Satellite flexible structure static deformation compensation method based on piezoelectric actuator

Abstract

The application discloses a satellite flexible structure static deformation compensation method based on a piezoelectric actuator and belongs to the technical field of spacecraft control. The operation steps of the method comprise the following steps: installing p piezoelectric actuators at different positions of a flexible structure; measuring the static deformation of the flexible structure at m different measuring points; calculating the modal coordinates corresponding to the static deformation under the first n modes of the flexible structure, wherein p>n; taking the piezoelectric actuators as external force excitation sources of the flexible structure to construct a dynamic equation of the flexible structure; calculating the input voltage of each piezoelectric actuator according to the dynamic equation and the modal coordinates, thereby controlling the piezoelectric actuators to compensate the static deformation of the flexible structure. The method can quickly realize accurate and efficient compensation of the static deformation of the satellite flexible structure.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft control technology, specifically relating to a method for compensating for static deformation of a satellite flexible structure based on a piezoelectric actuator. Background Technology

[0002] In recent years, spacecraft structures have been developing towards larger size, lower stiffness, and greater flexibility, which has led to some new technical challenges. For example, due to their low structural stiffness and dense modal frequencies, flexible structures on satellites are prone to static shape changes in the complex space environment. These changes in the static shape of flexible satellite structures reduce component performance and pointing accuracy, and severely impact satellite performance, such as causing blurred imaging and reduced resolution. Therefore, it is essential to compensate for (or control) the static deformation of flexible satellite structures.

[0003] Currently, commonly used static deformation compensation methods mainly include: using piezoelectric actuators to control the static deformation of the satellite reflector, using cable net length adjustment to control the antenna reflector shape, and adding actuators behind the satellite reflector for position and curvature adjustment, as detailed below:

[0004] (1) Cao Yuyan, Wang Zhi, Zhou Chao, et al. disclosed a method for static shape control of large-diameter reflective surfaces using piezoelectric actuators in their paper "Static Shape Control and Actuator Position Optimization of Piezoelectric Intelligent Reflective Surfaces". This method attaches the piezoelectric actuator to the outside of the skin of the reflective surface for actuation control and uses the finite element total displacement vector of the structure for deformation control calculation. The disadvantage of this method is that attaching the piezoelectric actuator to the outside of the skin of the reflective surface for actuation control requires the piezoelectric actuator to be attached to the planar structure, which is not suitable for flexible truss structures. Moreover, this method uses the finite element total displacement vector of the structure for deformation control calculation. For more complex structures, the dimension of the total displacement vector will increase, resulting in an increase in the amount of calculation.

[0005] (2) Patent No. CN201410175151.0, entitled "A Threaded Differential Cable Length Adjustment Device and Adjustment Method for Deployable Cable Net Reflector," describes a novel threaded differential cable length adjustment device for adjusting the surface accuracy of the cable net deployable antenna reflector. Wang Lamei, in her research "Research on Surface Accuracy Adjustment of Cable Net Reflector," used this threaded differential cable length adjustment device as the actuator. She used a 3D photography system to measure the coordinates of each node, and then used the measured node coordinates to perform surface fitting. The difference between each node and the fitted surface was used to determine the adjustment amount. After adjustment, measurements and surface fitting were performed again until the surface accuracy no longer improved. While this method can effectively improve the shape accuracy of the cable net reflector, the shape adjustment is manual and requires multiple iterations of measurement and manual adjustment. This results in low shape control efficiency and a complex system, making it suitable for precise ground-based adjustments but difficult to implement in orbit.

[0006] The paper JWST Mirror Production Status discloses a method for shape control of the James Webb Space Telescope. This method divides a large reflective surface into 18 mirrors, each with seven dedicated precision actuators. These actuators adjust the position and curvature of each mirror to achieve precise control of the overall mirror shape. However, this method requires a dedicated measurement system and actuators, and the large number of actuators and drive motors results in a complex and expensive control system. Summary of the Invention

[0007] In view of this, the present invention provides a method for compensating the static deformation of a satellite flexible structure based on a piezoelectric actuator. This method calculates the modal coordinates corresponding to the static deformation of the satellite flexible structure, and then obtains the input voltage of the piezoelectric actuator. It can quickly achieve accurate and efficient compensation for the static deformation of the satellite flexible structure with only conventional measurement systems on the satellite and a small number of actuators.

