Electromagnetic driving biaxial nano-positioning platform and design method thereof

Abstract

The application discloses an electromagnetic driving double-shaft nano-positioning platform and a design method thereof. The platform comprises a flexible constraint mechanism and a driving main body mechanism. The flexible constraint mechanism comprises a cover plate and a flexible mechanism installed in the cover plate. The driving main body mechanism comprises a stator, permanent magnets, exciting coils and an armature. The armature is located at the center of the stator. Four permanent magnets and four exciting coils are alternately arranged in a ring shape around the armature. The armature is fixed with a center moving platform in the flexible mechanism. Through such a guide mechanism, the platform realizes good flexibility, high precision and controllable motion trajectory. The platform is electromagnetically driven by adopting the Maxwell force principle, and exhibits wide driving capacity. The unique characteristics of the non-contact type, high force density and adjustable stroke have significant advantages in the hundred-micron multi-shaft driving. The bidirectional progressive structure topology optimization method is adopted in the design of the flexible mechanism, which is not only novel in structure, but also has lower difficulty in calculating multiple performance indexes.

Description

Technical Field

[0001] This invention belongs to the field of compliant mechanism topology optimization technology for nanopositioning platforms, specifically relating to an electromagnetically driven biaxial nanopositioning platform and its design method. Background Technology

[0002] In the field of precision engineering, nano-positioning systems have always been a research hotspot, especially with the continuous development of micro / nano manufacturing, precision instruments, and biomedical engineering, where the requirements for positioning accuracy and working stroke of related equipment are becoming increasingly stringent. Based on the principle of driving force, precision drive technology is mainly divided into: electromagnetic drive and non-electromagnetic drive.

[0003] For electromagnetic force drives requiring nanometer precision, voice coil motors are the most common type. Their working principle primarily relies on the Lorentz electromagnetic force generated by the coil cutting magnetic induction lines, perpendicular to both the magnetic field and current directions. Essentially, it's a shear stress-based driving force. Because Lorentz force actuators have a simple working principle, can achieve large strokes within a certain frequency range, and require no intermediate transmission mechanism, they are frequently used in nanometer positioning platform systems. However, due to their low force density, their response speed is relatively slow, limiting Lorentz force drives to millimeter or ten-millimeter level large-stroke drives. In contrast, piezoelectric actuators, as a non-electromagnetic drive method, offer advantages such as compact structure, high driving force, and fast response speed, and are widely used in tool servo nanometer positioning platforms. However, their micro-displacement output is extremely limited, typically within the range of tens of micrometers. Typically, to meet motion stroke requirements, micro-displacement outputs are often designed as cascaded displacement amplification mechanisms, such as lever mechanisms, bridge mechanisms, and Scott-Russell mechanisms, which can effectively increase the output stroke. However, the cascaded structure of flexible hinges not only increases the complexity of the mechanism but also increases the system's motion mass, thereby reducing the system's response speed and severely limiting its frequency response range. Therefore, non-electromagnetic technologies are usually applied in small-stroke fields, such as micro / nano-cutting and high-speed scanning in atomic force microscopy. Unlike traditional voice coil motor drives, electromagnetic drives based on Maxwell force actuators generate driving magnetic fields and bias magnetic fields on both sides of the mover through excitation coils and permanent magnets, respectively. The difference in electromagnetic attraction caused by the difference in the strength of the magnetic fields on both sides is used as the driving force. The permanent magnet bias magnetic field here achieves the theoretical linearization between the driving current and the driving force, which is essentially a driving force based on the principle of normal stress. Maxwell force drives have a wider range of driving capabilities. The same structural form can achieve small-stroke, high-bandwidth drives similar to piezoelectric actuators, or independently achieve millimeter-level large-stroke, low-bandwidth drives similar to voice coil motors. The above analysis shows that both piezoelectric and voice coil motor drives have inherent limitations, making it difficult to achieve high-performance multi-axis drives with strokes as small as 100 micrometers. While Maxwell force electromagnetic drives have received relatively little research in micro / nano drives, their unique properties, such as non-contact operation, high force density, and adjustable stroke flexibility, give them significant advantages in multi-axis drives with strokes as small as 100 micrometers, and provide a new solution for realizing high-performance multi-axis micro / nano drives.

