A satellite solar array deployment co-simulation method based on MATLAB and ADAMS

Abstract

The application discloses a satellite sailboard deployment joint simulation method based on MATLAB and ADAMS, first, a satellite sailboard deployment dynamics simulation system is constructed based on ADAMS software, second, a multi-board deployment simulation model of the satellite sailboard is established by using a second Lagrange method, and a dynamics model of the system is obtained; next, Runge-Kutta with four-order high precision is used to solve the dynamics differential equation by using MATLAB software, and a deployment process of the satellite sailboard and a response result of the satellite sailboard subjected to impact are given; then, ADAMS dynamics simulation results (S1) are compared with theoretical model results (S3), and the accuracy of the dynamics theoretical model is verified within a certain error; finally, an optimal control law of state linear feedback of the satellite sailboard deployment system is derived based on a linear quadratic regulator (LQR) control algorithm; the application solves the problem that the satellite sailboard cannot be quickly and stably deployed, and can provide certain theoretical guidance for the structural design and on-orbit deployment of the space satellite sailboard.

Description

Technical Field

[0001] This invention belongs to the field of aerospace technology, specifically relating to a joint simulation method for satellite solar panel deployment. Background Technology

[0002] In recent years, with the continuous development of aerospace technology, research on large-scale flexible space structures such as solar panels, synthetic aperture radar antennas, and tethered satellites has attracted widespread attention from scholars around the world. Among them, space solar panels, as an important energy guarantee component for satellites, are widely used in various spacecraft to provide energy for the normal operation of spacecraft and are the most common form of large-scale space structure. Such structures often have dynamic characteristics such as low frequency, dense modes, low damping, and strong nonlinearity. Inevitably, the solar panels are prone to small-scale vibrations during the deployment and locking process, which can then cause vibrations in the entire satellite through the connecting frame. At the same time, due to the lack of air damping in the external space environment, once vibrations occur, the solar panel structure is difficult to naturally decay in a short period of time, posing a certain challenge to the strength and attitude stability control of the overall structure.

[0003] Currently, space satellite dynamics models often employ rigid-flexible coupling (central rigid body and flexible solar panel) methods and are built using finite element method (FE) software. However, both methods have certain limitations under specific conditions. On one hand, the accuracy of rigid-flexible coupling models decreases significantly under conditions of excessive solar panel deformation and displacement, making them difficult to apply to situations with large vibration amplitudes. Furthermore, the presence of nonlinear coupling terms makes solving the equations very difficult. On the other hand, while finite element method (FE) software can more accurately simulate the deployment characteristics of satellite solar panels, it consumes a significant amount of computation time, and it also struggles to solve nonlinear problems.

[0004] In practical applications, existing space satellite solar panels and their connecting structures are extremely complex, with complex conditions involving mutual interference and coupling between the main body and the panels, as well as among the individual panels. Furthermore, during deployment or locking, satellite solar panels are highly susceptible to impacts and interference from the external space environment, such as high-energy radiation and high-speed particles. Simplifying the satellite solar panel model while maintaining a certain level of accuracy, and addressing the connecting structures to ensure they function effectively as connectors while efficiently controlling the rapid and stable deployment of the solar panels, are key technical challenges in this field. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, this invention provides a joint simulation method for satellite solar panel deployment based on MATLAB and ADAMS. First, a satellite solar panel deployment dynamics simulation system is constructed using ADAMS software. Second, a multi-panel deployment simulation model of the satellite solar panel is established using the second-order Lagrange method, yielding the system's dynamic model. Next, the dynamic differential equations are solved using MATLAB software with fourth-order high-precision Runge-Kutta calculus, providing the deployment process of the satellite solar panel and its impact response results. Then, the ADAMS dynamics simulation results (S1) are compared and analyzed with the theoretical model results (S3), verifying the accuracy of the dynamics theoretical model within a certain error range. Finally, based on the linear quadratic regulator (LQR) control algorithm, the optimal control law of the state linear feedback of the satellite solar panel deployment system is derived. This invention solves the problem of the inability of satellite solar panels to deploy quickly and stably, and can provide theoretical guidance for the structural design and on-orbit deployment of space satellite solar panels.

[0006] The technical solution adopted by this invention to solve its technical problem includes the following steps:

[0007] S1: Use ADAMS software to build a satellite solar panel deployment dynamics simulation system;

[0008] S11: Create a 3D geometric model of the satellite solar panel, add material properties, and define connections;

[0009] In ADAMS software, a three-dimensional geometric model of the satellite solar panel is constructed according to the input dimensions. Material properties are added to the structure, and fixed joints and revolute joints are set at the connection points between the ground and the satellite body, and between the satellite body and the solar panel, respectively.

[0010] S12: Set driving force;

[0011] The specific method of using a torsion spring damper to drive the deployment of the sail is to set a preload angle, and then use a preset rotation angle to fully deploy the sail. Therefore, the formula for calculating the torque provided by the torsion spring is:

[0012] W tor =k×(β-β) pre )

[0013] Among them: W tor Let β be the torque, k be the torsional stiffness, and β be the torsional angle; pre The preset angle is a positive value, indicating that the relative motion angle is clockwise; otherwise, it is counterclockwise.

