A Single Motor-Driven Focusing Mechanism with Flexure Hinges for Small Satellite Optical Systems

Featured Application: An optical system with a compact dimension for the small satellite, requiring less power consumption. Abstract: For earth observation, the optical systems in small satellites are crucial to obtain high-resolution images. However, the alignment between a primary and a secondary mirror in an optical system can be disturbed due to the harsh environments inside vehicles or space (i.e., vibrations, shock loading during launch, dramatic temperature changes, or high vacuum pressure in space). To compensate for such undesired deformations, a focusing mechanism should be embedded into the optical system. In this paper, we propose a novel Single Motor-Driven Focusing mechanism with Flexure Hinges (SMFH), allowing the Flexure Hinge (FlexHe) to displace in the longitudinal direction. The presented FlexHe incorporates radial zig-zag-patterned slits to achieve ﬂexibility, and preloading of the hinge structures to reduce the resulting hysteresis. To investigate an optimal conﬁguration of FlexHe, a numerical simulation is performed by means of ANSYS 19.2. The variation of Modulation Transfer Function (MTF), corresponding to an image resolution, is evaluated by using an optics simulation program (CODE-V). The experimental setups are built by exploiting the fabricated SMFH and ﬁve LVDT (Linear Variable Di ﬀ erential Transformer) sensors with a high resolution of 0.031 µ m. As a result, hysteresis can be reduced up to 6.52% with a pre-stretched length of 3 µ m. The proposed SMFH allows not only the De-space to displace up to 23.93 µ m, but also the De-center and the Tilt to achieve the desired displacements of 5.20 µ m and 88.45 µ rad, respectively. Conclusively, the SMFH shows promising characteristics to embed a feedback control, due to its high resolution (up to 0.1 µ m) for De-space with the MTF of 37%. De-space represents the undesired deformation of the secondary mirror that occurs parallel to the direction of the optical axis. De-center indicates an undesired lateral displacement perpendicular to the optical axis, and Tilt indicates an irregular angle or skews relative to the optical axis, measured as rotation about axes perpendicular to the optical axis.


Introduction
Although a small satellite has limited dimensions, it has shown promising potential to observe the earth [1][2][3] with many advantages, i.e., lower launching expenses [4], less energy consumption [5,6], and shorter lead-times [7], etc. Recently, small satellites have been able to accommodate an expanding

Definition of Mis-Alignments of the Schmidt-Cassegrain Optical System
Undesired mis-alignments cause the degradation of image resolution, which consists of independent parameters in three-dimensional cartesian space (i.e., De-space, De-center, and Tilt), as shown in Figure 1. De-space represents the undesired deformation of the secondary mirror that occurs parallel to the direction of the optical axis. De-center indicates an undesired lateral displacement perpendicular to the optical axis, and Tilt indicates an irregular angle or skews relative to the optical axis, measured as rotation about axes perpendicular to the optical axis.

Approximation of MTF Performance
The MTF is a function of spatial frequency, which correlates the resolution and the contrast for the image. The variation of MTF occurs due to the mis-alignments of optics. Indeed, these parameters cause diffraction of the light when it forms an image; thereby, image's quality and/or resolution (pixel) could be degraded. Among the mis-alignment parameters, the ranges of De-space and Tilt are crucial because variations in these values could degrade MTF. Therefore, we limited certain ranges for the De-space and the Tilt, and defined the De-center to correspond to a single pixel size of the conceptual detector (8.2 µm). Here, once the De-center is within a range of 8.2 µm, the variation of MTF does not exceed the requirements that we assigned in Table 1. To verify these, in this section, the variation in MTF values was demonstrated by using an optics simulation program (CODE-V). Here, we employed the specific frequency-Nyquist frequency corresponds to the pixel size of the sensor. Since the pixel size of the employed image sensor is 8.2 µm, the Nyquist frequency is 61.0 lp/mm. With this condition, a comparison group was made to figure out their proper ranges, as shown in Table 2. Each case corresponds to a variation in Tilt, ranging from 0 to 200 µrad, with steps of 100 µrad. Here, two axes are defined as the x and y axes, respectively. The Tilts that occur for each axis are defined as α and β-Tilts, respectively. When α and β-Tilts were 200 µrad (Case 3), respectively, the MTF was reduced by 5%, compared to Case 1 (without Tilts), as shown in Figure 2. On the other hand, once the De-space was over ±5 µm, the MTF degraded to below 30%, where the MTF value was a minimum. Furthermore, the degradation of MTF due to De-space is more significant than the degradation caused by Tilt.

