## 2 Rigid-Flexible Model for Spacecraft with Hoop-Truss Antenna
The velocity vector of the mass element \(m_{j}\) on the antenna can be expressed in \(O\)-\(XYZ\) as:  
\[\dot{\mathbf{R}}_{2}=\dot{\mathbf{R}}_{1}+\big{(}\dot{\mathbf{r}}_{3}+\dot{ \mathbf{r}}_{j}+\dot{\mathbf{u}}_{j}\big{)}^{b}+\mathbf{\omega}\times\big{(} \mathbf{r}_{3}+\mathbf{r}_{j}+\mathbf{u}_{j}\big{)} \tag{3}\]  
where the superscript '\(b\)' denotes derivative with respect to the \(O_{b}\)-\(x_{b}y_{b}z_{b}\). Since \(\big{(}\dot{\mathbf{r}}_{3}\big{)}^{b}=0\) and \(\big{(}\dot{\mathbf{r}}_{j}\big{)}^{a}=0\), we can obtain the velocity in matrix form as:  
\[\ddot{\mathbf{R}}_{2}=\mathbf{e}^{T}\bar{\mathbf{R}}_{1}+\mathbf{a}^{T}\big{(} \dot{\mathbf{u}}_{j}\big{)}^{a}-\mathbf{a}^{T}\big{(}\mathbf{r}_{j}^{X}+ \mathbf{u}_{j}^{X}\big{)}\mathbf{\omega}_{a}-\mathbf{b}^{T}\big{(}\mathbf{r}_{3}^ {X}+\mathbf{A}_{ab}^{T}\Big{(}\mathbf{r}_{j}^{X}+\mathbf{u}_{j}^{X}\big{)} \mathbf{A}_{ab}\big{)}\mathbf{\omega} \tag{4}\]  
where the subscript '\(a\)' denotes derivative with respect to the \(O_{a}\)-\(x_{a}y_{a}z_{a}\). \(\mathbf{A}_{ab}\), \(\mathbf{A}_{be}\), and \(\mathbf{A}_{ae}\) denote the transformation matrix from the \(O_{b}\)-\(x_{b}y_{b}z_{b}\) to \(O_{a}\)-\(x_{a}y_{a}z_{a}\), the \(O\)-\(XYZ\) to the \(O\)-\(XYZ\) to \(O_{b}\)-\(x_{b}y_{b}z_{b}\), and the \(O\)-\(XYZ\) to \(O_{a}\)-\(x_{a}y_{a}z_{a}\), respectively. The (\(\mathbf{X}\))\({}^{X}\) denotes \(\mathbf{X}\)'s anti-symmetric diagonal matrix. Then, the acceleration of \(m_{j}\) can be obtained as:  
\[\ddot{\mathbf{R}}_{2} =\mathbf{a}^{T}\mathbf{A}_{ae}\bar{\mathbf{R}}_{1}+\mathbf{a}^{T} \big{(}\ddot{\mathbf{u}}_{j}\big{)}^{a}-\mathbf{a}^{T}\big{(}\mathbf{r}_{j}^{ X}+\mathbf{u}_{j}^{X}\big{)}\dot{\mathbf{u}}_{a}-\mathbf{a}^{T}\big{(} \mathbf{A}_{ab}\mathbf{r}_{3}^{X}\mathbf{A}_{ba}+\mathbf{r}_{j}^{X}+\mathbf{u} ^{X}\big{)}\mathbf{A}_{ab}\dot{\mathbf{u}}\] \[-\mathbf{a}^{T}\dot{\mathbf{u}}_{j}^{X}\big{(}\mathbf{A}_{ab} \mathbf{\omega}+\mathbf{\omega}_{a}\big{)}-\mathbf{a}^{T}\big{[}\big{(}\mathbf{2}\mathbf{ A}_{ab}\mathbf{\omega}\big{)}^{X}+\mathbf{\omega}_{a}\big{]}\big{(}\mathbf{r}^{X}+ \mathbf{u}_{j}^{X}\big{)}\mathbf{\omega}_{a}\] \[-\mathbf{a}^{T}\big{(}\mathbf{A}_{ab}\mathbf{\omega}\big{)}^{X}\big{(} \mathbf{A}_{ab}\mathbf{r}_{3}^{X}\mathbf{A}_{ba}+\mathbf{r}_{j}^{X}+\mathbf{u} _{j}^{X}\big{)}\mathbf{A}_{ab}\mathbf{\omega}\]  
Neglecting the higher orders of \(\mathbf{\omega}\), \(\mathbf{\omega}_{a}\), \(\mathbf{u}_{j}\), we obtained,  
\[\ddot{\mathbf{R}}_{2} =\mathbf{a}^{T}\mathbf{A}_{ae}\bar{\mathbf{R}}_{1}+\mathbf{a}^{T} \big{(}\ddot{\mathbf{u}}_{j}\big{)}^{a}-\mathbf{a}^{T}\mathbf{r}_{j}^{X}\dot{ \mathbf{u}}_{a}-\mathbf{a}^{T}\big{(}\mathbf{A}_{ab}\mathbf{r}_{3}^{X}\mathbf{A }_{ba}+\mathbf{r}_{j}^{X}\big{)}\mathbf{A}_{ab}\dot{\mathbf{u}} \tag{6}\] Figure 1:0  
Figure 1:1 Coordinate systems. Similarly, for the mass element \(m_{k}\) of the rigid body, the velocity and acceleration are expressed as follows:  
Figure 1:2  
Figure 1:3  
In order to study the coupling dynamics of attitude and vibration, we express the elastic displacement \(\mathbf{u}_{j}\) in the form of independent modal coordinates as:  
Figure 1:4  
where \(\mathbf{Q}=[\mathbf{Q}_{1},\,\dots,\,\mathbf{Q}_{n}]\), \(\eta=[\mathbf{\eta}_{1},\,\dots,\,\mathbf{\eta}_{n}]^{T}\), \(\mathbf{\eta}_{i}\) is the \(i\)th modal coordinate, and \(\mathbf{Q}_{i}\) is the modal vector representing the \(i\)th modal shape of the structure. Since the structure of the hoop-truss antenna is relatively complex, it is difficult to find the appropriate assumed modal shapes. Both \(\mathbf{\eta}_{i}\) and \(\mathbf{Q}_{i}\) will be obtained by analytical finite element method in the next section. Substituting Equation (6), Equation (8) into Equation (1) yields the rotational equations of motion:  
Figure 1:5  
Substituting Equation (6), Equation (8) into Equation (2) yields an equation of motion for translational motion:  
\[