[0008] A method for compensating static deformation of a satellite flexible structure based on a piezoelectric actuator, comprising the following steps:

[0009] Install p p piezoelectric actuators at different locations on the flexible structure;

[0010] Measure the static deformation of the flexible structure at m different measurement points;

[0011] Calculate the modal coordinates corresponding to the static deformation in the first n modes of the flexible structure, where p>n;

[0012] Using the piezoelectric actuator as the external force excitation source of the flexible structure, the dynamic equation of the flexible structure is constructed;

[0013] Based on the dynamic equations and the modal coordinates, the input voltage of each piezoelectric actuator is calculated, thereby controlling the piezoelectric actuator to compensate for the static deformation of the flexible structure.

[0014] Furthermore, the relationship between the static deformation and the modal coordinates is as follows:

[0015]

[0016] Among them, w j Let η' be the static deformation three-dimensional displacement vector of the j-th measurement point, j = 1, 2, ..., m; i Let be the coordinates of the i-th modal corresponding to the static deformation, i = 1, 2, ..., n; Let be the mode shape of the flexible structure at the j-th measurement point and the i-th mode.

[0017] Furthermore, when 3m > n, the modal coordinates are calculated using the least squares method as follows:

[0018]

[0019] in:

[0020] Furthermore, the dynamic equation of the flexible structure is:

[0021]

[0022] in: Here is the modal damping matrix; Here is the modal stiffness matrix; ζ i ω is the damping ratio of the i-th mode; i η is the natural frequency of the i-th order vibration of the flexible structure; η is the modal coordinate of the flexible structure; Φ is the mode shape of the flexible structure; B is the position matrix composed of the direction cosines of p piezoelectric actuators; K v It is a diagonal matrix. V represents the control force generated by the l-th piezoelectric actuator under a unit voltage, where l = 1, 2, ..., p. piezo Let V be a column vector consisting of the input voltages of the piezoelectric actuator, and its components be denoted as V. l , representing the input voltage of the l-th piezoelectric actuator.

[0023] Furthermore, let

[0024] η=-η′

[0025] Therefore, the output voltage of the piezoelectric actuator is calculated as follows:

[0026]

[0027] in,

[0028] Furthermore, the input voltage is limited using the following formula:

[0029]

[0030] Where V l V is the input voltage of the l-th piezoelectric actuator after limiting, where l = 1, 2, ..., p; max V is the maximum operating voltage of the piezoelectric actuator. min This is the minimum operating voltage of the piezoelectric actuator.

[0031] Furthermore, the flexible structure is a flexible truss.

[0032] Furthermore, the piezoelectric actuator is a piezoelectric ceramic stacked actuator.

[0033] Furthermore, when measuring the static deformation of the flexible structure, a method of averaging multiple measurements is adopted.

[0034] Furthermore, the length of the flexible truss ranges from 50 to 300 m.

[0035] Beneficial effects:

[0036] 1. The method provided by this invention first calculates the modal coordinates corresponding to the static deformation of the satellite flexible structure, and then obtains the input voltage of the actuators installed at different positions of the satellite flexible structure according to the relationship between the dynamic equation of the satellite flexible structure and the modal coordinates. This achieves compensation for the static deformation of the satellite flexible structure, avoiding the direct calculation of the input voltage of the piezoelectric actuator through the static deformation displacement of the satellite flexible structure. The calculation is small, requiring only a conventional measurement system and a small number of actuators. It is also applicable to the compensation of static deformation of the flexible structure of on-orbit satellites.

[0037] 2. The method provided by this invention measures the static deformation of a satellite's flexible structure at different measurement points as a three-dimensional displacement vector, resulting in more accurate static deformation measurement results. This ensures the accuracy of subsequent piezoelectric actuator input voltage calculations and reduces errors in static deformation compensation. Furthermore, based on the relationship between the static deformation of the flexible structure and its modal coordinates, appropriate modal orders can be flexibly selected according to actual needs, further guaranteeing high accuracy in static deformation compensation.

[0038] 3. The method provided by this invention selects the least squares method to solve the modal coordinates based on the relationship between the static deformation and modal coordinates of the satellite flexible structure. The solution speed is fast, so that the static deformation of the satellite flexible structure can be quickly compensated.

[0039] 4. The method provided by this invention calculates the input voltage of the piezoelectric actuator and then limits it according to the corresponding formula, so that the input voltage of the piezoelectric actuator is always within its normal operating voltage range, ensuring that the piezoelectric actuator is not damaged and improving the safety performance of the satellite.

[0040] 5. The present invention employs a piezoelectric ceramic stacked actuator, commonly used in flexible truss vibration control, as an actuator for static deformation control of flexible trusses. This actuator is easy to install in truss members, avoiding the problems of existing methods that use piezoelectric ceramic sheets or piezoelectric polymers as actuators for static deformation compensation. In flexible truss deformation compensation, the piezoelectric sheets are inconvenient to install or require a large number of specially designed actuators, thus increasing the complexity of the system.