Approximation of MTF Performance
The MTF is a function of spatial frequency, which correlates the resolution and the contrast for the image. The variation of MTF occurs due to the mis-alignments of optics. Indeed, these parameters cause diffraction of the light when it forms an image; thereby, image's quality and/or resolution (pixel) could be degraded. Among the mis-alignment parameters, the ranges of De-space and Tilt are crucial because variations in these values could degrade MTF. Therefore, we limited certain ranges for the De-space and the Tilt, and defined the De-center to correspond to a single pixel size of the conceptual detector (8.2 µm). Here, once the De-center is within a range of 8.2 µm, the variation of MTF does not exceed the requirements that we assigned in Table 1. To verify these, in this section, the variation in MTF values was demonstrated by using an optics simulation program (CODE-V). Here, we employed the specific frequency-Nyquist frequency corresponds to the pixel size of the sensor. Since the pixel size of the employed image sensor is 8.2 µm, the Nyquist frequency is 61.0 lp/mm. With this condition, a comparison group was made to figure out their proper ranges, as shown in Table 2. Each case corresponds to a variation in Tilt, ranging from 0 to 200 µrad, with steps of 100 µrad. Here, two axes are defined as the x and y axes, respectively. The Tilts that occur for each axis are defined as α and β-Tilts, respectively. When α and β-Tilts were 200 µrad (Case 3), respectively, the MTF was reduced by 5%, compared to Case 1 (without Tilts), as shown in Figure 2. On the other hand, once the De-space was over ±5 µm, the MTF degraded to below 30%, where the MTF value was a minimum. Furthermore, the degradation of MTF due to De-space is more significant than the degradation caused by Tilt. Therefore, the De-space and α and β-Tilts should be limited by ±5 µm and 200 µrad, respectively. Also, the available range of MTF should be over 30%, which is the minimum value that would allow the optical system to produce a high-resolution image.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 16 Therefore, the De-space and α and β-Tilts should be limited by ±5 µm and 200 µrad, respectively. Also, the available range of MTF should be over 30%, which is the minimum value that would allow the optical system to produce a high-resolution image. Using a DC motor with a rotary encoder, as it will be detailed in the next section, the displacement resolution of the linear screw was identified to 0.1 µm. Since the control resolution for the De-space does not exceed 0.1 µm, a minimum MTF control step of 0.22% is identified.

Requirements for a Novel Focusing Mechanism
Given that the De-space significantly affects variations in MTF, the design of the focusing mechanism should ensure the restoration of optical alignment mostly by adjusting De-space, with limited adjustments to De-center and Tilt. As shown in Figure 2, by exploiting the relation between the De-space and MTF, the desired range of the De-space was defined to ±10 µm, and the desired Decenter and the Tilt were limited to 8.2 µm and 200 µrad, respectively. Accordingly, due to the assigned ranges for the De-space, De-center, and Tilt (See Table 3), the resulting potential range of MTF is greater than 0.35, and a pixel loss can be minimized (less than 1).

Concept Design and Working Principle of SMFH
Based on the requirements of the Schmidt-Cassegrain optics system, the focusing mechanism is conceptualized. As shown in Figure 3, the proposed SMFH consists of a. Secondary mirror dummy, b. Secondary mirror supporter, c. Flexure hinge, d. Rotation body, e. Motor screw, f. Linear screw, g. Motor, h. Roller, and i. Rotation shaft. Once the DC motor rotates in CW or CCW, the rotation body induces a deformation, and then each FlexHe is elongated. The detailed working principle is as follows: 1. Once the motor screw that connects the motor shaft rotates CW, it advances the linear screw, enabling the movement of the x-axis (−). 2. The rotation body attached to the end of the linear screw rotates CW. Using a DC motor with a rotary encoder, as it will be detailed in the next section, the displacement resolution of the linear screw was identified to 0.1 µm. Since the control resolution for the De-space does not exceed 0.1 µm, a minimum MTF control step of 0.22% is identified.

