# Coordinated Control after Grasping the Space Targets Using Controllable Damping Mechanism
## 3 Design of the Coordinated Controller  
According to Section 2, the six-dimensional damping force outputs by the FDCDM are equivalented to the active flexible forces and torques in the end joint, so a coordinated controller is proposed to optimize the momentum distribution and impact absorption in Section 3. The instantaneous impact may result in sudden change of base attitude, and the non-cooperative target increases the parameter uncertainty of the system. So, the design of the controller must have high robustness, high precision, and fast transient response. Let \(\mathbf{e}=\mathbf{x}-\mathbf{x}_{d}\) denote the trajectory tracking error, where \(\mathbf{x}_{d}\) denotes the desired trajectory. In order to obtain the above control performance, a backstepping nonsingular fast integral terminal sliding mode control (NFITSMC) is designed as:  
\[\mathbf{s}=\int\left(\mathbf{e}+c_{1}\mathbf{\varphi}(\mathbf{e})+\mathbf{c}_{2}\dot{\mathbf{e}}^{p/q}\right) \tag{22}\]  
where \(c_{1}=\text{diag}\{c_{1 ii}\}\in\Re^{n\times n}\), \(\mathbf{c}_{2}=\text{diag}\{c_{2 ii}\}\in\Re^{n\times n}\) are positive diagonal matrices, \(p\) and \(q\) are positive odd numbers satisfying the relation \(1<p/q<2\), and \(\mathbf{\varphi}(\mathbf{e})\) is designed as  
\[\varphi_{i}(e_{i})= \begin{cases}|e_{i}|^{\alpha_{1}}\text{sgn}(e_{i}),&|e_{i} |>u_{i},\\ |e_{i}|^{\alpha_{2}}\text{sgn}(e_{i}),&|e_{i}|<u_{i}.\end{cases}  \tag{23}\]  
In the case of \(|e_{i}|>u_{i}\), the second order derivative of \(s_{i}\) yields \(\bar{s}_{i}=\dot{e}_{i}+c_{1 ii}\alpha_{1}|e_{i}|^{\alpha_{1}-1}\dot{e}_{i}\)\(+c_{2 ii}(p/q)\ddot{e}_{i}^{p/q-1}\). Although \(\alpha_{1}-1<0\), but \(|e_{i}|>u_{i}\), so it does not cause singularity. When \(|e_{i}|<u_{i}\), the second order time derivative \(\ddot{s}_{i}=\dot{e}_{i}+c_{1 ii}\alpha_{2}|e_{i}|^{\alpha_{2}-1}\dot{e}_{i}+c_ {2 ii}(p/q)\ddot{e}_{i}^{p/q-1}\), note that \(\alpha_{2}-1>0\), even \(e_{i}=0\), there is no singularity. This design of \(\varphi(\mathbf{e})\) can improve the problem of the convergence speed slowing down when the terminal attractor \(\mathbf{e}\) is close to zero. Meanwhile, it can avoid the singularity problem. Therefore, the proposed sliding surface combines the advantages of the fast-sliding mode control and non-singular sliding mode control, giving the system features of fast finite-time convergence without the singular problem. Define \(t_{r}\) is the finite time form \(s(0)\
eq 0\) to \(s=0\), and \(t_{s}\) is the finite time taken to travel from \(e(t_{r})\
eq 0\) to \(e(t_{r}+t_{s})=0\), which is given by [39]:  
\[t_{s}=\frac{\frac{p}{q}|e(t_{r})|^{1-\frac{q}{q}}}{c_{1}\left(\frac{p}{q}-1 \right)}\cdot\Omega\left( \begin{array}{c}\frac{q}{q},\frac{\frac{p}{q}-1}{( \alpha_{1}-1)\frac{p}{q}},1+\frac{\frac{p}{q}-1}{(\alpha_{1}-1)\frac{p}{q}}\\ -c_{1}|e(t_{r})|^{\alpha_{1}-1}\end{array} \right) \tag{24}\]  
where \(\Omega\) denote Gauss' hypergeometric function. According to (18) and (19), the time derivative of sliding surface \(s\) yields  
\[ \begin{array}{l}\dot{\mathbf{s}}=\mathbf{e}+c_{1}\mathbf{\varphi}(\mathbf{e})+c_{2}\dot{\mathbf{ e}}^{p/q},\\ \ddot{\mathbf{s}}=\dot{\mathbf{e}}+c_{1}\dot{\mathbf{\varphi}}(\mathbf{e})+c_{2}(p/q)\mathbf{e}^{p/q -1}\cdot\ddot{\mathbf{e}},\\ \ddot{\mathbf{s}}=\frac{\dd}{\dd}\big{(}\mathbf{\Psi}(\mathbf{e},\dot{\mathbf{e}})\mathbf{+}\mathbf{ \Xi}(\dot{\mathbf{e}})\cdot(\mathbf{J}^{-1}(\mathbf{x}_{1})\mathbf{\tau+}\mathbf{C+}\Delta\!-\!\ddot{ x}_{d})\big{)}.\end{array}  \tag{25}\]where \(\mathbf{\varGamma}\big{(}e\dot{e}\dot{e}\big{)}=\dot{e}+c_{1}\dot{\mathbf{\varphi}}(e)\), \(\mathbf{\varXi}\big{(}\dot{e}\big{)}=c_{2}(p/q)\dot{e}^{(p/q)-1}\),  
\[\mathbf{C}=\sum_{k=1}^{7}\mathbf{V}_{k}\cdot m_{k}\cdot\mathbf{y}-\sum_{k=1}^{7}\mathbf{W}_{k} \cdot\left[\mathbf{\omega}_{k}^{0}\times\left(\mathbf{I}_{k}\mathbf{\omega}_{k}^{0}\right) \right]-\sum_{k=1}^{7}\mathbf{W}_{k}\cdot\mathbf{I}_{k}\cdot\dot{\mathbf{W}}_{k}\cdot y\]  
and \(\Delta\) denote the dynamic uncertainties and external disturbances.