# Coordinated Control after Grasping the Space Targets Using Controllable Damping Mechanism
## 2 Modeling of Spacecraft-Manipulator System with FDCDM
### Equivalent Kinematics Equation  
The six-dimensional damping force outputs by the FDCDM can be equivalent to the actuator outputs in the end joint, as shown in Figure 3. The relative linear velocities and angular velocities between adjacent rigid bodies \(k\) and \(k\)-1 are set as \(\mathbf{v}_{k}^{k-1}\) and \(\mathbf{\omega}_{k}^{k-1}\)  
Figure 1: 3D model of the full-dimensional controllable damping mechanism with gyro-structure [20]. Figure 2: The spacecraft-manipulator system with FDCDM (SMS-FDCDM). (\(k=1,\,2,\,3,\,\ldots,\,7\)), respectively. Furthermore, the relative line displacement and angular displacement are set as \(\mathbf{s}_{k}^{k-1}\) and \(\mathbf{q}_{k}^{k-1}\), respectively. We have:  
\[ \begin{array}{l}\mathbf{q}_{1}^{0}=\left[\begin{array}{cccc}\beta_{1}&\beta_{2}& \beta_{3}\end{array}\right]^{\mathrm{T}}&\mathbf{\omega}_{1}^{0}=\left[\begin{array} []{cccc}\alpha_{1}&\alpha_{2}&\alpha_{3}\end{array}\right]^{\mathrm{T}}\\ \mathbf{q}_{2}^{1}=\left[\begin{array}{cccc}0&0&\beta_{4}\end{array}\right]^{ \mathrm{T}}&\mathbf{\omega}_{1}^{2}=\left[\begin{array}{cccc}0&0&\alpha_{4} \end{array}\right]^{\mathrm{T}}\\ \mathbf{q}_{3}^{2}=\left[\begin{array}{cccc}\beta_{5}&0&0\end{array}\right]^{ \mathrm{T}}&\mathbf{\omega}_{3}^{2}=\left[\begin{array}{cccc}\alpha_{5}&0&0 \end{array}\right]^{\mathrm{T}}\\ \mathbf{q}_{4}^{3}=\left[\begin{array}{cccc}\beta_{6}&0&0\end{array}\right]^{ \mathrm{T}}&\mathbf{\omega}_{3}^{4}=\left[\begin{array}{cccc}\alpha_{6}&0&0 \end{array}\right]^{\mathrm{T}}\\ \mathbf{q}_{5}^{4}=\left[\begin{array}{cccc}0&\beta_{7}&0\end{array}\right]^{ \mathrm{T}}&\mathbf{\omega}_{5}^{4}=\left[\begin{array}{cccc}0&\alpha_{7}&0 \end{array}\right]^{\mathrm{T}}\\ \mathbf{q}_{6}^{5}=\left[\begin{array}{cccc}0&0&\beta_{8}\end{array}\right]^{ \mathrm{T}}&\mathbf{\omega}_{6}^{5}=\left[\begin{array}{cccc}0&0&\alpha_{8} \end{array}\right]^{\mathrm{T}}\\ \mathbf{s}_{1}^{0}=\left[\begin{array}{cccc}\beta_{9}&\beta_{10}&\beta_{11}\end{array} \right]^{\mathrm{T}}&\mathbf{v}_{1}^{0}=\left[\begin{array}{cccc}\alpha_{9}& \alpha_{10}&\alpha_{11}\end{array}\right]^{\mathrm{T}}\\ \mathbf{s}_{6}^{6}=\left[\begin{array}{cccc}0&0&\beta_{12}\end{array}\right]^{ \mathrm{T}}&\mathbf{v}_{6}^{6}=\left[\begin{array}{cccc}0&0&\alpha_{12}\end{array} \right]^{\mathrm{T}}\\ \mathbf{s}_{1}^{1}=\mathbf{s}_{2}^{2}=\mathbf{s}_{3}^{4}=\mathbf{s}_{4}^{4}=\mathbf{s}_{5}^{5}=\mathbf{s }_{6}^{5}=\left[\begin{array}{cccc}0&0&0\end{array}\right]^{\mathrm{T}}\\ \mathbf{v}_{2}^{1}=\mathbf{v}_{3}^{2}=\mathbf{v}_{4}^{3}=\mathbf{v}_{5}^{4}=\mathbf{v}_{6}^{5}=\left[ \begin{array}{cccc}0&0&0\end{array}\right]^{\mathrm{T}}\end{array}  \tag{1}\]  
where \(\mathbf{\alpha}\in\Re^{12\times 1}\) denote the generalized velocity and \(\mathbf{\beta}\in\Re^{12\times 1}\) denote the generalized position. Then, we can obtain the relative rotation matrix of the proprio-coordinate system in rigid bodies \(k\) and \((k-1)\) set as \(\mathbf{A}_{k}^{k-1}\in\Re^{3\times 3}\)  
\[ \begin{array}{l}\mathbf{A}_{0}^{1}=\mathrm{Rot}(X,\beta_{1})\cdot\mathrm{Rot}(Y, \beta_{2})\cdot\mathrm{Rot}(Z,\beta_{3})\\ \mathbf{A}_{2}^{1}=\mathrm{Rot}(Z,\beta_{4})\;\;,\;\;\mathbf{A}_{2}^{2}=\mathrm{Rot}(X, \beta_{5})\\ \mathbf{A}_{3}^{3}=\mathrm{Rot}(X,\beta_{6}),\;\;\;\;\mathbf{A}_{5}^{4}=\mathrm{Rot}(Y, \beta_{7})\\ \mathbf{A}_{6}^{5}=\mathrm{Rot}(Z,\beta_{8}),\;\;\;\;\;\mathbf{A}_{7}^{5}=\mathbf{I}\end{array}  \tag{2}\]  
\(\Sigma_{k}\) is the ontology coordinate system in rigid bodies \(k\), and \(\Sigma_{0}\) denote the inertial coordinate system. Then, we have the absolute rotation matrix of \(\Sigma_{k}\) and \(\Sigma_{0}\) as  
\[\mathbf{A}_{k}^{0}=\mathbf{A}_{1}^{0}\cdot\mathbf{A}_{2}^{1}\cdot\cdot\cdot\mathbf{A}_{k}^{k-1} \tag{3}\]  
In this paper, the base proprio-coordinate system \(\Sigma_{1}\) is XYZ Euler angles \(\gamma\) relative to the inertial coordinate system \(\Sigma_{0}\), denote \(\gamma=\left[ \begin{array}{ccc}\beta_{1}&\beta_{2}&\beta_{3}\end{array}  \right]^{T}\).