[0014] S2: Establish the simulation dynamic equations for the multi-plate deployment of satellite solar panels;

[0015] S21: Define the analysis model;

[0016] Assume the satellite solar panel system consists of N rectangular solar panels of equal size and weight. Define the connection point between the satellite body and the solar panels as the origin of the coordinate system. When the solar panels are fully deployed, the axial direction is the X-direction of the coordinate system, and the vertical direction is the Y-direction. Define the angles between the solar panels and the Y-direction as α1, α2, ..., α N That is, the generalized coordinates of the windsurfing board; the length of the windsurfing board is l, and the mass is m1, m2, ..., m. N The coordinates of the center of mass of the windsurfing board are C1(x1,y1), C2(x2,y2), ..., C N (x N ,y N The stiffness of the torsion spring at the connection point is k1, k2, ..., k N The damping is c1, c2, ..., c N ;

[0017] S22: Establish the dynamic equations;

[0018] Assuming the satellite body and solar panels are considered as rigid, deformation-free bodies, and the connection points are modeled as torsion spring dampers, the multi-panel deployment simulation model of the satellite solar panels is established using the second type of Lagrange method as follows:

[0019]

[0020] Where T represents the system's generalized coordinates α i and various generalized velocities The kinetic energy represented by Q; V is the system potential energy; Q is the system potential energy. i For the corresponding α i The generalized force; N is the number of sails, i.e., the number of degrees of freedom of the system, which is also equal to the number of point masses and the number of complete constraint equations of the system;

[0021] The total kinetic energy of the system is:

[0022]

[0023] Among them, a ij Let A be the elements of the square matrix A;

[0024] The total kinetic energy T of the system can be expressed as the sum of the elements of an N×N symmetric matrix A, and its expression is:

[0025]

[0026] For the last column of A, the value is ± when the number of windsurfing boards is even, and ± when it is odd.

[0027] The system potential energy is:

[0028]

[0029] The generalized force of the system is:

[0030]

[0031] Substituting equations (2)-(4) into the Lagrange equation (1), the multi-plate deployment dynamics equations of the satellite solar panels are obtained by taking the derivative with respect to time and the partial derivatives with respect to the generalized coordinates and generalized velocities.

[0032] S3: Solve the simulation dynamic equations for the deployment of the satellite solar panels;

[0033] S31: Solved using the fourth-order Runge-Kutta method;

[0034] The dynamic differential equations of motion for the multi-plate expansion derived by S2 are solved using the fourth-order Runge-Kutta formula.

[0035]

[0036] K1=f(x n ,y n )

[0037]

[0038]

[0039] K4=f(x n +h,y n +hK3) (6)

[0040] S32: A step function is used to simulate external shock interference;

[0041] The instantaneous force is simulated using the step function stepfun in MATLAB. The expression for the unit step function is:

[0042] f d =stepfun(t,t) d1 )-stepfun(t,t d2 (7)

[0043] Where t is the total time duration, t d1 ,t d2 These represent the start and end times of external disturbances acting on the windsurfing board, respectively.

[0044] Suppose there is a size F d An external force is applied to point D of the (i+1)th sail in the system, with its direction making an angle θ with the X-axis. The expressions for the torques acting on nodes i and i+1, where the i-th and (i+1)-th sails are located, are as follows:

[0045] F d(i) = -2×l×f d ×sin(α i+1 -θ) (8)

[0046] F d(i+1) =l d ×f d ×cos(α i+1 -θ) (9)

[0047] Among them, l d Let D be the length from point D to node i+1;

[0048] S4: Verification of the accuracy of the dynamic model;

[0049] Set the same parameters in the simulation software, compare and analyze the ADAMS dynamic simulation results with the theoretical model results in step S3, and verify the accuracy of the dynamic theoretical model within the set error range.

[0050] S5: Control algorithm design;

[0051] S51: Establish a state-space model

[0052] The multi-plate simulation model (1) is linearized near the equilibrium point, and the state-space expression is as follows:

[0053]

[0054] in:

[0055]

[0056]

[0057] C = [1 0 0] i 0 N 0 0 0 i 0 N ] 1×2N

[0058]

[0059]

[0060]

[0061] S52: LQR control algorithm design;

[0062] Based on the LQR algorithm, state feedback is introduced into the relevant state variables and their derivatives of the solar panel deployment mechanism to complete the control of the solar panel rotation angle and rotation angular velocity.

[0063] Define the quadratic performance index function J as:

[0064]

[0065] In the formula: Q and R are the positive definite weight matrix of the relevant state variables and the positive definite weight matrix of the control input, respectively;

[0066] According to LQR theory, the optimal control law that minimizes the above performance index is:

[0067] u = -Kx = -R -1 B T Px (12)

[0068] Where K is the state feedback matrix, P is a non-negative definite symmetric matrix, and P is the unique positive definite solution to the Riccati equation:

[0069] A T +PA-PBR -1 B T P+Q=0 8×8 (13)

[0070] After multiple simulation experiments, the simulation effects under different weights were compared, and the feedback controller K matrix corresponding to the optimal value was obtained.