[0004] The large stroke and high-frequency response of nanopositioning platform systems are often conflicting issues. Therefore, in practical applications, to prioritize ensuring the stroke of the nanopositioning system, high-frequency tracking capability usually needs to be sacrificed. To solve this problem, there is an urgent need to propose optimization design methods. Currently, the design schemes of flexible mechanisms are mainly developing in two directions: the pseudo-rigid body method and the topology optimization method. The former aims to separate form synthesis and scale synthesis, while the latter considers form synthesis and scale synthesis in a unified manner. The basic idea of ​​the pseudo-rigid body method is to replace the traditional connecting joints of existing rigid mechanisms with flexible hinges, thereby forming a flexible mechanism. Since its modeling can directly adopt the analysis methods of existing rigid mechanisms, this method is widely used in the mechanism design of precision positioning and operation fields. However, the drawback of this method is that it must start with a rigid linkage mechanism, which makes the design highly dependent on the designer's experience. The basic idea of ​​the topology optimization method is to seek the optimal material distribution within a given design domain to achieve the optimal performance for specific purposes.

[0005] Currently, topology optimization of flexible mechanisms has been widely applied in various fields, such as aerospace, energy systems, medical devices, and civil engineering. It has been comprehensively studied and various methods have been developed, including variable density methods, asymptotic structure optimization methods, and level set methods. Among these, the bidirectional asymptotic structure optimization method is widely welcomed due to its high computational efficiency, algorithmic robustness, simplicity, and ease of implementation. Summary of the Invention

[0006] The purpose of this invention is to provide an electromagnetically driven biaxial nanopositioning platform and its design method. This design method is based on a bidirectional progressive structural optimization method for the flexible structure design of the biaxial nanopositioning platform. It proposes a novel optimization synthesis method based on a two-layer flexible parallel mechanism and employs a multi-objective optimization design formula with a weighted sum of frequency and stiffness. This design method not only considers various factors affecting the design of the nanopositioning platform but also treats the flexible structural subchains as the design domain of a topology optimization problem. This invention achieves high-precision two-degree-of-freedom motion control, improves the accuracy and stability of the positioning system during motion, and brings significant technological progress and application prospects to related fields.

[0007] In a first aspect, the present invention provides an electromagnetically driven dual-axis nanopositioning platform, comprising a flexible constraint mechanism and a driving main body mechanism. The flexible constraint mechanism includes a cover plate and a flexible mechanism installed within the cover plate. The flexible mechanism includes a central moving platform and four flexible connecting branches distributed around the central moving platform; the two ends of each flexible connecting branch are fixed to the central moving platform and the cover plate, respectively.

[0008] The drive mechanism includes a stator, permanent magnets, excitation coils, and an armature; the armature is located at the center of the stator. Four permanent magnets and four excitation coils are fixed inside the stator and arranged alternately in a ring around the armature. The armature is fixed to the central moving platform of the flexible mechanism.

[0009] Preferably, there are two flexible constraint mechanisms. The driving main mechanism is positioned between the two flexible constraint mechanisms. The flexible mechanisms in the two flexible constraint mechanisms are aligned with each other.

[0010] Preferably, the cover plates in the two drive main bodies are aligned with each other and fixed by a first fixing screw; all the multiple connecting protrusions on the outer circumference of the stator; the connecting protrusions are arranged in the grooves on the opposite sides of the two cover plates and fixed by pre-tightening screws.

[0011] Preferably, the stator has four long arms and four short arms arranged alternately around its center. Four excitation coils are wound on the four long arms, and four permanent magnets are fixed to the end faces of the four short arms.

[0012] Preferably, the electromagnetically driven dual-axis nanopositioning platform further includes a displacement detection component. The displacement detection component includes multiple capacitive displacement sensors fixed to the cover plate. These multiple capacitive displacement sensors are used to detect displacements in different directions of the central moving platform.

[0013] Preferably, the flexible connecting branch comprises a plurality of flexible units connected in sequence.

[0014] Secondly, the present invention provides a design method for a dual-axis nanopositioning platform, which is used to design the aforementioned electromagnetically driven dual-axis nanopositioning platform; the design method is as follows:

[0015] Step 1: Define the size and volume of the optimal flexible connecting branch as constraints, and set the objective function to maximize the natural frequency.

[0016] Step 2: Determine the optimal number of branches and constrain the overall displacement of the synthesized double-layer parallel flexible constraint mechanism using the flexibility matrix method. For a flexible connecting branch, define a fixed boundary constraint at the end of the connecting cover plate and apply an external force to the end of the connecting central moving platform to perform topology optimization of the flexible mechanism.

