## 6 Trajectory-Planning-Based Deployment Dynamics
### Trajectory Planning
After measuring in _SolidWorks_, we obtain that \(I_{\mathrm{P}}=0.2210618\,\mathrm{kg}\,\mathrm{m}^{2}\); (3) the angle \(\gamma=\langle\mathbf{v},(0,0,-1)^{\mathrm{T}}\rangle=37.85^{\circ}\). method, the expression in Equation (21) is expanded at \(\theta_{2}=0\) using [PERSON]'s expansion as follows:  
\[\theta_{1}=\theta_{2}\cos\gamma+\frac{1}{3}\theta_{2}^{3}\Big{(}\cos\gamma-\cos^ {3}\gamma\Big{)}+O\Big{(}\theta_{2}^{4}\Big{)}, \tag{23}\]  
where the second and third term are truncated, i.e., take \(\theta_{1}=\theta_{2}\cos\gamma\). To explore the approximation performance, these two methods are visualized in Figure 12, from which it is concluded that the second liberalization method exhibits a better approximation effect and thus is taken as the preferable method. Meanwhile, the MOIs of the input and output rod are negligible because they are far lower than those of the petal, based on which the Lagrange function in Equation (19) is simplified to  
\[ \begin{cases}\mathcal{L}=\frac{1}{2}I_{\mathrm{P}}\theta_{2}^{2}-\frac{1}{2}k (\cos\gamma\theta_{2}-\cos\gamma\theta_{0})^{2}=\frac{1}{2}I_{\mathrm{P}} \theta_{2}^{2}-\frac{1}{2}k^{*}(\theta_{2}-\theta_{0})^{2},\\ \mathcal{Q}=\frac{1}{2}(c\cos^{2}\gamma)\theta_{2}^{2}=\frac{1}{2}c^{*}\theta _{2}^{2},\end{cases}  \tag{24}\]  
where \(k^{*}=k\cos^{2}\gamma\) and \(c^{*}=c\cos^{2}\gamma\). After conducting the above operations and according to Equation (24), the torsional spring can be considered to be mounted on the output rod, with a new stiffness coefficient \(k^{*}\) and damping coefficient \(c^{*}\). The mechanical system in Figure 10(a) is essentially simplified into a mass-spring-damper system. For clearer illustration, the tension spring is utilized as an analogy to the torsional one, and eventually the corresponding simplified dynamic model is demonstrated in Figure 10(b). According to Ref. [74] and Figure 10(b), the dynamic equation of the first deployment stage can be formulated as  
\[ \begin{cases}\dot{x}(t)+2\zeta\omega_{\mathrm{n}}\dot{x}(t)+\omega_{\mathrm{n }}^{2}x(t)=0,\\ x(0)=-\theta_{0},\\ \dot{x}(0)=0,\end{cases}  \tag{25}\]  
where \(\omega_{\mathrm{n}}=\sqrt{k^{*}/I_{\mathrm{P}}}\), \(\zeta=\frac{c^{*}}{2\sqrt{k^{*}/I_{\mathrm{P}}}}\) and hence the circular frequency \(\omega_{\mathrm{d}}=\sqrt{1-\zeta^{2}}\omega_{\mathrm{n}}\). Also, based on the geometric relation demonstrated in Figure 10(b), the angular displacement of the petal is  
\[\theta_{2}(t)=x(t)-\theta_{0}. \tag{26}\]  
By referring to Ref. [74], the exact solution of \(\theta_{2}\) can be obtained by simultaneously solving Equations (25) and (26) (see Equation (A2)), to which the solution is a function of \(k\) and \(c\). By substituting the solution of \(\theta_{2}\) into Equation (18), a system of equations (SOE) concerning \(k\) and \(c\) is thus obtained. This SOE is a nonlinear one and includes several trigonometric functions, so it may have multiple sets of solutions. For an intuitive insight into the distribution of these solutions, the SOE is drawn as implicit functions in Figure 13, where the red points represent the solutions. Therefore, these solutions can be obtained by  
Figure 12: Linearization of \(\theta_{2}\). This figure compares two kinds of linearization methods for \(\theta_{2}\). taking the points' rough locations as the iterative initial values and solving the SOE with [PERSON]'s method, which is reported in Table 6.  
Substituting these solutions back into Equation (A2) obtains the solutions of \(\theta_{2}\) with different \(k\) and \(c\), which are drawn in Figure 14. It can be found from the figure that all of the curves are oscillatory during the first deployment stage except for the first curve, amongst which each one of them appears as more noticeable than the next. The above fact can also be concluded from Table 6, in which only the semi-period of the first solution is greater than \(60\,\mathrm{s}\). For other solutions, the semi-period becomes smaller, causing more intensive oscillation.