Requirements for a Novel Focusing Mechanism
Given that the De-space significantly affects variations in MTF, the design of the focusing mechanism should ensure the restoration of optical alignment mostly by adjusting De-space, with limited adjustments to De-center and Tilt. As shown in Figure 2, by exploiting the relation between the De-space and MTF, the desired range of the De-space was defined to ±10 µm, and the desired De-center and the Tilt were limited to 8.2 µm and 200 µrad, respectively. Accordingly, due to the assigned ranges for the De-space, De-center, and Tilt (See Table 3), the resulting potential range of MTF is greater than 0.35, and a pixel loss can be minimized (less than 1).

Concept Design and Working Principle of SMFH
Based on the requirements of the Schmidt-Cassegrain optics system, the focusing mechanism is conceptualized. As shown in Figure 3, the proposed SMFH consists of a. Secondary mirror dummy, b. Secondary mirror supporter, c. Flexure hinge, d. Rotation body, e. Motor screw, f. Linear screw, g. Motor, h. Roller, and i. Rotation shaft. Once the DC motor rotates in CW or CCW, the rotation body induces a deformation, and then each FlexHe is elongated. The detailed working principle is as follows:

1.
Once the motor screw that connects the motor shaft rotates CW, it advances the linear screw, enabling the movement of the x-axis (−).

2.
The rotation body attached to the end of the linear screw rotates CW.

3.
A roller that is mounted at the tip of the rotation body positively displaces the secondary mirror along the z-axis (+), where it contacts the secondary mirror supporter. 4.
The three flexure hinges are elongated along the +z-axis, and then it generates the De-spacewith a restoring force. 5.
The motor rotates CCW, and then steps 1 to 4 repeat in the opposite direction/order. This process causes the secondary mirror to negatively displace in the z-axis (−).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 16 3. A roller that is mounted at the tip of the rotation body positively displaces the secondary mirror along the z-axis (+), where it contacts the secondary mirror supporter. 4. The three flexure hinges are elongated along the +z-axis, and then it generates the De-spacewith a restoring force. 5. The motor rotates CCW, and then steps 1 to 4 repeat in the opposite direction/order. This process causes the secondary mirror to negatively displace in the z-axis (−).

Design of Flexure Hinge (FlexHe)
The primary objective of FlexHe is to maximize the displacement in the z-axis (De-space), while at the same minimizing Tilt and De-center in the x and y axes. Accordingly, the structure should be reinforced to resist both bending and torsion, yet be compliant in tension and compression. Moreover, the dimensions should satisfy the requirements that were introduced in the aforementioned section. The presented FlexHe can reduce structural complexity, as well as achieve promising mechanical characteristics. The slits of the FlexHe are geometrically arranged in a zig-zag pattern. The best structure with a relatively high bending stiffness and a more compliant axial stiffness was identified by means of the Finite Element Method.

FEM Modeling of the FlexHe
A finite element (FE) was created in ANSYS 19.2 (ANSYS Inc, Canonsburg, PA, USA) with static structure analysis, and the mechanical characteristics were analyzed (i.e., strain versus stress, strain energy of structure through cyclic loading/un-loading, cyclic fatigue, and fracture). The analysis identified titanium alloys (Ti-6Al-4V) as having suitable material properties for the FlexHe due to its promising material properties for space use (i.e., specific strength, lightweight, low density, etc) [38].
Each FlexHe type was defined by the number of slits: types 1 to 6 were defined according to slit number [n], which ranged from 2 to 7, respectively. In addition to static characteristics, the fatigue analysis for the FlexHe was carried out under a fully reversed loading condition (tensioncompression loading, stress ratio R= −1). Here, the thermal and vibrational effects were neglected. The stress versus the number of cycles to failure (S-N curves) and endurance limits associated with high-cycle fatigue (HCF) were compiled as a reference in [39].
The simulation results showed different mechanical behaviors. First, the greater the number of slits, the less axial stiffness is observed, as depicted in Figure 4c,d. Such compliant characteristics cause the FlexHe to undergo undesired deformations (e.g., buckling or column squirm), since the correlation between the buckling pressure and the stiffness of the structure is proportional [40,41]. Indeed, the axial stiffness of Type 6 is relatively more compliant than Type 1, which indicates that Type 6 could be too delicate for the compression load. Similarly, the plotted relation of the strain energy versus deformation showed that Type 1 has excessive axial stiffness (Reaction force of 35.6 N). Comparing Type 1 with Type 6, the maximum reaction force of Type 6 was only 8 N, which is

