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Abstract
In order to compensate the optical system bias, which is caused by the change of elevation angle and thermal gradient during the optical alignment of the telescope, a novel high stiffness micro-nano positioning hexapod platform with flexure hinges is proposed in this paper. The novel flexure hinge has a mechanical limit, and its equivalent model is established and analyzed. In addition, in order to speed up the solution process, a novel simplified inverse kinematic model is developed based on the rigid body kinematic theory. Then, an effective rigid-flexible coupling simulation system is built to verify the correctness and applicability of the inverse kinematic model. Finally, a systematic experimental test method and a statistical-based data analysis theory are proposed. The experimental results show that the resolution and repeatability of translation and rotation and lateral stiffness are as follows: 0.3 mm and 0.5 arc sec, ± 0.5 µm and ±0.5 arc sec, 131.6N$\cdot$µm-1 and 133.0N$\cdot$µm-1. The proposed hexapod platform can be used to correct the optical system bias of large-aperture telescopes.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In large-aperture astronomical telescopes, the ability of the secondary mirror to maintain strict alignment in real time with respect to its primary mirror affects its imaging quality to a certain extent. However, the relative positions of the primary and secondary mirrors in engineering applications are not strictly aligned because large-aperture ground-based telescopes are subject to gravitational deformation and structural thermal deformation [1]. Since its primary mirror has the characteristics of large mass, large size and difficult movement, the mechanism of active adjustment of the secondary mirror is used to correct the aberration.
In order to actively compensate for the aforementioned optical deviations, researchers have carried out extensive research work. In the Gran Telescopio Canarias, Casalta et al. [2] have used the hexapod platform to adjust the secondary mirror's position and pose. The overall accuracy of the translational and rotational motions of this adjustment mechanism is 22 µm and 5.7 µrad, respectively. To reduce image blurring caused by wind vibration, a hexapod platform with the hardpoint flexure has been used in the Giant Magellan Telescope (GMT), whose high axial stiffness allows its mirror to have an intrinsic frequency higher than the minimum frequency of the telescope [3,4]. Yang et al. [5] have designed a hexapod platform with flexible hinges as an adjustment mechanism for telescopes with a translational repeatability of about half a micron. Similarly, the positioning system of the secondary mirror in the European Extremely Large Telescope (E-ELT) is a six-degree-of-freedom parallel adjustment mechanism, where the actuator chooses the worm wheel system as the solution and it has a resolution of 100 microns [6]. The hexapod platform is also used in VISTA telescopes for secondary mirror corrections, and it has a positioning accuracy of 1 µm and 0.79", respectively [7,8]. Yun et al. [9,10] have designed an isotropic Stewart platform with flexure hinges for the secondary mirror of the telescope, and the axial stiffness of the flexure hinge is 310N·µm-1. Of course, there are some secondary mirror adjustment mechanisms with excellent performance, such as JWST [11–13], HET [14], Gemini [15], etc. However, as the aperture of the secondary mirror increases, the weight of the optical element to be supported also increases. Currently, the weight of optical loads is well over one ton. And, the repeatability and resolution of the rotational motion are required to be no greater than ±0.5 arc sec and 0.5 arc sec, respectively. Then, the repeatability and resolution of the translational motion are no greater than ±0.5 µm and 0.5 µm, respectively. Therefore, the performance of these mechanisms is not outstanding.
It is worth noting that flexible joints are widely used in aerospace and other fields [16,17]. Cai et al. [18,19] have designed a 6-DOF precision positioning system with T-shape hinges and elliptical flexure hinges. Similarly, A PZT-driven six-axis high-precision positioning system consisting of three series-parallel mechanisms have been proposed, but the stroke is not satisfactory [20]. Then, Dan et al. [21] have designed a 6-DOF compliant parallel manipulator with spatial beam flexure hinges and derived the load-displacement relationship for closed PSS-type limbs. What is more interesting is that a novel flexible jointed precision parallel nano-positioning system has been proposed by Kang et al., and it has a motion resolution of 15 nm and a rotational resolution of 0.14 arc sec [22]. In addition, Du et al. [23] have proposed a piezo-actuated high-precision flexible parallel pointing mechanism, which features sub-micro-radian resolution and microradian repeatability. Then, Zhang et al. [24] have proposed a high-resolution, high-repeatability, and low-parasitic motion 6-DOF parallel positioning system. Lu et al. [25] have proposed a novel parallel precision platform with wide-range flexure hinges. Yun et al. [26] have designed a novel 6-DOF 8-PSS/SPS compatible dual-redundant parallel robot with wide-range flexure hinges, and established a kinematic model based on the stiffness model and the Newton-Raphson method. Although these parallel mechanisms have a very good positioning accuracy, the load capacity is not very satisfactory.
