# Coordinated Control after Grasping the Space Targets Using Controllable Damping Mechanism
## 3 Design of the Coordinated Controller
For the convenience of writing, we define that  
\[ \begin{array}{l}\dot{\mathbf{s}}=\dot{\mathbf{\sigma}}_{1},\\ \ddot{\mathbf{s}}=\ddot{\mathbf{\sigma}}_{1}=\ddot{\mathbf{\sigma}}_{2},\\ \ddot{\mathbf{s}}=\ddot{\mathbf{\sigma}}_{2}=\ddot{\mathbf{\sigma}}_{3}.\end{array}  \tag{26}\]  
Then, a backstepping algorithm is used to find a control input which is effective for the stable of the space robot after capturing. The following change of coordinate is launched:  
\[ \begin{array}{l}\mathbf{\mu}_{1}=\mathbf{\sigma}_{1},\\ \mathbf{\mu}_{2}=\mathbf{\sigma}_{2}-\mathbf{\alpha}_{1},\\ \mathbf{\mu}_{3}=\mathbf{\sigma}_{3}-\mathbf{\alpha}_{2}.\end{array}  \tag{27}\]  
where \(\alpha_{i}(i=1,2)\) are the virtual control laws in the first and second steps, respectively. Step 1: Differentiating \(\mathbf{\mu}_{1}\) in (27) with respect to time, we can obtain  
\[\dot{\mathbf{\mu}}_{1}=\dot{\mathbf{\sigma}}_{1}=\mathbf{\mu}_{2}+\mathbf{\alpha}_{1}. \tag{28}\]  
Therefore, a positive Lyapunov function is selected as  
\[\mathbf{V}_{1}=\frac{1}{2}\mathbf{\mu}_{1}\mathbf{\tau}\mathbf{\mu}_{1} \tag{29}\]  
Differentiating \(\mathbf{V}_{1}\) in (29) with respect to time yields  
\[\dot{\mathbf{V}}_{1}=\mathbf{\mu}_{1}(\mathbf{\mu}_{2}+\mathbf{\alpha}_{1}) \tag{30}\]  
where \(\alpha_{1}\) can be chosen as \(\mathbf{\alpha}_{1}=-\mathbf{\xi}_{1}\mathbf{\mu}_{1}\). \(\mathbf{\xi}_{1}\) is a positive design parameter. Substituting (30) into (29) we can obtain  
\[\dot{\mathbf{V}}_{1}=-\mathbf{\xi}_{1}|\mathbf{\mu}_{1}|^{2}+\mathbf{\mu}_{1}\mathbf{\mu}_{2} \tag{31}\]  
From (31), we can see that \(\dot{\mathbf{V}}_{1}=-\xi_{1}|\mathbf{\mu}_{1}|^{2}\leq 0\) once \(\mathbf{\mu}_{2}=\mathbf{O}_{n\times n}\). Step 2: A virtual control input \(\mathbf{\alpha}_{2}\) is generated in this step which drives the variable \(\mathbf{\mu}_{2}\to 0\). Differentiating \(\mathbf{\mu}_{2}\) in (27) with respect to time, we have  
\[\mathbf{\mu}_{2}=\dot{\mathbf{\sigma}}_{2}-\dot{\mathbf{\alpha}}_{1}=\mathbf{\mu}_{3}+\mathbf{ \alpha}_{2}+\xi_{1}\mathbf{\sigma}_{2} \tag{32}\]  
A Lyapunov function is choose as:  
\[\mathbf{V}_{2}=\mathbf{V}_{1}+\frac{1}{2}\mathbf{\mu}_{2}^{2} \tag{33}\]  
Differentiating \(\mathbf{V}_{2}\) in (33) with respect to time, we have  
\[\dot{\mathbf{V}}_{2}=\dot{\mathbf{V}}_{1}+\mathbf{\mu}_{2}\dot{\mathbf{\mu}}_{2}=-\xi_{1}|\mathbf{ \mu}_{1}|^{2}+\mathbf{\mu}_{1}\mathbf{\mu}_{2}+\mathbf{\mu}_{2}(\mathbf{\mu}_{3}+\mathbf{\alpha}_{ 2}+\xi_{1}\mathbf{\sigma}_{2}) \tag{34}\]  
Based on (34), the \(\mathbf{\alpha}_{2}\) is chosen as \(\mathbf{\alpha  
Step 3: This step we will contain the input torque \(\mathbf{\tau}\). The following Lyapunov function is defined  
\[\mathbf{V}_{3}=\mathbf{V}_{2}+\frac{1}{2}\mathbf{\mu}_{3}^{2} \tag{36}\]  
Differentiating \(\mathbf{V}_{3}\) with respect to time, considering the results in (25) and (27), we have  
\[\dot{\mathbf{V}}_{3}=\dot{V}_{2}+\mathbf{\mu}_{3}\dot{\mathbf{\mu}}_{3}=-\xi_{ 1}|\mathbf{\mu}_{1}|^{2}-\xi_{2}|\mathbf{\mu}_{2}|^{2}+\mathbf{\mu}_{2}\mathbf{\mu}_{3}+\mathbf{\mu} _{3}\times\left(\frac{\mathrm{d}}{\mathrm{d}t}\left(\mathbf{\mathbf{\tau}}\left(\mathbf{e},\dot{\mathbf{e}}\right)+\Xi\left(\dot{\mathbf{e}}\right)\cdot\right.\right. \tag{37}\]  
Therefore, a backstepping NFITSMC (BNFITSMC) is designed as  
\[\mathbf{\tau}=\Xi\left(\dot{\mathbf{e}}\right)^{-1}\mathbf{J}(\mathbf{x}_{1})\left(\mathbf{\tau}_{ n}-\Xi\left(\dot{\mathbf{e}}\right)\mathbf{\tau}_{s}\right) \tag{38}\]  
where \(\mathbf{\tau}_{n}=-\Xi\left(\dot{\mathbf{e}}\right)\left(\mathbf{C}-\ddot{\mathbf{x}}_{d} \right)-\mathbf{\mathbf{\tau}}\left(\mathbf{e},\dot{\mathbf{e}}\right)+\mathbf{\alpha}_{2}-\int( \xi_{3}\mu_{3}+\mu_{2}),\dot{\mathbf{r}}_{s}=(\mathbf{\Lambda}+\varsigma)\text{sign}( \mathbf{\mu}_{3})\), \(\mathbf{\Lambda}\) is select as \(\mathbf{\Lambda}\geq|\gamma|\), \(\gamma=(\mathrm{d}/\mathrm{d}t)\left(\Xi\left(\dot{\mathbf{e}}\right)\Delta\right)\) is a positive small constant.