## 6 Trajectory-Planning-Based Deployment Dynamics
### Trajectory Planning  
During deployment, the antenna will usually be deployed first by an angle with torsional springs to avoid dead points, and then deployed by motors [71]. This process can be approximated as the dynamic model shown in Figure 11a. It can be found from the figure that the petal is equivalent to a small ball anchored on the output rod with a mass \(M\) and distance \(l\) from the ball center to the output rod, and that the driving motor is connected with a reducer immediately linked to the input rod. In addition, a torsional spring is anchored at the input rod to provide the torque at the first deployment stage. Finally, assume that the reduction ratio is \(i_{\mathrm{r}}=50\), the preloading angle of the spring is \(\theta_{0}^{*}\) and its stiffness coefficient \(k\) and damping coefficient \(c\) are unknown parameters to be identified. At the first deployment stage, i.e., the deployment with torsional springs, the time taken is assigned as \([0,t_{\mathrm{f_{1}}}]\). As for the second stage, i.e., the deployment with motors, the time taken is assumed to be within \([t_{\mathrm{f_{1}}},t_{\mathrm{f_{2}}}]\), and the following values are taken: \(t_{\mathrm{f_{1}}}=60\,\mathrm{s}\), \(t_{\mathrm{f_{2}}}=180\,\mathrm{s}\) and \(t_{\mathrm{d}}=t_{\mathrm{f_{2}}}-t_{\mathrm{f_{1}}}=120\,\mathrm{s}\). Furthermore, for the first stage, the additional boundary conditions  
\[ \begin{cases}\theta_{2}(t_{\mathrm{f_{1}}},k,c)=\theta_{0},\\ \dot{\theta}_{2}(t_{\mathrm{f_{1}}},k,c)=v_{0},\end{cases}  \tag{18}\]  
are supplemented, where \(\theta_{0}=30^{\circ}\) and \(v_{0}=0.5^{\circ}/\mathrm{s}\). Also, assume that, at \(t=t_{\mathrm{f_{1}}}\), the spring is precisely in its equilibrium position. Referring to Ref. [72], the Lagrangian function of the mechanical system shown in Figure 11a can be expressed as  
\[\mathcal{L}=\mathcal{T}-\mathcal{V}=\frac{1}{2}I_{\mathrm{i}}\theta_{1}^{2}+\frac {1}{2}(I_{\mathrm{P}}+I_{\mathrm{o}})\theta_{2}^{2}-\frac{1}{2}k(\theta_{1}- \theta_{0}^{*})^{2}, \tag{19}\]  
where \(I_{\mathrm{i}}\), \(I_{\mathrm{P}}\) and \(I_{\mathrm{o}}\) are the MOIs of the input rod, petal and output rod, respectively. Additionally, the dissipation function is  
\[\mathcal{Q}=\frac{1}{2}c\theta_{1}^{2}. \tag{20}\]  
According to Ref. [73], we know that  
\[\theta_{1}=\arctan(\cos\gamma\tan\theta_{2}), \tag{21}\]  
from which we can obtain that \(\theta_{0}^{*}=24.51^{\circ}\). Substituting Equation (21) into Equations (20) and (19) yields a dissipation function \(\mathcal{Q}^{*}\) and Lagrangian function \(\mathcal{L}^{*}\), both of which containing only variable \(\theta_{2}\). As a result, the dynamic equation with respect to \(\theta_{2}\) is formulated as  
\[\frac{d}{dt}\frac{\partial\mathcal{L}^{*}}{\partial\theta_{2}}-\frac{\partial \mathcal{L}^{*}}{\partial\theta_{2}}+\frac{\partial\mathcal{Q}^{*}}{\partial \theta_{2}}=0. \tag{22}\]  
The explicit expression of Equation (22) is shown in Equation (A1), from which we know that it is a considerably complex nonlinear ordinary differential equation (ODE) and does not have an analytical solution. This causes difficulty in identifying the values of \(k\) and \(c\), which can be attributed to the nonlinearity of \(\theta_{1}\). To solve this problem, it is of vital necessity to linearize \(\theta_{1}\). One reasonable method is to suppose that the velocities of the input and output rod remain approximately the same, i.e., let \(\theta_{1}=\theta_{2}\). As for another  
Figure 11: Dynamic model. (**a**) Equivalent dynamic model of petal deployment. (**b**) Simplified dynamic model of petal deployment. (1) Variable \(\theta_{1}\) is the angular displacement of the input rod, variable \(\theta_{2}\) is the angular displacement of the output rod; (2) the moment of inertia (MOI) of the ball is \(I_{\mathrm{P}}=I_{\mathrm{c}}+Ml^{2}\) according to the parallel axis theorem, where \(I_{\mathrm{c}}\) is the MOI about the centroidal axis.