A Method for Modeling and Predicting the Performance of Fast Reflector Systems Based on the Transfer Matrix Method

Abstract

This invention relates to the field of precision optical instruments and dynamic modeling technology for mechatronic systems, and particularly to a method for modeling and predicting the performance of a fast mirror system based on the transfer matrix method. The method modularly decomposes the FSM system into basic components such as flexible hinges and rigid links; establishes the transfer matrix for each component to describe its input-output mechanical relationship; then assembles these matrices into a complete model according to the system topology; and by introducing boundary conditions, the mapping relationship between the driving displacement and the mirror pointing angle can be calculated quickly and accurately. This method significantly reduces the single analysis time while ensuring small calculation errors, achieving an order-of-magnitude improvement in efficiency.

Description

Technical Field

[0001] This invention belongs to the field of precision optical instruments and mechatronic system dynamics modeling technology, and particularly relates to a method for modeling and performance prediction of a fast reflector system based on the transfer matrix method. Background Technology

[0002] Fast mirrors (FSMs) are the core actuators for achieving rapid and precise beam pointing control, and their performance directly determines the ultimate capability limit of high-end optoelectronic systems. In space laser communication, FSMs are used to compensate for beam jitter caused by satellite platform vibrations, establishing and maintaining stable interplanetary links. In adaptive optics systems, FSMs act as wavefront correctors, correcting wavefront distortion caused by atmospheric turbulence in real time. In high-end imaging and tracking systems, FSMs achieve micro-motion scanning and high-precision stabilization of the line of sight. Furthermore, FSMs play an indispensable role in laser processing, biological microscopy imaging, and quantum communication. Therefore, key performance indicators of FSMs—such as the response bandwidth determining system response speed, pointing accuracy and resolution determining control limits, and working stroke determining the adjustment range—directly constitute the core bottlenecks of the entire optical system in terms of speed, accuracy, and stability. Their performance has become a key indicator of the advancement of an optoelectronic system.

[0003] Currently, the design and analysis of FSMs heavily rely on the Finite Element Analysis (FEA). Although FEA offers high computational accuracy, it has the following inherent limitations in engineering practice:

[0004] (1) Complex preprocessing: It requires detailed 3D modeling and mesh generation of complex geometric structures (especially flexible hinges), which is tedious and time-consuming.

[0005] (2) High computational resource consumption and low efficiency: A single static analysis usually takes more than 10 minutes, while dynamic or parametric analysis can take several hours or even days. This constitutes a serious efficiency bottleneck in the design optimization and parameter scanning stages that require a large number of iterations.

[0006] (3) Difficulty in model integration: FEA models usually exist as "black boxes", making it difficult to extract simple input-output relationships and unable to be directly embedded into the control system for controller design or real-time simulation.

[0007] (4) Lack of universality: FEA models for specific structures are difficult to be directly applied to FSMs of other configurations, resulting in poor model reusability.

[0008] Furthermore, although there are simplified methods such as the Pseudo-Rigid-Body Model (PRBM), which approximate distributed compliance with concentrated compliance and bring convenience to the analysis, it is difficult to guarantee modeling accuracy when dealing with complex flexible hinges (such as elliptical hinges) commonly used in high-performance FSMs, and the expression of physical meaning is not clear enough.

[0009] Therefore, this field has long faced a prominent contradiction: how to achieve rapid modeling and analysis of FSMs while meeting engineering accuracy requirements. For this reason, developing a new modeling method that combines high accuracy, high efficiency, and high versatility has become a crucial step in overcoming R&D bottlenecks. This is of significant engineering value and urgency for shortening R&D cycles, enabling parametric optimization design, and supporting advanced model-based control strategies. Summary of the Invention

[0010] In view of this, the present invention aims to provide a method for modeling and performance prediction of a fast mirror system based on the transfer matrix method. This method modularly decomposes the FSM system into basic components such as flexible hinges and rigid links, establishing a transfer matrix for each component to describe its input-output mechanical relationship. Then, based on the system topology, these matrices are assembled into a complete model. By introducing boundary conditions for solving, the mapping relationship between the driving displacement and the mirror pointing angle (i.e., the platform rotation angle) can be calculated quickly and accurately. This method significantly reduces the single analysis time while ensuring small calculation errors, achieving an order-of-magnitude improvement in efficiency.

[0011] To achieve the above objectives, the technical solution of the present invention is implemented as follows:

[0012] A method for modeling and predicting the performance of a fast reflector system based on the transfer matrix method includes the following:

[0013] Step 1: Construct a physical model of a generalized fast-reflecting mirror system. This physical model includes:

[0014] platform;

[0015] Multiple identical drive chains are connected in parallel between the platform and the fixed base. Each drive chain includes the following three types of components connected in series:

[0016] The driver is mounted on a fixed base;

[0017] First rigid link;

[0018] Flexible hinges are respectively connected between the driver output end and the first rigid link, and between the first rigid link and the platform;

[0019] Fixed base;

[0020] Step 2: Based on the dimensions of the flexible hinge, determine the transmission matrix of the flexible hinge; based on the motion state of the actuator and the first rigid link, determine the transmission matrix of the actuator and the first rigid link; based on the transmission matrix of the first rigid link and the motion state of the platform, determine the transmission matrix of the platform.

