A modeling method for a planar dual-axis micro-nano positioning platform based on adaptive meshing

Abstract

The present invention discloses a modeling method for a planar dual-axis micro-nano positioning platform based on adaptive mesh division; the simulation method is as follows: 1. Adaptively divide the planar dual-axis micro-nano positioning platform model into triangular meshes; 2. Construct a unit stiffness matrix for each triangular unit in the triangular mesh; obtain an overall stiffness matrix based on all unit stiffness matrices; 3. Obtain the displacement of the target point based on the overall stiffness matrix and the force of the force output node. The overall stiffness matrix constructed by the present invention comprehensively considers the stiffness of the flexible hinge part and the rigid body part, which helps to improve the modeling accuracy; and, based on the overall stiffness matrix, the present invention can obtain the stiffness matrix between any point and the force output point on the two-degree-of-freedom nanopositioning platform, thereby realizing convenient and accurate simulation of the motion simulation of any position on the two-degree-of-freedom nanopositioning platform.

Description

Technical Field

[0001] The present invention belongs to the technical field of micro-nano positioning simulation, and particularly relates to a planar dual-axis micro-nano positioning platform modeling method based on adaptive grid division. Background Art

[0002] With the rapid development of modern science and technology and high-end manufacturing, as well as the increasing demand for performance in industrial products, modern industry is increasingly driven by key concepts such as "energy efficiency," "cost," and "performance." In the field of precision equipment, nanopositioning platforms are particularly important tools. For example, atomic force microscopes require extremely high precision and stability for nanoscale measurements; micro- / nano-manipulation devices require high accuracy and sensitivity when manipulating micron- or nanometer-scale objects; nanoimprinting techniques demand high resolution and repeatability when fabricating nanostructures; and servo devices for ultra-precision machining must meet diverse performance and functional requirements under diverse operating environments. Achieving flexible design to meet complex requirements in diverse application scenarios within the precision equipment sector has become a key research priority. Therefore, minimizing cost, maximizing efficiency, and ensuring multifunctionality in the structural design of nanopositioning platforms has become a major challenge in the rapid development of modern industry. Consequently, the study of nanopositioning platform structural optimization methods has become one of the most critical topics in modern industrial product design.

[0003] For nanopositioning platforms, the mainstream drive sources are electromagnetic drives and piezoelectric actuators. Motion guidance mechanisms are mainly divided into flexible hinge mechanisms, high-precision air-bearing guides, ultrasonic suspension, and magnetic levitation. Considering factors such as structural size and cost, flexible hinge mechanisms are mainly used to achieve driving force transmission and motion guidance at the submillimeter and even millimeter scales. Currently, the theories and methods for flexible mechanism design mainly include: pseudo rigid body (PRB), integral Castigliano's second theorem (ICST), inverse kinematic modeling (IKM) of Lagrange's equations, screw theory (ST), and matrix-based compliance modeling (MCM).

[0004] However, kinematic methods often have inherent problems in modeling and analysis, such as cumbersome processes and fuzzy boundaries of coupling relationships between models. Although the FEA method is a very effective and convenient theoretical calculation method, compared with other theoretical methods, it has the advantages of a complete theoretical system, intuitive and clear physical meaning, convenient and fast modeling, and high accuracy of solution results. However, the traditional FEA method has the following shortcomings: 1) Although the output displacement and operating frequency can be directly obtained when simulating a two-dimensional positioning platform, it cannot provide specific input and output stiffness information, which is insufficient for the subsequent optimization of structural parameters and providing a more accurate reference for platform design. 2) When the three-dimensional model is changed, the three-dimensional model needs to be rebuilt and detailed simulation settings need to be performed again. This process involves relatively cumbersome intermediate links. Summary of the Invention

[0005] The present invention aims to provide a piezoelectrically actuated high-precision planar dual-axis micro-nano positioning platform simulation and structural optimization method.

[0006] In a first aspect, the present invention provides a method for modeling a planar dual-axis micro-nano positioning platform based on adaptive grid division, which comprises the following steps:

[0007] Step 1: Perform adaptive triangular meshing on the planar dual-axis micro-nano positioning platform model;

[0008] Step 2: Construct a unit stiffness matrix for each triangular element in the triangular mesh; obtain the overall stiffness matrix based on the unit stiffness matrices of all triangular elements;

[0009] Step 3: Set the target point for displacement simulation and one or more force output nodes in the planar dual-axis micro-nano positioning platform; obtain the displacement of the target point based on the overall stiffness matrix and the force of the force output node.