[0071] The beneficial effects of this invention are as follows:

[0072] 1) In the modeling process, this invention provides a general model and modeling method for satellite solar panel multi-panel systems. This universal model has stronger versatility and wider applicability; it solves the complexity problem brought about by the change in the number of solar panels in the modeling process and greatly reduces the amount of computation required to build the model.

[0073] 2) This invention adopts MATLAB and ADAMS joint simulation technology. The simulation results of the former software can guide the establishment of theoretical models in a timely, effective and accurate manner, thereby achieving the purpose of dynamic verification. The model accuracy is higher and the simulation accuracy can be sufficiently guaranteed.

[0074] 3) The LQR control law designed in this invention fully considers the energy consumed during the deployment of the satellite solar panel and the time required for the deployment to reach a stable state. Combining the simulation effects under different weights, the optimal control law is proposed. Compared with damper vibration suppression, the satellite solar panel can be deployed quickly and stably. Attached Figure Description

[0075] Figure 1 This is a flowchart illustrating the specific process of the present invention.

[0076] Figure 2This is a geometric diagram of the deployed satellite solar panel in this invention.

[0077] Figure 3 This is an ADAMS simulation diagram of the satellite solar panel deployment in this invention.

[0078] Figure 4 This is the satellite solar panel deployment analysis model in this invention.

[0079] Figure 5 This is a response curve of the satellite solar panel in this invention when it is deployed without interference.

[0080] Figure 6 This is the step function curve of the satellite solar panel in this invention under external impact.

[0081] Figure 7 This is a response curve of the satellite solar panel in this invention deploying under external impact.

[0082] Figure 8 This serves as a dynamic model verification for the deployment of the satellite solar panels in this invention.

[0083] Figure 9 This is a response curve diagram of the satellite solar panel deployment process controlled by LQR in this invention. Detailed Implementation

[0084] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0085] The purpose of this invention is to provide a joint simulation method for satellite solar panel deployment based on MATLAB and ADAMS. While ensuring simulation accuracy, this method can more accurately simulate the dynamic response of a space satellite's on-orbit deployment and its exposure to external disturbances such as impacts under real-world conditions. Furthermore, under the designed control law, it enables rapid and stable deployment. The implementation process of this method is as follows: Figure 1As shown. This method first constructs a satellite solar panel deployment dynamics simulation system based on ADAMS software. The satellite body and the solar panels, as well as the individual solar panels, are connected by revolute joints and driven by torsion springs with preset angles. The dynamic performance of the four solar panels during deployment is analyzed. Next, a multi-panel deployment simulation model of the satellite solar panels is established using the Lagrange method. Taking the four-panel system as an example, the dynamic model of the system is obtained based on the general model and analysis method. Then, using the fourth-order high-precision Runge-Kutta algorithm, the dynamic differential equations established in S2 are solved using MATLAB software, giving the deployment process of the satellite solar panels and... The system's response to impact was analyzed; then, the ADAMS dynamic simulation results (S1) were compared and analyzed with the theoretical model results (S3), verifying the accuracy of the dynamic theoretical model within a certain error range; finally, based on the LQR control algorithm, the optimal control law of the state linear feedback of the satellite solar panel deployment system was derived; this invention uses MATLAB and ADAMS joint simulation technology, ensuring sufficient model simulation accuracy, and provides analysis results of the satellite solar panel system deployment and its response to external disturbances such as impacts, solving the problem of the satellite solar panel's inability to deploy quickly and stably, and providing certain theoretical guidance for the structural design and on-orbit deployment of space satellite solar panels.

[0086] A method for joint simulation of satellite solar panel deployment based on MATLAB and ADAMS includes the following steps:

[0087] S1: Use ADAMS software to build a satellite solar panel deployment dynamics simulation system;

[0088] S11: Create a 3D geometric model of the satellite solar panel, add material properties, and define connections;

[0089] In ADAMS software, a three-dimensional geometric model of the satellite solar panel is constructed according to the input dimensions. Material properties are added to the structure, and fixed joints and revolute joints are set at the connection points between the ground and the satellite body, and between the satellite body and the solar panel, respectively.

[0090] S12: Set driving force;

[0091] The specific method of using a torsion spring damper to drive the deployment of the sail is to set a preload angle, and then use a preset rotation angle to fully deploy the sail. Therefore, the formula for calculating the torque provided by the torsion spring is:

[0092] W tor =k×(β-β) pre )

[0093] Among them: W tor Let β be the torque, k be the torsional stiffness, and β be the torsional angle; pre The preset angle is a positive value, indicating that the relative motion angle is clockwise; otherwise, it is counterclockwise.