[0021] Step 3: Assemble the transfer matrices of the flexible hinge, driver, and first rigid link obtained in Step 2 to obtain the transfer matrix of the drive chain. Based on the transfer matrix of the drive chain, obtain the overall transfer matrix of the generalized fast reflector system. Finally, calculate the unknown motion state of the platform based on the input and output relationship of the entire generalized fast reflector system.

[0022] Step two includes:

[0023] a. The transfer matrix of the flexible hinge as follows:

[0024] ;

[0025] The expressions for the non-zero, non-identity elements in the transfer matrix are as follows: ;

[0026] in , For Young's modulus, The length of the semi-major axis of the upper semi-elliptical region of the elliptical cross-section of the flexible hinge. The thickness of the flexible thin-walled region of the flexible hinge. The thickness of the flexible hinge. The length of the minor axis of the upper semi-elliptical region of the elliptical cross section of the flexible hinge;

[0027] b. The transmission matrix of the driver is the same as that of the first rigid link, and they are collectively referred to as the transmission matrix of the rigid link. The specific structure is as follows:

[0028] ;

[0029] in Let be the angular frequency of the rigid link element's rotation about its center of mass. For the mass of the rigid connecting rod, Let be the moment of inertia of a rigid connecting rod about its center of mass, and be the element. for:

[0030] ;

[0031] in Let the centroid of the rigid link element and the input point be in the local coordinate system of the rigid link. x Distance on the axis Let the centroid of the rigid link element and the input point be in the local coordinate system of the rigid link. y Distance on the axis The output and input points of the rigid link element are in the local coordinate system of the rigid link. x Distance on the axis The output and input points of the rigid link element are in the local coordinate system of the rigid link. y Distance on the axis;

[0032] c. The transfer matrix of the platform as follows:

[0033] ;

[0034] in The second input point of the platform and the first input point in the platform's local coordinate system x Distance along the axis, The second input point of the platform and the first input point in the platform's local coordinate system y Distance on the axis.

[0035] Step three includes:

[0036] Step 1, for the drive chain The drive chain is obtained by assembling the transfer matrices of each component. transfer matrix :

[0037] ;

[0038] in driving chain after coordinate transformation The transfer matrix of the first rigid link. driving chain after coordinate transformation The transfer matrix of the upper driver, driving chain after coordinate transformation The transfer matrix of the upper flexible hinge;

[0039] Step 2, determine the overall transfer matrix ;in:

[0040] ;

[0041] ;

[0042] ;

[0043] ;

[0044] ;

[0045] ;

[0046] yes The first 6×6 blocks, yes The last 6×3 blocks, This is the transfer matrix of the platform after coordinate transformation. , is the force extraction matrix, , is the displacement extraction matrix; Let be the displacement coordination matrix for the two input points on the platform. yes The first 3 lines;

[0047] Step 3: Construct the overall equation describing the input and output relationship of the entire generalized fast reflector system. ,in:

[0048] ;

[0049] , Let be the state vector of the input points of the first and second driving chains in the global coordinate system. This is the state vector of the platform's output points in the global coordinate system;

[0050] Step 4, calculate using the following formula The vector consists of all unknown displacement components and internal force components, including the platform rotation angle reflecting the direction of the reflector. :

[0051] ;

[0052] in for A vector composed of all known displacement components and internal force components. for Zhongyu The corresponding element, for Zhongyu The corresponding element.

[0053] The derivation process of steps 2-4 is as follows:

[0054] a. Establish the transmission relationship equations between the two drive chains in one rotation direction of the control platform:

[0055] ;

[0056] ;

[0057] The two drive chains in one rotation direction of the control platform are referred to as the first drive chain and the second drive chain, respectively. , Let be the state vector of the input points of the first and second driving chains in the global coordinate system. , This represents the state vectors of the output points of the first and second driving chains in the global coordinate system.

[0058] b. and Substitute the transitivity equations into 'a' and combine them to form the overall transitivity matrix. The specific process is as follows:

[0059] First drive chain input point state vector Output point state vector of the platform The dynamic contribution is adopted The calculation method is as follows: ;

[0060] Second drive chain input point state vector Output point state vector of the platform The dynamic contribution is adopted The calculation method is as follows: ;

[0061] First drive chain input point state vector How to affect its output point state vector The displacement is adopted The calculation method is as follows: ;

[0062] Second drive chain input point state vector How to affect its output point state vector The displacement is adopted The calculation method is as follows: ;

[0063] Combining the above matrices, we obtain the overall transfer matrix. ,in , ;

[0064] c. The vector composed of all known displacement and internal force components. The vector consisting of all unknown displacement components and internal force components Divide into blocks, Zhongyu The corresponding elements are divided into ,Bundle Zhongyu The corresponding elements are divided into ,get:

[0065] ;

[0066] Due to the overall equation Therefore, we get:

[0067] ;

[0068] Rearranged into a non-homogeneous linear system of equations:

[0069] ;

[0070] The solution yields all unknown vectors, including the platform rotation angle reflecting the direction the mirror points:

[0071] .

[0072] The Specifically:

[0073] ;

[0074] in The second input point of the platform and the first input point in the global coordinate system x Distance on the axis The second input point of the platform and the first input point in the global coordinate system y Distance on the axis.

[0075] The transformation matrix used for coordinate transformation in step 1 is: Specifically:

[0076] ;

[0077] in Local coordinate system x The axis in the global coordinate system xoy Projection on a plane and the global coordinate system x The included angle of the axis.

