## 4 Coupled Dynamic and Control for Thermally Induced Vibration and Attitude Motion
### Simulations  
We consider the mass of the central rigid body of the satellite M\({}_{b}\) to be equal to 100 kg. Based on the parameters in Table 4, the mass of the satellite antenna M\({}_{a}\) can be obtained as 12.3595 kg. Momentum wheels installed along body axes serve as the attitude actuator whose maximum torque output is 0.03 Nm. The remaining parameters used in Equation (24) can also be obtained by the parameters in Table 4.  
Based on Equations (22) and (23) and considering the initial attitude angle vector \(\theta\) and initial angular velocity \(\omega\) are both \(\omega\), the angular responses of the rigid-flexible coupled system and the single rigid body without control are given in Figure 14. It can be seen from Figure 14 that without the vibration, the attitude angles maintain zeros, and the thermal induced vibrations have great influence on the attitude angles which cause the attitude motion to diverge rapidly. In return, the variations of attitude angles also cause the model coordinates \(\eta_{a}\) to diverge which finally results in the large displacements for the node coordinates. A simple example of the displacements for node 25 is shown in Figure 15. The red line, black line, and blue dash-dotted line denote the uncoupled responses which are adopted from Figure 12 and shown as a comparison. The red, black, and blue dash lines denote the coupled responses based on Equations (22) and (23). Therefore, it is necessary to consider a control law to ensure a stable operation in space.  
Based on the control law for external moment \(\mathbf{T}_{c}\) in Equation (36) and assuming the command attitude angle \(\theta_{c}\) = 0, we can choose the control parameters to guarantee the convergence of the system. To verify the asymptotic stability of the controller, the optimization of the parameters was not included in this paper and we simply choose a feasible solution of the control parameter. The outer loop control coefficients \(\rho_{1}\) = 3, \(\mathbf{K}_{1}\) = diag(10,10,10), inner loop control coefficients \(\rho_{2}\) = 3, \(\mathbf{K}_{2}\) = diag[100,100,100], \(\mathbf{k}\) = diag[300,300,300], and the controlled results for attitude angles, angular velocities, and control moments are shown in Figures 16-18, respectively. It is found that the attitude angular velocity commands of the three axes gave out immediately along with the control commands and the bias of the attitude angle caused by the thermal-induced load can be restrained in short time under the action of the control torques. The stabilization time is within 1 s, and the control torque is within 0.03 Nm, which can fit the actual constraint requirements of the onboard actuators. During the process, the attitude angles slowly coincide with the command attitude  
Figure 14: Angular responses of the rigid-flexible coupled structure model without control.  
Figure 15: Comparison of vibration response of antenna’s node 25.  
angle \(\theta_{\rm c}\). The angular biases of three axes are within 1.313 \(\times\) 10\({}^{-3}\) deg, 6.569 \(\times\) 10\({}^{-4}\) deg, and 4.055 \(\times\) 10\({}^{-4}\) deg, respectively, at 1 s. The variations of the angular biases are listed in Table 5. It is found that after 1 s, the biases for three axes drop dramatically to the order of 10\({}^{-4}\), and the biases will keep dropping to the order of 10\({}^{-5}\) at 7 s. The angular velocity biases of the three axes are with 3.840 \(\times\) 10\({}^{-3}\) deg/s, 3.959 \(\times\) 10\({}^{-3}\) deg/s, and 5.570 \(\times\) 10\({}^{-3}\) deg/s from 0.5 s. Similarly, the variations of the angular velocity biases are listed in Table 6.  
Figure 16: The track diagram of roll angle, yaw angle, and pitch angle.  
Figure 17: Angular velocity commands of roll, yaw, and pitch angle.  
The simulation results show the effectiveness of the proposed controller which has the advantages of fast-tracking response, short adjustment time, and good convergence. The controller can accurately track the attitude angles and restrain the thermal vibrations of the spacecraft.