Design of Flexure Hinge (FlexHe)
The primary objective of FlexHe is to maximize the displacement in the z-axis (De-space), while at the same minimizing Tilt and De-center in the x and y axes. Accordingly, the structure should be reinforced to resist both bending and torsion, yet be compliant in tension and compression. Moreover, the dimensions should satisfy the requirements that were introduced in the aforementioned section. The presented FlexHe can reduce structural complexity, as well as achieve promising mechanical characteristics. The slits of the FlexHe are geometrically arranged in a zig-zag pattern. The best structure with a relatively high bending stiffness and a more compliant axial stiffness was identified by means of the Finite Element Method.

FEM Modeling of the FlexHe
A finite element (FE) was created in ANSYS 19.2 (ANSYS Inc, Canonsburg, PA, USA) with static structure analysis, and the mechanical characteristics were analyzed (i.e., strain versus stress, strain energy of structure through cyclic loading/un-loading, cyclic fatigue, and fracture). The analysis identified titanium alloys (Ti-6Al-4V) as having suitable material properties for the FlexHe due to its promising material properties for space use (i.e., specific strength, lightweight, low density, etc.) [38].
Each FlexHe type was defined by the number of slits: types 1 to 6 were defined according to slit number [n], which ranged from 2 to 7, respectively. In addition to static characteristics, the fatigue analysis for the FlexHe was carried out under a fully reversed loading condition (tension-compression loading, stress ratio R = −1). Here, the thermal and vibrational effects were neglected. The stress versus the number of cycles to failure (S-N curves) and endurance limits associated with high-cycle fatigue (HCF) were compiled as a reference in [39].
The simulation results showed different mechanical behaviors. First, the greater the number of slits, the less axial stiffness is observed, as depicted in Figure 4c,d. Such compliant characteristics cause the FlexHe to undergo undesired deformations (e.g., buckling or column squirm), since the correlation between the buckling pressure and the stiffness of the structure is proportional [40,41]. Indeed, the axial stiffness of Type 6 is relatively more compliant than Type 1, which indicates that Type 6 could be too delicate for the compression load. Similarly, the plotted relation of the strain energy versus deformation showed that Type 1 has excessive axial stiffness (Reaction force of 35.6 N). Comparing Type 1 with Type 6, the maximum reaction force of Type 6 was only 8 N, which is only 22% of Type 1. In other words, Type 1 needs a relatively strong force in order to achieve the desired deformation, and accordingly, this would consume a lot of energy in small satellite.
only 22% of Type 1. In other words, Type 1 needs a relatively strong force in order to achieve the desired deformation, and accordingly, this would consume a lot of energy in small satellite.
In the cases of Types 1 through 3, the relationship between deformation and strain energy showed a non-linear behavior. On the other hand, in cases of Types 4 through 6, we identified that the relationship showed a linear response. Due to this, the strain energy was not dramatically increased even though deformation is imposed. In light of these analyses, we determined that FlexHe Type 4, consisting of five slits, was the optimal structure for the operation. Indeed, as the graphs in Figure 4 demonstrate, Type 4 showed not only relatively high axial stiffness compared to Types 5 and 6, but also relatively compliant rather than Types 1 through 3. With the optimized FlexHe (Type 4), the constant amplitude load fatigue analysis was carried out. By using the displacement of 25 µm and the estimated reaction force (11.5 N) from the simulation study, various parameters (i.e., alternating equivalent stress, safety factor, biaxial indication, fatigue sensitivity, fatigue life) were evaluated. The local stress was determined by means of the biaxial stress. In general, the biaxiality indication refers to the ratio between the major and minor principal stresses. As a result of fatigue analysis, the biaxial indication ranged between −0.99996 and 0.99152, as shown in Figure 5. Thus, the maximum stress of the FlexHe occurred at the inside of each slit. However, since the minimum safety factor was 6.239 (Not less than 1), the available fatigue life extends the design life to 100,000 cycles at 11.5 N and avoids any critical regions that could cause a material failure and/or fracture. The fatigue sensitivity was computed to identify the variations in the fatigue life according to the loads. As a result, it showed that the fatigue life could be unlimited, even though the maximum load was increased by 150%. In the cases of Types 1 through 3, the relationship between deformation and strain energy showed a non-linear behavior. On the other hand, in cases of Types 4 through 6, we identified that the relationship showed a linear response. Due to this, the strain energy was not dramatically increased even though deformation is imposed. In light of these analyses, we determined that FlexHe Type 4, consisting of five slits, was the optimal structure for the operation. Indeed, as the graphs in Figure 4 demonstrate, Type 4 showed not only relatively high axial stiffness compared to Types 5 and 6, but also relatively compliant rather than Types 1 through 3.
With the optimized FlexHe (Type 4), the constant amplitude load fatigue analysis was carried out. By using the displacement of 25 µm and the estimated reaction force (11.5 N) from the simulation study, various parameters (i.e., alternating equivalent stress, safety factor, biaxial indication, fatigue sensitivity, fatigue life) were evaluated. The local stress was determined by means of the biaxial stress. In general, the biaxiality indication refers to the ratio between the major and minor principal stresses. As a result of fatigue analysis, the biaxial indication ranged between −0.99996 and 0.99152, as shown in Figure 5. Thus, the maximum stress of the FlexHe occurred at the inside of each slit. However, since the minimum safety factor was 6.239 (Not less than 1), the available fatigue life extends the design life to 100,000 cycles at 11.5 N and avoids any critical regions that could cause a material failure and/or fracture. The fatigue sensitivity was computed to identify the variations in the fatigue life according to the loads. As a result, it showed that the fatigue life could be unlimited, even though the maximum load was increased by 150%. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 16

