Reliability analysis method and system for the deployment and locking function of the deployable mechanism of spaceborne SAR antenna

Abstract

This invention discloses a method and system for reliability analysis of the deployment locking function of a deployable mechanism for spaceborne SAR antennas. It relates to the technical field of deployable mechanisms for spaceborne antennas, establishing a dynamic model of the deployment process. This model can quantitatively analyze the deployment locking characteristics of the mechanism under the influence of factors such as rod length error and driving torque fluctuation, providing solid theoretical support for comprehensively judging the deployment reliability of the spaceborne SAR antenna. By analyzing the kinematic characteristics of the deployable mechanism under different errors, it can also effectively guide the optimization of the deployment mechanism, improving assembly efficiency while ensuring assembly quality. In particular, the numerical iterative solution algorithm in this invention can effectively obtain the dynamic response of the underdetermined drive deployment process, and this algorithm also has reference value for solving dynamic problems under nonholonomic constraints.

Description

Technical Field

[0001] This invention relates to the field of spaceborne antenna deployable mechanism technology, specifically to a method and system for reliability analysis of the deployment locking function of a spaceborne SAR antenna deployable mechanism. Background Art

[0002] Due to the volume limitations of launch vehicles, large spaceborne SAR antennas need to be retracted during launch and deployed after entering orbit. Therefore, the success or failure of the satellite launch mission is directly determined by the deployable mechanism's ability to both deploy and lock. However, under the combined influence of various factors such as component assembly errors and driving torque fluctuations, the deployment and locking reliability of the deployable mechanism often faces significant technical risks. Unlike general linkage mechanisms, the spaceborne SAR antenna deployable mechanism not only possesses characteristics such as "multi-loop closed chains, variable degrees of freedom, underdetermined actuation, and strong nonlinearity," but also requires some hinges to reliably lock according to a predetermined deployment strategy. This presents a severe challenge to the theoretical modeling and analysis of the mechanism's deployment and locking process. Therefore, engineers have to repeatedly perform folding and unfolding tests during ground assembly to ensure its normal deployment and locking function. However, repeated folding and unfolding can damage weak components such as coatings and coil springs, potentially triggering other potential accidents. Therefore, quantitatively evaluating the reliability of the deployment and locking function of the spaceborne SAR antenna deployable mechanism is particularly important.

[0003] A review of existing methods revealed that no reliable analysis methods for the deployment locking function of deployable mechanisms for spaceborne SAR antennas have been reported in China. Firstly, there are few modeling methods that reveal the dynamic characteristics of multi-ring closed-chain variable degree-of-freedom mechanisms from a mechanistic perspective. Secondly, classical solution methods (such as direct integration and reduced-order integration) are difficult to incorporate the driving characteristics and constraints of actual mechanisms, thus failing to guide engineering practice. Summary of the Invention

[0004] The purpose of this invention is to provide a reliability analysis method and system for the deployment locking function of a deployable mechanism of a spaceborne SAR antenna, so as to overcome the technical defects of existing purely experimental verification methods. This invention fully considers the characteristics of the deployable mechanism, such as multi-loop closed chain, variable degrees of freedom, underdetermined actuation, and strong nonlinearity, and abandons the general probabilistic reliability evaluation method. It performs functional reliability assessment based on dynamic analysis, which can better meet actual needs and effectively guide engineering practice.

[0005] A reliability analysis method for the deployment locking function of a deployable mechanism for a spaceborne SAR antenna includes the following steps:

[0006] S1. Establish a mathematical model of the nine-bar linkage mechanism according to the actual configuration of the deployable mechanism of the spaceborne SAR antenna. Divide the deployment stages according to the driving strategy of the deployable mechanism, calculate the degrees of freedom of each deployment stage, and determine whether each deployment stage is an underactuated deployment process or a well-posed deployment process based on the degrees of freedom of each deployment stage.

[0007] S2. Establish the coordinate system of the deployable mechanism, determine the parameters of the deployable mechanism, list the geometric ring constraint equations corresponding to the nine-bar linkage mathematical model, and characterize the rotation angle of each driven member as a function of the rotation angle of the driving member according to the geometric ring constraint equations, and establish the kinematic model of the deployment process.

[0008] S3. Solve the position of the center of mass, angular velocity and angular acceleration of each component according to the kinematic model of the deployment process, calculate the kinetic energy and elastic potential energy of the entire deployable mechanism, and establish the dynamic model of each deployment stage of the deployable mechanism based on the Lagrange equation.