[0041] 6. The method provided by this invention takes into account that the deformation at the measurement point of the satellite's flexible structure changes with time during satellite flight, and that errors are inevitable in measuring static deformation displacement. Therefore, when measuring the static deformation displacement of the satellite's flexible structure, the method of averaging multiple measurements is adopted to obtain the average static deformation displacement vector of each measurement point. Moreover, the measurement does not require the satellite's flexible structure to be completely stationary; it only needs to be carried out during the stage when its static deformation is basically stable. The measurement results are highly reliable, ensuring the accuracy of static deformation compensation of the flexible structure.

[0042] 7. This invention is applicable to static deformation compensation of large flexible satellite structures up to 100 meters long. Using the method provided by this invention, only twelve piezoelectric ceramic stacked actuators are arranged at different locations on the flexible truss, and the static deformation of the flexible truss is measured at twenty measurement points. Considering the first four modes of the flexible truss, the static deformation of the flexible truss is rapidly compensated, and after compensation, the remaining deformation of the flexible truss is reduced to 0.9 mm, demonstrating high efficiency. Attached Figure Description

[0043] Figure 1 This is a schematic diagram showing the installation position of the piezoelectric ceramic stacked actuator in the flexible truss in Embodiment 1 of the present invention.

[0044] Figure 2 This is a schematic diagram of the static deformation displacement values ​​of each measurement point before static deformation compensation of the flexible truss in Embodiment 1 of the present invention.

[0045] Figure 3 This is a schematic diagram of the residual deformation values ​​at each measurement point after static deformation compensation of the flexible truss in Embodiment 1 of the present invention. Detailed Implementation

[0046] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0047] This invention provides a method for compensating for static deformation of a satellite flexible structure based on a piezoelectric actuator, comprising the following steps:

[0048] Install p p piezoelectric actuators at different locations on the flexible structure;

[0049] Measure the static deformation of a flexible structure at m different measurement points;

[0050] Based on the relationship between static deformation and the first n modal coordinates of the flexible structure, calculate the modal coordinates corresponding to the first n modes of the flexible structure under static deformation;

[0051] Using piezoelectric actuators as the external force excitation source for flexible structures, the dynamic equations of flexible structures are constructed.

[0052] Based on the dynamic equations and modal coordinates, the input voltage of each piezoelectric actuator is calculated, thereby compensating for the static deformation of the flexible structure by controlling the piezoelectric actuators.

[0053] It is worth noting that in order to calculate the input voltage of the piezoelectric actuator, this method should satisfy p>n.

[0054] As can be seen, this method first calculates the modal coordinates corresponding to the static deformation of the satellite's flexible structure, and finally obtains the input voltage of the actuators installed at different positions on the satellite's flexible structure based on the relationship between the dynamic equation of the satellite's flexible structure and the modal coordinates. This achieves compensation for the static deformation of the satellite's flexible structure, avoiding the direct calculation of the piezoelectric actuator's input voltage through the static deformation displacement of the satellite's flexible structure. The computational load is small, requiring only a conventional measurement system and a small number of actuators. It is also applicable to the static deformation compensation of the flexible structure of on-orbit satellites.

[0055] In this method, specifically, the relationship between static deformation and modal coordinates is shown in the following formula (1):

[0056]

[0057] Among them, w j Let η' be the static deformation three-dimensional displacement vector of the j-th measurement point, where j = 1, 2, ..., m; i Let be the coordinates of the i-th modal corresponding to the static deformation, i = 1, 2, ..., n; Let i be the i-th mode shape of the flexible structure at the j-th measurement point.

[0058] It can be seen that when measuring the static deformation of the satellite's flexible structure at different measurement points, this method transforms the static deformation into a three-dimensional displacement vector, resulting in more accurate static deformation measurement results. This, in turn, ensures the accuracy of subsequent piezoelectric actuator input voltage calculation and reduces the error in static deformation compensation. Furthermore, based on the relationship between the static deformation of the flexible structure and the modal coordinates, the appropriate modal order can be flexibly selected according to actual needs, further ensuring the high accuracy of static deformation compensation.

[0059] More specifically, the matrix form of the above formula (1) is shown in the following formula (2):

[0060]

[0061] in:

[0062] When 3m > n, the modal coordinates are calculated using the least squares method as shown in formula (3).

[0063]

[0064] in, It is Φ measure The transpose of .