[0078] The flexible hinge is an elliptical flexible hinge, and the rotation axes of the two flexible hinges on the same drive chain are orthogonal in space.

[0079] The platform and the fixed base have the same shape and are aligned. The platform, the fixed base, and the four drive chains fixed between them together form a spatial cube. The two drive chains located at opposite corners of the spatial cube form a group and are used to control the rotation of the platform in one direction. The two drive chains at opposite corners are symmetrically arranged.

[0080] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0081] (1) Revolutionary improvement in computational efficiency: This invention reduces the single-condition analysis time of the FSM system from tens of minutes in FEA to about ten milliseconds, improving computational efficiency by about four orders of magnitude (tens of thousands of times). This makes it possible to complete tens of thousands of design parameter scans and optimization iterations within a day, greatly shortening the R&D cycle.

[0082] (2) Accuracy meets high-standard engineering requirements: Through direct comparison with traditional high-precision FEA results, the relative error of the pointing angle calculated by the method of this invention is strictly controlled within 0.5%. This level of accuracy fully meets the design and performance prediction requirements of FSM in precision optical systems.

[0083] (3) Inherent universality and flexibility: The method of this invention is universal and is not targeted at a single product model. By changing or modifying the transfer matrix of basic components (such as flexible hinges with different geometric parameters), FSM models with different topologies and sizes can be quickly constructed and analyzed, realizing "one-time derivation, multiple applications", which significantly reduces the modeling cost of new products.

[0084] (4) Laying the foundation for real-time control and simulation: Since the model of this invention has analytical / semi-analytical form and extremely high calculation speed, it can be easily integrated into the system simulation environment (such as Matlab / Simulink), and can even be used as the controlled object model directly for the design and online operation of advanced control algorithms such as model predictive control (MPC) and feedforward compensation, which is something that the FEA model cannot achieve.

[0085] (5) Clear physical meaning, easy to understand and optimize: The transfer matrix method clearly reveals the transfer path of load and deformation in the system, helping designers to intuitively understand the mechanical behavior of the system, making it easier to identify performance bottlenecks and key design parameters, and thus carry out targeted optimization. Attached Figure Description

[0086] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0087] Figure 1This is a schematic diagram of the physical model structure of the universal fast-reflecting mirror system of the present invention;

[0088] Figure 2 This is a schematic diagram of the elliptical cross-section of the flexible hinge of the present invention;

[0089] Figure 3 This is a schematic diagram of the single-input single-output rigid linkage unit of the present invention;

[0090] Figure 4 This is a schematic diagram of the dual-input single-output rigid linkage unit of the present invention;

[0091] Figure 5 This is a comparison chart of the embodiments of the present invention and the simulation experiment results.

[0092] The reference numerals in the accompanying drawings include: 1. Platform; 2. Flexible hinge; 3. First rigid link; 4. Driver; 5. Fixed base. Detailed Implementation

[0093] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not constitute a limitation thereof. Similar elements in different embodiments are referred to by associated similar element reference numerals. In the following embodiments, many details are described to facilitate a better understanding of the invention. However, those skilled in the art will readily recognize that some features may be omitted in different situations, or may be replaced by other elements, materials, or methods. In some cases, certain operations related to the invention are not shown or described in the specification. This is to avoid obscuring the core parts of the invention with excessive description. For those skilled in the art, detailed description of these related operations is not necessary; they can fully understand the related operations based on the description in the specification and general technical knowledge in the art.

[0094] It should be noted that, unless otherwise specified, the embodiments and features described in this invention can be combined to form various implementations. Furthermore, the order of the steps or actions in the method description can be changed or adjusted in a manner readily apparent to those skilled in the art. Therefore, the various orders in the specification and drawings are merely for the clear description of a particular embodiment and do not imply a mandatory order, unless otherwise stated that a particular order must be followed.

[0095] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," and "counterclockwise," etc., indicating orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the invention. Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined with "first," "second," etc., may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0096] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art will understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0097] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0098] Please see Figure 1 A method for modeling and predicting the performance of a fast mirror system based on the transfer matrix method is proposed. This method treats the complex FSM system as a combination of standard mechanical components connected by a specific topology. By establishing and assembling the transfer matrices of each component, the input-output model of the entire system is constructed. Specifically, it includes the following:

[0099] (1) Construct a physical model of a generalized fast reflector (FSM) system;

[0100] This model is a modular, general physical model for a four-point driven FSM system, such as... Figure 1 As shown. The model includes, from top to bottom:

[0101] Platform 1, which is used to mount the reflector, is the end effector, specifically the back plate used to mount the reflector, and is the final output component of the system. In the modeling, it is treated as a rigid body.

[0102] Multiple identical drive chains are connected in parallel between the platform 1 and the fixed base 5. Each drive chain includes the following three types of components connected in series:

[0103] The driver 4 is mounted on the fixed base 5;

[0104] The first rigid connecting rod 3, which serves as a transmission component, is relatively thin, with a thickness of 1mm;

[0105] Flexible hinges 2 are respectively connected between the output end of the driver 4 and the first rigid link 3, and between the first rigid link 3 and the platform 1;

[0106] Two flexible hinges 2 are respectively connected to the upper and lower ends of the first rigid link 3. That is, the lower flexible hinge 2 is connected between the output end of the driver 4 and the first rigid link 3, and the upper flexible hinge 2 is connected between the first rigid link 3 and the platform 1.