[0009] S4. Based on the kinematic model of the deployment process and the dynamic model of each deployment stage of the deployable mechanism, given the initial motion conditions of each component and the driving strategy of the active component, the dynamic response characteristics of each component during the underactuated deployment process are iteratively solved based on the principle of constant velocity and acceleration within two subdivided time intervals; the dynamic response characteristics of each component during the well-posed drive deployment process are obtained by directly solving the Lagrange dynamic equations.

[0010] S5. Based on the obtained motion laws of each deployment process, check whether there is motion interference or singularity in the deployable mechanism. If there is motion interference or singularity, it means that the deployment locking function is unreliable; otherwise, calculate the final deployment angle of each locking hinge, and determine whether the deployment locking function of the mechanism is safe and reliable according to the engineering practice criteria based on the final deployment angle of each locking hinge.

[0011] Preferably, if the number of main drives is less than the system degrees of freedom, it is an underactuated deployment process; if the number of main drives is equal to the system degrees of freedom, the entire mechanism is in a well-done deployment process.

[0012] Preferably, the unfolding process of the deployable mechanism is divided into three stages. According to the theory of degrees of freedom of planar mechanisms, the degrees of freedom of the mechanism in the three stages are as follows:

[0013]

[0014] Where: DOF i (i = 1, 2, 3) represents the degrees of freedom in the i-th stage; N represents the number of components; P L P is a low-order subnumber in the plane; H It represents the plane height subnumber.

[0015] Preferably, each component is equivalent to a point mass at its midpoint. The total kinetic energy of the system is obtained by expressing the rotational kinetic energy of each component. The elastic potential energy of each auxiliary coil spring is calculated according to the torque-rotation angle relationship. Then, the dynamic model of the system under three different deployment stages is established using the Lagrange method.

[0016] Preferably, the three loops are modeled separately using the closed-loop vector method:

[0017] For the loop ABCHGA, there are equality constraints in the horizontal and vertical directions:

[0018]

[0019] For the loop CIFHC, we have the following equation:

[0020]

[0021] Similarly, for the loop HCDEJFH, we can obtain:

[0022]

[0023] In the above three formulas: l i (i = 1, 2, ..., 13) represents the length of the i-th segment; This represents the angle between the j-th segment and the horizontal direction;

[0024] After differentiating both sides of equations (2), (3), and (4) with respect to time, using... And its angular velocity, other angular velocities can be expressed as:

[0025]

[0026]

[0027]

[0028]

[0029]

[0030]

[0031] In the formula for The corresponding angular velocity; further differentiation yields the relevant acceleration, i.e.:

[0032]

[0033]

[0034]

[0035]

[0036]

[0037]

[0038] In the formula for The corresponding angular acceleration.

[0039] Preferably, the generalized input force of the system is the input torque τ at hinge G and hinge J. J and τ G With no external force input at hinge A, the dynamic model of each deployment stage of the deployable mechanism is as follows:

[0040]

[0041] E k and E p These are the system's kinetic energy and elastic potential energy, respectively.

[0042] A reliability analysis system for the deployment locking function of a spaceborne SAR antenna deployable mechanism includes:

[0043] Degrees of freedom calculation module: Establish a mathematical model of the nine-bar linkage mechanism according to the actual configuration of the deployable mechanism of the spaceborne SAR antenna. Divide the deployment stage according to the driving strategy of the deployable mechanism, calculate the degrees of freedom of each deployment stage, and determine whether each deployment stage is an underactuated deployment process or a well-actuated deployment process based on the degrees of freedom of each deployment stage.

[0044] Kinematic model building module: Establish the coordinate system of the deployable mechanism, determine the parameters of the deployable mechanism, list the geometric ring constraint equations corresponding to the nine-bar linkage mathematical model, and characterize the rotation angle of each driven member as a function of the rotation angle of the driving member according to the geometric ring constraint equations, and establish the kinematic model of the deployment process.

[0045] Dynamics model building module: Based on the kinematic model of the deployment process, solve for the position of the center of mass, angular velocity and angular acceleration of each component, calculate the kinetic energy and elastic potential energy of the entire deployable mechanism, and establish the dynamics model of each deployment stage of the deployable mechanism based on the Lagrange equation;

[0046] Dynamic response characteristics calculation module: Based on the kinematic model of the deployment process and the dynamic model of each deployment stage of the deployable mechanism, given the initial motion conditions of each component and the driving strategy of the active component, the dynamic response characteristics of each component during the underactuated deployment process are iteratively solved based on the principle of constant velocity and acceleration within two subdivided time intervals; the dynamic response characteristics of each component during the well-posed drive deployment process are obtained by directly solving the Lagrange dynamic equations.

