# A Single Motor-Driven Focusing Mechanism with Flexure Hinges for Small Satellite Optical Systems
## 3 Experimental Results  
As a result of the rotation body measurements, the relation between the input of the linear screw and the displacements of the rotation body showed a linear behavior. Based on this relationship, the dynamic characteristics of the SMFH were identified by using the SMFH characterization platform. As shown in Figure 8, the hysteresis of the De-space was observed to 8.73%. Such a hysteresis of a flexible structure has been commonly observed, yet it can be reduced by means of a preload in the opposite direction of deformation. Here, the stretched length of 5 \(\upmu\)m ensures the unlimited fatigue cycle described in the fatigue analysis section. To investigate the variation of hysteresis, a cyclic loading/unloading was performed. Six comparison groups were made, with steps of 1 \(\upmu\)m pre-stretched lengths, ranging from 0 to 5 \(\upmu\)m. Here, since the cyclic traveling begins from the offset origin due to its pre-stretched length, the initial point (\(P_{i}\)) of the secondary mirror was defined. Then, starting from the initial point (\(P_{i}\)), a top point (\(P_{t}\)) and a bottom point (\(P_{b}\)) were defined. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control Parameter** & **P** & **I** & **D** \\ \hline Value & 47 & 188 & 61 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The employed control parameters to implement the feedback system. Figure 8: Optimal pre-stretching experiment results. Where \(P_{t}\) and \(P_{b}\) indicate that the De-space is maximized (Positive) and minimized (Negative) from the initial point. Based on these two points, a single cycle was divided into four steps, from Travel 1 to 4, and each phase is as follows: the secondary mirror moves from \(P_{i}\) to \(P_{t}\) (Travel 1), back from \(P_{t}\) to \(P_{i}\) (Travel 2), and moves from \(P_{i}\) to \(P_{b}\) (Travel 3), back again from \(P_{b}\) to \(P_{i}\) (Travel 4). Namely, Travels 1 and 4 represent loading, while Travels 2 and 3 represent unloading. As shown in Figure 8, Case 4 showed a minimum hysteresis of 6.52% at a pre-stretched length of 3 \(\upmu\)m, which represents 25.3% lower hysteresis than Case 1, a non-pre-stretched condition. The maximum error for the loading and unloading cycles was only 1.44 \(\upmu\)m. Therefore, the optimal pre-stretched length for the secondary mirror supporter was defined to 3 \(\upmu\)m (Case 4). For all cases, the hysteresis and the displacements were summarized in Table 5.  
In addition to the investigations of cyclic travel, an evaluation in a zero-gravity environment was performed, as addressed in Supplementary Material (Text S1). Indeed, since the objects on earth are influenced by gravity, SMFH can deflect under their own weight (See Figure S2). These undesired deformations may affect the mechanical behavior of the structure. Accordingly, by compensating for gravity, it is possible to mimic a zero-gravity space environment in the experiment. The derived equation with the defined parameters (See Table S1) eliminates the deflections of SMFH that could occur along the \(x\), \(y\), and \(z\)-axes, respectively. To obtain each deflection measurement, the cyclic experiments were performed in 10 trials on each axis. As shown in Figure 9a, the De-space can range from \(-10.93\) to \(13\)\(\upmu\)m with a hysteresis of 10.46%, and the maximum error for loading and unloading was 2.47 \(\upmu\)m. The loading and the unloading cycles were characterized by means of 5 th polynomial fitting, in order to avoid the possible errors between the desired and the current De-space due to the hysteresis. As summarized in Table 6, the agreement between the interpolated curve and the experimental curve was very good, with R-square of 99.99%. In the case of De-center, its distribution can range from 0 to 5.2 \(\upmu\)m, and the hysteresis was 41.60% (Figure 9b). Conversely, Tilt showed a relatively large hysteresis of 73.63%, yet its response achieved the range of between 0 to 88.45 \(\upmu\)rad (Figure 9c).