In general, the micro-nano positioning platform need to be designed based on the requirements of load capacity and pointing accuracy. However, there are very few existing adjustment mechanisms that can achieve both large load capacity and high pointing accuracy. Therefore, it is very difficult to develop the mechanism with a load of 1.2 tons, which can have the advantage of micro-nano positioning.
In addition, the control based on the compliance model is also very complex. The kinematic model and dynamic model of the 3-PUPU dual parallel manipulator are established using the stiffness equation and Kane’s method, respectively [27]. Rouhani et al. [28] have proposed a solution to the inverse kinematics of microhexapods with flexure joints of varying rotation center based on elastokinematic model. Similarly, Hou et al. [29] have analyzed and derived an inverse kinematics model based on elastokinematic analysis. It is worth noting that the calculation time of the kinematic model based on the stiffness model exceeds 1 min in Du's work [30]. However, minute-level time is unacceptable, which cannot satisfy the real-time control. Therefore, an efficient kinematic model also needs to be improved.
Based on the previous contributions, a novel high stiffness micro-nano positioning hexapod platform with flexure hinges is proposed. Then, the novel flexure hinge has a mechanical limit, and its equivalent model is established and analyzed. In addition, a novel simplified inverse kinematic model is established for the hexapod platform based on the rigid body kinematic theory, which speeds up the solution process. And a rigid-flexible coupling simulation system is established to verify the correctness and applicability of the simplified kinematic model. Finally, the performance of hexapod platform is tested. Moreover, the resolution of the platform is analyzed based on statistical theory.
2. System architecture design
To compensate for the optical system deviations caused by the change of elevation angle and thermal gradient, a novel high stiffness hexapod platform is designed, as shown in Fig. 1. Its optical loading capacity is up to 1.2 tons. The hexapod platform is mounted at the center of the top of the telescope truss, and it mainly comprises of a moving platform, actuators, and a fixed base. As shown in Fig. 2, the actuator components include flexure hinges, incremental encoder, brushless torque motor, harmonic gear drive, planetary roller screw (40 mm diameter and 1 mm lead), hall switch, and magnetic grid. Piezoelectric motors are neglected due to short stroke and small output force.
It is worth noting that the load transfer element of the planetary roller screw is a threaded roller, which is typically a linear contact. In contrast, the load transfer element of the ball screw is a ball, which is in point contact. Therefore, the planetary roller screw has the advantages of high load capacity (both static and dynamic loads) and long service life compared to the ball screw. Compared with conventional gearboxes, harmonic drive gear systems offer the advantages of zero backlash, excellent positioning accuracy and repeatability. The flexure hinge has the advantages of no mechanical friction, no backlash, high motion sensitivity and no assembly, etc. Its application will greatly improve the indexes of stiffness, load capacity and positioning accuracy of the hexapod platform.
Then, the degrees of freedom of the platform can be expressed as
where F is the calculated degree of freedom of the platform, the number of components in the system n = 14, the number of joints g = 18, and the allowable degree of freedom of the j-th joint is fj. The planetary roller screw and the flexure hinge each provide 2 degrees of freedom. Therefore, the degree of freedom of the hexapod platform is 6.3. Simulation design of the flexure hinge
As shown in Figs. 3(a)-(c), the working principle of the flexure hinge is very similar to that of the traditional Hooke hinge. However, it has higher precision and higher stiffness. The bending deformation of the flexure hinge is very small when the parallel adjustment mechanism moves in a small range of motion. Therefore, the hinge shaft can be regarded as fixed in a small range of motion. To facilitate kinematic modeling and control, the fixed hinge shaft is set at its geometric center [31].
Fig. 3. The flexure hinge: (a) 3D model; (b) equivalent flexure hinge (Similar Hook hinges); (c) finished prototype.