[0047] Reliability Analysis Module: Based on the obtained motion laws of each deployment process, check whether there is motion interference or singularity in the deployable mechanism. If motion interference or singularity exists, it indicates that the deployment locking function is unreliable; otherwise, calculate the final deployment angle of each locking hinge, and determine whether the deployment locking function of the mechanism is safe and reliable according to engineering practice criteria based on the final deployment angle of each locking hinge.

[0048] Preferably, if the number of main drives is less than the system degrees of freedom, it is an underactuated deployment process; if the number of main drives is equal to the system degrees of freedom, the entire mechanism is in a well-done deployment process.

[0049] Preferably, the unfolding process of the deployable mechanism is divided into three stages. According to the theory of degrees of freedom of planar mechanisms, the degrees of freedom of the mechanism in the three stages are as follows:

[0050]

[0051] Where: DOF i (i = 1, 2, 3) represents the degrees of freedom in the i-th stage; N represents the number of components; P L P is a low-order subnumber in the plane; H It represents the plane height subnumber.

[0052] Preferably, the three loops are modeled separately using the closed-loop vector method:

[0053] For the loop ABCHGA, there are equality constraints in the horizontal and vertical directions:

[0054]

[0055] For the loop CIFHC, we have the following equation:

[0056]

[0057] Similarly, for the loop HCDEJFH, we can obtain:

[0058]

[0059] In the above three formulas: l i (i = 1, 2, ..., 13) represents the length of the i-th segment; This represents the angle between the j-th segment and the horizontal direction;

[0060] After differentiating both sides of equations (2), (3), and (4) with respect to time, using... And its angular velocity, other angular velocities can be expressed as:

[0061]

[0062]

[0063]

[0064]

[0065]

[0066]

[0067] In the formula for The corresponding angular velocity; further differentiation yields the relevant acceleration, i.e.:

[0068]

[0069]

[0070]

[0071]

[0072]

[0073]

[0074] In the formula for The corresponding angular acceleration.

[0075] Compared with the prior art, the present invention has the following beneficial technical effects:

[0076] This invention presents a reliability analysis method for the deployment and locking function of a deployable mechanism for a spaceborne SAR antenna. It establishes a dynamic model of the deployment process, enabling quantitative analysis of the mechanism's deployment and locking characteristics under the influence of factors such as rod length error and driving torque fluctuation. This provides solid theoretical support for comprehensively determining the deployment reliability of the spaceborne SAR antenna. By analyzing the kinematic characteristics of the deployable mechanism under different errors, it can also effectively guide the optimization of the deployment mechanism's assembly and adjustment, improving assembly efficiency while ensuring assembly quality. In particular, the numerical iterative solution algorithm in this invention can effectively obtain the dynamic response of the underdetermined drive deployment process, and this algorithm also has reference value for solving dynamic problems under nonholonomic constraints.

[0077] This invention fully considers the characteristics of deployable mechanisms, such as multi-loop closed chains, variable degrees of freedom, underdetermined actuation, and strong nonlinearity, and provides a numerical iterative algorithm for effectively solving the kinematics and dynamics problems of deployable mechanisms. It abandons the general probabilistic reliability evaluation method and performs functional reliability assessment based on dynamic analysis, thus better meeting practical needs and effectively guiding engineering practice. Attached Figure Description

[0078] Figure 1 This is a flowchart of the analysis method in an embodiment of the present invention.

[0079] Figure 2 This is a schematic diagram of the planar configuration of the deployable mechanism and the position of the hinge in an embodiment of the present invention.

[0080] Figure 3 This is a schematic diagram of the coordinate system and symbol representation of the unfolding process of the deployable mechanism in an embodiment of the present invention.

[0081] Figure 4 This is a flowchart of the numerical iterative solution algorithm in an embodiment of the present invention.