[0017] The objective function for topology optimization of flexible mechanisms is:

[0018]

[0019] st:KU=F

[0020]

[0021] 0 <x min ≤x i ≤1; i=1,2,…,n; m=1,2…N dof

[0022] Among them, f obj The objective function of the optimization problem; x i Let be a design variable, representing the density of the i-th cell in the flexible connected branched mesh; Φ represents the eigenvalues; U is the global displacement matrix of the flexible connection branch; K and M represent the global stiffness matrix and mass matrix of the flexible connection branch, respectively; F represents the external force; Φ m V represents the m-th structural mode; i V represents the material volume of the i-th cell in the mesh; * The total required material volume; u in,x u in,y These represent the displacement components in the x and y directions, respectively; Indicates the input displacement constraint condition; ε * x is the constraint exponent for the input displacement; min Design variables for empty volume elements; n is the number of elements; N dof The natural frequency order.

[0023] Preferably, the flexibility expression of the flexible mechanism is:

[0024] C s =[(C p ) -1 +[T z (π / 2)C p T y (π / 2) T ] -1 ] -1

[0025] Among them, C p The flexibility of one set of relatively flexible connecting branches (11); T y (·), T z (·) are the compliance transformation matrices for rotations about the y-axis and z-axis, respectively.

[0026] Softness C p The expression is as follows:

[0027] C p =[(C l ) -1 +[T y (π)C l T y (π) T ] -1 ] -1

[0028] Among them, C l The flexibility matrix of the flexible connecting branch (11).

[0029] Preferably, the drive mechanism satisfies the following conditions:

[0030]

[0031] Where N is the number of turns of the excitation coil; I m and |d m | represents the maximum current and displacement, respectively. u0 is the free permeability; d0 represents the initial gap between the armature and the excitation coil; B is the bias DC magnetic flux generated by the permanent magnet. sat is the saturation magnetic flux density.

[0032] As a preferred approach, the sensitivity of the objective function and all its constraints is handled using a gradient optimization algorithm.

[0033] The present invention has the following beneficial effects:

[0034] 1. This invention employs Maxwell's force electromagnetic drive, possessing unique advantages such as non-contact operation, high force density, and adjustable stroke, making it outstanding in multi-axis drive applications up to 100 micrometers in size, and providing a new solution for driving high-performance dual-axis micro / nano positioning platforms. Through a bidirectional progressive topology optimization method, the integrated design of configuration and parameters is effectively solved, further balancing stroke and operating bandwidth, and simplifying the complexity of multi-performance design calculations.

[0035] 2. This invention uses Maxwell's electromagnetic drive, which utilizes the difference in electromagnetic attraction caused by the difference in magnetic field strength on both sides as the driving force. The permanent magnet bias magnetic field realizes the theoretical linearization between the driving current and the driving force, making this invention have a wider driving capability. The same structure can realize small stroke and high bandwidth drive similar to piezoelectric actuators, or independently realize millimeter-level large stroke and low bandwidth drive similar to voice coil motors.

[0036] 3. This invention utilizes a flexible structure topology optimization design method to design a nano-positioning platform, overcoming the difficulties of traditional multi-degree-of-freedom structure design and improving structural performance from a conceptual design perspective. Its advantages lie in its novel structure, excellent flexibility, precise control of micro-displacement motion, and flexible adjustment of the platform's trajectory in space. Simultaneously, the double-layer flexible parallel mechanism design effectively solves the torsion phenomenon of single-layer flexible structures during the driving process, improving positioning accuracy. The double-layer flexible structure is arranged in a cross shape, allowing independent driving in the same direction, eliminating coupling, enhancing control precision, and the motion trajectory can be achieved by adjusting the Maxwell force electromagnetic drive signal parameters.

[0037] 4. This invention employs a typical PID controller with a feedforward compensator, significantly improving the system's tracking performance. By calculating the system error and adjusting the Maxwell force electromagnetic drive voltage signal, the drive system can effectively compensate for motion errors caused by external disturbances. Attached Figure Description

[0038] To more clearly explain this invention, the accompanying drawings of the embodiments will be briefly described below. Obviously, these drawings only represent specific embodiments of the invention. Those skilled in the art can derive other related illustrations based on these drawings without additional creative effort.

[0039] Figure 1 This is a planar schematic diagram and an equivalent DC magnetic flux model diagram of the driving main mechanism designed in Embodiment 2 of the present invention.

[0040] Figure 2 This is the design domain and theoretical design diagram of the flexible constraint mechanism designed in Embodiment 2 of the present invention.

[0041] Figure 3 This is a schematic diagram of the overall structure of the biaxial nanopositioning platform provided in Embodiment 1 of the present invention.