[0010] As a preferred method, the process of obtaining the displacement of the target point in step 3 is as follows: Extract the stiffness matrix K between each force output node and the target point s ; The force F output by each force output node s Divide by the corresponding stiffness matrix K s Then superimpose to get the displacement of the target point.

[0011] Preferably, the triangular mesh generated in step one is optimized by a two-dimensional Gaussian filter; the mesh density near the boundary area is greater than the mesh density far from the boundary area, so that the mesh density at the flexible hinge of the planar dual-axis micro-nano positioning platform model is greater than the mesh density at other positions.

[0012] Preferably, in step 1, a point set of a triangular mesh is established and some points are deleted and supplemented according to different densities, thereby effectively avoiding singularity problems and jagged appearance to improve the quality of the volume fitting mesh.

[0013] Preferably, the element stiffness matrix The expression is:

[0014]

[0015] in, and They represent the constitutive matrix and strain-displacement matrix of the unit entity material respectively; t is the thickness of the planar biaxial micro-nano positioning platform; x and y represent the coordinate values ​​of the two axes.

[0016] Preferably, the overall stiffness matrix The expression is:

[0017]

[0018] in, is a diagonal matrix used to store the area of ​​triangular cells; is the density value of each triangle unit; is the Laplace matrix; n is the number of triangular units.

[0019] As an option, in step 2, a unit mass matrix is ​​constructed for each triangular unit. , and according to the element mass matrix of all triangular elements , get the overall mass matrix ;

[0020] The element mass matrix The expression is:

[0021]

[0022] Where ρ is the density of the planar dual-axis micro-nano positioning platform; N is the shape function matrix; x and y represent the coordinate values ​​of the two axes.

[0023] In a second aspect, the present invention provides a method for optimizing the structure of a planar dual-axis micro-nano positioning platform, comprising the following steps:

[0024] Step 1: Perform adaptive triangular meshing on the planar dual-axis micro-nano positioning platform model.

[0025] Step 2: Construct a unit stiffness matrix for each triangular unit in the triangular mesh; obtain the overall stiffness matrix based on the unit stiffness matrices of all triangular units.

[0026] Step 3: Based on the overall stiffness matrix, determine whether the output stiffness of the planar dual-axis micro-nano positioning platform meets the load design requirements; if not, adjust the dimensional parameters of each component in the planar dual-axis micro-nano positioning platform and re-obtain the overall stiffness matrix.

[0027] Step 4: Repeat step 3 until the output stiffness of the planar dual-axis micro-nano positioning platform meets the load design requirements.

[0028] In a third aspect, the present invention provides a planar dual-axis micro-nano positioning platform, which is optimized by the aforementioned planar dual-axis micro-nano positioning platform structure optimization method.

[0029] Preferably, the planar biaxial micro-nano positioning platform includes a frame, and four drive units and an output platform installed on the frame; the four drive units surround the output platform; the drive units include a piezoelectric actuator, a bridge amplifying structure and an end connection structure; the piezoelectric actuator is installed in the bridge amplifying structure; the amplified output position of the bridge amplifying structure is connected to one side edge of the output platform through the end connection structure; it is characterized in that: the end connection structure includes a lateral constraint component and an absorption coupling structure; the lateral constraint component includes an output block and a straight flexible hinge; the output block is fixed to the amplified output position of the bridge amplifying structure; the two sides of the output block are connected to the two opposite sides of the frame respectively through straight flexible hinges; the side of the output block facing away from the bridge amplifying structure is connected to the output platform through an absorption coupling structure; the absorption coupling structure includes a plurality of serpentine flexible hinges arranged side by side; the two ends of each serpentine flexible hinge are connected together.

[0030] Preferably, the serpentine flexible hinge includes a plurality of transverse hinge plates and a plurality of longitudinal hinge plates that are alternately connected in sequence; the transverse hinge plates are perpendicular to the direction from the output block to the output platform; the longitudinal hinge plates are parallel to the direction from the output block to the output platform; and two adjacent longitudinal hinge plates are respectively connected to opposite ends of the transverse hinge plates.