[0082] In the diagram, 1 is the first satellite connecting rod, 2 is the second satellite connecting rod, 3 is the inner support rod, 4 is the first middle support rod, 5 is the second middle support rod, 6 is the first outer support rod, 7 is the second outer support rod; 8 is the inner antenna plate, 9 is the outer antenna plate; hinge A; hinge C; hinge E; hinge F; hinge H; 180° locking hinge B; 180° locking hinge D; 180° locking hinge I; 180° locking hinge J; 90° locking hinge G. Detailed Implementation

[0083] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0084] It should be noted that the terms "first", "second", etc. in the description and claims of the present invention and the above-mentioned drawings are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that the numbers used in this way can be interchanged where appropriate, so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions. For example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

[0085] like Figure 1 As shown, the reliability analysis method for the deployment locking function of the spaceborne SAR antenna deployment mechanism of the present invention includes the following steps:

[0086] S1. Establish a mathematical model of the nine-bar linkage mechanism according to the actual configuration of the deployable mechanism of the spaceborne SAR antenna. Divide the deployment stages according to the driving strategy of the deployable mechanism, calculate the degrees of freedom of each deployment stage, and determine whether each deployment stage is an underactuated deployment process or a well-posed deployment process based on the degrees of freedom of each deployment stage.

[0087] Specifically, the mathematical model of the nine-bar linkage is represented by seven supporting rods and two antenna panels. Although the deployable mechanism of the spaceborne SAR antenna is a spatial mechanism, all components only undergo planar motion during deployment. Therefore, the nine-bar linkage formed by projecting it onto this plane of motion encompasses the kinematic characteristics of the entire deployment process.

[0088] The determination of the underactuated deployment process and the well-actuated deployment process is based on whether the number of main drives is less than the system's degrees of freedom. If the number of main drives is less than the system's degrees of freedom, the dynamic constraint equations of the entire mechanism are less than the number of unknowns and cannot be solved directly, which is an underactuated deployment process. If the number of main drives equals the system's degrees of freedom, the entire mechanism is in a well-actuated deployment process, and the dynamic constraint equations equal the number of unknowns and can be solved directly.

[0089] S2. Establish the coordinate system of the deployable mechanism, determine the parameters of the deployable mechanism, list the geometric ring constraint equations corresponding to the nine-bar linkage mathematical model, and characterize the rotation angle of each driven member as a function of the rotation angle of the driving member according to the geometric ring constraint equations, and establish the kinematic model of the deployment process.

[0090] The kinematic model of the unfolding process is obtained by dividing the mathematical model of the nine-bar linkage into three closed loops, constructing the loop constraint equations using the closed loop vector method, and then decomposing them along the two axes (i.e., the X-axis and the Y-axis) of the planar coordinate system of the deployable mechanism.

[0091] S3. Solve the position of the center of mass, angular velocity and angular acceleration of each component according to the kinematic model of the deployment process, calculate the kinetic energy and elastic potential energy of the entire deployable mechanism, and establish the dynamic model of each deployment stage of the deployable mechanism based on the Lagrange equation.

[0092] The dynamic model of the unfolding process treats each component as a point mass at its midpoint, obtains the total kinetic energy of the system by expressing the rotational kinetic energy of each component, and calculates the elastic potential energy of each auxiliary coil spring according to the torque-rotation angle relationship. Then, the Lagrange method is used to establish the dynamic model of the system under three different unfolding stages, including a Lagrange dynamic model of one underactuated unfolding process and two well-determined driven unfolding processes.

[0093] S4. Based on the kinematic model of the deployment process and the dynamic model of each deployment stage of the deployable mechanism, given the initial motion conditions (center of mass position, angular velocity and angular acceleration) of each component and the driving strategy of the active component, the dynamic response characteristics (motion laws) of each component during the underactuated deployment process are iteratively solved based on the principle of constant velocity and acceleration within two subdivided time intervals; the dynamic response characteristics (motion laws) of each component during the well-posed drive deployment process are obtained by directly solving the Lagrange dynamic equations.

[0094] For the underactuated deployment process, the motion law of each component is obtained by iteratively solving the problem based on the principle that the velocity and acceleration remain constant in the two time intervals when the time is infinitely subdivided and combined with the initial motion parameters. For the well-determined deployment process, the dynamic response state of the previous stage is used as the boundary condition, and the complete motion trajectory of the deployable mechanism is obtained by solving the Lagrange dynamic equation.

[0095] S5. Based on the obtained motion laws of each deployment process, check whether there is motion interference or singularity in the deployable mechanism. If there is motion interference or singularity, it means that the deployment locking function is unreliable; otherwise, calculate the final deployment angle of each locking hinge, and determine whether the deployment locking function of the mechanism is safe and reliable according to the engineering practice criteria based on the final deployment angle of each locking hinge.

[0096] The assessment of the safety and reliability of the deployment locking function is not a probabilistic score, but rather a determination of "failure" or "non-failure" based on dynamic prediction results from two aspects: first, whether there is motion interference or even motion singularities during the entire deployment process; and second, whether the final deployment angle of the locking hinge meets the design requirements.