It should be noted that the stiffness and the rotation limit of the flexure hinge affect the positioning accuracy and the workspace or compensation capacity of the hexapod platform, respectively. And the hexapod platform must be protected using mechanical limits, so the designed flexure hinge needs to have the structural characteristics of a safe limit (limit value is 2.5°). As shown in Fig. 3(a), the design of the limiting groove ensures that the flexure hinge does not deform plastically or break at the limit position. The flexure hinge can be manufactured by low speed one-way walk wire cut electrical discharge machining process (WEDM-LS). Along the direction of the boundary line of the notch, cut off two symmetrical pieces of material in the closed area enclosed by the limit groove to get it. Initially, the material used for the flexure hinge is titanium alloy, but its stiffness and rotational capacity are difficult to meet the needs of engineering applications. Alloy 17-4 PH is a precipitation hardening martensitic stainless, which has high strength and good corrosion resistance. Mechanical properties can be optimized with heat treatment. Very high yield strength up to 1100-1300 MPa can be achieved. The material property parameters can be expressed in Table 1.
Table 1. Material property parameters
In addition, the desired safety factor of the hinge is 1.4. The condition that the flexure hinge does not deform plastically or break at the limit position can be expressed as
As shown in Fig. 4, the axial stiffness and bending stiffness of the flexure hinge can be obtained from the finite element analysis (FEA). An axial force of 1 × 104 N and a bending moment of 100N·m are applied to the free end of the hinge, respectively. The results show that the axial stiffness and bending stiffness of the hinge are 3.15 × 108 N·m-1 and 1.17 × 103 N·m·rad-1, respectively. When the angle of rotation is 2.5°, the von Mises stress of the hinge is 574 MPa. The result shows that the flexure hinge does not deform plastically or break at the limit position.
In addition, to describe the position of the hinge shaft, the assumption is verified by observing the position of the point F (where OF = 64.1 mm.), as shown in Fig. 3(b). As shown in Fig. 3(a), the rotation angles around the y-axis and z-axis are Φy and Φz, respectively. The results of the finite element analysis are shown in Fig. 5. The ideal results are in general agreement with the simulated results. Therefore, the setting of the hinge shaft can be considered reasonable within the rotation range of 2.5°.
4. Kinematics analysis
In order to avoid complex deformation analysis of the flexure hinge and speed up the solution process, this section presents a simplified inverse kinematic model of the hexapod platform based on the rigid body kinematics theory.
In addition, a novel simplified model of the hexapod platform is given based on the working principle of the flexure hinge for a better description, as shown in Fig. 6. It should be pointed out that the hinge shaft is initially set at its geometric center. Therefore, the simplified hexapod platform is almost identical to the conventional hexapod platform.
The kinematic model of the simplified hexapod platform can be expressed in Fig. 7, where P-xAyAzA and O-xByBzB are the moving and fixed coordinate systems, respectively. It should be noted that Ai and Bi (i = 1, …, 6) are the centers of the upper and lower hinge shafts, respectively. The structural configuration of the hexapod platform is expressed in Table 2.
Table 2. Main geometric parameters
Assuming that the position OP and pose BRA of the hexapod platform are determined, the purpose of the inverse kinematic modeling is to obtain the length of the limb. Then, the vector li (i = 1, …, 6) of the limb can be expressed as
The rotation matrix BRA can be expressed as
It should be noted that α, β and γ are the angles of rotation of the mobile coordinate system around the x, y and z axes of the fixed coordinate system, respectively. In addition, c is the abbreviation of cos and s is the abbreviation of sin.
Further, the length of the limb simply is expressed as
According to the analysis of the internal structure of the limb, it is known that during the hexapod platform movement, the way the limb moves actually contains two completely different spiral motions. One is the spiral motion δL1 from the limb (It is defined as active spiral motion). The other one is the relative spiral motion δL2 of the screw and the screw nut during the movement of the upper platform (Similarly, it is defined as passive spiral motion). Therefore, the length variation δL of the limb can be expressed as
According to the previous analysis, the flexure hinge is simplified to a conventional Hooke hinge. Then, the flexure hinge in the kinematic model can also be considered as a Hooke hinge for calculation, as shown in Fig. 6. In the fixed coordinate system O-xByBzB, ki, ji, mi, and ni are the axial unit direction vectors of the hinge shaft in the upper and lower flexure hinge, respectively. According to the structural analysis of the flexure hinge, it is known that ki⊥ji, ki⊥li, mi⊥ni, mi⊥li. Therefore, the vectors ji, ki, and mi can be expressed as
In addition, in the moving coordinate system P-xAyAzA, the axial unit direction vector of the hinge shaft of the upper flexure hinge can be expressed as Bji. Therefore, the angle consisting of the vectors ki and mi can be expressed as
It should be noted that θi ∈[0°, 180°], which indicates that the default positive direction of the initial vector ni needs to be modified. Therefore, the angle θi is normalized to [0°, 90°] when the moving platform is located at the initial position. Finally, the vector ni (i = 1, …, 6) with the modified default positive direction can be expressed as
It needs to be noted that θi0 is the angle formed by the vector ki and mi when the hexapod platform is in the initial position. The change in the length of the limb caused by the passive spiral motion is related to the relative rotation angle of the screw and the screw nut. Therefore, defining the direction of counterclockwise rotation of the screw with respect to the screw nut as the positive direction, the relative rotation angle θi can be expressed as
Thus, δL2 can be expressed as
where n = 2 if the roller screw is rotating to the right screw. Otherwise, n = 1. In addition, P is the lead of the screw.The above solution describes the typical inverse kinematics of a hexapod platform to find the length of its six limbs from a given position and pose of the moving platform. In addition, the calculation time of the simplified inverse kinematic model is less than 50 ms, which can meet the requirements of real-time control.