[0094] S2: Establish the simulation dynamic equations for the multi-plate deployment of satellite solar panels;

[0095] S21: Define the analysis model;

[0096] Assume the satellite solar panel system consists of N rectangular solar panels of equal size and weight. Define the connection point between the satellite body and the solar panels as the origin of the coordinate system. When the solar panels are fully deployed, the axial direction is the X-direction of the coordinate system, and the vertical direction is the Y-direction. Define the angles between the solar panels and the Y-direction as α1, α2, ..., α N The windsurfing boards are all of length l and have masses m1, m2, ..., m N The coordinates of the center of mass of the windsurfing board are C1(x1,y1), C2(x2,y2), ..., C N (x N ,y N The stiffness of the torsion spring at the connection point is k1, k2, ..., k N The damping is c1, c2, ..., c N ;

[0097] S22: Establish the dynamic equations;

[0098] Assuming the satellite body and solar panels are considered as rigid, deformation-free bodies, and the connection points are modeled as torsion spring dampers, the multi-panel deployment simulation model of the satellite solar panels is established using the second type of Lagrange method as follows:

[0099]

[0100] Where T represents the system's generalized coordinates α i and various generalized velocities The kinetic energy represented by Q; V is the system potential energy; Q is the system potential energy. i For the corresponding α i The generalized force; N is the number of sails, i.e., the number of degrees of freedom of the system, which is also equal to the number of point masses and the number of complete constraint equations of the system;

[0101] The total kinetic energy of the system is:

[0102]

[0103] For ease of expression, the total kinetic energy of the system is represented as the sum of the elements of an N×N symmetric matrix A, and its expression is:

[0104]

[0105] For the last column of A, the value is ± when the number of windsurfing boards is even, and ± when it is odd.

[0106] The system potential energy is:

[0107]

[0108] The generalized force of the system is:

[0109]

[0110] Substituting equations (2)-(4) into the Lagrange equation (1), the multi-plate deployment dynamics equations of the satellite solar panel are obtained by taking the derivative with respect to time and the partial derivatives with respect to the generalized coordinates and generalized velocities. Taking a four-plate satellite solar panel system as an example, based on the above general model and modeling method, let N=4 to derive the dynamics equations of the four-plate satellite solar panel system.

[0111] S3: Solve the simulation dynamic equations for the deployment of the satellite solar panels;

[0112] S31: Solved using the fourth-order Runge-Kutta method;

[0113] The transient motion differential equations of the satellite solar panel system proposed by S2 are difficult to solve analytically, and can only be studied through numerical simulation. Specifically, the fourth-order Runge-Kutta method is used to solve them; a classic fourth-order Runge-Kutta formula is given here to solve the dynamic differential equations.

[0114]

[0115] K1=f(x n ,y n )

[0116]

[0117]

[0118] K4=f(x n +h,y n +hK3) (6)

[0119] S32: A step function is used to simulate external shock interference;

[0120] The instantaneous force is simulated using the step function stepfun in MATLAB. The expression for the unit step function is:

[0121] f d =stepfun(t,t) d1 )-stepfun(t,t d2 (7)

[0122] Where t is the total time duration, t d1 ,td2 These represent the start and end times of external disturbances acting on the windsurfing board, respectively.

[0123] Taking N=4 as an example, suppose there is a size F d An external force is applied to point D of the tail section of the system's sail, with its direction making an angle θ with the X-axis. The expressions for the torques acting on nodes 3 and 4, where the third and fourth sails are located, are as follows:

[0124] F d3 = -2×l×f d ×sin(α4-θ) (8)

[0125] F d4 =l d ×f d ×cos(α4-θ) (9)

[0126] Among them, l d The distance from point D to node 4;

[0127] S4: Verification of the accuracy of the dynamic model;

[0128] To verify the accuracy of the theoretical model, the same parameters were set in the simulation software, and the ADAMS dynamic simulation results were compared and analyzed with the theoretical model results in step S3. The accuracy of the dynamic theoretical model was verified within the set error range.

[0129] S5: Control algorithm design;

[0130] S51: Establish a state-space model

[0131] The multi-plate simulation model (1) is linearized near the equilibrium point, and the state-space expression is as follows:

[0132]

[0133] y=CX+DU (10)

[0134] in:

[0135]

[0136]

[0137] C = [1 0 0 0 0 0 0 0]

[0138]

[0139]

[0140]

[0141] S52: LQR control algorithm design;

[0142] Based on the LQR algorithm, state feedback is introduced into the relevant state variables and their derivatives of the solar panel deployment mechanism to control the solar panel rotation angle and rotation angular velocity. The mathematical meaning of the LQR optimal algorithm is to find a state feedback matrix K that enables the system to have good tracking performance and minimizes the quadratic performance index function J.

[0143] Define the quadratic performance index function J as:

[0144]

[0145] In the formula: Q and R are the positive definite weight matrix of the relevant state variables and the positive definite weight matrix of the control input, respectively;

[0146] According to LQR theory, the optimal control law that minimizes the above performance index is:

[0147] u = -Kx = -R -1 B T Px (12)

[0148] Where K is the state feedback matrix, P is a non-negative definite symmetric matrix, and P is the unique positive definite solution to the Riccati equation:

[0149] A T +PA-PBR -1 B T P+Q=0 8×8 (13)

[0150] As the weight of a corresponding element in the Q matrix increases, the impact of the change in the state variable corresponding to that element on the system increases. While this speeds up the system's settling time and the time to reach equilibrium, it also causes more intense oscillations. Therefore, the values ​​of Q and R determine the system's dynamic performance. After multiple simulation experiments and comparing the simulation effects under different weights, it was found that Q provides better control. Based on this, the LQR function in MATLAB was used to solve for the feedback controller K matrix. By comprehensively balancing the equilibrium time and the maximum amplitude of oscillations, the optimal feedback controller K matrix was obtained. Specific implementation examples:

[0152] like Figure 1 The diagram illustrates the simulation method for satellite solar panel deployment based on MATLAB and ADAMS provided by this invention, which includes the following steps:

[0153] S1. Construct a satellite solar panel deployment dynamics simulation system using ADAMS software; specifically including the following steps:

[0154] S11. Establish a three-dimensional geometric model of the satellite solar panel.