Experimental Investigation for the FlexHe
Based on the criteria established through the simulation results, the FlexHe was fabricated and experimentally investigated. As shown in Figure 6a,b, the extension test for a single FlexHe was performed by using a Micro stage with a LVDT sensor, and the reaction force was obtained by using a load cell (Dacell, CB1, South Korea). As per the plotted graph in Figure 6c, the obtained reaction force at a deformation of 23.5 µm was 11.05 N, and the error rate was 4.03% with respect to the simulation result (10.62 N). The relation of the reaction force versus strain showed a linear behavior, and the deformation followed the elastic region of the structure. For a single FlexHe, hysteresis was not observed in both of the experimental and simulation results.
Through the experimental investigation, we identified that the pushing force of the linear screw should be 30 N, so that three FlexHes displace up to 20 µm. With a safety factor of 1.5, a pushing force of 45 N was obtained. Then, according to the loading or unloading phases, the desired torques of the geared motor were theoretically derived, as below: where F is the required force of the linear screw to push the rotation body, dm is the diameter of the linear screw, f is the friction coefficient, and l is the pitch of the linear screw, respectively. The required torques for the loading and unloading were identified as 28.8 mN·m and 20.6 mN·m, respectively. The geared motor (Maxon Motor ® , A-Max 16, customized, Sachseln, Switzerland) with a maximum torque of 230 mN·m was employed by using a safety factor of 8. Also, the employed encoder provides 512 PPR (Pulses/Rev) of resolution; thereby, the system provides the resolution up to 55,296 PPR. Accordingly, considering the pitch of the linear screw (0.5 mm) and the system resolution, the geared motor can be controlled by 0.01 µm, theoretically. However, we observed the undesired error (±5 PPR) during the position control, and thus, the controllable resolution for producing the De-space

Experimental Investigation for the FlexHe
Based on the criteria established through the simulation results, the FlexHe was fabricated and experimentally investigated. As shown in Figure 6a,b, the extension test for a single FlexHe was performed by using a Micro stage with a LVDT sensor, and the reaction force was obtained by using a load cell (Dacell, CB1, South Korea). As per the plotted graph in Figure 6c, the obtained reaction force at a deformation of 23.5 µm was 11.05 N, and the error rate was 4.03% with respect to the simulation result (10.62 N). The relation of the reaction force versus strain showed a linear behavior, and the deformation followed the elastic region of the structure. For a single FlexHe, hysteresis was not observed in both of the experimental and simulation results.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 16 was determined to 0.1 µm. Therefore, the embedded DC geared motor with the encoder can minimize the error rate of De-space and enables highly precise control of the MTF at increments of 0.22%.

