## 3 Controller Design
### Controller for the Case without Radial Control  
Define the desired relative state of motion as \(\mathbf{X}_{d}=\left[\mathbf{\rho}_{d}^{\rm T}\quad\dot{\mathbf{\rho}}_{d}^{\rm T}\right]^ {\rm T}\), where \(\mathbf{\rho}_{d}=\left[x_{d}\quad y_{d}\quad z_{d}\right]^{\rm T}\) and \(\dot{\mathbf{\rho}}_{d}=\left[\dot{x}_{d}\quad\dot{y}_{d}\quad\dot{z}_{d}\right]^ {\rm T}\) are the desired relative position vector and relative velocity vector, respectively, and the desired dynamics equation for the elliptical orbital underactuated formation can be formulated as follows:  
\[\dot{\mathbf{X}}_{id}=\mathbf{AX}_{id},i=1\,,\,2. \tag{6}\]  
The error dynamics system obtained by making the difference between Equations (1) and (6) is given by:  
\[\dot{\mathbf{e}}_{i} =\mathbf{A}\mathbf{e}_{i}+m^{-1}(\mathbf{B}_{i}\mathbf{U}_{i}+\mathbf{D}_{i}),\;i=1\,, \,2. \tag{7}\]  
where \(\mathbf{e}_{i}=\mathbf{X}_{i}-\mathbf{X}_{id}\) is the error state vector. Then the error dynamics equation in the absence of radial control can be rewritten as [13]. \[\left\{ \begin{array}{l}\dot{\mathbf{e}}_{1u}=\mathbf{A}_{11}\mathbf{e}_{1u}+\mathbf{A}_{12} \mathbf{e}_{1a}+m^{-1}\mathbf{d}_{1u},\\ \dot{\mathbf{e}}_{1a}=\mathbf{A}_{13}\mathbf{e}_{1u}+\mathbf{A}_{14}\mathbf{e}_{1a}+m^{-1}(\mathbf{U}_ {1}+\mathbf{d}_{1a}),\end{array} \right. \tag{8}\]with  
\[ \begin{split} A_{11}&=\left[\begin{array}{cccc}0&0&0&1 \\ 0&0&0&0\\ 0&0&0&0\\ \omega_{C}^{2}+2\eta_{C}^{2}&\alpha_{C}&0&0\end{array}\right],\quad A_{12}= \left[\begin{array}{cc}0&0\\ 1&0\\ 0&1\\ 2\omega_{C}&0\end{array}\right],\\ A_{13}&=\left[\begin{array}{cccc}-\alpha_{C}&\omega_{C}^{ -}-\eta_{C}^{2}&0&-2\omega_{C}\\ 0&0&-\eta_{C}^{2}&0\end{array}\right],\quad A_{14}=\left[\begin{array}{cc}0 &0\\ 0&0\end{array}\right].\end{split}  \tag{9}\]  
where \(e_{1u}=\left[e_{x}\quad e_{y}\quad e_{z}\quad\dot{e}_{x}\right]^{\text{T}}\), \(e_{1a}=\left[\dot{e}_{y}\quad\dot{e}_{z}\right]^{\text{T}}\), \(\mathbf{d}_{1u}=\left[\mathbf{0}_{1\times 3}\quad d_{y}\right]^{\text{T}}\) and \(\mathbf{d}_{1a}=\left[d_{y}\quad d_{z}\right]^{\text{T}}\). Note that \(e_{1u}\in\mathbb{R}^{4}\) and \(\mathbf{e}_{1a}\in\mathbb{R}^{2}\), we make a linear transformation of \(\mathbf{e}_{1u}\) to obtain a new variable \(\mathbf{\bar{e}}_{1u}=\mathbf{C}_{11}\mathbf{e}_{1u}\in\mathbb{R}^{2}\), where \(\mathbf{C}_{11}\in\mathbb{R}^{2\times 4}\) is the design parameter matrix denoting the linear transformation for \(\mathbf{e}_{1u}\) from four dimensions to two dimensions,  
\[\mathbf{C}_{11}= \begin{bmatrix}k_{11}&k_{12}&0&k_{13}\\ 0&0&1&0\end{bmatrix}  \tag{10}\]  
where \(k_{11}\), \(k_{12}\) and \(k_{13}\) are the controller design parameters. Then the dynamics of \(\mathbf{\bar{e}}_{1u}\) is given by:  
\[\dot{\mathbf{\bar{e}}}_{1u}=(\mathbf{C}_{12}+\dot{\mathbf{C}}_{11})\mathbf{e}_{1u}+\mathbf{C}_{13} \mathbf{e}_{1a}+m^{-1}\mathbf{C}_{11}\mathbf{d}_{1u} \tag{11}\]  
with  
\[\mathbf{C}_{12}=\mathbf{C}_{11}\mathbf{A}_{11},\quad\mathbf{C}_{13}=\mathbf{C}_{11}\mathbf{A}_{12}= \begin{bmatrix}k_{12}+2\omega_{C}k_{13}&0\\ 0&1\end{bmatrix} . \tag{12}\]  
The sliding surface is then chosen as:  
\[\mathbf{s}_{1}=\left[s_{11}\quad s_{12}\right]^{\text{T}}=\alpha_{1}\mathbf{\bar{e}}_ {1u}+[(\mathbf{C}_{12}+\dot{\mathbf{C}}_{11})\mathbf{e}_{1u}+\mathbf{C}_{13}\mathbf{e}_{1a}]\approx \alpha_{1}\mathbf{\bar{e}}_{1u}+\dot{\mathbf{\bar{e}}}_{1u}, \tag{13}\]  
where \(\alpha_{1}>0\) is a positive constant, the control law is designed as:  
\[\mathbf{U}_{1}=\mathbf{u}_{1 eq}+\mathbf{u}_{1s}, \tag{14}\]  
where \(\mathbf{u}_{1 eq}\) is the equivalent control and \(\mathbf{u}_{1s}\) is the reaching law, respectively designed as:  
\[ \begin{split}\mathbf{u}_{1 eq}&=-\dot{m}\overline{A}_{1}- \delta_{1},\\ \mathbf{u}_{1s}&=\dot{\mathbf{s}}_{1}=-\dot{m}\mathbf{C}_{13}^{- 1}[\mathbf{K}_{1}\mathbf{s}_{1}+\eta_{1}\text{sig}^{\sigma_{1}}(\mathbf{s}_{1})],\end{split}  \tag{15}\]  
with  
\[