[0065] In fact, different methods can be used to calculate modal coordinates. Based on the relationship between the static deformation of the satellite's flexible structure and the modal coordinates, when the number of piezoelectric actuators and the modal order satisfy 3m>n, the least squares method is chosen to solve the modal coordinates. This method is fast and allows the static deformation of the satellite's flexible structure to be quickly compensated.

[0066] More specifically, the dynamic equation of the above-mentioned flexible structure is shown in equation (4):

[0067]

[0068] in: Here is the modal damping matrix; Here is the modal stiffness matrix; ζ i ω is the damping ratio of the i-th mode; i η is the natural frequency of the i-th order vibration of the flexible structure; η is the modal coordinate of the flexible structure; Φ is the mode shape of the flexible structure; B is the position matrix composed of the direction cosines of p piezoelectric actuators; K v It is a diagonal matrix. V represents the control force generated by the l-th piezoelectric actuator under a unit voltage, where l = 1, 2, ..., p. piezo Let V be a column vector consisting of the input voltages of the piezoelectric actuator, and its components be denoted as V. l , representing the input voltage of the l-th piezoelectric actuator.

[0069] The above matrix definition is consistent with the definition on pages 129-131 of the literature "Li Dongxu, Dynamic Analysis and Fuzzy Vibration Control of Large Flexible Space Truss Structures, Beijing, Science Press, 2008".

[0070] In theory, in order for the piezoelectric actuator to fully compensate for the static deformation of the flexible structure, formula (5) should be satisfied:

[0071] η=-η′ (5)

[0072] Therefore, the input voltage mentioned above can be calculated according to formula (6):

[0073]

[0074] in, It is K piezo The transpose of .

[0075] As an improvement, the input voltage of the piezoelectric actuator calculated by formula (6) can also be limited according to formula (7):

[0076]

[0077] Where V l The input voltage of the l-th piezoelectric actuator after limiting, l = 1, 2, ..., p; V max V is the maximum operating voltage of the piezoelectric actuator. min This is the minimum operating voltage of the piezoelectric actuator.

[0078] This ensures that the input voltage of the piezoelectric actuator is always within its normal operating voltage range, preventing damage to the piezoelectric actuator and improving the safety performance of the satellite.

[0079] Example 1:

[0080] Based on the above method, this embodiment compensates for the static deformation of a 100m long flexible truss finite element model with a symmetrical structure.

[0081] like Figure 1 As shown, in a certain symmetrical flexible truss with a length of 100m (due to the symmetry of the flexible truss), Figure 1The diagram shows only half of the flexible truss. A total of twelve piezoelectric ceramic stacked actuators are installed (p=12, of which six are labeled ① and ②). The number of nodes where the piezoelectric ceramic stacked actuators are installed is also given. The piezoelectric ceramic stacked actuators, which are commonly used in the vibration control of flexible trusses, are used as the actuators for static deformation control of flexible trusses. They are easy to install in the truss members. This avoids the problem that existing methods using piezoelectric ceramic sheets or piezoelectric polymers as actuators for static deformation compensation are not suitable for installing piezoelectric sheets in flexible truss deformation compensation, or require a large number of specially designed actuators to perform deformation compensation, thus increasing the complexity of the system.

[0082] Specifically, because the flexible truss has a triangular cross-section, four piezoelectric ceramic stacked actuators are installed along the longitudinal direction of the flexible truss on each of the three long sides to better compensate for the static deformation of the flexible truss.

[0083] Step 1: Calculation of modal coordinates corresponding to static deformation

[0084] Twenty measuring points (m=20) were set up on the flexible truss. Before compensating for the static deformation of the flexible truss, the static deformation displacement values ​​at the measuring points were as follows: Figure 2 As shown, the flexible truss exhibits significant deformation in the Y direction, with a maximum static deformation displacement of 6.16 mm (this static deformation displacement was calculated using a finite element model; during actual satellite in-orbit measurements, the static deformation displacement was obtained by averaging multiple deformation displacement measurements).

[0085] Based on the mode shape at the measurement point, considering the first four modes (n=4) of the flexible truss, the model coordinates corresponding to the static deformation are calculated as follows:

[0086]

[0087] Step 2: Calculation of input voltage for piezoelectric ceramic stacked actuator

[0088] Based on the finite element model of the flexible truss, matrix K is calculated. piezo The values ​​are shown in Table 1 below:

[0089]

[0090] The required control voltage V for the piezoelectric ceramic stack actuator is calculated based on the formula (7) above. piezo The 12 components are (unit: V): 392.19, -173.46, 250.80, 333.60, -218.96, 207.15, 90.45, 16.20, 19.40, 78.63, 13.66, 16.09.

