## 2 Related Works and Problem Statement
### Position Controller Design  
For the position subsystem, define error variables \(\mathbf{z}_{iA}\), \(\mathbf{z}_{i5}\), \(\mathbf{z}_{i6}\) as follows:  
\[\left\{ \begin{array}{l}\mathbf{z}_{i4}=\sum\limits_{j=1}^{N}a_{ij}\big{[}(P_{i}- \Delta_{i})-\big{(}P_{j}-\Delta_{j}\big{)}\big{]}+d_{i}(P_{i}-\Delta_{i}-P_{r}) \\ \mathbf{z}_{i5}=\overline{\alpha}_{i2}-\alpha_{i2}\\ \mathbf{z}_{i6}=U_{i}-\overline{\alpha}_{i2}\end{array} \right. \tag{55}\]where \(\alpha_{i2}\) is a virtual control law. \(\overline{\alpha}_{i2}\) is a filter control law. \(\overline{\alpha}_{i2}\) is obtained by \(\alpha_{i2}\) through a first-order filter. \(\chi_{i}\) is given by the corresponding diagonal element and is a diagonal positive definite matrix. \[\chi_{i}\dot{\overline{\alpha}}_{i2}+\overline{\alpha}_{i2}=\alpha_{i2} \tag{56}\]  
The derivative of the error variable \(z_{i4}\), \(z_{i5}\), \(z_{i6}\) can be obtained by  
\[\left\{ \begin{array}{l}\dot{z}_{i4}=\sum\limits_{j=1}^{N}a_{ij}R_{i}(\theta _{i})\bigg{[}\begin{array}{c}z_{i5}+z_{i6}+\alpha_{i2}-U_{r}\\ -\big{(}z_{i5}+z_{j6}+\alpha_{j2}-U_{r}\big{)}\end{array}\bigg{]}+d_{i}R_{i}( \theta_{i})(z_{i5}+z_{i6}+\alpha_{i2}-U_{r})\\ \dot{z}_{i5}=-\chi_{i}^{-1}z_{i5}-\dot{\alpha}_{i2}\\ \dot{z}_{i6}=\frac{1}{m_{i1}^{\rm T}}\tau_{i}^{X}+\bar{f}_{i}^{X}+\chi_{i}^{- 1}z_{i5}\end{array} \right. \tag{57}\]  
where \(\dot{z}_{4}\) is expressed by the Kronecker product as \(\dot{z}_{4}=\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}}\otimes I_{1} \big{)}(z_{5}+z_{6}+\mathbf{\alpha}_{2}-\mathbf{U}_{r})\), \(\mathcal{R}=\text{diag}\{R_{i}(\theta_{i})\}\). Consider the candidate [PERSON] function \(V_{4}\) as follows:  
\[V_{4}=\frac{1}{2}z_{4}^{\rm T}z_{4} \tag{58}\]  
The derivative of \(V_{4}\) with respect to time is  
\[\dot{V}_{4}=z_{4}^{\rm T}\Big{[}\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{ L}}\otimes I\big{)}(z_{5}+z_{6}+\mathbf{\alpha}_{2}-\mathbf{U}_{r})\Big{]} \tag{59}\]  
Construct the virtual control law as  
\[\alpha_{i2}=-\mathbf{M}_{i1}z_{i4}-\mathbf{M}_{i0}\text{tanh}\bigg{(}\frac{Nz_{i4}}{ \delta}\bigg{)} \tag{60}\]  
where \(\mathbf{M}_{i1}\mathbf{M}_{i0}\in\mathbb{R}^{3\times 3}\) is a diagonal matrix. \(\delta\) is a variable positive constant. Definition \(z_{4}=[z_{14},\dots,z_{N4}]^{\rm T}\), \(z_{5}=[z_{15},\dots,z_{N5}]^{\rm T}\), \(z_{6}=[z_{16},\dots,z_{N6}]^{\rm T}\), \(\mathbf{M}_{0}=\text{diag}\{\mathbf{M}_{i0}\}\), \(\mathbf{M}_{1}=\text{diag}\{\mathbf{M}_{i1}\}\), and \(\mathbf{\chi}=\text{diag}\{\chi_{i}\}\), \(\mathbf{\alpha}_{2}=[\alpha_{i2},\dots,\alpha_{N2}]^{\rm T}\). Therefore, the virtual control law \(\alpha_{i2}\) can be written in the following form:  
\[\mathbf{\alpha}_{2}=-\mathbf{M}_{1}z_{4}-\mathbf{M}_{0}\text{tanh}\bigg{(}\frac{Nz_{4}}{ \delta}\bigg{)} \tag{61}\]  
Substituting (61) into (59), we have  
\[ \begin{array}{l}\dot{V}_{4}=z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(} \overline{\mathcal{L}}\otimes I\big{)}(z_{5}+z_{6})-z_{4}^{\rm T}\mathcal{R}^{ \rm T}\big{(}\overline{\mathcal{L}}\otimes I\big{)}\mathbf{M}_{1}z_{4}\\ -\mathbf{M}_{0}z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}} \otimes I\big{)}\text{tanh}\bigg{(}\frac{Nz_{4}}{\delta}\bigg{)}-z_{4}^{\rm T} \mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}}\otimes I_{1}\big{)}\mathbf{U}_{r} \end{array}  \tag{62}\]  
According to Lemma 3, it can be obtained as  
\[