[0107] Fixed base 5 provides a fixed mounting foundation and motion reference system for the entire system;

[0108] The flexible hinge 2 is an elliptical flexible hinge, and the rotation axes of the two flexible hinges 2 on the same drive chain are orthogonal in space, thereby converting the linear output of the driver 4 into the pitch and yaw motion of the platform 1 around two orthogonal axes.

[0109] The platform 1 and the fixed base 5 have the same shape and are aligned. The platform 1, the fixed base 5, and the four drive chains fixed between them together form a spatial cube. The two drive chains located at opposite corners of the spatial cube form a group and are used to control the rotation of the platform 5 in one direction. The two drive chains at opposite corners are symmetrically arranged, that is, after one drive chain at the corner rotates 180 degrees around the center of the platform 1, it can coincide with the other drive chain at the corner, so as to achieve decoupling of motion.

[0110] The first rigid link 3 and each flexible hinge 2 on the same drive chain are coaxial.

[0111] The fixed base 5 does not participate in the actual displacement calculation during the subsequent transfer matrix construction process; the driver 4 is used to provide input displacement, which is equivalent to a rigid link in this invention, referred to as the second rigid link. Its output end drives the first rigid link 3 through a flexible hinge 2. The first rigid link 3 is connected to the platform 1 through another flexible hinge 2, thereby transmitting the motion of the driver 4 to the platform 1; the driver 4 is a voice coil motor (VCM) or piezoelectric ceramic motor (PZT) commonly used in the industry; the flexible hinge 2 is responsible for releasing deformation in a specific direction (in this article, it is the bending deformation in two mutually perpendicular directions), while the first rigid link 3 located in the middle is designed to avoid stress concentration at the four corners of the flexible hinge 2, and is used to accurately transmit the displacement of the lower flexible hinge 2 to the upper flexible hinge 2.

[0112] (2) Define the state vector of the physical model of the mirror system;

[0113] To fully describe the mechanical state at any point on the physical model of the mirror system, a six-dimensional state vector is defined for any point. ,in:

[0114] ;

[0115] in , For any point in the corresponding coordinate system x Axial direction and y Displacement in the axial direction; For any point in the corresponding coordinate system, around z Angular displacement of the axis of rotation; , Each point in the corresponding coordinate system is along... x shaft and y Force exerted in the axial direction; This represents the torque experienced at any point. This state vector contains both displacement and force information, and can completely characterize the motion and force state at any location.

[0116] (3) Derive the transmission matrix of key basic components: driver 4, flexible hinge 2, first rigid link 3 and platform 1;

[0117] 1. The transfer matrix of the flexible hinge 2 is used to characterize the overall distributed flexibility characteristics of the flexible hinge 2. It can accurately reflect the distributed flexibility characteristics of the flexible hinge 2, specifically:

[0118] The center of the output end of flexible hinge 2 in the local coordinate system of the flexible hinge The state vector at that location, The center of the input end of flexible hinge 2 in the local coordinate system of the flexible hinge The state vector at that point; the two are related through a transfer matrix, that is:

[0119] ;

[0120] The derivation process of the transfer matrix of flexible hinge 2 is as follows (e.g.) Figure 2 (as shown)

[0121] Physical modeling: The elliptical cross-section (i.e., hourglass-shaped cross-section) of flexible hinge 2 in its natural, unforced state is used as the model. xoy Surface, flexible hinge 2 input end center in With the origin as the axis and the flexible hinge as the axis of 2, x A local coordinate system for the flexible hinge is established along the axis; the following derivation of the transfer matrix for flexible hinge 2 is based on this coordinate system.

[0122] In the local coordinate system plane, the flexible hinge 2 is discretized into a series of Euler-Bernoulli beams with varying cross-sections, the cross-section height of which is... along x The axis changes continuously according to the following ellipse equation:

[0123] ;

[0124] in for The minimum value, that is, the thickness of the flexible thin-walled region of flexible hinge 2, The length of the major semi-elliptical region on the upper half-elliptical section is denoted as the elliptical half-width. Let be the length of the minor axis of the semi-elliptical region above the elliptical cross section, denoted as the length of flexible hinge 2;

[0125] Matrix derivation:

[0126] Based on the differential equilibrium equations and deformation compatibility relations of Euler-Bernoulli beams, matrix differential equations are established. ,in This refers to an intermediate result obtained during numerical integration. This is the state vector of any point in the local coordinate system of the flexible hinge. The matrix differential equation is then processed using the fourth-order Runge-Kutta method. arrive Numerical integration is performed to obtain the transfer matrix characterizing flexible hinge 2. The specific solution process is as follows:

[0127] a. Initialization and Discretization:

[0128] The length of flexible hinge 2 exist x Uniformly discretized on the axis Each micro-segment has a step size (i.e., the length of the micro-segment) of 1. Discrete points are denoted as ;in , , ,and ;

[0129] Initialize a 6×6 initial transfer matrix. Set it as a 6th order identity matrix This matrix represents the position The cumulative transfer matrix at that point, ,So , For discrete points The state vector at that location;

[0130] b. Runge-Kutta iteration (achieving higher accuracy through multi-step prediction and correction):