[0097] Example

[0098] Analysis of degrees of freedom in each stage of the unfolding process:

[0099] Although the deployable mechanism of a spaceborne SAR antenna is essentially a spatial mechanism, during its deployment, each point on the component moves only on the plane formed by the deployment direction and the perpendicular direction of the antenna panel. Therefore, as... Figure 2 The planar configuration shown encompasses the deployment motion characteristics of this space support mechanism. Here, 1 is the first satellite connecting rod, 2 is the second satellite connecting rod, 3 is the inner support rod, 4 is the first middle support rod, 5 is the second middle support rod, 6 is the first outer support rod, and 7 is the second outer support rod; 8 is the inner antenna plate, and 9 is the outer antenna plate; hinges A, C, E, F, and H; 180° locking hinge B; 180° locking hinge D; 180° locking hinge I; 180° locking hinge J; and 90° locking hinge G. Hinges A and 90° locking hinge G are fixed to the payload bay. Drive motors are located at 90° locking hinge G and 180° locking hinge J, serving as the main drive source. Furthermore, to ensure smooth deployment and reliable locking of the mechanism, coil springs are added at 180° locking hinges B, D, and I. From the perspective of maintaining a stable deployment configuration, the deployment reliability requirement of the deployable mechanism is that after the motor stops running, all of the aforementioned 90° locking hinges G, J, D, I, and B must be locked. Considering that the drive hinge at the motor is fully locked by the drive strategy, the essence of reliable deployment lies in whether the remaining three hinges—180° locking hinges B, D, and I—can be reliably locked. Based on the design structure of the locking hinges, this means whether the deployment angles of 180° locking hinges B, D, and I fall within 180°. ° Within ±0.2°.

[0100] According to the driving strategy designed for the project, the deployment process of the deployable mechanism is divided into the following three stages:

[0101] (1) The 90° drive hinge and the 180° drive hinge work simultaneously, and the 180° locking hinge of the first outer support rod 6 and the second outer support rod 7 locks first.

[0102] (2) When the inner plate 8 and the outer plate 9 of the antenna are spread out to 180° (i.e. the inner and outer plates are on the same straight line), the 180° locking hinge J is locked, and the 180° locking hinge I of the first middle support rod 4 and the second middle support rod 5 are locked at the same time.

[0103] (3) Continue to push the mechanism to unfold. When the inner plate 8 of the antenna moves to the horizontal position (i.e., the 90° locking hinge G rotates 90°), the drive motor stops working, and the 180° locking hinge B between the first star connecting rod 1 and the second star connecting rod 2 locks at the same time.

[0104] According to the theory of degrees of freedom of planar mechanisms, the degrees of freedom of the above three stages of the mechanism are as follows:

[0105]

[0106] Where: DOF i (i = 1, 2, 3) represents the degrees of freedom in the i-th stage; N represents the number of components; P L P is a low-order subnumber in the plane; H It represents the plane height subnumber.

[0107] Since the three-degree-of-freedom mechanism only has two drive sources during the initial deployment, the first stage is an underactuated deployment process.

[0108] Establishment of the kinematic model for the unfolding process:

[0109] In the mathematical model of the nine-bar linkage, each link is represented as a displacement vector, with the starting point of the vector being one end of the link and the ending point being the other end. According to... Figure 3 The coordinate system and symbols shown are used to model the three loops respectively using the closed-loop vector method.

[0110] For the loop ABCHGA, there are equality constraints in the horizontal and vertical directions:

[0111]

[0112] For the loop CIFHC, we have the following equation:

[0113]

[0114] Similarly, for the loop HCDEJFH, we can obtain:

[0115]

[0116] In the above three formulas: l i (i = 1, 2, ..., 13) represents the length of the i-th segment; This represents the angle between the j-th segment and the horizontal direction.

[0117] After differentiating both sides of equations (2), (3), and (4) with respect to time, using... And its angular velocity, other angular velocities can be expressed as:

[0118]

[0119]

[0120]

[0121]

[0122]

[0123]

[0124] In the formula for The corresponding angular velocity. Based on this, the relevant acceleration can be obtained by further differentiation, that is:

[0125]

[0126]

[0127]

[0128]

[0129]

[0130]

[0131] In the formula for The corresponding angular acceleration.

[0132] Establishment of dynamic models for each deployment stage of the deployable mechanism:

[0133] A dynamic model of each deployment stage of the deployable mechanism is established using the Lagrange method. The Lagrange function of each deployment stage of the deployable mechanism is defined as the difference between the kinetic energy and potential energy of the system. The Lagrange equation for the system at this stage is:

[0134]

[0135] In the formula, F is the generalized force, and E k and E p These are the system's kinetic energy and elastic potential energy, respectively.

