## 3 Controller Design
### Controller for the Case without In-Track Control  
Similar to the approach in the case without radial control, for the case without in-track control, the error dynamics system can be rewritten as:  
\[\left\{ \begin{array}{l}\dot{\mathbf{e}}_{2u}=A_{21}\mathbf{e}_{2u}+A_{22}\mathbf{e}_{2u }+m^{-1}\mathbf{d}_{2u},\\ \dot{\mathbf{e}}_{2u}=A_{23}\mathbf{e}_{2u}+A_{24}\mathbf{e}_{2u}+m^{-1}(\mathbf{U}_{2}+\mathbf{d }_{2u}),\end{array} \right. \tag{26}\]  
with  
\[\mathbf{A}_{21}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ -\alpha_{C}&\omega_{C}^{2}-n_{C}^{2}&0&0\\ \end{array} \right],\qquad\mathbf{A}_{22}=\left[ \begin{array}{cccc}1&0\\ 0&0\\ 0&1\\ -2\omega_{C}&0\\ \end{array} \right], \tag{27}\] \[\mathbf{A}_{23}=\left[ \begin{array}{cccc}\omega_{C}^{2}+2n_{C}^{2}& \alpha_{C}&0&2\omega_{C}\\ 0&0&-n_{C}^{2}&0\\ \end{array} \right],\quad\mathbf{A}_{24}=\left[ \begin{array}{cccc}0&0\\ 0&0\\ \end{array} \right],\]  
where \(\mathbf{e}_{2u}=\left[ \begin{array}{cccc}e_{x}&e_{y}&\dot{e}_{z}&\dot{e}_{y} \end{array} \right]^{\rm T}\), \(\mathbf{e}_{2u}=\left[ \begin{array}{cccc}\dot{e}_{x}&\dot{e}_{z}\end{array}  \right]^{\rm T}\), \(\mathbf{d}_{2u}=\left[ \begin{array}{cccc}\mathbf{0}_{1\times 3}&\mathbf{d}_{y}\end{array} \right]^{\rm T}\) and \(\mathbf{d}_{2a}=\left[ \begin{array}{cccc}\mathbf{d}_{x}&\mathbf{d}_{z}\end{array} \right]^{ \rm T}\). The control input is \(\mathbf{U}_{2}=\left[ \begin{array}{cccc}\mathbf{U}_{x}&\mathbf{U}_{2}\end{array} \right]^{ \rm T}\). Similarly, note that \(\mathbf{e}_{2u}\in\mathbb{R}^{4}\) and \(\mathbf{e}_{2u}\in\mathbb{R}^{2}\), a linear transformation \(\mathbf{\bar{e}}_{2u}=\mathbf{C}_{21}\mathbf{e}_{2u}\in\mathbb{R}^{2}\) is performed, and the matrix \(\mathbf{C}_{21}\) is given by,  
\[\mathbf{C}_{21}= \begin{bmatrix}0&k_{21}&0&k_{22}\\ 0&0&1&0\end{bmatrix} . \tag{28}\]  
The dynamic equations of \(\mathbf{\bar{e}}_{2u}\) can be expressed as:  
\[\ddot{\mathbf{\bar{e}}}_{2u}=(\dot{\mathbf{C}}_{21}+\mathbf{C}_{22})\mathbf{e}_{2u}+\mathbf{C}_{23} \mathbf{e}_{2u}+m^{-1}\mathbf{C}_{21}\mathbf{d}_{2u}, \tag{29}\]  
with  
\[\mathbf{C}_{22}=\mathbf{C}_{21}\mathbf{A}_{21},\quad\mathbf{C}_{23}=\mathbf{C}_{21}\mathbf{A}_{22}= \begin{bmatrix}-2\omega_{C}k_{22}&0\\ 0&1\end{bmatrix} . \tag{30}\]  
The sliding surface is then chosen as:  
\[\mathbf{s}_{2}=\left[ \begin{array}{cc}s_{21}&s_{22}\end{array} \right]^{\rm T}= \alpha_{2}\mathbf{\bar{e}}_{2u}+\left[(\mathbf{P}_{22}+\dot{\mathbf{P}}_{21})\mathbf{e}_{2u}+\bm {P}_{23}\mathbf{e}_{2u}\right]\approx\alpha_{2}\mathbf{\bar{e}}_{2u}+\dot{\mathbf{\bar{e}}}_ {2u}, \tag{31}\]  
where \(\alpha_{2}>0\) is a positive constant. The control law is designed as:  
\[\mathbf{U}_{2}=\mathbf{u}_{2 eq}+\mathbf{u}_{2s}, \tag{32}\]where \(\mathbf{u}_{2 eq}\) is the equivalent control and \(\mathbf{u}_{2s}\) is the reaching law designed as:  
\[ \begin{split}&\mathbf{u}_{2 eq}=-\hat{m}\overline{A}_{2}-\delta_{2},\\ &\mathbf{u}_{2s}=\hat{\mathbf{s}}_{2}=-\hat{m}\mathbf{C}_{2s}^{-1}[\mathbf{K}_{2} s_{2}+\eta_{2}\text{sig}^{\alpha_{2}}(\mathbf{s}_{2})],\end{split}  \tag{33}\]  
with  
\[ \begin{split}&\overline{A}_{2}=C_{23}^{-1}\Big{[}(\dot{\mathbf{C}}_{2 2}+\ddot{\mathbf{C}}_{21})\mathbf{e}_{2u}+\dot{\mathbf{C}}_{23}\mathbf{e}_{2u}+(\mathbf{C}_{22}+ \dot{\mathbf{C}}_{21})(A_{21}\mathbf{e}_{2u}+A_{22}\mathbf{e}_{2u})+C_{23}\mathbf{e}_{2u}) \Big{]}\\ &\quad+(A_{23}\mathbf{e}_{2u}+A_{24}\mathbf{e}_{2u})+\alpha_{2}\mathbf{C}_{23 }^{-1}\big{[}(\mathbf{C}_{22}+\dot{\mathbf{C}}_{21})\mathbf{e}_{2u}+\mathbf{C}_{23}\mathbf{e}_{2u} \Big{]},\\ &\delta_{2}=\mathbf{C}_{23}^{-1}\Big{[}(\mathbf{C}_{22}+\dot{\mathbf{C}}_{21} )\mathbf{d}_{2u}\Big{]}+\mathbf{d}_{2u}+\alpha_{2}\mathbf{C}_{23}^{-1}C_{21}\mathbf{d}_{2u}, \end{split}  \tag{34}\]  
where \(K_{2}>0,\eta_{2}>0\) are the positive constants.