[0155] First, in the Object—Basic Shape module, select design points and add them to the ground. Determine the reference positions of the satellite body and the four solar panels using the coordinates of the design points. Then, create cubes to represent the satellite body and the solar panels. The positions of the satellite body and solar panels are determined by the previously created design points. The satellite body has dimensions of 1.5 × 1 × 0.7 m, and each solar panel has dimensions of 1 × 0.5 × 0.05 m.

[0156] S12, Add material properties

[0157] In the Modify-Classification interface, select Mass Characteristics, define the satellite body as infinite mass, and the mass of each solar panel m = 15 kg;

[0158] S13, Define component connections

[0159] In the Connection-Kinematic Pairs module, select the fixed pair to connect the satellite body to the ground, meaning the satellite body is fully constrained with no rigid body motion by default. Also, set revolute pairs to be applied between the satellite body and the solar panels, and between adjacent solar panels. The revolute pairs constrain the three translational degrees of freedom and two rotational degrees of freedom between the connected components, retaining only the rotational degree of freedom of the solar panels in the deployment direction. The satellite solar panel deployment geometry is as follows: Figure 2 As shown;

[0160] S14, Set driving force

[0161] In the force-flexible connection module, a torsion spring damper is selected as the driving force. The torsion spring is installed on the revolute joint, and its direction is consistent with the direction of rotation. The torsional stiffness coefficient of the spring damper is k = 50 N·m / rad and the torsional damping coefficient is c = 50 N·m·s / rad.

[0162] The specific method for using design variables is as follows: Create two new design variables in the Design Exploration—Design Variables module, selecting torsional stiffness and torsional damping as the units, respectively. Set the standard value of the design variables to be the specific value adopted by the current simulation model. Select the absolute maximum and absolute minimum values ​​for the range of the design variable values. Then, select these two design variables as the stiffness coefficient and damping coefficient in the properties of the torsion spring. The advantages of using design variables to control the stiffness and damping coefficient properties of the torsion spring damper are twofold: firstly, when there are multiple solar panel components, the corresponding properties of each solar panel connector can be modified simultaneously by modifying the design variables; secondly, the nonlinearity of stiffness or damping can be achieved by creating functions using design variables.

[0163] The specific method of using a torsion spring damper to drive the deployment of the sail is to set a preload angle, and then use a preset rotation angle to fully deploy the sail. Therefore, the formula for calculating the torque provided by the torsion spring is:

[0164] Wtor =k×(β-β) pre )

[0165] Among them: W tor Let β be the torque, k be the torsional stiffness, and β be the torsional angle. pre The preset angle is a positive value, indicating that the relative motion angle is clockwise; otherwise, it is counterclockwise.

[0166] S15, Simulation Analysis

[0167] Right-click on the corresponding rotational hinge and select "Measurement and Design Exploration" to create a measurement function to measure the system's dynamic characteristics, such as rotational displacement. During the simulation, the dynamic characteristics of the system that need to be monitored will be tracked and measured. Figure 3 Simulation diagrams of satellite solar panels deployed at times t = 0, 1, 2, 10, 40, and 100 s are shown.

[0168] S2. Establish the simulation dynamic equations for the multi-plate deployment of the satellite solar panel; specifically, this includes the following steps:

[0169] S21. Define the analysis model

[0170] Assume the satellite solar panel system consists of N rectangular panels of equal size and weight. Define the connection point between the satellite body and the solar panels as the origin of the coordinate system. When the solar panels are fully deployed, the axial direction is the X-direction of the coordinate system, and the vertical direction is the Y-direction. Define the angles between the solar panels and the Y-direction as α1, α2, ..., α... N The length of each windsurfing board is l = 1m, and its mass is m1 = m2 = m N = m, the coordinates of the center of mass of the windsurfing are C1(x1,y1), C2(x2,y2), ..., C N (x N ,y N The stiffness of the torsion spring at the connection is k1 = k2 = k N =k Damping is c1=c2=c N =c; Specific analysis model as follows Figure 4 As shown;

[0171] S22. Establish the dynamic equations

[0172] Assuming the satellite body and solar panels are considered as rigid, undeformed bodies, and the connection points are modeled as torsion spring dampers, the multi-panel deployment simulation model of the satellite solar panels is established using the second type of Lagrange method as follows:

[0173]

[0174] Where T represents the system's generalized coordinates α i and various generalized velocities The kinetic energy represented by Q; V is the system potential energy; Q is the system potential energy.i For the corresponding α i The generalized force; N is the system's degrees of freedom, i.e., the number of solar panels, which is equal to the number of point masses and the number of complete constraint equations in the system;

[0175] The total kinetic energy of the system is:

[0176]

[0177] For ease of expression, the total kinetic energy of the system is represented as the sum of the elements of a symmetric square matrix A (N×N), and its expression is:

[0178]

[0179] Of particular note is the last column of matrix A, where the value is ± when the number of windsurfing boards is even, and ± when it is odd.