[0131] Using the fourth-order Runge-Kutta method through matrix differential equations Update the initial transfer matrix to complete. After iteration of each micro-segment, in The matrix is ​​obtained at the location That is, the transfer matrix of the flexible hinge 2. Its composition is as follows:

[0132] ;

[0133] The non-zero, non-unit elements in the transfer matrix reflect the flexibility characteristics of the hinge, expressed as: ;

[0134] This is for the purpose of simplifying the description. , , To characterize the Young's modulus of the flexible hinge 2 material, The thickness of flexible hinge 2 is a parameter that needs to be assigned a value; that is, the position of flexible hinge 2 perpendicular to the coordinate system of the flexible hinge. xoy face z Thickness in the axial direction;

[0135] 2. The derivation process of the transfer matrix of the rigid link is as follows, such as... Figure 3 As shown:

[0136] Based on the motion characteristics, the first rigid link 3 is simplified into a single-input single-output (SISO) rigid link unit moving in the same plane; the driver 4 is equivalent to a rigid link, denoted as the second rigid link, which is also simplified into a single-input single-output rigid link unit moving in the same plane; therefore, the derivation process of the transfer matrix is ​​consistent for both, as detailed below:

[0137] Based on rigid body kinematics (displacement transfer) and the Newton-Euler equations (force / torque balance), the mass of the rigid link is considered. and the moment of inertia of a rigid connecting rod about its center of mass Derive its transfer matrix in the rigid link element coordinate system;

[0138] Establish a local coordinate system for the single-input, single-output rigid linkage element: Since the single-input, single-output rigid linkage element moves within the same plane, the input and output points of the element are also in the same plane. Within this plane, construct a rectangular plane using the line connecting the input and output points as the endpoints of the diagonal. Use this rectangular plane as the coordinate system... xoy A plane, with the input point as the origin and one side of the rectangular plane as... x axis, one side is y A local coordinate system is established for the single-input, single-output rigid link element; the following derivation of the transfer matrix of the rigid link is based on this coordinate system.

[0139] Under the assumption of simple harmonic motion, this transfer matrix includes an inertial term. and ,in Let be the angular frequency of a rigid link element rotating about its center of mass. Consider a single-input, single-output rigid link element moving in a plane, with its input point being... i The output point is o The center of mass is C The specific derivation process is as follows:

[0140] A. Derivation of kinematic relationships:

[0141] The displacement and rotation angle of the output point of a rigid linkage element can be determined by the motion state of the input point:

[0142] Output point at x Displacement in the axial direction Let this be denoted as Formula 1;

[0143] in The angle by which the output point rotates about the input point. For the input point at x Displacement in the axial direction, For output points y Axis coordinates For input points y Axis coordinates; For the output point and input point at y Distance on the axis; at this time, the output point state vector for ;

[0144] Output point at yDisplacement in the axial direction This is denoted as Formula 2;

[0145] in For the input point at y Displacement in the axial direction, For output points x Axis coordinates For input points x Axis coordinates For the output point and input point at x The distance on the axis; for a rigid body in rotation, all points rotate at the same angle, therefore the angle of rotation of the input point about the output point is... This is denoted as Formula 3. At this time, the input point state vector... for ;

[0146] B. Derivation of dynamic relationships:

[0147] a) Centroid displacement relationship:

[0148] The center of mass of the rigid link unit is at x Displacement in the axial direction and in y Displacement in the axial direction Represented by the displacement and rotation angle of the input point:

[0149] ;

[0150] ;

[0151] in The center of mass of the rigid link element y Axis coordinates For the center of mass of the rigid link element and the input point at x Distance on the axis For the center of mass of the rigid link element and the input point at y Distance on the axis The center of mass of the rigid link element x Axis coordinates, outputting the angle of rotation of the point about its center of mass. At this time, the state vector of the centroid for ;

[0152] b) Equations for equilibrium of forces:

[0153] According to Newton's second law, consider the inertial force under simple harmonic motion:

[0154] x Directional force balance: This is denoted as Formula Four;

[0155] yDirectional force balance: This is denoted as Formula 5;

[0156] in For the output point at x Force in the axial direction, For the input point at x Force in the axial direction, For the output point at y Force in the axial direction, For the input point at y Force exerted in the axial direction;

[0157] c) Torque balance equation:

[0158] Taking the moment at the output point, considering inertial force and moment of inertia, the torque at the output point... ;

[0159] in The torque at the input point;

[0160] Substituting and rearranging the relationship of centroid displacement in (a) as follows:

[0161] Expanding and combining like terms yields:

[0162] ,

[0163] This is denoted as Formula Six;

[0164] Based on the above derivation, and considering the relationships between the elements in the state vectors of the output and input points...