[0042] Figure 4 This is a schematic diagram of the driving main body mechanism in Embodiment 1 of the present invention.

[0043] Figure 5 This is a schematic diagram of the flexible mechanism in Embodiment 1 of the present invention.

[0044] Figure 6 This is a schematic diagram of the structure of the first cover plate connecting flexible mechanism in Embodiment 1 of the present invention.

[0045] Figure 7 This is a schematic diagram of the structure of the flexible mechanism connecting the second cover plate in Embodiment 1 of the present invention.

[0046] Figure 8 This is a schematic diagram of the internal structure of the biaxial nanopositioning platform provided in Embodiment 1 of the present invention.

[0047] Figure 9 This is a system control block diagram of the biaxial nanopositioning platform provided in Embodiment 2 of the present invention. Detailed Implementation

[0048] To clearly describe the purpose, technical solution, and advantages of the present invention, the embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0049] Example 1

[0050] See Figure 3As shown, a biaxial nanopositioning platform consists of three layers: upper, middle, and lower. It includes two flexible constraint mechanisms and a drive mechanism positioned between them. The two flexible constraint mechanisms are referred to as the first flexible constraint mechanism and the second flexible constraint mechanism, respectively.

[0051] See Figure 6 As shown, the first flexible constraint mechanism serves as the upper layer of the biaxial nanopositioning platform, including a first cover plate 2 and a flexible mechanism 12 installed inside the first cover plate 2.

[0052] See Figure 4 As shown, the driving main mechanism, as the middle layer of the dual-axis nanopositioning platform, includes a stator 8, a permanent magnet 10, an excitation coil 13, and an armature 14.

[0053] See Figure 7 As shown, the second flexible constraint mechanism serves as the lower layer, including a second cover plate 7 and a flexible mechanism 12 installed inside the second cover plate 7.

[0054] The second cover plate 7 is also equipped with a displacement detection assembly. The displacement detection assembly includes a capacitive displacement sensor 1, a capacitive displacement sensor 6, and a capacitive displacement sensor 9; the second cover plate 7 has fixing through holes for mounting the capacitive displacement sensors 1, 6, and 9. The three capacitive displacement sensors are used to detect the x and y displacements and the z-axis rotation angle of the central moving platform of the flexible mechanism 12, respectively.

[0055] The first cover plate 2, the stator 8, and the second cover plate 7 are pre-tightened and fixed respectively by the pre-tightening screw 4 and the first fixing screw 3. The second fixing screw 5 is used to fix the three-layer mechanism to the high-precision optical vibration isolation air-floating platform.

[0056] In both the first cover plate 2 and the second cover plate 7, the connection structure of the flexible mechanism 12 is identical. The flexible mechanism 12 includes a central moving platform and four flexible connecting chains 11 surrounding the central moving platform. The central moving platform has a cross-shaped hollow structure. The four ends of the central moving platform are connected to the inner circumferential surface of either the first cover plate 2 or the second cover plate 7 via the flexible connecting chains 11. (See also...) Figure 5 As shown, the flexible connecting branches 11 are distributed in a cross shape on the flexible mechanism 12.

[0057] The specific structure of the flexible connecting branch 11 was obtained by solving the biaxial nanopositioning platform design method in Example 2.

[0058] See Figure 6 and 7 As shown, the two flexible mechanisms 12 are located in the middle of the first cover plate 2 and the second cover plate 7, respectively, and are distributed symmetrically from top to bottom.

[0059] See Figure 4 As shown, in the drive main mechanism, four excitation coils 13 are respectively wound on four long arms in the vertical and horizontal directions of the inner ring of the stator 8, and permanent magnets 10 are adhered and fixed to the end faces of the four short arms of the inner ring of the stator 8; the connecting protrusions on the outer circumference of the stator 8 are placed in the grooves on the opposite sides of the first cover plate 2 and the second cover plate 7, and are fixed by pre-tightening screws 4.

[0060] See Figure 6 and 7 As shown, the four excitation coils 13 are arranged orthogonally; specifically, the four excitation coils 13 are evenly distributed around the center point of the stator 8, and the axes of the four excitation coils 13 all pass through the center point of the stator 8; a corresponding permanent magnet 10 is inserted between any two adjacent excitation coils 13; the permanent magnet 10 is connected to the stator by structural adhesive.

[0061] See Figure 8 As shown, the armature 14 is disposed between two flexible mechanisms 12 and fixed by structural adhesive; four excitation coils 13 and four permanent magnets 10 surround the armature 14 and do not contact the armature 14.