[0180] The system's elastic potential energy is:

[0181]

[0182] The system is not a force:

[0183]

[0184] Substituting equations (2)-(4) into the second type of Lagrange equation (1), and after a series of calculations such as differentiating with respect to time and partial derivatives with respect to generalized coordinates and generalized velocities, the multi-plate deployment dynamics equation of the satellite solar panel can be obtained.

[0185] Taking a four-plate satellite solar panel system as an example, based on the above general model and modeling method, the dynamic equations of motion for the four-plate satellite solar panel system are obtained as follows:

[0186]

[0187]

[0188]

[0189]

[0190] S3. Solve the simulation dynamic equations for the deployment of the satellite solar panels; specifically, this includes the following steps:

[0191] S31. Solve using the fourth-order Runge-Kutta method.

[0192] For the transient motion differential equations of a typical satellite solar panel system, it is difficult to obtain analytical solutions, and currently, numerical methods are the only way to study them through simulation. Existing research shows that when the parameters are chosen appropriately, the Runge-Kutta method is unconditionally stable, and the calculation process is relatively simple with high accuracy. Therefore, the Runge-Kutta method is used to solve the time-domain response of the satellite solar panel deployment system.

[0193] Here we present a classic fourth-order Runge-Kutta formula for solving the dynamic differential equation.

[0194]

[0195] K1=f(x n ,y n )

[0196]

[0197]

[0198] K4=f(x n +h,y n +hK3) (10)

[0199] The above formula is the Runge-Kutta formula for solving a system of first-order ordinary differential equations, but the dynamic differential equations (5)-(8) are second-order. Therefore, it is necessary to reduce the order of the higher-order ordinary differential equations to obtain the corresponding system of first-order ordinary differential equations. For ease of solution, the order reduction step is omitted, and the Runge-Kutta formula for directly solving the system of second-order differential equations is given here: Let the system of second-order differential equations to be solved be:

[0200]

[0201] The corresponding Runge-Kutta formula is:

[0202]

[0203]

[0204] Where y n+1,1 (i) and y n+1,2 (i) represent the y values ​​obtained up to the (n+1)th step. i and y′ i , where i is the index of the dependent variable y, and there are N dependent variables in total. In the formula:

[0205] K 11 (i)=y n,2 (i)

[0206] K 12 (i)=f i (x n ,y n,1 (1),y n,1 (2),…,y n,1 (N),y n,2 (1),y n,2 (2),…,y n,2 (N))

[0207]

[0208]

[0209]

[0210]

[0211]

[0212] K 42 (i)=f i (x n +h / 2,y n,1 (1)+hK 31 (1),y n,1 (2)+hK 31 (2),…,y n,1 (N)+hK 31 (N),y n,2 (1)+hK 32 (1),y n,2 (2)+hK 32 (2),…,y n,2 (N)+hK 32 (N))

[0213] The fourth-order Runge-Kutta method was used to solve the problem, and the curves of the rotation angles of panels 1, 2, 3, and 4 of the satellite solar panel as a function of time under undisturbed conditions were obtained. For example... Figure 5 As shown, each plate rotates and unfolds from its initial position under the drive of the torsion spring preload. The rotational angular velocity reaches its maximum value at the instant of release and then gradually decreases. Each plate reaches its maximum rotation angle value, approximately |α|, at t = 9.30, 2.45, 9.35, and 2.40 s, respectively. max=1.78, 1.79, 2.00, 1.91 rad. Then, under the action of torsion spring damping, the energy of the system vibration is dissipated, and the vibration velocity gradually decreases. The vibration velocity reaches zero in about 80s, and each solar panel tends to stabilize. It is in a state of slight vibration for 30-80s. This is still not feasible for spacecraft with high precision requirements. Therefore, it is necessary to clarify the dynamic mechanism of the solar panel deployment structure and apply an effective controller in order to significantly reduce the vibration amplitude of the solar panels and shorten the time to reach stability.

[0214] S32. Using a step function to simulate external shock interference.

[0215] Satellite solar panels are highly susceptible to impacts and interference from high-energy rays and high-speed particles in the complex space environment during deployment / locking. Because these high-speed particles are extremely small and have negligible mass, their impact on the solar panels produces a transient impact effect with a very short duration. The transient force is simulated using the STEPFUN function in MATLAB. Figure 6 The step function force curve of the satellite solar panel under external impact is shown. The expression for the unit step function (unit force) is:

[0216] f d =stepfun(t,t) d1 )-stepfun(t,t d2 (13)

[0217] Where t = 150s is the total time duration, t d1 =80s,t d2 =81s represents the start and end times of external disturbances acting on the sail, respectively.