5. Realization of co-simulation of rigid-flexible coupling system
Since the inverse kinematic model described in the previous section is done based on rigid body kinematics, the real deformation of the flexure hinge is not considered. Moreover, whether the hinge shaft of the flexure hinge can be set approximately at its geometric center has still not been confirmed intuitively and effectively. Furthermore, during the research of the hexapod platform with flexure hinges, the correctness and applicability of this kinematic modeling approach has hardly been verified by researchers. In addition, it is also necessary to analyze whether the configuration of the hexapod platform and the stiffness of the flexure hinge are reasonable. The related analysis is the guarantee of the positioning accuracy of the platform. Therefore, this approximate kinematic modeling approach still needs to be analyzed intuitively and effectively.
To demonstrate the correctness and applicability of the kinematic modeling theory, the flexure hinges are considered as flexible bodies and the remaining the components are considered as rigid bodies. Moreover, a rigid-flexible coupling simulation system is built using PATRAN & NASTRAN and ADAMS software to investigate the performance of the hexapod platform.
Firstly, all the flexure hinges in the hexapod platform are individually imported into PATRAN for meshing and the connection points are set using DOF lists. Secondly, the modal analysis of all flexure hinges is performed using NASTRAN software to export the MNF modal neutral file supported by ADAMS. Then, the hexapod platform assembly model is imported into ADAMS and all flexure hinges in the hexapod platform are replaced with the MNF modal neutral file. Finally, the connections of the hexapod platform are checked to complete the rigid-flexible coupling simulation model. The specific operation is shown in Fig. 8.
In order to study the performance of the hexapod platform in more depth, it is necessary to analyze it according to the requirements of engineering applications. In order to verify the correctness and applicability of the kinematic modeling theory, The analysis is based on a 1.2-ton load and a no-load case, respectively.
As can be learned from the Table 3, the output data of rotational motion is relatively accurate, while the output data of translational motion shows discrepancies relative to the desired data. The maximum error and parasitic rotation are less than 4.0 µm and 0.6 arc sec, respectively, when the translational motion in each of the three directions is set to 6 mm. When the rotational motion in each of the three directions is set to 0.6°, the maximum error and the parasitic translational motion are less than 0.65 arc sec and 1.3 µm, respectively. Similarly, the maximum deviation of the translational and rotational motions under an optical load of 1.2 tons is shown. When the translational motion in each of the three directions is set to 6 mm, the maximum error and parasitic rotation are less than 17.20 µm and 0.65 arc sec, respectively. When the rotational motion in each of the three directions is set to 0.6°, the maximum error and parasitic translation are less than 0.65 arc sec and 17.16 µm, respectively. It is worth noting that the maximum and parasitic error of the z-directional translational motion incorporates the deformation of the hexapod platform when an optical load is applied. This deformation is related to the configuration of the hexapod platform and the stiffness of the flexure hinge. Based on the analysis of the output data, the kinematic properties of the platform are not significantly affected. Therefore, the inverse kinematic model is still valid.
Table 3. Desired data and error (length unit: µm; angle unit: arc sec)
In order to illustrate the kinematic limitation, a trajectory needs to be planned. Then, the position OP = [x y z]’ and pose BRA of the moving platform can be represented as
As shown in Fig. 9, the simulation results show that the error increases as the range of motion increases. Therefore, the kinematic theory is only applicable to a small range of motion.
In most common cases, the smaller the discrepancy between the output and the desired data, the more the correctness and applicability of the kinematic modeling theory can be demonstrated. There are three main reasons for the mismatch of simulation results here. One is that the difference between the flexure hinge and the traditional Hooke hinge is ignored. The difference is mainly reflected in the fact that the hinge shaft of the flexure hinge is idealized to be fixed. The second may be the poor quality of the mesh in the finite element model. The third is related to the stiffness of the flexure hinge and the configuration of the platform.
