## 2 Related Works and Problem Statement
### Attitude Controller Design  
For the attitude subsystem, define the error variables \(z_{i1}\), \(z_{i2}\), \(z_{i3}\) as follows:  
\[\left\{ \begin{array}{l}z_{i1}=\theta_{i}-\theta_{j}\\ z_{i2}=\overline{\alpha}_{i1}-\alpha_{i1}\\ z_{i3}=r_{i}-\overline{\alpha}_{i1}\end{array} \right. \tag{14}\]  
where \(\alpha_{i1}\) is a virtual control law. \(\overline{\alpha}_{i1}\) is a filter control law. \(\overline{\alpha}_{i1}\) is obtained by \(\alpha_{i1}\) through a first-order filter. Time constant \(\gamma_{i}\) is given by the corresponding diagonal element and is a diagonal positive definite matrix. \[\gamma_{i}\dot{\overline{\alpha}}_{i1}+\overline{\alpha}_{i1}=\alpha_{i1} \tag{15}\]  
The derivative of the error variable \(z_{i1}\), \(z_{i2}\), \(z_{i3}\) can be obtained as follows:  
\[\left\{ \begin{array}{l}\dot{z}_{i1}=z_{i2}+z_{i3}+\dot{\alpha}_{i1}-\dot{r}_ {j}\\ \dot{z}_{i2}=-\gamma_{i}^{-1}z_{i2}-\dot{\alpha}_{i1}\\ \dot{z}_{i3}=\frac{1}{m_{i}^{3}}\tau_{i}^{\sigma}+\boldsymbol{f}_{i}+\gamma_ {i}^{-1}\boldsymbol{z}_{i2}\end{array} \right. \tag{16}\]  
In this paper, DSC is adopted to deal with the complex differential problem in the calculation process of virtual control law \(\alpha_{i1}\), which further improves the practicability of the controller. Firstly, a Lyapunov function is selected as follows:  
\[V_{1}=\frac{1}{2}\tau_{1}^{\mathsf{T}}z_{1} \tag{17}\]  
The derivative of \(V_{1}\) with respect to time is  
\[\dot{V}_{1}=\boldsymbol{z}_{1}^{\mathsf{T}}\big{[}\boldsymbol{z}_{2}+ \boldsymbol{z}_{3}+\boldsymbol{\alpha_{1}}-\dot{\boldsymbol{r}}_{j}\big{]} \tag{18}\]  
Construct the virtual control law as  
\[\alpha_{i1}=-\boldsymbol{K}_{i1}\boldsymbol{z}_{i1}-\boldsymbol{K}_{i0} \text{tanh}\bigg{(}\frac{N\boldsymbol{z}_{i1}}{\varepsilon}\bigg{)} \tag{19}\]  
where \(\boldsymbol{K}_{i1}\boldsymbol{K}_{i0}\in\mathbb{R}^{3\times 3}\) is a diagonal matrix. \(\varepsilon\) is a variable positive constant. Definition \(\boldsymbol{z}_{1}=[z_{11},\ldots,z_{N1}]^{\mathsf{T}}\), \(\boldsymbol{z}_{2}=[\boldsymbol{z}_{12},\ldots,\boldsymbol{z}_{N2}]^{\mathsf{ T}}\), \(\boldsymbol{z}_{3}=[\boldsymbol{z}_{13},\ldots,\boldsymbol{z}_{N3}]^{\mathsf{T}}\), \(\boldsymbol{K}_{0}=\text{diag}\{\boldsymbol{K}_{i0}\}\), \(\boldsymbol{K}_{1}=\text{diag}\{\boldsymbol{K}_{i1}\}\), and \(\boldsymbol{\gamma}=\text{diag}\{\gamma_{i}\}\),\(\alpha_{1}=[\alpha_{i1},\ldots,\alpha_{N1}]^{\mathsf{T}}\). Therefore, the virtual control law \(\alpha_{i1}\) can be written in the following form:  
\[\boldsymbol{\alpha}_{1}=-\boldsymbol{K}_{1}\boldsymbol{z}_{1}-\boldsymbol{K} _{0}\text{tanh}\bigg{(}\frac{N\boldsymbol{z}_{1}}{\varepsilon}\bigg{)} \tag{20}\]  
Substituting (21) into (19),  
\[\dot{V}_{1}=\boldsymbol{z}_{1}^{\mathsf{T}}(\boldsymbol{z}_{2}\boldsymbol{+} \boldsymbol{z}_{3})-\boldsymbol{z}_{1}^{\mathsf{T}}\boldsymbol{K}_{1} \boldsymbol{z}_{1}-\boldsymbol{K}_{0}\boldsymbol{z}_{1}^{\mathsf{T}}\text{ tanh}\bigg{(}\frac{N\boldsymbol{z}_{1}}{\varepsilon}\bigg{)}-\boldsymbol{z}_{1}^{ \mathsf{T}}\dot{\boldsymbol{r}}_{j} \tag{21}\]  
According to Lemma 3, it can be obtained as follows:  
\[ \begin{array}{rl}\dot{V}_{1}&\leq z_{1}^{\mathsf{T}}(z_{2}+z_{3})-z_{1}^{ \mathsf{T}}\boldsymbol{K}_{1}\boldsymbol{z}_{1}-\boldsymbol{K}_{0}\boldsymbol{ z}_{1}^{\mathsf{T}}\text{tanh}\bigg{(}\frac{N\boldsymbol{z}_{1}}{\varepsilon} \bigg{)}+\left\lVert\boldsymbol{z}_{1}\right\rVert_{\infty}N\rho_{M}\\ &\leq z_{1}^{\mathsf{T}}(z_{2}+z_{3})-z_{1}^{\mathsf{T}}\boldsymbol{K}_{1} \boldsymbol{z}_{1}-\boldsymbol{K}_{0}\boldsymbol{z}_{1}^{\mathsf{T}}\text{ tanh}\bigg{(}\frac{N\boldsymbol{z}_{1}}{\varepsilon}\bigg{)}+N\rho_{M}k_{p} \varepsilon+N^{2}\rho_{M}\boldsymbol{z}_{1}^{\mathsf{T}}\text{tanh}\,\!( \frac{N\boldsymbol{z}_{1}}{\varepsilon})\end{array}  \tag{22}\]Let \(\lambda(K_{0})_{\min}>N^{2}\rho_{M}\), and we can further conclude that \(N^{2}\rho_{M}\mathbf{z}_{1}^{\mathrm{T}}\mathrm{tanh}\left(\frac{Nz_{1}}{ \varepsilon}\right)\), \(-K_{0}\mathbf{z}_{1}^{\mathrm{T}}\mathrm{tanh}\left(\frac{Nz_{1}}{\varepsilon} \right)<0\).