Fabrication of SMFH and Experimental Setups
Due to the high resolution of displacement control, the proposed SMFH can be controlled at the sub-micrometer level. However, the precision and the repeatability of these adjustments could be Through the experimental investigation, we identified that the pushing force of the linear screw should be 30 N, so that three FlexHes displace up to 20 µm. With a safety factor of 1.5, a pushing force of 45 N was obtained. Then, according to the loading or unloading phases, the desired torques of the geared motor were theoretically derived, as below: where F is the required force of the linear screw to push the rotation body, d m is the diameter of the linear screw, f is the friction coefficient, and l is the pitch of the linear screw, respectively. The required torques for the loading and unloading were identified as 28.8 mN·m and 20.6 mN·m, respectively. The geared motor (Maxon Motor ® , A-Max 16, customized, Sachseln, Switzerland) with a maximum torque of 230 mN·m was employed by using a safety factor of 8. Also, the employed encoder provides 512 PPR (Pulses/Rev) of resolution; thereby, the system provides the resolution up to 55,296 PPR. Accordingly, considering the pitch of the linear screw (0.5 mm) and the system resolution, the geared motor can be controlled by 0.01 µm, theoretically. However, we observed the undesired error (±5 PPR) during the position control, and thus, the controllable resolution for producing the De-space was determined to 0.1 µm. Therefore, the embedded DC geared motor with the encoder can minimize the error rate of De-space and enables highly precise control of the MTF at increments of 0.22%.

Fabrication of SMFH and Experimental Setups
Due to the high resolution of displacement control, the proposed SMFH can be controlled at the sub-micrometer level. However, the precision and the repeatability of these adjustments could be diminished due to rough machining tolerances in the fabrication phase. Also, the backlash that occurs between the teeth of different stages within the gearbox and the linear screw could degrade the control precision. Accordingly, we first employed restrictive machining tolerances for each component and polished the surfaces where the sensors contact the secondary mirror supporter, in order to minimize the measurement error. Then, it was integrated with the linear screw and the geared DC motor, as shown in Figure 7.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 16 was determined to 0.1 µm. Therefore, the embedded DC geared motor with the encoder can minimize the error rate of De-space and enables highly precise control of the MTF at increments of 0.22%.

Fabrication of SMFH and Experimental Setups
Due to the high resolution of displacement control, the proposed SMFH can be controlled at the sub-micrometer level. However, the precision and the repeatability of these adjustments could be diminished due to rough machining tolerances in the fabrication phase. Also, the backlash that occurs between the teeth of different stages within the gearbox and the linear screw could degrade the control precision. Accordingly, we first employed restrictive machining tolerances for each component and polished the surfaces where the sensors contact the secondary mirror supporter, in order to minimize the measurement error. Then, it was integrated with the linear screw and the geared DC motor, as shown in Figure 7. As shown in Figure 7b, five LVDT sensors were used to coordinate the geometry of the secondary mirror due to the deformation of FlexHes. First, to calculate the orthonormal vector of the plane (De-space and Tilt), three LVDT sensors were arranged at 120° from each other and were attached at the secondary mirror supporter. By using the orthonormal vectors obtained by these As shown in Figure 7b, five LVDT sensors were used to coordinate the geometry of the secondary mirror due to the deformation of FlexHes. First, to calculate the orthonormal vector of the plane (De-space and Tilt), three LVDT sensors were arranged at 120 • from each other and were attached at the secondary mirror supporter. By using the orthonormal vectors obtained by these sensors, the initial position of the secondary mirror was identified. The other two LVDT sensors were used to obtain the De-center, which was placed at both the x and y-axis of the secondary mirror supporter, respectively. The description of data obtaining spots and vector directions are shown in Figure S1. The employed LVDT sensor provides a high resolution of 0.031 µm with a fast sampling time of 1 ms. Secondly, the DC motor (Maxon Motor ® , A-Max 16, customized, Sachseln, Switzerland), having a reduction ratio of 108:1, was integrated. The geared DC motor included a rotary encoder, and each encoder pulse could precisely measure the angle of the motor shaft down to 0.00637 • . The motor's speed was fixed to 50 RPM, and the applied PID values are summarized in Table 4. Then, by converting the encoder data into the displacement of the secondary mirror supporter, input control functions were determined. Here, to reduce the errors due to the backlash, the EPOS2 controller (Maxon Motor ® , EPOS2 24/2, Sachseln, Switzerland) that provides a dual loop position control was employed. The control architecture features a proportional controller, and the gain scheduler makes a P gain for the main loop controller to track the errors that could occur between the current and the desired load position. Accordingly, the embedded feedback control enables the system to avoid chattering or hunting due to the backlash. Furthermore, the obtained data from the LVDT sensors, De-space, De-center, and Tilt were compiled by means of MATLAB (MathWorks ® , R2018a, Natick, MA, USA).

