# Coordinated Control after Grasping the Space Targets Using Controllable Damping Mechanism
## 2 Modeling of Spacecraft-Manipulator System with FDCDM
### Partial Velocity Equation  
The angular velocity of bodies \(k\) in inertial coordinate system \(\Sigma_{0}\) is written as  
\[\mathbf{\omega}_{k}^{0}=\mathbf{\omega}_{1}^{0}+\sum_{i=1}^{k-1}\mathbf{A}_{i}^{0}\mathbf{ \omega}_{i+1}^{i} \tag{7}\]  
The partial angular velocity is defined as  
\[\mathbf{\omega}_{kl}=\frac{\partial\mathbf{\omega}_{k}^{0}}{\partial\mathbf{y}_{l}}\quad l =1,2,3,\cdots,12 \tag{8}\]  
Substituting the (7) into (8), we can obtain the partial angular velocity \(\mathbf{W}\in\Re^{21\times 12}\) as  
\[\mathbf{W}=\begin{bmatrix}\mathbf{I}_{3\times 3}&\mathbf{O}_{3\times 1}&\mathbf{O}_{3\times 1}&\mathbf{O }_{3\times 1}&\mathbf{O}_{3\times 1}&\mathbf{O}_{3\times 4}\\ \mathbf{I}_{3\times 3}&\mathbf{A}_{13}^{0}&\mathbf{O}_{3\times 1}&\mathbf{O}_{3\times 1}&\mathbf{O}_{3 \times 1}&\mathbf{O}_{3\times 4}\\ \mathbf{I}_{3\times 3}&\mathbf{A}_{13}^{0}&\mathbf{A}_{21}^{0}&\mathbf{O}_{3\times 1}&\mathbf{O}_{3 \times 1}&\mathbf{O}_{3\times 1}&\mathbf{O}_{3\times 4}\\ \mathbf{I}_{3\times 3}&\mathbf{A}_{13}^{0}&\mathbf{A}_{21}^{0}&\mathbf{A}_{31}^{0}&\mathbf{O}_{3 \times 1}&\mathbf{O}_{3\times 4}\\ \mathbf{I}_{3\times 3}&\mathbf{A}_{13}^{0}&\mathbf{A}_{21}^{0}&\mathbf{A}_{31}^{0}&\mathbf{A}_{2}^{0}& \mathbf{O}_{3\times 1}&\mathbf{O}_{3\times 4}\\ \mathbf{I}_{3\times 3}&\mathbf{A}_{13}^{0}&\mathbf{A}_{21}^{0}&\mathbf{A}_{31}^{0}&\mathbf{A}_{2}^{0}& \mathbf{A}_{3\times 4}\\ \mathbf{I}_{3\times 3}&\mathbf{A}_{13}^{0}&\mathbf{A}_{21}^{0}&\mathbf{A}_{31}^{0}&\mathbf{A}_{42}^{0}& \mathbf{A}_{53}^{0}&\mathbf{O}_{3\times 4}\\ \mathbf{I}_{3\times 3}&\mathbf{A}_{13}^{0}&\mathbf{A}_{21}^{0}&\mathbf{A}_{31}^{0}&\mathbf{A}_{42}^{0}& \mathbf{A}_{53}^{0}&\mathbf{O}_{3\times 4}\\ \end{bmatrix} \tag{9}\]  
where \(\mathbf{A}_{ij}^{0}\in\Re^{3\times 1}\) is the \(j\)th column of \(\mathbf{A}_{i}^{o}\), \(\mathbf{I}_{3\times 3}\) represents the identity matrix of \(3\times 3\), \(\mathbf{O}_{3\times 1}\) denotes the zero vector of \(3\times 1\).  
The position vector of the mass center of the rigid body \(k\) in the inertial frame \(\Sigma_{0}\) is  
\[\mathbf{p}_{k}^{0}=\mathbf{s}_{1}^{0}+\sum_{i=1}^{k-1}\mathbf{A}_{i}^{0}\Big{(}\mathbf{d}_{i} +\mathbf{s}_{i+1}^{i}\Big{)}+\mathbf{A}_{k}^{0}\mathbf{r}_{k} \tag{10}\]  
where \(\mathbf{d}_{i}\) denotes the position vector of the rigid bodies \(k\) in the coordinate system \(\Sigma_{i}\), \(\mathbf{r}_{k}\) represents the centroid position vector of the rigid bodies \(k\) in the coordinate system \(\Sigma_{k}\).  
Differentiating \(\mathbf{p}_{k}^{0}\) in (10) with respect to time, we can obtain the velocity of the mass center of the rigid bodies \(k\) in inertial system as  
\[\mathbf{v}_{k}^{0}=\mathbf{s}_{1}^{0}+\sum_{i=1}^{k-1}\left[\hat{\mathbf{A}}_{i}^{0}\Big{(} \mathbf{d}_{i}+\mathbf{s}_{i+1}^{i}\Big{)}+\mathbf{A}_{i}^{0}\mathbf{s}_{i+1}^{i}\right]+\hat{ \mathbf{A}}_{k}^{0}\mathbf{r}_{k} \tag{11}\]  
The partial linear velocity is defined as  
\[\mathbf{v}_{kl}=\frac{\partial\mathbf{v}_{k}^{0}}{\partial\mathbf{y}_{l}}\quad l=1,2,3, \cdots,12 \tag{12}\]Substituting the (11) into (12), we can obtain the partial angular velocity \(\mathbf{V}\in\Re^{21\times 12}\) as  
\[\mathbf{V}_{kl}=\left\{\begin{array}{ll}\sum\limits_{i=1}^{k-1}\bigl{[}\mathbf{\omega} _{il}\times\mathbf{A}_{i}^{0}\bigl{(}\mathbf{d}_{i}+\mathbf{s}_{i+1}^{i}\bigr{)}\bigr{]}+ \mathbf{\omega}_{kl}\times\mathbf{A}_{k}^{0}\mathbf{r}_{k}\quad\quad(l\leq k+2)\\ 0\quad\quad\quad(k+2<l\leq 8)\\ \mathbf{\omega}_{k(l-8)}\quad\quad\quad(8<l\leq 11)\\ 0\quad\quad\quad\bigl{(}11<l\leq 12\ \,\quad k<7\ \bigr{)}\\ \mathbf{A}_{63}^{0}\quad\quad\quad\bigl{(}11<l\leq 12\,\quad k=7\ \bigr{)}\end{array}\right. \tag{13}\]  
where \(\mathbf{V}_{kl}\) is the \(k\)-th row and the \(l\)-th column of matrix \(\mathbf{V}\).