[0165] By comparing the relationships and using the six formulas (formulas one through six) concerning the output point, the transfer matrix is ​​assembled to obtain the transfer matrix of the rigid link. :

[0166] ;

[0167] To simplify the description, the elements in the matrix are... According to the definition of dynamic equilibrium conditions:

[0168] ;

[0169] 3. Platform 1 is simplified into a dual-input single-output (DISO) rigid linkage unit that moves in the same plane, such as... Figure 4 As shown, consider a rigid link element moving in a plane, with two input points... and state vector and With output point state vector They are linked together through a transfer matrix, in the form of: The transfer matrix of platform 1 The derivation process is the same as that of the single-input single-output rigid linkage unit, which is also based on rigid body kinematics and static equilibrium. The input point... For output points The contribution of the state vector is the same as that in the single-input single-output rigid linkage element. It can be seen that the transfer matrix of platform 1 only needs to add the input point characterization function compared to the transfer matrix of the single-output rigid element. For output points The three columns of the state vector contribution are sufficient; therefore... The complete matrix is ​​as follows:

[0170] ;

[0171] in The second input point of the platform, i.e., the input point The first input point is the input point. exist x Distance along the axis, For the platform's second input point and the first input point at y Distance on the axis; element The expression is the same as the corresponding content in the transfer matrix of a single-input single-output rigid element, but the coordinate parameters need to be determined relative to the selected reference point;

[0172] The transfer matrix of platform 1 The derivation is based on the local coordinate system of the dual-input single-output rigid linkage unit, i.e., the platform's local coordinate system. This coordinate system is defined as follows: Since the dual-input single-output rigid linkage unit moves in the same plane, the two input points and the output point within the unit are also in the same plane. Taking this plane as... xoy Surface, with input point Origin and input points and input point The direction of the connection is x Establish a local coordinate system for a dual-input, single-output rigid linkage element;

[0173] 4. Assembly and solution of the overall transfer matrix of the universal fast-reflecting mirror system, the specific content of which is as follows:

[0174] Driver chain transfer matrix assembly: For any driver chain The transfer matrices of each component are multiplied by their corresponding transformation matrices to convert them into transfer matrices in the global coordinate system. Finally, the transfer matrices in the global coordinate system are multiplied together in order from input to output to obtain the driving chain. transfer matrix :

[0175] ;

[0176] in driving chain after coordinate transformation The transfer matrix of the first rigid link 3. driving chain after coordinate transformation The transfer matrix of the second rigid link. driving chain after coordinate transformation The transfer matrix of the upper flexible hinge 2;

[0177] The transformation matrix used for the coordinate transformation as follows:

[0178] ;

[0179] in For the corresponding local coordinate system x The axis in the global coordinate system xoy Projection on a plane and the global coordinate system x The angle between the axes;

[0180] Introducing geometric compatibility conditions: Since platform 1 at the end is rigid, the state vectors of its two corresponding input points can be obtained through a displacement compatibility matrix. Interrelated, that is The displacement coordination matrix of the second input point relative to the first input point The coordinates of the second input point relative to the first input point Decision, among which The second input point of the platform and the first input point in the global coordinate system x Distance on the axis The second input point of the platform and the first input point in the global coordinate system y Distance on the axis;

[0181] ;

[0182] Establish the transmission relationship equations for the two drive chains (i.e., two drive chains located diagonally opposite each other, denoted as the first drive chain and the second drive chain) in one rotation direction of control platform 1:

[0183] ;

[0184] ;

[0185] in , Let be the state vector of the first drive chain input point and the second drive chain input point (i.e., the driver input point of the first drive chain and the driver input point of the second drive chain) in the global coordinate system. , The first drive chain output point and the second drive chain output point (i.e., the top flexible hinge output point of the first drive chain and the top flexible hinge output point of the second drive chain) are the state vectors in the global coordinate system. The first drive chain output point and the second drive chain output point are also two input points of platform 1. To control platform 1, the transfer matrix of the first drive chain in one of the two drive chains in a rotational direction, The transfer matrix of the second drive chain in one of the two drive chains in a rotational direction of the control platform 1.

[0186] Constructing the overall system equations This describes the input-output relationship of the entire system, with the core idea being to satisfy the following two conditions:

[0187] A. Dynamics (Platform Transfer Matrix) How the output point of platform 1 is ultimately driven by the output points of the two drive chains that control one direction of rotation of platform 1 (these two points are also the connection points between the drive chains and platform 1) (force balance and transmission); that is... How to , Received;

[0188] B. Kinematics (Displacement Coordination Matrix) Platform 1 and the output ends of the two drive chains must move synchronously as the same rigid body (geometric constraint).

[0189] The solution is divided into the following three steps:

[0190] a. Assemble the overall transfer matrix :

[0191] Based on the two conditions A and B above, and Substituting the equations for the transmission relationships of the two driving chains and combining them, we can form a 9×18 overall transmission matrix. Overall transfer matrix Center first 6 lines (From Condition A - Dynamics) describes the state vectors of the two drive chain input points. and How to ultimately generate the state vector of the platform 1 output point? :

[0192] ;

[0193] in This is the transfer matrix of platform 1 after coordinate transformation;

[0194] Overall matrix The last three lines (From Condition B - Kinematics) describes how to satisfy rigid body constraints, and The geometric relationship that must be satisfied between them is:

[0195] ;

[0196] Here , , , The combination matrix was introduced to simplify the expression:

[0197] Dynamics section ( and ):

[0198] State vector representing the input point of the first drive chain Output point state vector for platform 1 The dynamic contribution.

[0199] Calculation method: Soon As Substitution Obtained from;

[0200] in yes The first 6×6 blocks.

[0201] State vector representing the input point of the second drive chain State vector of output point of platform 1 The dynamic contribution.

[0202] The calculation method is the same: ;

[0203] in yes The last 6×3 blocks, It is the force extraction matrix.

[0204] Kinematics section ( and ):

[0205] State vector representing the input point of the first drive chain How to kinematically influence the output point state vector The displacement.

