# Adaptive Deployable Thin Spherical Shell Reflectors
## 5 Petal Reflector  
The petal configuration suggested in Figure 1 allows a simple deployment scenario, and also reduces the excessive hoop stiffness of the full spherical shell, simplifying the control of some optical aberrations, as investigated in [9]. In this section, we consider the same spherical shell as in the previous section, except that it is formed of six petals as shown in Figure 9. It is also controlled with a set of 120 electrodes. To give an idea of the authority of the piezoelectric actuation, the application of 100 V to all the 20 electrodes of one petal produces a tip displacement of 250 \(\upmu\)m.  
Figure 8: Effect of a thermal gradient of \(\mp 15^{\circ}\) K on the full reflector with balanced layer design and near zero CTE substrate (**a**) Thermal displacements along \(z\) without control. (**b**) Residual displacements after control. (**c**) Cross section at 30\({}^{\circ}\) of the displacements before (blue) and after control (red). (**d**) Map of the correcting voltages.  
Figure 9: (**a**) Geometry of the petal reflector (the boundary conditions are the same as in Figure 6). (**b**) Layout of the 120 electrodes.  
Table 4 shows the surface figure error induced by the same linear thermal gradient of \(\mp 15^{\circ}\)K, before and after control with the 120 piezoelectric actuators, for the same three layer configurations considered in Table 3. One sees that, in absence of control, the distortion of the unimorph reflector is significantly larger than with the full reflector, while the residual surface figure error after control of the balanced configurations are even smaller than those of the full reflector and obtained with a smaller actuator budget.  
Figure 10 illustrates the case of the balanced layer design with near zero CTE substrate (last line in Table 4). Figure 10a shows the displacements along the \(z\) axis induced by the thermal gradient while Figure 10b shows the residual error when the control has been applied. Figure 10c shows the thermal displacements (in blue) and the residual error (in red) along the 30\({}^{\circ}\) cross section indicated on Figure 10a,b. Figure 10d shows the map of the correcting control voltages. The thermal displacement, residual error and voltage maps are similar to those of Figure 10 for both the unimorph and the thermally balanced high CTE design, except for the amplitudes which are given in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Petal Reflector** & **PV** & **RMS** & **PV (Control)** & **RMS (Control)** & **Voltage \(\Delta\)V** \\ \hline Nominal & 1681 & 218 & 20 & 1.6 & 967 \\ \hline Balanced & & & & & \\ High CTE & 25.8 & 4.5 & 2 & 0.22 & 33 \\ \hline Balanced & & & & & \\ Low CTE & 1.4 & 0.25 & 0.11 & 0.01 & 1.9 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Surface figure error of the petal reflector due to a thermal gradient of \(\mp 15^{\circ}\)K, before and after control with an array of 120 electrodes. All amplitudes are in \(\upmu\)m and voltages in V.  
Figure 10: Effect of a thermal gradient of \(\mp 15^{\circ}\) K on the petal reflector with balanced layer design and near zero CTE substrate. (**a**) Thermal displacements along \(z\) without control. (**b**) Residual displacements after control. (**c**) Cross section at 30\({}^{\circ}\) of the displacements before (blue) and after control (red). (**d**) Map of the correcting voltages.