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 µm 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 µm pre-stretched lengths, ranging from 0 to 5 µm. 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. Where Pt and Pb 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 Pi to Pt (Travel 1), back from Pt to Pi (Travel 2), and moves from Pi to Pb (Travel 3), back again from Pb to Pi (Travel 4). Namely, Travels 1 and 4 represent loading, while Travels 2 and 3 represent unloading. 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 µm, 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 µm. Therefore, the optimal pre-stretched length for the secondary mirror supporter was defined to 3 µm (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 µm with a hysteresis of 10.46%, and the maximum error for loading and unloading was 2.47 µm. The loading and the unloading cycles were characterized by means of 5th 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 µm, 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 µrad (Figure 9c).

Conclusions and Discussion
In this paper, the SMFH was proposed to correct the mis-alignments of an optical system in small satellites. The presented SMFH corrects the alignment of a secondary mirror in a Schmidt-Cassegrain optical system. A novel structure with radial slits was proposed to reinforce the bending stiffness and maintain compliant axial stiffness. To investigate its mechanical characteristics, a static structural analysis was carried out. As a result, the FlexHe that features five slits was identified as the optimal design. Then, an extension test for a single optimized FlexHe was performed to determine its reaction force. The reaction force applied on the three FlexHes was measured to 30 N. The desired pushing force for the linear screw was measured at approximately 45 N, while multiplying a safety factor of 1.5. Based on the obtained reaction force, a geared motor having a torque of 230 mN·m was employed. The SMFH integrating three FlexHes was fabricated. To identify the proper preload, optimization was performed through an experimental study. By using six different pre-stretched lengths, we obtained a different hysteresis for each case. Among them, Case 4, having a pre-stretched length of 3 µm, showed a minimized hysteresis of 6.52%. For further investigation, the cyclic compression/extension test was repeated in ten trials, and we numerically compensated for any structural deformations caused by gravity. As a result, the proposed SMFH was not only able to generate a 23.93 µm stroke in a longitudinal direction (De-space), but also to achieve a maximum De-center and Tilt of 5.20 µm and 88.45 µrad, respectively. Moreover, given the resolution of the geared motor, the control system allowed the proposed device to achieve a high resolution of De-space control at a maximum of 0.1 µm. The SMFH ensures the requirement of optical alignment. The control resolution for the De-space was not over 0.1 µm; thereby, the minimum control step of MTF identified 0.22%, allowing the maximum MTF to achieve 37%.
Consequently, the proposed SMFH showed promising feasibilities. Among them, the primary contribution of this study is to present a simplified structure that embeds a single motor, and it is anticipated to consume less satellite power. Moreover, in light of our findings on the interpolated relation of the input versus the De-space, a simple yet precise feedback control system can be achieved, without embedding exteroceptive sensors. These promising advantages could provide opportunities that the proposed SMFH will be able to perform the desired objectives passively, and to complete an advanced mechanism ensuring low-cost and simplicity.
However, the need still remains for further studies on the mechanical behaviors of the structure in extreme space environments (i.e., extremely high/low temperature, vacuum, vibration, etc.). With this in mind, our future works will focus on furthering the objectives of simplifying SMFH, reducing satellite size, power consumption, and expense.
Supplementary Materials: The following are available online at http://www.mdpi.com/2076-3417/10/20/7087/s1, Text S1: Gravity compensation, Figure S1: (a) Five spots for obtaining displacement data and (b) notation of coordinate and vector directions, Figure S2: Examples of the SMFH orientation with the gravity that acts along (a) -z-direction and (b) -x-direction, Table S1: Parameters for gravity compensation.