[0206] Calculation method: Soon As Substitution Obtained from;

[0207] in yes The first 3 rows of the matrix;

[0208] Represents the state vector of the input point of the second drive chain. How to kinematically influence the output point state vector The displacement.

[0209] Calculation method: ;

[0210] in It is the displacement extraction matrix.

[0211] b. Establish the overall equation:

[0212] Combine the above matrices and substitute them into the overall equation. .

[0213] The overall transfer matrix It is a 9×18 vector, and It is an 18×1 vector.

[0214] Solve based on the following boundary conditions:

[0215] Input: Driver 4 applies a specified displacement, therefore the displacement components in the state vectors of the two input points on the two drive chains... Given the internal force components unknown;

[0216] Output end (Platform 1): Considered as a free end, therefore the internal force components in the state vector of the output point of Platform 1 are... =0 (known), displacement component unknown;

[0217] Therefore, in Of the 18 variables, there are 9 unknowns and 9 knowns.

[0218] Will The vector composed of all known displacement and internal force components. (9×1) and the vector consisting of all unknown displacement components and internal force components Divide the data into blocks of (9×1) and... Zhongyu The corresponding elements are divided into ,Bundle Zhongyu The corresponding elements are divided into ,get:

[0219] ;

[0220] in and All are 9×9 matrices.

[0221] Substituting into the overall equation, we get:

[0222] ;

[0223] Rearranged into a non-homogeneous linear system of equations:

[0224] ;

[0225] The solution yields all unknown vectors, including the rotation angle of platform 1, which reflects the direction the mirror points. :

[0226] .

[0227] This invention utilizes the transfer matrix method to construct a generalized FSM theoretical model, in which all parameters can be adjusted at any time, such as the geometric parameters of the flexible hinge 2 and the rigid link (e.g., , , The parameters and material parameters (elastic modulus E, density ρ, Poisson's ratio υ, etc.) can be customized and replaced according to the actual structure; and the accurate pointing orientation of the end platform 1 under the current parameters can be quickly calculated, and the pointing error can be obtained. Therefore, this model has great guiding significance for the early design of FSM. Since the assembly method of the system's total transfer matrix can be freely matched and replaced according to the actual structure, it is possible to analyze the structure suitable for different working conditions through different parameter combinations; in addition, the extremely high computational efficiency is also very beneficial for the real-time dynamic pose adjustment of the reflector in actual engineering, so as to improve the dynamic response capability of FSM.

[0228] To demonstrate the reliability of this invention, the physical model of the constructed generalized fast-reflecting mirror system was imported into the finite element simulation software Patran using existing technology. A finite element mesh (mesh element width 0.5 mm) was generated, and element material properties were assigned (E=71 GPa, υ=0.33, ρ=2770 kg / m³). Boundary conditions were given, and different working conditions were established. The model was then submitted to Nastran software for simulation calculations. The calculation results were returned to Patran software for structural post-processing, displaying displacement contour plots. Displacement data from two sets of 21 sampling points each located in orthogonal directions on platform 1 were exported. Linear fitting of discrete points was performed in the data processing software R to calculate the actual pointing azimuth angle of platform 1 under the finite element results.

[0229] Under all operating conditions, such as Figure 5 As shown, the pointing angle calculated using this method is very close to the results of the finite element simulation experiment, with the relative error controlled within 0.5% (the maximum error being 0.1714%). This fully demonstrates that the established transfer matrix model has high computational accuracy and can accurately predict the static pointing performance of the FSM under different driving conditions. In addition, the calculation time for a single operating condition is reduced from 10 to 30 minutes (excluding finite element preprocessing, post-processing, and analysis) to about 10 milliseconds.

[0230] In summary, the above description is merely a preferred embodiment of this specification and is not intended to limit the scope of protection of this specification. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this specification should be included within the scope of protection of this specification.

Claims

1. A fast mirror system modeling and performance prediction method based on transfer matrix method, characterized in that, The application includes the following contents: Step 1, constructing a physical model of a general fast steering mirror system, the physical model comprising: a platform; a plurality of same driving chains, each driving chain being connected in parallel between the platform and a fixed base, each driving chain comprising the following three types of elements connected in series: a driver arranged on the fixed base; a first rigid link; a flexible hinge connected between the output end of the driver and the first rigid link and between the first rigid link and the platform, respectively; a fixed base; Step 2, determining the transfer matrix of the flexible hinge based on the size of the flexible hinge, determining the transfer matrix of the driver and the first rigid link based on the motion state of the driver and the first rigid link, and determining the transfer matrix of the platform based on the transfer matrix of the first rigid link and the motion state of the platform; Step 3, assembling the transfer matrices of the flexible hinge, the driver and the first rigid link obtained in Step 2 to obtain the transfer matrix of the driving chain, obtaining the overall transfer matrix of the general fast steering mirror system based on the transfer matrix of the driving chain, and finally obtaining the motion state of the platform which is unknown according to the input and output relationship of the entire general fast steering mirror system.