[0062] Optionally, the flexible connecting branches 11 in the two flexible constraint mechanisms are parallel to each other, forming a two-degree-of-freedom closed system, and the output end of the flexible mechanism 12 can realize not only the x-axis direction movement, but also the y-axis direction movement.

[0063] Through the above technical solutions, the Maxwell force electromagnetic drive biaxial nanopositioning platform of the present invention can not only effectively seek the optimal mechanism configuration-parameters, but also achieve optimal mechanism performance through optimal drive design.

[0064] This embodiment provides a novel linear variable magnetoresistive driven dual-axis nanopositioning platform, achieving high bandwidth and a large workspace. Overall, the main contributions include: in terms of precision actuation, a dual-axis nanopositioning platform based on Maxwell force electromagnetic actuation is proposed. Simultaneously, the non-contact actuation at the input effectively avoids cross-coupling and protects the actuator. Furthermore, the nanopositioning platform provided in this embodiment employs a double-layer flexible parallel mechanism design, effectively solving the torsion phenomenon of a single flexible structure during the actuation process, thereby improving the positioning accuracy of the nanopositioning platform.

[0065] Example 2

[0066] A design method for a biaxial nanopositioning platform includes the following steps:

[0067] Maxwell's electromagnetic actuator problem description: such as Figure 1As shown in part (a), when the excitation coil 13 is energized in the horizontal direction, the windings of the permanent magnet 10 and the excitation coil 13 will respectively generate bias DC magnetic flux in the circuit. and alternating magnetic flux Therefore, within the two working gaps of the moving armature 14, due to the difference in magnetic flux density, the total magnetic flux generated is: and Thus, the difference in excitation force between the left and right poles of armature 14 ultimately generates the total excitation force F. x The detailed derivation of the total incentive is as follows:

[0068]

[0069] in Because the magnetic reluctance of the permanent magnet 10 is significantly greater than that of air during the working interval, the total bias flux generated by the movement of the armature 14 is... It can be considered a constant. Here, Represents the residual magnetic flux of permanent magnet 10, where and A pm These represent the magnetic flux and pole area of ​​the permanent magnet 10, respectively. d0 and d represent the initial working air gap and armature 14 displacement on each side, respectively. Figure 1 As shown in (b), α is the magnetic leakage coefficient, and its expression is:

[0070]

[0071] Where R pm =L pm / (u0A pm ) and R L These are the internal magnetic resistance and leakage magnetic resistance of the permanent magnet 10, respectively. L The results can be obtained through finite element analysis. The magnetic reluctances of the two working air gaps between the armature 14 and the permanent magnet 10 are R1 = (d0 - d) / (u0A) and R2 = (d0 + d) / (u0A), respectively. Here, u0 is the permeability of free space, and A represents the pole area of ​​the stator 8 core. Furthermore, R... r =d1 / (u0A) represents the magnetic reluctance of the air gap between the permanent magnet 10 and the armature 14, where d1 is the length of the air gap between the permanent magnet 10 and the armature 14. u0 is modeled as magnetic reluctance R. r .

[0072] When the excitation coil 13 operates independently, the total excitation coil NI generates an alternating magnetic flux in the circuit. Alternating current magnetic flux It can be represented as Therefore, based on Maxwell's stress tensor theory, the DC and AC magnetic fluxes will be superimposed. The excitation force F can be obtained from the magnetic flux B1 in the right air gap and the magnetic flux B2 in the left air gap. a as follows:

[0073]

[0074] Considering the saturation magnetic flux density (B) of the armature of the magnetic material sat To obtain the maximum excitation force in both directions simultaneously, the following conditions must be met:

[0075]

[0076] Among them, I m and |d m These represent the maximum current and displacement, respectively.

[0077] Description of the topology optimization problem for flexible mechanisms: Figure 2 Part (a) shows the synthesis process of the optimal flexible connecting branch 11, which mainly consists of two steps. In step I, the size and volume of the optimal branch are defined as constraints, and the objective function is set to maximize the natural frequency. Γ is a fixed boundary constraint. In step II, the number of optimal branches is predetermined, and the overall displacement of the synthesized double-layer flexible parallel mechanism is constrained by the flexibility matrix method. In step II, Γ is redefined as a fixed boundary constraint at the left end, and an external force F is applied to the single-layer flexible connecting branch 11 in the x-axis and y-axis directions at the right end. x and F y To achieve the optimal structural optimization result for the single-layer flexible connecting branch 11, this embodiment employs a bidirectional progressive structural optimization method for the design of the flexible mechanism of the related nano-positioning platform, aiming to maximize the fundamental frequency.