[0136] The system has only three coil springs, namely:

[0137]

[0138] Specifically, the expressions for each elastic potential energy are as follows:

[0139]

[0140] Where: K T Let θ be the spring constant, and θ1, θ2, and θ3 be the initial torsion angles of the three springs, respectively.

[0141] Since the system has 3 degrees of freedom, the generalized input force of the system is the input torque τ at hinge G and hinge J. J and τ G There is no external force input at hinge A.

[0142] Therefore, substituting into the Lagrange equations, we obtain the dynamic models for each stage of the unfolding mechanism:

[0143]

[0144] Numerical iterative solution:

[0145] Based on the kinematic and dynamic models of the deployment process, there are 27 unknown parameters, but only 21 independent equations. Therefore, the number of unknown parameters in the first underactuated stage is less than the number of system equations, and the motion laws cannot be obtained using traditional methods. To address this, this application proposes a numerical iterative algorithm for solving the kinematic and dynamic problems of deployable mechanisms. Given the initial motion conditions of each component (center of mass position, angular velocity, and angular acceleration) and the driving strategy of the active component, the algorithm iteratively solves the dynamic response characteristics (motion laws) of each component during the underactuated deployment process based on the principle of constant velocity and acceleration within two subdivided time intervals, directly obtaining the kinematic and dynamic characteristics of each component. and The initial values ​​of the angle, angular velocity, and angular acceleration are determined, which reduces the number of unknown parameters to 21, thus equaling the number of equations. Therefore, the values ​​of all variables at the initial moment can be solved.

[0146] If from t i-1 Time to t i Since the time interval δt is infinitesimally small, the velocity and acceleration of the rod can be considered constant within this infinitesimally small interval, equal to the velocity and acceleration of the previous moment within that interval. However, the velocity and acceleration at different moments within the interval change abruptly, i.e.:

[0147]

[0148] Based on the above expression, the first stage deployment process of the deployable mechanism can be numerically iteratively analyzed. The numerical iterative solution algorithm flow is as follows: Figure 4 As shown, the algorithm solves both the kinematic and dynamic equations simultaneously.

[0149] After hinge D is locked, the system applies another constraint. The underactuated deployment process is complete; in the next stage, the number of degrees of freedom is the same as the number of driving sources, and the kinematic and dynamic models of the deployment process can be solved using the following Lagrange equations:

[0150]

[0151] For hinge I, there are two possible scenarios: one is that hinge I and hinge D are completely self-locking, in which case constraint equations are added to the system. Second, when τ J When the angle between links CI and IF is not 180° at time 0, this angle remains constant after the system enters the third stage because the two links do not move relative to each other. Therefore, the third stage can also be analyzed using the traditional Lagrange method. Since hinge J is locked, its dynamic equations degenerate into:

[0152]

[0153] Determine the final unfolding angle of the locking hinge:

[0154] Based on the obtained motion law of the full unfolding process, check whether there is motion interference or singularity in the deployable mechanism. Calculate whether the final unfolding angle of each locking hinge is within ±0.2° of its respective locking angle (180° for a 180° locking hinge and 90° for a 90° locking hinge). If the final unfolding angle of each locking hinge is within ±0.2° of its respective locking angle, it indicates that the mechanism's unfolding and locking function is reliable; otherwise, it is unreliable.

[0155] In the embodiments of this application, when the final unfolding angle of 180° locking hinge B, 180° locking hinge D, 180° locking hinge I, and 180° locking hinge J is 180° ° ±0.2°, while the 90° locking hinge G's final unfolding angle is at 90°. ° Within ±0.2°, the mechanism's deployment and locking function is determined to be safe and reliable; if any locking hinge's final deployment angle is outside the ±0.2° range of its respective locking angle, the mechanism's deployment and locking function is deemed unreliable.

[0156] This invention establishes a dynamic model of the deployment process of a deployable mechanism, enabling quantitative analysis of the mechanism's deployment and locking characteristics under the influence of factors such as rod length error and driving torque fluctuation. This provides solid theoretical support for comprehensively determining the deployment reliability of the mechanism for spaceborne SAR antennas. Furthermore, by analyzing the kinematic characteristics of the deployable mechanism under different errors, it can effectively guide the optimization of the deployment mechanism's assembly and adjustment, improving assembly efficiency while ensuring assembly quality. In particular, the numerical iterative solution algorithm in this invention can effectively obtain the dynamic response of the underdetermined drive deployment process, and this algorithm also has reference value for solving dynamic problems under nonholonomic constraints.