In general, since the parasitic motions are small compared to the main motions, the effect of parasitic motions on the positioning accuracy of the mechanism can be neglected in small motions. Therefore, if the flexure hinge has a relatively small range of motion with reasonably stiffness, and the configuration of the platform is excellent, the kinematic modeling theory still applies to the hexapod platform with the same type of flexure hinges. On the contrary, the position of the hinge shaft needs to be redefined. Or if the relationship between the displacement and parasitic motion is known, then the positioning accuracy of the platform can be further improved by error compensation.
Finally, the lateral forces Fx and Fy are applied to the moving platform respectively. The stiffness is initially estimated by observing the displacement of the moving platform. The results show that the transverse stiffness of the platform is 139.71 N·µm-1.
6. Prototype fabrication and experimental investigation
6.1 Experimental system
In order to investigate the performance of the hexapod platform in more depth and to identify any manufacturing or assembly errors, its resolution, repeatability, and stiffness have to be tested. Prototype system of the hexapod platform is shown in Fig. 10 and experimental system is shown in Fig. 11. In addition, the hexapod platform is required to be able to perform well under an optical load of 1.2 tons. Hence, the resolution and repeatability of the hexapod platform need to be tested under loaded conditions.
Then, the platform is controlled in open loop by inverse kinematics, and each limb is controlled in closed loop, as shown in Fig. 12. The desired position and pose [xr, yr, zr, αr, βr, γr] of the moving platform need to be given explicitly. Then, the length li of each limb is calculated based on the inverse kinematics. The final output position and pose [xo, yo, zo, αo, βo, γo] of the moving platform can be obtained by controlling the motor.
It is worth noting that the position and pose of the moving platform can be measured using a laser displacement sensor. It has an absolute error of less than ±0.03% and a resolution of 0.03 µm within a linear range of 2 mm. In addition, translational motion can be measured directly. However, the rotational motion cannot be measured directly by the laser displacement sensor. The error in inverse kinematics is ignored because the range of motion is relatively small compared to a conventional parallel positioning system. Therefore, the rotational motion can be measured using the approximation principle, as shown in Fig. 13. The rotational motion of the moving platform around the x-axis is specified as α, and the reference point is point 1. However, the final measured distance is the distance between point 2 and point 1.
The same is true for the other two directions of rotational motion. when α, β, and γ are very small. Then, α, β, and γ can be approximated as
Then, α, β, and γ can be expressed as
It is worth noting that Rx, Ry and Rz are the measured radius of point 1, point 2 and point 3, respectively. And Hx, Hy and Hz are the measured distances of the three points, respectively.
6.2 Stiffness evaluation
It is worth noting that the flexure hinge exhibits high axial stiffness and low bending stiffness. The lateral stiffness is an important index to determine the stability of hexapod platform. The higher the lateral stiffness, the stronger the anti-interference ability of the system. As shown in Fig. 11, a push-pull gauge and laser displacement sensor are used to measure the force and displacement of the moving platform in real time.
The results of the test are shown in Fig. 14, where two straight lines are obtained by linear fitting of the experimental data. The slope of the straight lines is the lateral stiffness of the platform along the x-axis and y-axis directions, respectively. The results show that the lateral stiffness along the x-axis and y-axis are as high as 131.6 N$\cdot$µm-1 and 133.0 N$\cdot$µm-1, respectively. Based on the analysis in section 5, the relative errors of the simulation results and the experimental results are 5.80% and 4.80%, respectively. The desired lateral stiffness of the platform is not less than 80 N$\cdot$µm-1. This experimental result shows that the designed hexapod platform has very good lateral stiffness and configuration.
6.3 Positioning resolution evaluation
Resolution in a positioning system is the ability to determine the minimum increment of motion allowed by the hexapod platform, which is the smallest mechanical step in the positioning system for point-to-point motion. As determined by the analysis of the requirements for the use of the secondary mirror, if the hexapod platform is able to achieve resolutions of 0.5 µm and 0.5 arc sec, the deviations in the optical system that have an impact on the image quality are well compensated.
It is important to note that the measurement method of the laser displacement sensor is non-contact and real-time. In order for the data obtained from the test to be stable and credible, the duration of the test needs to be extended. Each segment of test data is considered as a number of discrete data points, which is not convenient to measure the performance of the platform. Importantly, gross errors caused by mechanical shocks, external vibrations and other changes in measurement conditions are inevitable during the test, which can produce significant distortions in the measurement results. In order to eliminate data points containing gross errors in the measurement data, the data points of each segment are processed based on the 3σ principle.
