## 6 Trajectory-Planning-Based Deployment Dynamics
### Trajectory Planning
In summary, only the first solution is the feasible one, denoted as \(\theta_{2}^{(1)}(t)\). After the first deployment stage is accomplished, the action on the petal of the torsional springs is released, at which moment the driving motor's torque is exerted to the petal immediately, thereby initiating the second deployment stage. In this phase, the planned trajectory of the petal is prescribed as a cubic curve, and the duration is \(t_{\mathrm{d}}=120\,\mathrm{s}\); hence, its expression, denoted as \(\theta_{2}^{(2)}(t)\), can be written as \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & \(k\) **[Nmm/rad]** & \(c\) **[Nmms/rad]** &
\begin{tabular}{c} **Semi-Period** \\ \(\pi t/u_{\mathrm{d}}\)**[s]** \\ \end{tabular} \\ \hline
1 & 0.398 & 8.139 & 97.703 \\
2 & 6.596 & 23.943 & 23.241 \\
3 & 19.921 & 30.166 & 13.161 \\
4 & 40.555 & 34.168 & 9.165 \\
5 & 68.545 & 37.123 & 7.027 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Solution of \(k\) and \(c\). Figure 14: Behavior of solution. The first curve is the expected one. Figure 13: Points of intersection. The rough intersection locations are used as the initial values of [PERSON]’s method. \[\theta_{2}^{(2)}(t)=\theta_{0}+v_{0}t+\left[\frac{3}{t_{\rm d}^{2}}(\theta_{\rm d}- \theta_{0})-\frac{2}{t_{\rm d}}v_{0}\right]t^{2}+\left[-\frac{2}{t_{\rm d}^{3}}( \theta_{\rm d}-\theta_{0})+\frac{1}{t_{\rm d}^{2}}v_{0}\right]t^{3}\quad 0\leq t \leq t_{\rm d}, \tag{27}\]  
according to Ref. [75]. As a result, during the entire deployment process, the expression of the planned trajectory can be written as  
\[\theta_{2}(t)= \begin{cases}\theta_{2}^{(1)}(t)\quad 0\leq t\leq t_{\rm f_{1}},\\ \theta_{2}^{(2)}(t-t_{\rm f_{1}})\quad t_{\rm f_{1}}<t\leq t_{\rm f_{2}},\end{cases}  \tag{28}\]  
to which its motion curves, including displacement, velocity and acceleration, are shown in Figure 15. In the figure, the red line represents the theoretically estimated result from Equation (28), and the green dashed line represents the result obtained by the commercial software _Automatic Dynamic Analysis of Mechanical Systems (ADAMS)_ employing the same \(k\) and \(c\). According to Ref. [73], the correlation between the input angle \(\theta_{1}\) and the output angle \(\theta_{2}\) can be described by  
\[\theta_{1}(t)= \begin{cases}\arctan\Bigl{(}\cos\gamma\tan\theta_{2}^{(2)} \Bigr{)}\quad 0<t\leq t_{90},\\ \pi+\arctan\Bigl{(}\cos\gamma\tan\theta_{2}^{(2)}\Bigr{)}\quad t_{90}<t\leq t _{\rm f_{2}},\end{cases}  \tag{29}\]  
where \(t_{90}\) is the time when \(\theta_{1}=90^{\circ}\). This expression along with its first-order and second-order derivatives, as well as the simulated results obtained by _ADAMS_, is drawn in Figure 16.  
It can be observed from Figures 15 and 16 that these two kinds of curves, i.e., the theoretically estimated one and the one obtained by _ADAMS_, al  
Figure 16: Input motion. (1) Required input motions of the torsional spring and the driving motor, respectively; (2) comparison between the two input motions obtained by theoretical estimation and by _ADAMS_. other, which verifies the effectiveness of the planned deployment. Owing to this fact, it suffices for us to rely solely on the theoretical method to conduct trajectory planning of the antenna. Considering the fact that the displacement and the velocity are continuous whereas the acceleration is not, soft shock occurs at the moment when the torsional-spring-driven deployment switches to the motor-driven one. Fortunately, the soft shock does not impose a perceptible influence on the whole system because the deployment process is considerably slow.