[0031] The beneficial effects of the present invention are:

[0032] 1. The present invention divides the planar two-axis micro-nano positioning platform into triangular meshes and calculates the unit stiffness matrix of each triangular unit to conveniently obtain the overall stiffness matrix of the two-degree-of-freedom nanopositioning platform. Then, the stiffness matrix between any point and the force output point on the two-degree-of-freedom nanopositioning platform can be obtained, thereby realizing convenient and accurate simulation of the motion simulation of any position on the two-degree-of-freedom nanopositioning platform.

[0033] 2. The present invention establishes a complete performance model for the planar dual-axis micro-nano positioning platform, which can also incorporate the rigid body part connected to the flexible mechanism into the performance modeling; at the same time, the present invention uses an adaptive triangular mesh generation method in the output stiffness modeling to avoid the appearance of rough boundaries; thereby optimizing the overall static and dynamic modeling of the mechanism and improving the final displacement simulation accuracy.

[0034] 3. By establishing a stiffness simulation model that fully considers the rigid body and flexible hinge parts of the planar dual-axis micro-nano positioning platform, the present invention can accurately optimize the dimensions of each component in the planar dual-axis micro-nano positioning platform, thereby effectively improving the output stiffness of the output platform and improving the load-bearing capacity of the planar dual-axis micro-nano positioning platform. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] In order to better interpret the patent of the present invention, the following will briefly introduce the implementation of the technical solution of the present invention with drawings. Obviously, the following drawings are only embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0036] Figure 1 This is a grid division diagram of the simulation method provided in Example 1 of the present invention.

[0037] Figure 2 1 is an overall schematic diagram of a simulated planar dual-axis micro-nano positioning platform according to embodiment 1 of the present invention.

[0038] Figure 3 2 is a schematic diagram of the back side of the simulated planar dual-axis micro-nano positioning platform according to Example 1 of the present invention.

[0039] Figure 4 Schematic diagram of the connection between the four driving units and the output platform 2 in the simulated planar dual-axis micro-nano positioning platform according to embodiment 1 of the present invention.

[0040] Figure markings: 1. Frame; 2. Output platform; 3. Detection component; 3-1. Capacitive sensor; 3-2. Target detection block; 4. Piezoelectric actuator; 5. Bridge amplification structure; 6. End connection structure; 6-1. Lateral constraint component; 6-2. Absorption coupling structure. DETAILED DESCRIPTION

[0041] In order to more clearly illustrate the objectives, technical solutions and advantages of the present invention, embodiments of the present invention are now described in detail with reference to the accompanying drawings.

[0042] Example 1

[0043] The invention discloses a planar dual-axis micro-nano positioning platform modeling method based on adaptive grid division, which is used to perform motion simulation on a planar dual-axis micro-nano positioning platform with two degrees of freedom.

[0044] The method for modeling a planar dual-axis micro-nano positioning platform based on adaptive grid division includes the following steps:

[0045] Step 1: Mesh Generation: Use the bisection method to redistribute solid and void volumes, and determine material boundaries based on the zero-level isosurface. Use the adaptive triangulated meshing method to mesh the planar dual-axis micro-nano positioning platform. Simultaneously, to further smooth the material boundaries within the design domain, a two-dimensional Gaussian filter is used to optimize the mesh. The specific process is as follows:

[0046] Create a fixed node set in the plane ; Fixed node set Including the four corner points of the rectangle that completely covers the planar dual-axis micro-nano positioning platform; using the adaptive triangular mesh as the volume fitting mesh under the two-dimensional plane condition, using the contour function in Matlab software to generate the point set on the boundary H=[ ℎ 1 , ℎ 2 , ℎ 3 ,…, ℎ n ] ; These nodes are fixed and defined as unit vertices, which are used to describe the boundary in the proposed method. However, considering that the points on the contour are too densely distributed, it may lead to triangular elements with small angles, which may cause singularity problems; and if the distance between adjacent points is too large, the generated adaptive mesh cannot accurately express the smooth boundary. To this end, it is necessary to merge points that are too close to each other and perform a point set The redistribution of , thus effectively avoiding singularity problems and jagged appearance to improve the quality of volume fitting mesh.

[0047] In this embodiment, additional points are added between adjacent points with larger distances, so that the adjacent points and The distance between Need to adjust to appropriate values:

[0048] Formula (1)

[0049] in, and Respectively represent the minimum and maximum distances allowed between adjacent points. are adjacent points and midpoint.