[0218] Suppose there is a size F d An external force of 50 N is applied to point D of the end solar panel of the system, with its direction making an angle of θ = 30° with the X-axis. The expressions for the torques acting on nodes 3 and 4, where the third and fourth solar panels are located, are as follows:

[0219] F d3 = -2×l×f d ×sin(α4-θ) (14)

[0220] F d4 =l d ×f d ×cos(α⁴-θ) (15)

[0221] Among them, l d =0.5m is the length from point D to node 4;

[0222] Figure 7The response curve of the satellite solar panels to external impacts during deployment is shown. The first 80 seconds of deployment are the same as step S31. By the 80th second, each solar panel has reached a stable state. At this time, when the end solar panel is subjected to an instantaneous impact within the time period of 80-81 seconds, the four solar panels have an instantaneous rotational speed, causing the system to produce a small vibration. Under the action of the original spring damping, the system takes 50 seconds to recover to a stable state.

[0223] S4. Verification of the accuracy of the dynamic model

[0224] In step S1, dynamic simulation was performed using ADAMS software to analyze the response curves of the four satellite solar panels as they unfolded over time. To verify the accuracy of the theoretical model, the same parameters were set in the simulation software. The results of the ADAMS dynamic simulation and the theoretical analysis model (S3) were compared as follows: Figure 8 As shown; taking board 1 and board 2 as examples, it can be seen that the response curves of the two modeling methods during the deployment process of the solar panel are roughly the same under the same parameters, the deployment trajectory is basically the same, the time for deployment to be completed and stabilized is roughly the same, about 80s, and the maximum error corresponding to the maximum rotation angle value is within the acceptable range, at 6.0% and 14.7% respectively. Therefore, the accuracy of the satellite solar panel dynamic model established in step S2 is also verified.

[0225] S5, Control Algorithm Design

[0226] S51. Establish a state-space model

[0227] Since the dynamic equations (5)-(8) are nonlinear, they need to be linearized near the equilibrium point. Ignoring all higher-order terms and coupling terms of the derivatives, the state-space expressions derived from equation (1) are as follows:

[0228]

[0229] y = CX + DU (16)

[0230] in:

[0231]

[0232]

[0233] C = [1 0 0 0 0 0 0 0]

[0234]

[0235]

[0236]

[0237] S52 and LQR control algorithm design

[0238] Based on the LQR algorithm, state feedback is introduced into the relevant state variables and their derivatives of the solar panel deployment mechanism to control the rotation angle and angular velocity of the solar panel. The mathematical meaning of the LQR optimal algorithm is to find a state feedback matrix K that enables the system to have good tracking performance and minimizes the quadratic performance index function J.

[0239]

[0240] In the formula: Q and R are the positive definite weight matrices of the relevant state variables and the positive definite weight matrices of the control input, respectively. According to LQR theory, the optimal control law that minimizes the above performance index is:

[0241] u = -Kx = -R -1 B T Px (18)

[0242] Where P is a non-negative definite symmetric matrix and is the unique positive definite solution to the Riccati equation:

[0243] A T +PA-PBR -1 B T P+Q=0 8×8 (19)

[0244] When the weight of a corresponding element in the Q matrix increases, the impact of the change in the state variable corresponding to that element on the system increases. Although this speeds up the system's adjustment time and the time to reach equilibrium, it also causes the system to produce stronger vibrations. Therefore, the values ​​of Q and R determine the dynamic performance of the system. After multiple simulation experiments and comparing the simulation effects under different weights, it was found that Q provides a better control effect for the system. Based on this, the LQR function in MATLAB was used to solve for the feedback controller K matrix. Considering both the equilibrium time and the maximum amplitude of vibration, the optimal feedback controller K matrix was obtained as follows: K1 = 0.3053, K2 = 0.1383, K3 = -0.6125, K4 = -0.0066, K5 = 0.0705, K6 = 0.0793, K7 = 0.1219, K8 = 0.0028.

[0245] Figure 9The deployment response curves of the satellite solar panels under LQR control and spring damper control are shown. Under the action of the spring damper, the system stabilizes at about 80s. Under the control of the above algorithm, the deployment-stabilization time is about 50s, which is reduced by 37.5%. During the deployment process, the maximum amplitude of each solar panel is reduced by 9.66%, 26.85%, 5.00%, and 17.86%, respectively. This proves the effectiveness of the control algorithm. At the same time, the results are not the same when the rotation angle of each panel is weighted by different coefficients in equation (17). The larger the weight coefficient of the tilt angle of panel 1, the shorter the time required for the system to balance. On the contrary, the maximum amplitude of vibration during the balancing process increases. Therefore, reducing the stabilization time and reducing the vibration intensity cannot be achieved at the same time. The weight coefficients should be adjusted in a timely manner according to the specific actual working conditions such as the satellite solar panels themselves and mission requirements to obtain a more effective controller.

