathbf{\alpha}+\mathbf{v}_{1}^{*}-J_{1}^{T}\dot{\mathbf{\eta}}_{ g}+\dot{M}\Big{)}+\frac{1}{2m}\mathbf{z}_{1}^{T}\Lambda\mathbf{z}_{2}+\frac{1}{2m}\mathbf{z}_{2}^ {T}\mathbf{M}_{a}\dot{\mathbf{z}}_{2}\\ &=\mathbf{z}_{1}^{T}\Big{(}-\mathbf{\omega}\mathbf{z}_{1}+\Lambda\mathbf{v}_{e}+\Lambda\mathbf{ \alpha}+\mathbf{v}_{1}^{*}-J_{1}^{T}\dot{\mathbf{\eta}}_{g}+\dot{M}\Big{)}+\frac{1}{2m} \mathbf{z}_{1}^{T}\Lambda\mathbf{z}_{2}+\frac{1}{2m}\mathbf{z}_{2}^{T}\big{(}m^{-1}(\mathbf{ \tau}_{a}-\mathbf{f}_{a}+\Theta)\big{)}\end{array}  \tag{45}\]  
where \(m>m_{2},\Theta=m^{-1}\big{(}-\mathbf{C}_{a}\mathbf{v}-\mathbf{D}_{a}\mathbf{v}-\mathbf{G}_{a}-\mathbf{ M}_{a}\dot{\mathbf{v}}_{d}\big{)}\). By Assumption 3, \(\Theta\) satisfies the following properties:  
\[ \begin{array}{rl}\parallel\Theta\parallel\leq m^{-1}\Big{(}c\parallel\mathbf{v }\parallel^{2}+d_{1}\parallel\mathbf{v}\parallel+d_{2}\parallel\mathbf{v}\parallel^ {2}+g+m_{2}\dot{\mathbf{v}}_{d}\Big{)}\leq l_{0}+l_{1}\parallel\mathbf{v}\parallel+l_ {2}\parallel\mathbf{v}\parallel^{2}\end{array}  \tag{46}\]  
\(l_{0},\ l_{1},\ l_{2}\) is the unknown positive parameter; \(\mathbf{\tau}_{a}\) is the controller to be designed; and the design of the actual control law is \(\mathbf{\tau}_{a}\).