[0050] Based on a fixed node set and regularly distributed non-fixed points to generate a volume fitting grid; using the matrix Represents the coordinates of all nodes in the design domain h =[ x , y ] .in, and Respectively indicate that all nodes are included and Vector of coordinates. Matrix is the updated point set The matrix formed by the coordinates of .

[0051] Manually define the original length when the triangle mesh is not expanded , which is used to control the density or sparsity of the mesh. Taking into account the repulsive force in the triangular mesh, the force-displacement relationship can be expressed as follows:

[0052] Formula (2)

[0053] in, is the current length after deformation; is the force between two nodes in the mesh; is the structural elastic modulus.

[0054] Since the internal forces are derived from the difference between the current length l and the original unextended length l0, external forces are used to prevent the device from moving out of the design domain. and external forces , the force-displacement function of the rod in equilibrium can be expressed as:

[0055] Formula (3)

[0056] The positions of the nodes can be obtained by solving the static force balance equation.

[0057] The approximate solution can be iteratively updated using the forward Euler method starting from regularly spaced nodes:

[0058] Formula (4)

[0059] in, is the step size; s is the current number of iterations.

[0060] Location is known, so the structural topology (i.e. triangulation) of the current point set is also known. The external reaction force works as follows: Update to During the process, any points that extend beyond the boundary are moved back to the nearest boundary point. This satisfies the requirement that forces act perpendicular to the boundary. Points can move along the boundary but cannot exceed it. This approach results in a natural volume-fitting mesh that accurately captures the boundaries of solids and voids. The volume-fitting mesh density varies within the design domain, generating a denser adaptive triangulated mesh around the boundaries of solids and voids, ensuring smoothness and accuracy. Simultaneously, a sparser volume-fitting mesh is generated in areas away from the boundaries.

[0061] Since the width of the flexible hinge is small and the entire hinge is close to the boundary, the grid density at the flexible hinge is greater than that of the rigid body part on the planar dual-axis micro-nano positioning platform. Therefore, the modeling method provided in this embodiment not only considers the stiffness of the rigid body part and the flexible hinge part at the same time, thereby improving the accuracy of modeling and simulation; but also can focus the calculation of the stiffness matrix modeling on the flexible hinge part that has a greater impact on the stiffness.

[0062] Step 2: Calculation of the structural characteristics of the planar biaxial micro-nano positioning platform: First, for the planar triangular unit in the triangular mesh generated in step 1, the unit stiffness matrix can be calculated according to the two-degree-of-freedom structure of each node. List form, namely:

[0063] Formula (5)

[0064] Among them, k ij is the influence stiffness of the i-th degree of freedom on the j-th degree of freedom, which reflects the elastic restoring force of the i-th degree of freedom under the unit displacement of the j-th degree of freedom. The diagonal elements represent the direct stiffness of each degree of freedom, and the non-diagonal elements represent the influence stiffness between different degrees of freedom; i = 1, 2, ..., 6; j = 1, 2, ..., 6. t is the thickness of the planar biaxial micro-nano positioning platform; and They represent the constitutive matrix and strain-displacement matrix of the unit solid material respectively.

[0065] Element mass matrix List sub-blocks, namely:

[0066] Formula (6)

[0067] Among them, m ijis the influence inertia of the i-th degree of freedom on the j-th degree of freedom, reflecting the elastic restoring force of the i-th degree of freedom under unit displacement of the j-th degree of freedom. The diagonal elements represent the direct inertia of each degree of freedom, and the off-diagonal elements represent the influence inertia between different degrees of freedom. i = 1, 2, ..., 6; j = 1, 2, ..., 6. ρ is the density of the planar dual-axis micro-nanopositioning platform; N is the shape function matrix, which describes the relationship between any point in the grid and the node displacement.

[0068] In order to obtain the overall stiffness matrix and the overall mass matrix , firstly transform the local element stiffness matrix and the local element mass matrix Converted into global stiffness matrix and the overall mass matrix ; Therefore, the diffusion term in the triangular mesh is calculated, and the overall stiffness matrix and the overall mass matrix The models are expressed as:

[0069] Formula (7)

[0070] Where, Indicates a A diagonal matrix used to store the area of ​​each triangular unit; Indicates the density value of each triangle unit, contained in a Vector R =[ ρ 0 , ρ 1 , ρ 2 ,…, ρ 7 ] ; express The Laplace matrix of .