Each set of measurement data is divided into j (j = 1, …, 7) segments. It is assumed that the value of each data point is xji (i = 1, …, Nj) and the number of data points is Nj. The average value of the test data can be expressed as
Next, the residual error for each test data point can be derived from
Then, the standard deviation σj of the sample data can be expressed as
According to the discrimination criteria of the 3σ principle, the residual errors of the data points containing gross errors can be expressed as
It is assumed that the number of data points satisfying Eq. (20) is Mj. After eliminating the above data points, the value of the data points in the new sample is x′j. At this point, the average value and residual error of the data in the new sample can be expressed as
According to the Bessel formula, the standard deviation of the data in the new sample can be expressed as
To avoid wasting a lot of time solving for the standard deviation, Eq. (23) can be expressed as
To evaluate the step size of each data set, each segment of the processed data is expressed as
where, xtj is the theoretical value of each segment of data.Then, the standard deviation of each data set can be expressed as
The actual step size of each data set can be expressed as
The test results are shown in Figs. 15, 16. The results of the resolution analysis are shown in Table 4. It should be noted that the minimum mechanical steps of translational and rotational motions need to be no greater than 0.3 µm and 0.5 arc sec, respectively. Although the test data contain some noise, the results still show that the platform can be used to correct the optical system bias.
Table 4. Resolution test
6.4 Positioning repeatability evaluation
Repeatability of motion is the ability of a hexapod platform to repeat its motion. Or its ability to reach the same position. It can be evaluated by the error when moving to the same position several times in succession within the coverage of the positioning system. Repeatability tests are performed by moving from the zero position to the negative position, back to zero, to the positive position, back to zero and repeating the sequence several times. The repeatability of the hexapod platform is measured by the periodic error, using the data points at the four position moments as a period for comparison.
The step size for this test is chosen to be 0.5 mm and 1250 arc sec, as this represents a larger displacement compared to the size of the canonical shift during actual telescope operation. It is important to note that the desired repeatability of translational and rotational motions is required to be no greater than ±0.5 µm and ±0.5 arc sec, respectively. As shown in Fig. 17, the repeatability of the hexapod platform is ±0.5 µm for bi-directional translational motion and ±0.5 arc sec for bi-directional rotational motion. These results are considered excellent performance and are sufficient to meet the repeatability requirements of the adjustment mechanism. Therefore, the results show that the platform can be used to improve the optical imaging quality.
7. Conclusion
In this paper, a novel high stiffness micro-nano positioning hexapod platform with flexure hinges is proposed to ensure the accurate alignment of the secondary mirror. The designed novel flexure hinge has a mechanical limit and high stiffness, allowing the platform to achieve high accuracy.
Then, a simplified inverse kinematic model of the hexapod platform based on the rigid body kinematics theory of the parallel mechanism is proposed, which speeds up the solution process. And an effective rigid-flexible coupling simulation system is established to derive and verify the simplified inverse kinematic model. The results show that if the flexure hinge has a relatively small range of motion with reasonably stiffness, and the configuration of the platform is excellent, the simplified inverse kinematic model still applies to the hexapod platform with the same type of flexure hinges. On the contrary, the position of the hinge shaft needs to be redefined.
In addition, a more systematic test method and analysis theory are proposed. The experimental results show that the translation resolution and rotation resolution of the hexapod platform are 0.3 mm and 0.5 arc sec. The bi-directional repeatability of translation and rotation can reach ±0.5 µm and ±0.5 arc sec, respectively. And the lateral stiffness along the x-axis and y-axis are as high as 131.6N$\cdot$µm-1 and 133.0N$\cdot$µm-1, respectively. The results show that the designed platform can be used to correct the optical system bias.
The research in this paper will provide suggestions for the optimization of the structure and control algorithm of parallel mechanisms with similar flexure hinges to achieve better performance. And if the relationship between the displacement and parasitic motion is known, then the positioning accuracy of the platform can be further improved by error compensation.
Funding
National Natural Science Foundation of China (11972343, 12133009, 62235018).
Acknowledgments
This work was supported by the National Nature Science Foundation of China (11972343, 12133009, and 62235018). The author would like to thank Hang Li for her help in the revision.
Disclosures
The authors declare no conflicts of interest.
Data availability
Datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.
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