## 2 Rigid-Flexible Model for Spacecraft with Hoop-Truss Antenna  
In this section, a spacecraft similar to NASA's Soil Moisture Active Passive (SMAP) is considered, which is composed of a rigid main body, a flexible antenna, and rigid connecting booms, as shown in Figure 1a. The antenna generally consists of the mesh, front-net, tension-tie, rear-net, and truss, as shown in Figure 1b. In addition, the hoop-truss antenna consists of several elastic beams and the connecting boom serves only as a connection whose mass and shape are ignored. The gravitational effect is also neglected for simplicity. Only the truss is considered for the following modeling and analysis. To describe the rigid-body motion of the spacecraft and the elastic displacement of the flexible hoop-truss, three coordinate systems are defined and shown in Figure 2. The \(O\)-\(XYZ\) is the inertial coordinate system with its origin located at the center of the Earth. The \(\mathbf{e}=[\mathbf{e}_{x}\ \mathbf{e}_{y}\ \mathbf{e}_{z}]^{\mathrm{T}}\) denotes the unit vectors. The \(O_{b}\)-\(x_{b}y_{b}z_{b}\) is defined as a body-fixed coordinate system and denoted by \(\mathbf{b}=[\mathbf{b}_{x}\ \mathbf{b}_{y}\ \mathbf{b}_{z}]^{\mathrm{T}}\) which rotates around \(O\)-\(XYZ\) with angular velocity \(\mathbf{\omega}\). \(O_{b}\) is the center of mass of the rigid body, and three axes are all parallel to the principal axes of inertia. The antenna coordinate system \(O_{a}\)-\(x_{a}y_{a}z_{a}\) denoted by \(\mathbf{a}=[\mathbf{a}_{x}\ \mathbf{a}_{y}\ \mathbf{a}_{z}]^{\mathrm{T}}\) is located at the point \(O_{a}\) which is the junction point of the rigid body and the antenna. Three axes are parallel to the axes of the \(O_{b}\)-\(x_{b}y_{b}z_{b}\) coordinate system. The antenna's angular velocity with respect to \(O_{b}\)-\(x_{b}y_{b}z_{b}\) is \(\mathbf{\omega}_{a}\). In Figure 2, \(\mathbf{R}_{1}\), \(\mathbf{R}_{2}\), and \(\mathbf{R}_{3}\) represent the vector from the Earth's center to the mass center of the rigid body \(O_{b}\), mass element \(m_{k}\) of the  
Figure 1: The SMAP satellite. (**a**) Configuration of the satellite. [[https://www.semanticscholar.org/paper/Design-and-performance-of-Astromesh-reflector-Soil-Mobrem-Kuehn/1489](https://www.semanticscholar.org/paper/Design-and-performance-of-Astromesh-reflector-Soil-Mobrem-Kuehn/1489) cfc0 bb5 cf16b3]([https://www.semanticscholar.org/paper/Design-and-performance-of-Astromesh-reflector-Soil-Mobrem-Kuehn/1489](https://www.semanticscholar.org/paper/Design-and-performance-of-Astromesh-reflector-Soil-Mobrem-Kuehn/1489) cfc0 bb5 cf16b3) 89830a33 cc1f5 fb4 bcc296(accessed on 17 January 2022) (**b**) Configuration of the antenna [20]. rigid body, and mass element \(m_{j}\) of the antenna, respectively. \(\mathbf{r}_{3}\) represents the vector from \(O_{b}\) to \(O_{a}\), \(\mathbf{r}_{j}\) represents the vector from \(O_{a}\) to \(m_{j}\), \(\mathbf{r}_{j}\) represents the vector from \(O_{b}\) to \(m_{k}\), and \(\mathbf{u}_{j}\) is the displacement of the mass element \(m_{j}\). It can be found that \(\mathbf{R}_{2}=\mathbf{R}_{1}+\mathbf{r}_{3}+\mathbf{r}_{j}+\mathbf{u}_{j}\) and \(\mathbf{R}_{3}=\mathbf{R}_{1}+\mathbf{r}_{k}\). Based on the momentum and angular momentum principle, the rotational dynamic equation and the translational dynamic equation can be written as:  
\[\sum_{A}m_{j}\big{(}\mathbf{r}_{3}+\mathbf{r}_{j}+\mathbf{u}_{j}\big{)}\times \bar{\mathbf{R}}_{2}+\sum_{B}\mathbf{r}_{k}\times m_{k}\bar{\mathbf{R}}_{3}= \mathbf{W} \tag{1}\]  
\[\sum_{A}m_{j}\bar{\mathbf{R}}_{2}+\sum_{B}m_{k}\bar{\mathbf{R}}_{3}=\mathbf{F} \tag{2}\]  
where \(\mathbf{W}\) denotes the total external moments and \(\mathbf{F}\) denotes the total external forces. \(A\), \(B\) denote the sum with respect to the antenna and the satellite body, respectively.