[0157] This invention fully considers the characteristics of deployable mechanisms, such as multi-loop closed chains, variable degrees of freedom, underdetermined actuation, and strong nonlinearity, and provides a numerical iterative algorithm for effectively solving the kinematics and dynamics problems of deployable mechanisms. It abandons the general probabilistic reliability evaluation method and performs functional reliability assessment based on dynamic analysis, thus better meeting practical needs and effectively guiding engineering practice.

Claims

1. A reliability analysis method for the deployment locking function of a deployable mechanism for a spaceborne SAR antenna, characterized in that, Includes the following steps: S1. Establish a mathematical model of the nine-bar linkage mechanism according to the actual configuration of the deployable mechanism of the spaceborne SAR antenna. Divide the deployment stages according to the driving strategy of the deployable mechanism, calculate the degrees of freedom of each deployment stage, and determine whether each deployment stage is an underactuated deployment process or a well-posed deployment process based on the degrees of freedom of each deployment stage. S2. Establish the coordinate system of the deployable mechanism, determine the parameters of the deployable mechanism, list the geometric ring constraint equations corresponding to the nine-bar linkage mathematical model, and characterize the rotation angle of each driven member as a function of the rotation angle of the driving member according to the geometric ring constraint equations, and establish the kinematic model of the deployment process. S3. Solve the position of the center of mass, angular velocity and angular acceleration of each component according to the kinematic model of the deployment process, calculate the kinetic energy and elastic potential energy of the entire deployable mechanism, and establish the dynamic model of each deployment stage of the deployable mechanism based on the Lagrange equation. S4. Based on the kinematic model of the deployment process and the dynamic model of each deployment stage of the deployable mechanism, given the initial motion conditions of each component and the driving strategy of the active component, the dynamic response characteristics of each component during the underactuated deployment process are iteratively solved based on the principle of constant velocity and acceleration within two subdivided time intervals; the dynamic response characteristics of each component during the well-posed drive deployment process are obtained by directly solving the Lagrange dynamic equations. S5. Based on the obtained motion laws of each unfolding process, check whether there is motion interference or singularity in the deployable mechanism. If there is motion interference or singularity, it means that the unfolding locking function is unreliable. Otherwise, calculate the final unfolding angle of each locking hinge and determine whether the unfolding locking function of the mechanism is safe and reliable according to the engineering practice criteria based on the final unfolding angle of each locking hinge. Each component is equivalent to a point mass at its midpoint. The total kinetic energy of the system is obtained by expressing the rotational kinetic energy of each component. The elastic potential energy of each auxiliary coil spring is calculated according to the torque-rotation angle relationship. Then, the dynamic model of the system under three different unfolding stages is established using the Lagrange method. The three loops are modeled separately using the closed-loop vector method: For the loop ABCHGA, there are equality constraints in the horizontal and vertical directions: (2) For the loop CIFHC, we have the following equation: (3) Similarly, for the loop HCDEJFH, we can obtain: (4) Of the three formulas above: l i ( i =1, 2,…,13) represents the first, second,…,th,th. i The length of the segment; φ j ( j =1, 2,…,10) represents the first, second, and third digits. j The angle between the segment and the horizontal direction; After differentiating both sides of equations (2), (3), and (4) with respect to time, using... φ 1、 φ 5、 φ 6 and its angular velocity, other angular velocities can be expressed as: (5) (6) (7) (8) (9) (10) In the formula ( i =1, 2,…,9) is φ i The corresponding angular velocity; further differentiation yields the relevant acceleration, i.e.: (11) (12) (13) (14) (15) (16) In the formula ( i =1, 2,…,9) is φ i The corresponding angular acceleration.

2. The reliability analysis method for the deployment locking function of the deployable mechanism of the spaceborne SAR antenna according to claim 1, characterized in that, If the number of main drivers is less than the system's degrees of freedom, it is an underactuated expansion process; If the number of main drives equals the system's degrees of freedom, the entire mechanism is in a well-posed drive deployment process, which is called a well-posed drive deployment process.

3. The reliability analysis method for the deployment locking function of the deployable mechanism of the spaceborne SAR antenna according to claim 2, characterized in that, The unfolding process of a deployable mechanism consists of three stages. According to the theory of degrees of freedom of planar mechanisms, the degrees of freedom of the mechanism in the three stages are as follows: (1) In the formula: DOF i ( i =1, 2, 3) is the first i The degree of freedom of the stage; N The number of components; P L It is a low-order sub-number in the plane; P H It represents the plane height subnumber.