[0078] like Figure 2 As shown in section (b), a single flexible mechanism 12 connects four orthogonally arranged branches. This flexible mechanism 12 is made of aluminum alloy with a Young's modulus of 71 GPa and a density of 2.7 × 10³ kg / m³. 3 2.7×10 3 kg / m 3 The Poisson's ratio is 0.33. The design requirements for this nanopositioning platform are as follows:

[0079] ① The objective function of flexible connection branch 11 is to maximize the natural frequency.

[0080] ② The required material volume for the flexible connecting branch 11: 30%.

[0081] ③ Desired working space of flexible connecting branch 11: 240×240μm 2 .

[0082] ④ Overall dimensions of the mechanism: 125×35mm 2 .

[0083] Structural arrangement: In order to accommodate local displacement u j Constraints converted to global displacement Δ x and Δ y Constraints allow the flexible connecting branch 11 to be modified to a flexibility matrix C. According to Hooke's law, the governing equation for the elastic deformation of the flexible connecting branch 11 can be expressed as:

[0084]

[0085] Where δ and F represent the displacement vector and load vector, respectively. θ z This is the global rotation angle around the z-axis; These represent the translational compliance along the x, y, and z axes, and the rotational compliance about the z-axis, respectively.

[0086] The flexible mechanism 12 includes four parallel flexible connecting branches 11. Based on the flexibility matrix method, O xyz The flexibility C of the left and right flexible connecting branches 11 in the system p It can be represented as:

[0087] C p =[(C l ) -1 +[T y (π)C l T y (π) T ] -1 ] -1 (6)

[0088] Among them, T Z (0) indicates that the flexible connection branch 11 local system is connected to O. xyz The system's compliance transformation matrix has an angle of 0°; C l =T Z (0)CT Z (0) T This indicates that the flexible connecting branch 11 is in O xyz The compliance matrix in T; y (π) represents the area around O. y The compliance transformation matrix for a rotation of the axis by an angle of π.

[0089] Similarly, the flexibility of the flexible mechanism 12 can be expressed as:

[0090] C s =[(C p ) -1 +[T z (π / 2)C p Ty (π / 2) T ] -1 ] -1 (7)

[0091] Among them, T z (π / 2) represents the area around O. z The compliance transformation matrix for a rotational operation of the axis with an angle of π / 2.

[0092] Because the dual-layer flexible mechanism 12 of the biaxial nanopositioning platform is configured in parallel, it is in O xyz The flexibility of a system can be defined as:

[0093] K in =(C s ) -1 +(C s ) -1 (8)

[0094] Among them, K in It is the input stiffness of the nano-positioning platform.

[0095] Therefore, the displacement constraint u of the biaxial nanopositioning platform in the x-axis and y-axis directions in,x u in,y It can be represented as:

[0096] u in,x =(K in,x ) -1 F in,x ,u in,y =(K in,y ) -1 F in,y (9)

[0097] Among them, driving force F in,x =F in,y =F a .

[0098] Performance Modeling: In this optimization, volume constraints apply to the flexible connecting branch 11, while displacement constraints apply to the biaxial nanopositioning platform. The objective function focuses on maximizing the natural frequency of the flexible connecting branch 11, rather than the natural frequency of the entire nanopositioning platform. The maximum natural frequency of the entire nanopositioning platform is then determined through frequency calculation. The motion stroke and natural frequency of the nanopositioning platform will be described in detail below.

[0099] The optimization problem described above requires solving an eigenvalue problem. The governing equations of the finite element model of the dynamic system can be expressed as:

[0100]

[0101] Where M, C, K and F(t) represent the mass matrix, damping matrix, stiffness matrix and external force, respectively; Let and u(t) represent the vectors of acceleration, velocity, and transient displacement, respectively; v0 and u0 represent the initial velocity and initial displacement, respectively. When the damping matrix C = 0 and the external force F(t) = 0, this equation is the equation for undamped free vibration. By solving this equation, the natural frequencies and modal shapes of the final eigenvalue problem can be defined as:

[0102]

[0103] Where m is the set of modes corresponding to the structural degrees of freedom, Φ m It is the m-th eigenvalue eigenvectors.

