# Coordinated Control after Grasping the Space Targets Using Controllable Damping Mechanism
## 2 Modeling of Spacecraft-Manipulator System with FDCDM
### Dynamic Equations  
Ignoring microgravity in space, the six-dimensional damping force outputs by the FDCDM on both sides of the \(k\)-th rigid body can be obtained as  
\[\left\{ \begin{array}{ll}\mathbf{M}_{k}=\mathbf{A}_{k}^{0}\cdot\bigl{(}\mathbf{K}_{k} \cdot\mathbf{q}_{k+1}^{k}+\mathbf{B}_{k}\cdot\mathbf{\omega}_{k+1}^{k}\bigr{)},\quad\quad k =1,2,\cdots,5\\ \mathbf{F}_{6}=\mathbf{A}_{6}^{0}\cdot\bigl{(}\mathbf{K}_{6}\cdot\mathbf{s}_{7}^{6}+\mathbf{B}_{6} \cdot\mathbf{v}_{7}^{6}\bigr{)}\end{array} \right. \tag{14}\]  
where \(\mathbf{M}_{k}\) is the flexible torque acting on the right of rigid body \(k\); \(\mathbf{K}_{k}\) is the stiffness parameter of the torsion spring on the right of rigid body \(k\); \(\mathbf{B}_{k}\) is the damping parameter of the torsion spring on the right of rigid body \(k\); \(\mathbf{F}_{6}\) is the lifreible force acting on the right of rigid body \(6\); rigid body \(6\); \(\mathbf{K}_{6}\) is the stiffness parameter of the torsion spring on the right of rigid body \(6\); and \(\mathbf{B}_{6}\) is the damping parameter of the torsion spring on the right of rigid body \(6\). The equivalent active force \(\mathbf{F}_{kc}\) and the active torque of mass center of each rigid body \(\mathbf{M}_{kc}\) are:  
\[\left\{ \begin{array}{ll}\mathbf{F}_{1c}=\mathbf{O}_{3\times 1}\\ \mathbf{M}_{1c}=\mathbf{M}_{b}+\mathbf{M}_{1},\quad\quad\left\{\begin{array}{ll}\mathbf{F}_ {kc}=\mathbf{O}_{3\times 1}\\ \mathbf{M}_{1c}=\mathbf{M}_{k}-\mathbf{M}_{k-1}\end{array}\right.\quad,\quad k=2,3,4,5; \\ \mathbf{F}_{6c}=\mathbf{F}_{6}\\ \mathbf{M}_{6c}=-\mathbf{M}_{5}+\mathbf{A}_{6}^{0}(\mathbf{d}_{6}-\mathbf{c}_{6})\times\mathbf{F}_{6},\\ \mathbf{F}_{7c}=\mathbf{F}_{e}-\mathbf{F}_{6}\\ \mathbf{M}_{7c}=\mathbf{M}_{e}+\mathbf{A}_{7}^{0}((\mathbf{d}_{7}-\mathbf{c}_{7})\times\mathbf{F}_{e}- \mathbf{c}_{7}\times\mathbf{F}_{6})\end{array} \right. \tag{15}\]  
where \(\mathbf{M}_{b}\) represents the torque of base flywheel; \(\mathbf{c}_{i}\) is the centroid position vector of the rigid body \(i\) in the coordinate system \(\Sigma_{i}\); and \(\mathbf{F}_{e}\) and \(\mathbf{M}_{e}\) are the impact force and torque acting on the end joint, respectively. The equivalent inertia force \(\mathbf{F}_{kc}^{*}\) and the equivalent inertia torque \(\mathbf{M}_{kc}^{*}\) of rigid body \(k\) can be expressed as  
\[\left\{ \begin{array}{ll}\mathbf{F}_{kc}^{*}=-\mathbf{m}_{k}\mathbf{a}_{kc}^{0}\\ \mathbf{M}_{kc}^{*}=-\mathbf{I}_{k}\mathbf{\dot{\omega}}_{k}^{0}-\mathbf{\omega}_{k}^{0}\times \bigl{(}\mathbf{I}_{k}\mathbf{\omega}_{k}^{0}\bigr{)}\end{array} \right. \tag{16}\]  
where \(\mathbf{m}_{k}\) denotes the mass of rigid body \(k\), and \(\mathbf{I}_{k}\) is the inertia tensor of rigid body \(k\). \(\mathbf{\dot{\omega}}_{k}^{0}\) denotes the angular acceleration of rigid body \(k\), and \(\mathbf{a}_{kc}^{0}\) is the centroid acceleration of rigid body \(k\). In this paper, the spacecraft-manipulator system with FDCDM (SMS-FDCDM) can be equivalent to a tandem mechanism with 7 rigid body segments. Therefore, the generalized active force and generalized inertial force are set as \(\mathbf{F}\in\Re^{12\times 1}\) and \(\mathbf{F}^{*}\in\Re^{12\times 1}\), respectively. Then, the Kane dynamic equations can be described by  
\[\mathbf{F}_{l}+\mathbf{F}_{l}^{*}=\sum\limits_{k=1}^{7}\left(\mathbf{F}_{kc}\cdot\mathbf{V}_{ kl}+\mathbf{M}_{kc}\cdot\mathbf{W}_{kl}\right)+\sum\limits_{k=1}^{7}\left(\mathbf{F}_{kc}^{*} \cdot\mathbf{V}_{kl}+\mathbf{M}_{kc}^{*}\cdot\mathbf{W}_{kl}\right)=0 \tag{17}\]  
where \(k=1,2,\cdots,7;l=1,2,\cdots,12\). Substituting (9), (13), (15), and (16) into (17), we can obtain the nonlinear ordinary differential equations of SMS-FDCDM system  
\[\left\{ \begin{aligned} \dot{\mathbf{x}}&=\mathbf{D}\cdot\mathbf{y}\\ \dot{\mathbf{y}}&=\mathbf{f}^{-1}\cdot\mathbf{f}\end{aligned} \right. \tag{18}\]  
where  
\[\mathbf{D}= \begin{bmatrix}\mathbf{H}_{3\times 3}&\mathbf{O}_{3\times 9}\\ \mathbf{O}_{9\times 3}&\mathbf{I}_{9\times 9}\end{bmatrix} \in\Re^{12\times 12}, \mathbf{H}_{3\times 3}=\tfrac{1}{c\beta_{2}}