Claims

1. A joint simulation method for satellite solar panel deployment based on MATLAB and ADAMS, characterized in that, Includes the following steps: S1: Use ADAMS software to build a satellite solar panel deployment dynamics simulation system; S11: Create a 3D geometric model of the satellite solar panel, add material properties, and define connections; In ADAMS software, a three-dimensional geometric model of the satellite solar panel is constructed according to the input dimensions. Material properties are added to the structure, and fixed joints and revolute joints are set at the connection points between the ground and the satellite body, and between the satellite body and the solar panel, respectively. S12: Set driving force; The specific method of using a torsion spring damper to drive the deployment of the sail is to set a preload angle, and then use a preset rotation angle to fully deploy the sail. Therefore, the formula for calculating the torque provided by the torsion spring is: W tor =k×(β-β pre ) Among them: W tor Let β be the torque, k be the torsional stiffness, and β be the torsional angle; pre The preset angle is a positive value, indicating that the relative motion angle is clockwise; otherwise, it is counterclockwise. S2: Establish the simulation dynamic equations for the multi-plate deployment of satellite solar panels; S21: Define the analysis model; Assume the satellite solar panel system consists of N rectangular solar panels of equal size and weight. Define the connection point between the satellite body and the solar panels as the origin of the coordinate system. When the solar panels are fully deployed, the axial direction is the X-direction of the coordinate system, and the vertical direction is the Y-direction. Define the angles between the solar panels and the Y-direction as α1, α2, ..., α N That is, the generalized coordinates of the windsurfing board; the length of the windsurfing board is l, and the mass is m1, m2, ..., m. N The coordinates of the center of mass of the windsurfing board are C1(x1,y1), C2(x2,y2), ..., C N (x N ,y N The stiffness of the torsion spring at the connection point is k1, k2, ..., k N The damping is c1, c2, ..., c N ; S22: Establish the dynamic equations; Assuming the satellite body and solar panels are considered as rigid, deformation-free bodies, and the connection points are modeled as torsion spring dampers, the multi-panel deployment simulation model of the satellite solar panels is established using the second type of Lagrange method as follows: Where T represents the system's generalized coordinates α i and various generalized velocities The kinetic energy represented by Q; V is the system potential energy; Q is the system potential energy. i For the corresponding α i The generalized force; N is the number of sails, i.e., the number of degrees of freedom of the system, which is also equal to the number of point masses and the number of complete constraint equations of the system; The total kinetic energy of the system is: Among them, a ij Let A be the elements of the square matrix A; The total kinetic energy T of the system can be expressed as the sum of the elements of an N×N symmetric matrix A, and its expression is: For the last column of A, the value is ± when the number of windsurfing boards is even, and ± when it is odd. The system potential energy is: The generalized force of the system is: Substituting equations (2)-(4) into the Lagrange equation (1), the multi-plate deployment dynamics equations of the satellite solar panels are obtained by taking the derivative with respect to time and the partial derivatives with respect to the generalized coordinates and generalized velocities. S3: Solve the simulation dynamic equations for the deployment of the satellite solar panels; S31: Solved using the fourth-order Runge-Kutta method; The dynamic differential equations of motion for the multi-plate expansion derived by S2 are solved using the fourth-order Runge-Kutta formula. K1=f(x n ,y n ) K4=f(x n +h,y n +hK3) (6) S32: A step function is used to simulate external shock interference; The instantaneous force is simulated using the step function stepfun in MATLAB. The expression for the unit step function is: f d =stepfun(t,t d1 )-stepfun(t,t d2 ) (7) Where t is the total time duration, t d1 ,t d2 These represent the start and end times of external disturbances acting on the windsurfing board, respectively. Suppose there is a size F d An external force is applied to point D of the (i+1)th sail in the system, with its direction making an angle θ with the X-axis. The expressions for the torques acting on nodes i and i+1, where the i-th and (i+1)-th sails are located, are as follows: F d(i) =-2×l×f d ×sin(a i+1 -i) (8) F d(i+1) =l d ×f d ×cos(α) i+1 -i) (9) Among them, l d Let D be the length from point D to node i+1; S4: Verification of the accuracy of the dynamic model; Set the same parameters in the simulation software, compare and analyze the ADAMS dynamic simulation results with the theoretical model results in step S3, and verify the accuracy of the dynamic theoretical model within the set error range. S5: Control algorithm design; S51: Establish a state-space model The multi-plate simulation model (1) is linearized near the equilibrium point, and the state-space expression is as follows: in: C＝[1 0 0 i 0 N 0 0 0 i 0 N ] 1×2N S52: LQR control algorithm design; Based on the LQR algorithm, state feedback is introduced into the relevant state variables and their derivatives of the solar panel deployment mechanism to complete the control of the solar panel rotation angle and rotation angular velocity. Define the quadratic performance index function J as: In the formula: Q and R are the positive definite weight matrix of the relevant state variables and the positive definite weight matrix of the control input, respectively; According to LQR theory, the optimal control law that minimizes the above performance index is: u=-Kx=-R -1 B T Px (12) Where K is the state feedback matrix, P is a non-negative definite symmetric matrix, and P is the unique positive definite solution to the Riccati equation: A T +PA-PBR -1 B T P+Q=0 8×8 (13) After multiple simulation experiments, the simulation effects under different weights were compared, and the feedback controller K matrix corresponding to the optimal value was obtained.

Images