[0104] Based on the solid isotropic material penalty model, the alternative material interpolation method can be defined as:

[0105]

[0106] Wherein, E(x) i ) represents the interpolated Young's modulus; ρ(x) i ) represents the design variable corresponding to the interpolated density; p = 3 is a penalty factor used to prevent intermediate densities; E0 and ρ0 represent the Young's modulus and density of the solid material, respectively. Design variable x i This represents the density of the i-th cell, and a small value x. min .

[0107] Using interpolation methods, the global stiffness matrix K and the global mass matrix M can be defined as follows:

[0108]

[0109] Among them, K i and M i Let represent the element stiffness matrix and element mass matrix of the solid element, respectively. Let D represent the element elasticity matrix under plane stress conditions. Let B and t represent the strain-displacement matrix and plane thickness, respectively, while ρ and N represent the material density and shape function, respectively.

[0110] Therefore, the stiffness matrix K and the mass matrix M can be expressed in finite element analysis as:

[0111]

[0112] The corrected objective function can be defined as:

[0113]

[0114] Meanwhile, the overall displacement constraint of the nano-positioning platform can be described as follows:

[0115]

[0116] ε * It is a very small positive number, therefore if This indicates that the displacement constraint is satisfied; otherwise, it indicates that the displacement constraint is not satisfied.

[0117] Optimization Model: The topology optimization problem for constructing a flexible mechanism with optimal flexible connection branch 11 can be formulated as follows:

[0118]

[0119] Among them, f obj The objective function represents the optimization problem. Design variable x i The density of the i-th grid cell within the design domain of the flexible connection branch 11 is represented by a small value x. min (For example, 0.001 instead of 0 is used to represent a hollow element) as a design variable for the optimization problem. λ j =ω j 2 , where ω j It is the j-th natural frequency. K and M represent the global stiffness matrix and mass matrix of the structure, respectively; V i and V * These represent the total material volume and the required material volume, respectively. in and These represent the magnitude of the input displacement and its constraints, respectively. n and N dof The number of units and the natural frequency order are respectively expressed.

[0120] Sensitivity Analysis: Dynamic topology optimization problems with multiple constraints are typically handled using gradient optimization algorithms to address the sensitivity of the objective function and all its constraints. Therefore, the objective function f... obj The sensitivity can be expressed by equation (18), which has the following form:

[0121]

[0122] By using the eigenvector Φ m Normalized relative to the mass matrix M, the sensitivity of the i-th natural frequency in the solid-hollow design can be calculated as follows:

[0123]

[0124] In the bidirectional asymptotic structural optimization method, only two discrete design variables are used: solid element 1 and empty element x. minMeanwhile, the sensitivity values ​​used represent the element sensitivity. Therefore, the sensitivity values ​​for solid-hollow elements can be expressed as:

[0125]

[0126] When x min When the value approaches 0, the sensitivity value of the solid-empty unit of the objective function can be simplified to:

[0127]

[0128] Changes in displacement can be estimated based on changes in design variables. Therefore, for displacement constraints, sensitivity analysis can be expressed as:

[0129]

[0130] Among them, U i It is the displacement vector caused by the i-th element, which is given by U = K. -1 F i Obtain; U ij It is the virtual displacement vector of the i-th element caused by the virtual load. The j-th component of the virtual load is equal to 1, and all other components are zero.

[0131] Therefore, the displacement u in the next iteration in i+1 The displacement u in the current iteration can be used in i An approximate estimate is made, expressed as:

[0132]

[0133] Among them, u in i and u in i+1 These represent the displacements for the current iteration and the next iteration, respectively. For a detailed description, please refer to relevant literature.

[0134] When x min When the value approaches 0, the sensitivity value of the displacement-constrained solid-empty element can be simplified to:

[0135]

[0136] The above describes the actuator design and flexible mechanism optimization model for the dual-axis nanopositioning platform. Through the above process, the dual-axis nanopositioning platform of this invention can be effectively designed, such as... Figure 3 As shown.

[0137] In addition, see Figure 9As shown, this invention patent applies a typical PID controller with a feedforward compensator to a dual-axis nano-positioning platform, effectively improving the overall system's precision motion tracking and positioning performance.

[0138] according to Figure 1 When current is applied to the left and right coils, the generated Maxwell electromagnetic force will drive the end effector to move left and right; when current is applied to the upper and lower coils, the end effector can move up and down.

[0139] The above description is merely an illustrative illustration of the present invention, and those skilled in the art can understand and implement it without any creative effort. Through this description, those skilled in the art can understand various implementation methods and realize the present invention using necessary software and hardware. Based on this understanding, the contribution of the technical solution can be embodied in the form of a software product, which can be stored in a computer-readable storage medium.

