## 3 Controller Design
### Controller for the Case without Radial Control
\begin{split}&\overline{A}_{1}=\mathbf{C}_{13}^{-1}\Big{[}(\dot{\mathbf{ C}}_{12}+\ddot{\mathbf{C}}_{11})\mathbf{e}_{1u}+\dot{\mathbf{C}}_{13}\mathbf{e}_{1a}+(\mathbf{C}_{12}+ \ddot{\mathbf{C}}_{11})(\mathbf{A}_{11}\mathbf{e}_{1u}+\mathbf{A}_{12}\mathbf{e}_{1a})+\mathbf{C}_{13} \mathbf{e}_{1a})\Big{]}\\ &\quad+(\mathbf{A}_{13}\mathbf{e}_{1u}+\mathbf{A}_{14}\mathbf{e}_{1a})+\mathbf{\alpha }_{1}\mathbf{C}_{13}^{-1}\Big{[}(\mathbf{C}_{12}+\ddot{\mathbf{C}}_{11})\mathbf{e}_{1u}+\mathbf{C}_{ 13}\mathbf{e}_{1a}\Big{]},\\ &\delta_{1}=\mathbf{C}_{13}^{-1}\Big{[}(\mathbf{C}_{12}+\ddot{\mathbf{C}}_{11} )\mathbf{d}_{1u}\Big{]}+\mathbf{d}_{1a}+\mathbf{a}_{1}\mathbf{C}_{13}^{-1}\mathbf{C}_{11}\mathbf{d}_{1u},\\ &\quad\text{sig}^{\sigma_{1}}(\mathbf{s}_{1})=\big{[}\ \ |s_{11}|^{\sigma_{1}} \text{sgn}(s_{11})\quad|s_{12}|^{\sigma_{1}}\text{sgn}(s_{12})\ \ \big{]}^{\text{T}},\end{split}  \tag{16}\]  
where \(\dot{m}\) is the estimated value of the mass, and \(\delta_{1}\) is the total disturbance. The total disturbance \(\delta_{1}\) can be approximated by RBFNN in the active region. Substituting Equation (4) into Equation (14) yields. \[\mathbf{U}_{1}=-\dot{m}\overline{A}_{1}-\mathbf{W}_{1}^{\text{T}}\mathbf{h}(\mathbf{X})-\mathbf{ \varepsilon}_{1}+\mathbf{u}_{1s}, \tag{17}\]  
where \(\mathbf{X}=\big{[}\ x\,y\,z\,\dot{x}\,\dot{y}\,\dot{z}\ \big{]}^{\text{T}}\) is the input vector. Although the RBFNN has the ability to approximate the disturbance with high accuracy, the approximation error still exists. In order to reduce the approximation error, the upper bound of the error needs to be updated and estimated by the adaptive law. Defining \(\phi_{i}\) as the upper bound of the estimation error \(\epsilon_{i}\), denote \(\mathbf{\hat{\phi}}_{i}=[\hat{\phi}_{11},\hat{\phi}_{12}]^{\mathrm{T}}\) as the estimate of \(\phi_{i}\), and \(\mathbf{\hat{W}}_{1}\) as the estimate of the neural network weights, Equation (17) can be rewritten as:  
\[\mathbf{U}_{1}=-\hat{m}\overline{\mathbf{A}}_{1}-\hat{W}_{1}^{\mathrm{T}}\mathbf{h}(\mathbf{X} )-\mathrm{diag}(\mathbf{\hat{\phi}}_{11},\mathbf{\hat{\phi}}_{12})\mathrm{sgn}(\mathbf{s}_ {1})\,+\mathbf{u}_{1s}. \tag{18}\]  
The adaptive sliding mode control method is used to automatically adjust the control gain to compensate for the uncertainty in the upper bound outside the active region, and the control law is designed as:  
\[\mathbf{U}_{1}=-\hat{m}\overline{\mathbf{A}}_{1}-\hat{\delta}_{1}+\mathbf{u}_{1s}, \tag{19}\]  
where \(\hat{\delta}_{1}\) is an estimate of the unknown upper bound of perturbations. A nonlinear function \(\hat{h}_{a,b}(\mathbf{e}_{p})\) is chosen to construct the switching function [26]. \[\hat{h}_{a,b}(\mathbf{e}_{i})= \begin{cases}0\,,&\|\mathbf{e}_{i}\|\leq a,\\ 1-\cos^{2}\biggl{(}\frac{\pi}{2}\sin^{2}\biggl{(}\frac{\pi}{2}\frac{\|\mathbf{e}_ {i}\|^{2}-a^{2}}{b^{2}-a^{2}}\biggr{)}\biggr{)},&a<\|\mathbf{e}_{i}\|<b,\\ 1\,,&\|\mathbf{e}_{i}\|\geq b,\end{cases}  \tag{20}\]  
where \(0<a<b\) are design parameters to determine the neural active region. The input to the switching function is the error state \(\mathbf{e}_{i}\). The switching control law combining the neural network control law and the adaptive sliding mode control law is designed as:  
\[\mathbf{U}_{1}=-\hat{m}\overline{\mathbf{A}}_{1}+(1-\hat{h}_{a,b}(\mathbf{e}_{p}))\mathbf{u}_ {n1}+\,\hat{h}_{a,b}(\mathbf{e}_{p})\mathbf{u}_{s1}+\mathbf{u}_{1s}, \tag{21}\]  
with  
\[ \begin{array}{l}\mathbf{u}_{n1}=-\hat{W}_{1}^{\mathrm{T}}\mathbf{h}(\mathbf{X})-\mathrm{ diag}(\mathbf{\hat{\phi}}_{11},\mathbf{\hat{\phi}}_{12})\mathrm{sgn}(\mathbf{s}_{1}),\\ \mathbf{u}_{s1}=-\hat{\delta}_{1}.\end{array}  \tag{22}\]  
The adaptive laws of \(\hat{m}\), \(\mathbf{\hat{W}}_{1}\), \(\mathbf{\hat{\phi}}_{1}\) and \(\delta_{1}\) in Equation (21) are designed as:  
\[\left\{