[0071] In finite element analysis, the vector F =[ F 1 , F 2 , F 3 ,…, F n ] Represents the external force applied to the boundary. U =[ u 1 , u 2 , u 3 ,…, u n ] Indicates that each triangular element is subjected to external force The relationship between force and displacement can be expressed as:

[0072] Formula (8)

[0073] In addition, the natural frequency and mode of the system are inherent characteristics determined by the stiffness and mass of the system. Therefore, based on the Rayleigh quotient principle, the first-order eigenvalue (objective function) of the system motion equation can be expressed as:

[0074] Formula (9)

[0075] Where, It is with Order eigenvalue The associated eigenvector.

[0076] Step 3: Set the force output nodes to be simulated. The force output nodes are the two end points of the piezoelectric actuator 4. Enter the initial coordinates of the target point. Extract the stiffness matrix K between each force output node and the target point s The force F output by each force output node s Divide by the corresponding stiffness matrix K s Then superimpose to get the displacement of the target point.

[0077] In the overall mass matrix Extract the mass matrix M between each force output node and the target point s According to formula (9), the stiffness matrix K corresponding to each force output node is s , mass matrix M s And the output force F s , calculate and obtain the resonance frequency of the target point.

[0078] Table 1 below compares the simulation results of this embodiment with those of standard finite element analysis (FEA). As can be seen from Table 1, this embodiment achieves simulation accuracy approaching that of FEA. Because FEA often requires repeated manual adjustments of mesh density to improve accuracy in complex situations such as nonlinear problems, stress concentration areas, and crack tips, this embodiment, with its ease of operation, allows for rapid and convenient high-precision simulation of a planar piezoelectric micro-motion platform while fully accounting for the structural stiffness of all components, including the rigid body.

[0079] Table 1 Comparison of FEA and adaptive meshing simulation results

[0080]

[0081] In summary, this embodiment applies the adaptive finite element method to the design of a piezoelectrically actuated dual-axis micro-nanopositioning platform, effectively overcoming the limitations of traditional FEA and providing a more optimal numerical calculation method for engineering analysis of complex structures and high-precision requirements. Therefore, this invention provides important reference and guidance for a variety of fields, including precision engineering, nanopositioning platforms, and topology optimization design of flexible mechanisms.

[0082] Example 2

[0083] A method for optimizing a planar dual-axis micro-nano positioning platform comprises the following steps:

[0084] Step 1: Obtain the overall stiffness matrix of the planar dual-axis micro-nano positioning platform by using the planar dual-axis micro-nano positioning platform modeling method based on adaptive grid division provided in Example 1. .

[0085] Step 2: According to the overall stiffness matrix , judge whether the output stiffness of the planar dual-axis micro-nano positioning platform meets the load design requirements. If not, adjust the size parameters of each component in the planar dual-axis micro-nano positioning platform and re-obtain the overall stiffness matrix .

[0086] Step 3: Repeat step 2 until the output stiffness of the planar dual-axis micro-nano positioning platform meets the load design requirements.

[0087] This embodiment also provides a specific structure of an optimized planar dual-axis micro-nano positioning platform, which includes a frame 1, four drive units mounted on the frame 1, an output platform 2, and a detection assembly 3. The four drive units surround the output platform 2 and are used to drive the output platform 2 to move within a plane with two degrees of freedom; the detection assembly 3 is used to detect the position of the output platform 2.

[0088] See also Figure 4 As shown, the drive unit includes a piezoelectric actuator 4, a bridge-type amplifying structure 5, and an end connection structure 6. The piezoelectric actuator 4 is mounted within the bridge-type amplifying structure 5 and preloaded and secured with bolts. The amplified output portion of the bridge-type amplifying structure 5 is connected to a side edge of the output platform 2 via the end connection structure 6.

[0089] The end connection structure 6 includes a lateral constraint assembly 6-1 and an absorption coupling structure 6-2. The lateral constraint assembly 6-1 includes an output block and two sets of straight flexible hinges. The output block is fixed to the amplified output position of the bridge-type amplification structure 5. The two sides of the output block are connected to two opposing sides of the frame 1 via straight flexible hinges. The straight flexible hinges are specifically dual parallel straight flexible hinges. The two straight flexible hinges help to stabilize the movement direction of the output block.