[0140] The above are merely preferred embodiments of the present invention and do not limit the scope of the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention. The scope of protection should be determined based on the content described in the claims.

[0141] In summary, this invention provides a Maxwell force electromagnetically driven dual-axis nanopositioning platform and its working mode, which has wide applicability, can achieve high-precision motion control and decoupled motion, and has significant technological progress and application prospects for the development of related fields.

Claims

1. A design method for a biaxial nanopositioning platform, characterized in that: The designed dual-axis nanopositioning platform includes a flexible constraint mechanism and a drive body mechanism; the flexible constraint mechanism includes a cover plate and a flexible mechanism (12) installed in the cover plate; the flexible mechanism (12) includes a central moving platform and four flexible connecting branches (11) distributed around the central moving platform; the two ends of the flexible connecting branches (11) are fixed to the central moving platform and the cover plate respectively; The drive mechanism includes a stator (8), permanent magnets (10), excitation coils (13), and an armature (14); the armature (14) is located at the center of the stator (8); the four permanent magnets (10) and the four excitation coils (13) are all fixed inside the stator (8) and are arranged in a ring around the armature (14); the armature (14) is fixed to the central moving platform of the flexible mechanism (12); The design method of this biaxial nanopositioning platform is as follows: Step 1: Define the size and volume of the optimal flexible connecting branch as constraints, and set the objective function to maximize the natural frequency; Step 2: Determine the optimal number of branches and constrain the overall displacement of the synthesized double-layer parallel flexible constraint mechanism using the flexibility matrix method; for a flexible connecting branch, define a fixed boundary constraint at the end of the connecting cover plate and apply an external force to the end of the connecting central moving platform to perform topology optimization of the flexible mechanism. The objective function for topology optimization of flexible mechanisms is: in, Represent the objective function of the optimization problem; Design variables; For eigenvalues; , and These represent the global displacement matrix, stiffness matrix, and mass matrix, respectively; F represents the external force. For structural modes; Represents the material volume of a mesh cell; The total required material volume; , These represent the displacement components in the x and y directions, respectively; Indicates the input displacement constraint conditions; The constraint index is the displacement at the input end; Design variables for empty volume elements; n is the number of elements; The natural frequency order.

2. The design method for a biaxial nanopositioning platform according to claim 1, characterized in that: There are two flexible constraint mechanisms; the driving main mechanism is located between the two flexible constraint mechanisms; the flexible mechanism (12) in the two flexible constraint mechanisms is aligned with each other.

3. The design method for a biaxial nanopositioning platform according to claim 2, characterized in that: The cover plates in the two drive main bodies are aligned with each other and fixed by the first fixing screw (3); all multiple connecting protrusions on the outer circumference of the stator (8); the connecting protrusions are set in the grooves on the opposite sides of the two cover plates (7) and fixed by the pre-tightening screw (4).

4. The design method for a biaxial nanopositioning platform according to claim 1, characterized in that: The stator (8) has four long arms and four short arms arranged alternately around the center of the stator (8); four excitation coils (13) are wound on the four long arms respectively; and four permanent magnets (10) are fixed on the end faces of the four short arms respectively.

5. The design method for a biaxial nanopositioning platform according to claim 1, characterized in that: It also includes a displacement detection component; the displacement detection component includes multiple capacitive displacement sensors fixed on the cover plate; the multiple capacitive displacement sensors are used to detect the displacement of the central moving platform in different directions.

6. The design method for a biaxial nanopositioning platform according to claim 1, characterized in that: The flexible connecting branch (11) includes multiple flexible units connected in sequence.

7. The design method for a biaxial nanopositioning platform according to claim 1, characterized in that: The flexibility expression of the flexible mechanism (12) is: in, The flexibility of one set of relatively flexible connecting branches (11); , These are the compliance transformation matrices for rotations about the y-axis and z-axis, respectively; Softness The expression is as follows: in, The flexibility matrix of the flexible connecting branch (11).

8. The design method for a biaxial nanopositioning platform according to claim 1, characterized in that: The drive mechanism satisfies the following conditions: in, The number of turns of the excitation coil (13); and These are the maximum current and displacement, respectively. Permeability of free space; This indicates the initial gap between the armature (14) and the excitation coil (13); The bias DC flux generated by the permanent magnet (10); is the saturation magnetic flux density.

9. The design method for a biaxial nanopositioning platform according to claim 1, characterized in that: The sensitivity of the objective function and all its constraints is handled through gradient optimization algorithms.

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