[0091] Step 3: Limiting the input voltage of the piezoelectric ceramic stacked actuator

[0092] In this embodiment, the maximum operating voltage V of the piezoelectric ceramic stack actuator is... max With a minimum operating voltage of -150V and a voltage of 500V, the input voltage is limited, resulting in the following input voltages (in V) for the 12 piezoelectric ceramic stacked actuators: 392.19, -150.00, 250.80, 333.60, -150.00, 207.15, 90.45, 16.20, 19.40, 78.63, 13.66, and 16.09.

[0093] like Figure 3 The diagram shows the residual deformation values ​​at various measurement points after static deformation compensation of the flexible truss. The residual deformation is reduced to less than 0.9 mm after compensation. It can be seen that this method effectively compensates for the static deformation of the flexible truss.

[0094] It is worth noting that this method is applicable to flexible trusses ranging from 50 to 300 meters. Moreover, when conducting ground tests, only conventional laser displacement sensors are needed to measure the static deformation of the flexible truss, while during on-orbit measurements, conventional lasers and position-sensitive detectors (PSDs) on satellites can be used to measure the static deformation of the flexible truss.

[0095] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for compensating static deformation of a satellite flexible structure based on a piezoelectric actuator, characterized in that: Will A piezoelectric actuator is installed at different locations on the flexible structure; Measuring the flexible structure at Static deformation at different measurement points; Calculate the static deformation in front of the flexible structure The modal coordinates corresponding to the first mode, where ; Using the piezoelectric actuator as the external force excitation source of the flexible structure, the dynamic equation of the flexible structure is constructed; Based on the dynamic equations and the modal coordinates, the input voltage of each piezoelectric actuator is calculated, thereby controlling the piezoelectric actuator to compensate for the static deformation of the flexible structure.

2. The method for compensating static deformation of a satellite flexible structure based on a piezoelectric actuator as described in claim 1, characterized in that, The relationship between the static deformation and the modal coordinates is as follows: in, For the first The static deformation three-dimensional displacement vector of each measurement point =1,2,… ; The first corresponding to static deformation First modal coordinates, =1,2,… ; For flexible structures in the first The measurement point is the first Mode shapes under first-order modes.

3. The static deformation compensation method for satellite flexible structures based on piezoelectric actuators as described in claim 2, characterized in that, When 3 > When the modal coordinates are calculated using the least squares method, the following is given: in: , .

4. The static deformation compensation method for satellite flexible structures based on piezoelectric actuators as described in claim 1 or 2, characterized in that, The dynamic equation of the flexible structure is: in: Here is the modal damping matrix; Here is the modal stiffness matrix; For the first i First-order modal damping ratio; For flexible structures i The natural frequency of the first-order vibration; The modal coordinates of the flexible structure; For flexible structures; for p A position matrix composed of the direction cosines of each piezoelectric actuator; It is a diagonal matrix. , For the first The control force generated by a piezoelectric actuator under a unit voltage. =1,2,…p; Let be a column vector consisting of the input voltages of the piezoelectric actuator, and let its components be denoted as . , indicating the first The input voltage of a piezoelectric actuator.

5. The static deformation compensation method for satellite flexible structures based on piezoelectric actuators as described in claim 4, characterized in that: Therefore, the output voltage of the piezoelectric actuator is calculated as follows: in, .

6. The static deformation compensation method for satellite flexible structures based on piezoelectric actuators as described in claim 1, 2, or 5, characterized in that, The input voltage is limited using the following formula: in For the first time after the limit The input voltage of a piezoelectric actuator =1,2,… ; This is the maximum operating voltage of the piezoelectric actuator. This is the minimum operating voltage of the piezoelectric actuator.

7. The static deformation compensation method for satellite flexible structures based on piezoelectric actuators as described in claim 1, 2, or 5, characterized in that, The flexible structure is a flexible truss.

8. The method for compensating for static deformation of a satellite flexible structure based on a piezoelectric actuator as described in claim 1, 2, or 5, characterized in that, The piezoelectric actuator is a piezoelectric ceramic stacked actuator.

9. The method for compensating for static deformation of a satellite flexible structure based on a piezoelectric actuator as described in claim 1, 2, or 5, characterized in that, When measuring the static deformation of the flexible structure, an averaging method is used.

10. The method for compensating static deformation of a satellite flexible structure based on a piezoelectric actuator as described in claim 7, characterized in that, The length of the flexible truss ranges from 50 to 300 meters.

Images