[0090] The side of the output block facing away from the bridge-type amplification structure 5 is connected to the output platform 2 via an absorption coupling structure 6-2. The absorption coupling structure 6-2 comprises multiple serpentine flexible hinges (two in this embodiment) arranged side by side; the two ends of each serpentine flexible hinge are connected together. The serpentine flexible hinges include multiple transverse hinge plates and multiple longitudinal hinge plates that are alternately connected in sequence. The transverse hinge plates are perpendicular to the direction from the output block to the output platform 2. The longitudinal hinge plates are parallel to the direction from the output block to the output platform 2. Two adjacent longitudinal hinge plates are connected at opposite ends of the transverse hinge plates.

[0091] On the one hand, the serpentine flexible hinge can transmit displacement in a direction parallel to the moving direction of the output block, and on the other hand, it can prevent the displacement of the output platform 2 perpendicular to the moving direction of the output block from affecting the output block, and prevent the piezoelectric actuator 4 from being damaged by the tangential force; therefore, the end connection structure 6 can effectively absorb motion coupling.

[0092] The detection assembly 3 includes two capacitive sensors perpendicular to each other, which are used to detect the displacement components of the target detection block fixed on the output platform 2 along the X-axis and the Y-axis respectively.

[0093] When two of the piezoelectric actuators 4 facing each other are excited, the driving force they generate can drive the output platform 2 to move in one direction; similarly, when the other two piezoelectric actuators 4 facing each other are excited, the driving force they generate can drive the output platform 2 to move in the other direction.

[0094] The planar dual-axis micro-nanopositioning platform in this embodiment can flexibly select operating modes based on specific application requirements, enabling both decoupled motion in different directions and high-precision linkage control. This feature demonstrates broad applicability across multiple fields, driving advancements in related technologies and expanding practical application possibilities.

[0095] The foregoing is merely an illustrative description of the present invention, and those skilled in the art will be able to understand and implement the present invention without inventive effort. This description will enable those skilled in the art to grasp various implementation methods and apply them in conjunction with the necessary software and hardware. Furthermore, the contributions of this technical solution may also be embodied in the form of a software product, which may be stored on a computer-readable storage medium.

[0096] It should be emphasized that the above is only the best embodiment of the present invention and does not limit its scope of protection. Any modification, equivalent replacement or improvement made within the spirit and principle of the present invention shall be included in the scope of protection of the present invention, and the specific scope shall be subject to the content of the claims.

[0097] In summary, the present invention provides a piezoelectrically driven dual-axis nanopositioning platform and its working mode, which has wide applicability, can achieve high-precision motion control and decoupled motion, and has important technical driving role and application prospects for the development of related fields.

Claims

1. A modeling method for a planar dual-axis micro-nano positioning platform based on adaptive grid division, characterized by: The following steps are involved: Step 1: Perform adaptive triangular meshing on the planar dual-axis micro-nano positioning platform model; The planar biaxial micro-nano positioning platform comprises a frame (1), and four drive units and an output platform (2) mounted on the frame (1); the four drive units surround the output platform (2); the drive unit comprises a piezoelectric actuator (4), a bridge amplification structure (5) and an end connection structure (6); the piezoelectric actuator (4) is mounted in the bridge amplification structure (5); the amplified output position of the bridge amplification structure (5) is connected to a side edge of the output platform (2) via the end connection structure (6); the end connection structure (6) comprises a lateral constraint The lateral constraint component (6-1) includes an output block and a straight flexible hinge; the output block is fixed to the amplified output position of the bridge-type amplifying structure (5); the two sides of the output block are connected to the two opposite sides of the frame (1) through the straight flexible hinge; the side of the output block facing away from the bridge-type amplifying structure (5) is connected to the output platform (2) through the absorption coupling structure (6-2); the absorption coupling structure (6-2) includes a plurality of serpentine flexible hinges arranged side by side; the two ends of each serpentine flexible hinge are connected together; The serpentine flexible hinge comprises a plurality of transverse hinge plates and a plurality of longitudinal hinge plates which are alternately connected in sequence; the transverse hinge plates are perpendicular to the direction from the output block to the output platform (2); the longitudinal hinge plates are parallel to the direction from the output block to the output platform (2); two adjacent longitudinal hinge plates are respectively connected to opposite ends of the transverse hinge plates; Step 2: Construct a unit stiffness matrix for each triangular element in the triangular mesh; obtain the overall stiffness matrix based on the unit stiffness matrices of all triangular elements; Step 3: Set the target point of the displacement simulation and one or more force output nodes in the planar dual-axis micro-nano positioning platform; The displacement of the target point is obtained based on the global stiffness matrix and the force at the force output node.

