## 6 Trajectory-Planning-Based Deployment Dynamics
### Driving Torque  
The effectiveness in kinematics is based on the assumption that the antenna can be successfully deployed by torsional springs and driving motors. This requires that the effectiveness in kinetics shall be confirmed and validated; in other words, the driving force or the torque must be proven as reasonable. At the first deployment stage, for each petal, we have  
\[T_{\rm i}^{(1)}(t)=k(\theta_{0}^{*}-\theta_{1})=k\cos\gamma(\theta_{0}-\theta _{2})\quad 0\leq t\leq t_{\rm f_{1}}. \tag{30}\]  
where \(T_{\rm i}^{(1)}\) denotes the torque of each torsional spring at the first deployment stage. On the other hand, at the second stage, following the kinetic energy theorem for each petal, we arrive at  
\[\int_{t_{\rm f_{1}}}^{t}T_{\rm i}^{(2)}\dot{\theta}_{1}\mathrm{d}t=\frac{1}{2 }I_{\rm p}\dot{\theta}_{2}^{2}-\frac{1}{2}I_{\rm p}v_{0}^{2}=\frac{1}{2}I_{\rm p }(\dot{\theta}_{2}^{2}-v_{0}^{2})\quad t_{\rm f_{1}}\leq t\leq t_{\rm f_{2}}, \tag{31}\]  
if only considering the MOI of the petal and neglecting all the frictions. Thereinto, \(T_{\rm i}^{(2)}\) denotes the torque exerted on the input rod at the second deployment stage. By differentiating both sides of Equation (31), we have  
\[T_{\rm i}^{(2)}\dot{\theta}_{1}=I_{\rm p}\dot{\theta}_{2}\ddot{\theta}_{2}. \tag{32}\]  
Hence, the torque of the input rod is  
\[T_{\rm i}^{(2)}(t)=\frac{I_{\rm p}\dot{\theta}_{2}\ddot{\theta}_{2}}{\dot{ \theta}_{1}}=\frac{I_{\rm p}\dot{\theta}_{2}\ddot{\theta}_{2}}{\dot{\theta}_{2} \cos\gamma}=\frac{I_{\rm p}\ddot{\theta}_{2}}{\cos\gamma}\quad t_{\rm f_{1}} \leq t\leq t_{\rm f_{2}}, \tag{33}\]  
and the total estimated driving torque throughout the deployment process can be expressed as  
\[T_{\rm i}(t)=\begin{cases}k\cos\gamma(\theta_{0}-\theta_{2})\quad 0\leq t\leq t _{\rm f_{1}},\\ \frac{I_{\rm p}\ddot{\theta}_{2}}{\cos\gamma}\quad t_{\rm f_{1}}<t\leq t_{\rm f _{2}}.\end{cases} \tag{34}\]  
Consequently, the driving torque of the driving motor is  
\[T_{\rm m}(t)=\frac{I_{\rm p}\ddot{\theta}_{2}}{i_{\rm r}\cos\gamma}\quad t_{ \rm f_{1}}\leq t\leq t_{\rm f_{2}} \tag{35}\]  
By differentiating Equation (28) and substituting the result into Equations (34) and (35), the explicit expressions of \(T_{\rm i}\) and \(T_{\rm m}\) can be obtained. However, these explicit expressions are not presented here due to their complexity. Ultimately, the theoretically estimated result and _ADAMS_-simulated one are presented in Figures 17 and 18.  
It can be observed from Figures 17 and 18 that the torques of the input rod and of the driving motor obtained by theoretical estimation and _ADAMS_, respectively, are in good consistency with each other, thereby verifying the correctness of our trajectory planning. In addition, the varying tendency of the corresponding curves is similar to that in Figure 15c. This phenomenon is in agreement with our expectation since the torque is proportional to the angular acceleration, to which the petal contributes most of the MOI. According to Figure 18, the maximum torque is approximately \(3\times 10^{-3}\) Nmm and the minimum is about \(-4\times 10^{-3}\) Nmm, both of which staying within a reasonable range of magnitude for deploying a 0.2 kg petal with an MOI of 0.22 kg m\({}^{2}\). Based on the resultsin Figures 15-18, we argue that the antenna can be successfully deployed and exhibits excellent dynamical quality.  
Based on the research in this paper, our future work will focus on prototype fabricating, experiments of deployment dynamics and deployment control, etc., to verify the effectiveness of the proposed RRSA further.