4. The reliability analysis method for the deployment locking function of the deployable mechanism of the spaceborne SAR antenna according to claim 1, characterized in that, The generalized input force of the system is the input torque at hinge G and hinge J. τ J and τ G With no external force input at hinge A, the dynamic model of each deployment stage of the deployable mechanism is as follows: (20) E k and E p These are the system's kinetic energy and elastic potential energy, respectively.

5. A reliability analysis system for the deployment locking function of a spaceborne SAR antenna deployable mechanism, characterized in that, include: Degrees of freedom calculation module: Establish a mathematical model of the nine-bar linkage mechanism according to the actual configuration of the deployable mechanism of the spaceborne SAR antenna. Divide the deployment stage according to the driving strategy of the deployable mechanism, calculate the degrees of freedom of each deployment stage, and determine whether each deployment stage is an underactuated deployment process or a well-actuated deployment process based on the degrees of freedom of each deployment stage. Kinematic model building module: Establish the coordinate system of the deployable mechanism, determine the parameters of the deployable mechanism, list the geometric ring constraint equations corresponding to the nine-bar linkage mathematical model, and characterize the rotation angle of each driven member as a function of the rotation angle of the driving member according to the geometric ring constraint equations, and establish the kinematic model of the deployment process. Dynamics model building module: Based on the kinematic model of the deployment process, solve for the position of the center of mass, angular velocity and angular acceleration of each component, calculate the kinetic energy and elastic potential energy of the entire deployable mechanism, and establish the dynamics model of each deployment stage of the deployable mechanism based on the Lagrange equation; Dynamic response characteristics calculation module: Based on the kinematic model of the deployment process and the dynamic model of each deployment stage of the deployable mechanism, given the initial motion conditions of each component and the driving strategy of the active component, the dynamic response characteristics of each component during the underactuated deployment process are iteratively solved based on the principle of constant velocity and acceleration within two subdivided time intervals; the dynamic response characteristics of each component during the well-posed drive deployment process are obtained by directly solving the Lagrange dynamic equations. Reliability Analysis Module: Based on the obtained motion laws of each deployment process, check whether there is motion interference or singularity in the deployable mechanism. If motion interference or singularity exists, it indicates that the deployment locking function is unreliable; otherwise, calculate the final deployment angle of each locking hinge, and determine whether the deployment locking function of the mechanism is safe and reliable according to engineering practice criteria based on the final deployment angle of each locking hinge.

6. The reliability analysis system for the deployment locking function of the deployable mechanism of the spaceborne SAR antenna according to claim 5, characterized in that, If the number of main drivers is less than the system's degrees of freedom, it is an underactuated expansion process; If the number of main drives equals the system's degrees of freedom, the entire mechanism is in a well-posed drive deployment process, which is called a well-posed drive deployment process.

7. The reliability analysis system for the deployment locking function of the deployable mechanism of the spaceborne SAR antenna according to claim 5, characterized in that, The unfolding process of a deployable mechanism consists of three stages. According to the theory of degrees of freedom of planar mechanisms, the degrees of freedom of the mechanism in the three stages are as follows: (1) In the formula: DOF i ( i =1, 2, 3) is the first i The degree of freedom of the stage; N The number of components; P L It is a low-order sub-number in the plane; P H It represents the plane height subnumber.

8. The reliability analysis system for the deployment locking function of the deployable mechanism of the spaceborne SAR antenna according to claim 5, characterized in that, The three loops are modeled separately using the closed-loop vector method: For the loop ABCHGA, there are equality constraints in the horizontal and vertical directions: (2) For the loop CIFHC, we have the following equation: (3) Similarly, for the loop HCDEJFH, we can obtain: (4) Of the three formulas above: l i ( i =1, 2,…,13) represents the first, second,…,th,th. i The length of the segment; φ j ( j =1, 2,…,10) represents the first, second, and third digits. j The angle between the segment and the horizontal direction; After differentiating both sides of equations (2), (3), and (4) with respect to time, using... φ 1、 φ 5、 φ 6 and its angular velocity, other angular velocities can be expressed as: (5) (6) (7) (8) (9) (10) In the formula ( i =1, 2,…,9) is φ i The corresponding angular velocity; further differentiation yields the relevant acceleration, i.e.: (11) (12) (13) (14) (15) (16) In the formula ( i =1, 2,…,9) is φ i The corresponding angular acceleration.

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