2. The method for modeling a planar dual-axis micro-nano positioning platform based on adaptive grid division according to claim 1, characterized in that: The process of obtaining the displacement of the target point in step 3 is as follows: Extract the stiffness matrix K between each force output node and the target point s ; The force F output by each force output node s Divide by the corresponding stiffness matrix K s Then superimpose to get the displacement of the target point.

3. The method for modeling a planar dual-axis micro-nano positioning platform based on adaptive grid division according to claim 1, characterized in that: The triangular mesh generated in step 1 is optimized using a two-dimensional Gaussian filter; the mesh density near the boundary area is greater than the mesh density far from the boundary area, so that the mesh density at the flexible hinge of the planar dual-axis micro-nano positioning platform model is greater than the mesh density at other locations.

4. The method for modeling a planar dual-axis micro-nano positioning platform based on adaptive grid division according to claim 1, characterized in that: The element stiffness matrix The expression is: ； in, and They represent the constitutive matrix and strain-displacement matrix of the unit entity material respectively; t is the thickness of the planar biaxial micro-nano positioning platform; x and y represent the coordinate values ​​of the two axes.

5. The method for modeling a planar dual-axis micro-nano positioning platform based on adaptive grid division according to claim 4, characterized in that: The overall stiffness matrix The expression is: ； in, is a diagonal matrix used to store the area of ​​triangular cells; is the density value of each triangle unit; is the Laplace matrix; n is the number of triangular units.

6. The method for modeling a planar dual-axis micro-nano positioning platform based on adaptive grid division according to claim 1, characterized in that: In step 2, the unit mass matrix is ​​also constructed for each triangular unit , and according to the element mass matrix of all triangular elements , get the overall mass matrix ; The element mass matrix The expression is: ； Where ρ is the density of the planar dual-axis micro-nano positioning platform; N is the shape function matrix; x and y represent the coordinate values ​​of the two axes.

7. A method for optimizing the structure of a planar dual-axis micro-nano positioning platform, characterized by: The following steps are involved: Step 1: Perform adaptive triangular meshing on the planar dual-axis micro-nano positioning platform model; The planar biaxial micro-nano positioning platform comprises a frame (1), and four drive units and an output platform (2) mounted on the frame (1); the four drive units surround the output platform (2); the drive unit comprises a piezoelectric actuator (4), a bridge amplification structure (5) and an end connection structure (6); the piezoelectric actuator (4) is mounted in the bridge amplification structure (5); the amplified output position of the bridge amplification structure (5) is connected to a side edge of the output platform (2) via the end connection structure (6); the end connection structure (6) comprises a lateral constraint The lateral constraint component (6-1) includes an output block and a straight flexible hinge; the output block is fixed to the amplified output position of the bridge-type amplifying structure (5); the two sides of the output block are connected to the two opposite sides of the frame (1) through the straight flexible hinge; the side of the output block facing away from the bridge-type amplifying structure (5) is connected to the output platform (2) through the absorption coupling structure (6-2); the absorption coupling structure (6-2) includes a plurality of serpentine flexible hinges arranged side by side; the two ends of each serpentine flexible hinge are connected together; The serpentine flexible hinge comprises a plurality of transverse hinge plates and a plurality of longitudinal hinge plates which are alternately connected in sequence; the transverse hinge plates are perpendicular to the direction from the output block to the output platform (2); the longitudinal hinge plates are parallel to the direction from the output block to the output platform (2); two adjacent longitudinal hinge plates are respectively connected to opposite ends of the transverse hinge plates; Step 2: Construct a unit stiffness matrix for each triangular element in the triangular mesh; obtain the overall stiffness matrix based on the unit stiffness matrices of all triangular elements; Step 3: According to the overall stiffness matrix, determine whether the output stiffness of the planar dual-axis micro-nano positioning platform meets the load design requirements; If not, adjust the size parameters of each component in the planar dual-axis micro-nano positioning platform and re-obtain the overall stiffness matrix; Step 4: Repeat step 3 until the output stiffness of the planar dual-axis micro-nano positioning platform meets the load design requirements.

8. A planar dual-axis micro-nano positioning platform, characterized by: It is obtained by optimizing the structure optimization method of a planar dual-axis micro-nano positioning platform as described in claim 7.

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