2. The fast mirror system modeling and performance prediction method based on the transfer matrix method according to claim 1, wherein, The step 2 comprises: a. the transfer matrix of the flexible hinge As follows: ； The non-zero non-unitary element expression in the transfer matrix is as follows: ; wherein , is the Young's modulus, is the length of the major axis of the semi-elliptical area on the elliptical cross-section of the flexible hinge, is the thickness of the flexible thin-walled region of the flexible hinge, is the thickness of the flexible hinge, is the length of the minor axis of the semi-elliptical area on the elliptical cross-section of the flexible hinge; b. the driver is identical to the transfer matrix of the first rigid link, collectively referred to as the transfer matrix of the rigid link The specific structure is as follows: ； wherein is the angular frequency of rotation of the rigid link unit about its center of mass, is the mass of the rigid link, is the moment of inertia of the rigid link about the center of mass, element is: ； wherein is the distance between the rigid link unit mass center and the input point in the rigid link local coordinate system x axis, is the distance between the rigid link unit mass center and the input point in the rigid link local coordinate system y axis, is the distance between the rigid link unit output point and the input point in the rigid link local coordinate system x axis, is the distance between the rigid link unit output point and the input point in the rigid link local coordinate system y axis; c. the transfer matrix of the platform As follows: ； wherein is the distance between the second input point and the first input point in the platform local coordinate system x is the distance between the second input point and the first input point in the platform local coordinate system is the distance between the second input point and the first input point in the platform local coordinate system y is the distance between the second input point and the first input point in the platform local coordinate system 3. The fast mirror system modeling and performance prediction method based on the transfer matrix method according to claim 2, characterized in that, The step 3 comprises: Step 1, assembling the transfer matrix of each component in the drive chain Step 2, assembling the transfer matrix of the drive chain :​ ； wherein is the transfer matrix of the upper first rigid link after coordinate transformation, is the transfer matrix of the upper first rigid link after coordinate transformation, is the transfer matrix of the upper first rigid link after coordinate transformation, is the transfer matrix of the upper first rigid link after coordinate transformation, is the transfer matrix of the upper first rigid link after coordinate transformation, is the transfer matrix of the upper first rigid link after coordinate transformation, Step 2, determining the overall transfer matrix ; wherein: ； ； ； ； ； ； is the first 6x6 block of , is the last 6x3 block of , is the transfer matrix of the platform after coordinate transformation, is the force extraction matrix, is the displacement extraction matrix; is the displacement coordination matrix of the two input points on the platform, is the first 3 rows of ; Step 3, Constructing the overall equation describing the input and output relationships of the entire generalized fast steering mirror system where: ； , is the state vector of the first drive chain, the second drive chain input point in the global coordinate system, is the state vector of the platform output point in the global coordinate system; Step 4, calculate as follows The vector consisting of all unknown displacement components and internal force components including the platform corner reflecting the pointing of the mirror : ； wherein is a vector of all known displacement components and internal force components in is corresponding elements in corresponding elements in is corresponding elements in corresponding elements in 4. The fast mirror system modeling and performance prediction method based on the transfer matrix method according to claim 3, characterized in that, The derivation process of the steps 2-4 is as follows: a. establish the transfer relationship equation of the two driving chains for controlling the rotation direction of the platform: ； ； Wherein two driving chains of one rotating direction of the control platform are respectively denoted as a first driving chain and a second driving chain, 、 is a state vector of the input point of the first driving chain and the second driving chain in the global coordinate system, 、 is a state vector of the output point of the first driving chain and the second driving chain in the global coordinate system. b. The and substituted into the transfer relationship equation in a, and combined into the overall transfer matrix The specific process is as follows: First drive chain input point state vector Platform output point state vector The dynamics contribution of the platform output point state vector is denoted by ; Second drive chain input point state vector Platform output point state vector Kinetic contribution to the platform output point state vector is represented by ; First drive chain input point state vector How to influence its output point state vector The displacement of is denoted by ; Second drive chain input point state vector How to influence its output point state vector The displacement of is denoted by ; Combining the above matrices gives the overall transfer matrix where , ; c. The vector of all known displacement components and internal force components and the vector of all unknown displacement components and internal force components are partitioned into and corresponding elements of are partitioned into and corresponding elements of , yielding ； Because of the total equation , we have: ； arrange into a non-homogeneous linear equation group: ； solve to obtain all unknown vectors, including the platform rotation angle reflecting the pointing of the mirror: 。 5. The fast mirror system modeling and performance prediction method based on transfer matrix method of claim 2, wherein, In particular: ； wherein is the distance between the second input point of the platform and the first input point in the global coordinate system x on the axis, is the distance between the second input point of the platform and the first input point in the global coordinate system y on the axis.

6. The fast mirror system modeling and performance prediction method based on the transfer matrix method of claim 3, wherein, The conversion matrix used in the coordinate conversion in step 1 is , specifically: ； wherein is a local coordinate system x is the angle between the axes in the global coordinate system xoy is the angle between the axes in the global coordinate system x is the angle between the axes in the global coordinate system 7. The fast mirror system modeling and performance prediction method based on the transfer matrix method of claim 1, wherein, The flexible hinge is an elliptical flexible hinge, and the rotation axes of the two flexible hinges on the same driving chain are in orthogonal relationship in space.

8. The fast mirror system modeling and performance prediction method based on the transfer matrix method of claim 1, wherein, The platform and the fixed base are the same shape and are arranged in alignment, the platform, the fixed base and the four driving chains fixed therebetween jointly form a space cube, wherein the two driving chains located on the opposite corners of the space cube form a group for controlling the rotation of the platform in one direction, and the two driving chains on the opposite corners are symmetrically arranged.

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